Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation in Northeastern Brazil

*Natalia Fedorova, Vladimir Levit and Lucas Carvalho Vieira Cavalcante*

## **Abstract**

Tropical cyclone (TC) impacts on adverse phenomena in the tropical region of Northeastern Brazil (NEB) have been analyzed. TC influence on fog and rain formation was not described in the previous papers. The main goal of the chapter is to evaluate the existence of such influence and thus to improve the weather forecasting in this area. TC information from the NHC of the NOAA was used. METAR and SYNOP data were used for the adverse phenomena study. Analysis of the synoptic systems was based on different maps at the pattern levels and on satellite images. These maps were elaborated using reanalysis data from the ECMWF. Thermodynamic analysis was also used. Middle tropospheric cyclonic vortexes (MTCV) in the tropical region of the Southern Atlantic were described recently. Five from 10 MTCVs were associated with tropical cyclones and disturbances in the Northern Atlantic. Circulation patterns between TC and synoptic systems at the NEB are described. These circulations create sinking over the BNE and, as a result, form fog, mist and weak rain in the BNE during TC days. Mechanisms of TC influence on weather formation in the BNE are presented. This information is important for improving weather forecasting methods.

**Keywords:** tropical cyclones, Northeast Brazil, adverse phenomena

## **1. Introduction**

Fog and mist events are rare and show significant variation of frequency in the tropical region of the Brazilian Northeast (BNE) [1, 2]. Radiosonde data in the central and south regions of the BNE, i.e., Recife, Pernambuco state (8o S, 35o W), and Salvador, Bahia state (13o S, 39o W), registered fog (haze) on average in 13 (13) and 37 (41) days per year, respectively [3]. On the other hand, in the first study of fog formation in Maceio, Alagoas State (9o S, 39o W) [4], only two fog events (moderate and weak) during 1996 (both in winter) were found. Fog does not exist in the semiarid region [3]. In the northern region of the BNE, i.e., Belem, Para State (1o S, 48o W), and Quixeramobim, Ceara State (5o S, 39o W), fog (haze) was registered in

24 (1) and 4 (28) events/year, respectively. More frequently, fog was observed in Barra do Corda, Maranhao State (5<sup>o</sup> S, 45o W), in 48 days/year, but haze was very rare, only 4 days/year. Also, fog duration was very short, 1–2 h on average in the coastal region of the BNE [5].

Moreover, the physical mechanism of fog formation on the northern coast was atypical (typical radiation or advection fog does not occur typically) [6]. Summarizing the results of this chapter, it was possible to make the following conclusion regarding the processes of fog formation in the tropical region of the BNE. A weak confluence at the low-level trough (wave disturbances in trade winds—WDTW) contributes to weak pressure anomaly and creates the conditions for moisture convergence as well. A change in wind direction and air current from the river region contributes additional humidity for fog formation. Also, precipitation occurred before the fog events, contributing to humidification of the air through evaporation of the humid surface and raindrops. A warmer sea surface contributes to more evaporation and, as a consequence, increases the amount of water vapor in the surrounding air at the low levels near the coast. Positive latent heat flux shows a humidity increase and, therefore, moisture accumulation in the coastal region. Negative sensible heat flux results in air cooling, possible water vapor condensation, and, finally, fog formation.

All this information about rare fog events with a short duration and atypical physical mechanisms of their formation demonstrates the numerous problems for fog/haze forecasting.

An intertropical convergence zone (ITCZ), a South Atlantic subtropical high (SASH), trade winds, and an upper tropospheric cyclonic vortex are typical synoptic scale systems associated with different weather conditions in the BNE [7].

For the first time, the influence of wave disturbances in trade winds (WDTW) on rain formation in the BNE was shown in Molion and Bernardo [8]. The relationship between different types of troughs and adverse meteorological phenomena (fog and thunderstorms) in Alagoas State was studied by Rodrigues et al. [9]. It was noted that 87% of the troughs were associated with wave disturbances in trade winds (WDTW) on the northwestern periphery of the subtropical South Atlantic High. Rare stratus cloud events with a duration of 4 days each on the northeast coast of Brazil also were formed in WDTW [10]. Fog formation is usually associated with WDTW in Maceio (Alagoas State) [1, 5, 6, 11].

The influence of the Brazilian Northeast Jet Stream (BNEJS) on weather in the BNE was described recently [12–14]. Particularly, the results of this study show that the period from April to October (a rainy period and the transition to a dry season in the coastal region) was characterized by a rather high number of fast BNEJSs, with high wind speed in the core, a predominant northwesterly direction, and the location of BNEJS between the upper tropospheric trough (UTT) and the SASH.

Upper tropospheric cyclonic vortices (UTCVs) are important synoptic scale systems in this region; the greatest UTCV frequency occurs during the austral summer period, with the maximum frequency in January [15–18]. Intensive cloud development with precipitation was observed on the UTCV periphery and cloudlessness in their center as a result of a downward motion in its cold center and upward motion on the periphery. Rao and Bonatti [18] conjectured that barotropic instability of the regional mean basic winds at the upper levels could be one cause for UTCV formation. It was suggested that these air streams are associated with the subtropical jet stream and contribute to UTCV development. A strong positive shear zone was formed within the South Atlantic trough before the formation of the UTCV [19]. The same paper showed that the barotropic process dominates over the baroclinic

**29**

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation…*

one. They showed that barotropic energy conservation is the energy source for the growth of wave motion and wave amplitude and also that barotropic instability of

Middle tropospheric cyclonic vortices (MTCVs) located between 700 and 400 hPa in the BNE were described recently [20, 21]. About 232 MTCVs were observed each year over the BNE and adjacent ocean region. MTCVs were predominantly short and lasted less than 12 h. The vortices persisting longer than 30 h were

Frontal zones are observed regularly in the southern region of the Brazilian Northeast in the Southern Hemisphere (SH) [22, 23]. The influence of a western edge of a frontal cloud band on the weather conditions was found in the central region of the Brazilian Northeast SH and was described by Fedorova et al. [24]. The western edge of a frontal cloud band rarely passes across Alagoas State in the BNE. Only two to five frontal zones per year which directly affected the weather conditions were observed in 2004–2006. Nonetheless, Reeder and Smith [25] quoted different papers (including [26, 27]) where a cold front crossing the equator and penetrating the Northern Hemisphere (NH) tropics was documented. These fronts can initiate severe convection in the subtropic SH; however, they generally

A SASH was observed in the South Atlantic during the whole year and was located at the low levels in the eastern part of the South Atlantic, on average

S in February [29]. A SASH in summer is frequently split and is normally

Some studies show the influence of synoptic scale systems of the Northern Hemisphere on BNE weather. The influence of tropical cyclone Dany-15 on intensive fog formation in the NEB was descried by Fedorova and Levit [1]. But the information about tropical cyclone (TC) impact on BNE weather over many years has not yet been presented. Thus, the main aim of this chapter is to analyze the impact

W, 27o

W, on fog, mist, and weak rain formation in the BNE.

S in August, to 5<sup>o</sup>

N and the equator and

N and the

W have been studied, using data from

W,

S [7, 28]. It migrated from 15<sup>o</sup>

of all tropical cyclones formed from 2013 to 2015 between 20o

**2.1 Selection and classification of the tropical cyclones**

*Hurricane* (*H*). Winds greater than or equal to 33 m/s.

greater (*MH* was not observed during the study period).

equator whose centers passed through 35–50o

All tropical cyclones (TC) formed from 2013 to 2015 between 20o

the National Hurricane Center's Tropical Cyclone Reports https://www.nhc.noaa.

*Tropical depression* (*TDep*). Winds are less than 17 m/s, but there is a closed

*Tropical storm* (*TS*). Winds are greater than or equal to 17 m/s but less than 33 m/s with a definite closed circulation. The storm is usually assigned a name.

*Major hurricane* (*MH*). Typhoon with 1 min of sustained winds of 65 m/s or

TC has been classified according to their sustained wind speed [30] as follows: *Tropical disturbance* (*TD*). In this stage, winds are less than 17 m/s with open

detected more frequently in the summer and rarely in autumn.

*DOI: http://dx.doi.org/10.5772/intechopen.88804*

the shear zone can excite the UTCV.

tend to suppress convection in the ITCZ.

W and 30o

which passed through 35–50o

**2. Data and methodology**

circulation (no closed isobars).

circulation (closed isobars).

between 20o

weaker [7].

gov/data/tcr/.

33<sup>o</sup>

#### *Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation… DOI: http://dx.doi.org/10.5772/intechopen.88804*

one. They showed that barotropic energy conservation is the energy source for the growth of wave motion and wave amplitude and also that barotropic instability of the shear zone can excite the UTCV.

Middle tropospheric cyclonic vortices (MTCVs) located between 700 and 400 hPa in the BNE were described recently [20, 21]. About 232 MTCVs were observed each year over the BNE and adjacent ocean region. MTCVs were predominantly short and lasted less than 12 h. The vortices persisting longer than 30 h were detected more frequently in the summer and rarely in autumn.

Frontal zones are observed regularly in the southern region of the Brazilian Northeast in the Southern Hemisphere (SH) [22, 23]. The influence of a western edge of a frontal cloud band on the weather conditions was found in the central region of the Brazilian Northeast SH and was described by Fedorova et al. [24]. The western edge of a frontal cloud band rarely passes across Alagoas State in the BNE. Only two to five frontal zones per year which directly affected the weather conditions were observed in 2004–2006. Nonetheless, Reeder and Smith [25] quoted different papers (including [26, 27]) where a cold front crossing the equator and penetrating the Northern Hemisphere (NH) tropics was documented. These fronts can initiate severe convection in the subtropic SH; however, they generally tend to suppress convection in the ITCZ.

A SASH was observed in the South Atlantic during the whole year and was located at the low levels in the eastern part of the South Atlantic, on average between 20o W and 30o S [7, 28]. It migrated from 15<sup>o</sup> W, 27o S in August, to 5<sup>o</sup> W, 33<sup>o</sup> S in February [29]. A SASH in summer is frequently split and is normally weaker [7].

Some studies show the influence of synoptic scale systems of the Northern Hemisphere on BNE weather. The influence of tropical cyclone Dany-15 on intensive fog formation in the NEB was descried by Fedorova and Levit [1]. But the information about tropical cyclone (TC) impact on BNE weather over many years has not yet been presented. Thus, the main aim of this chapter is to analyze the impact of all tropical cyclones formed from 2013 to 2015 between 20o N and the equator and which passed through 35–50o W, on fog, mist, and weak rain formation in the BNE.

## **2. Data and methodology**

## **2.1 Selection and classification of the tropical cyclones**

All tropical cyclones (TC) formed from 2013 to 2015 between 20o N and the equator whose centers passed through 35–50o W have been studied, using data from the National Hurricane Center's Tropical Cyclone Reports https://www.nhc.noaa. gov/data/tcr/.

TC has been classified according to their sustained wind speed [30] as follows: *Tropical disturbance* (*TD*). In this stage, winds are less than 17 m/s with open circulation (no closed isobars).

*Tropical depression* (*TDep*). Winds are less than 17 m/s, but there is a closed circulation (closed isobars).

*Tropical storm* (*TS*). Winds are greater than or equal to 17 m/s but less than 33 m/s with a definite closed circulation. The storm is usually assigned a name.

*Hurricane* (*H*). Winds greater than or equal to 33 m/s.

*Major hurricane* (*MH*). Typhoon with 1 min of sustained winds of 65 m/s or greater (*MH* was not observed during the study period).

*Current Topics in Tropical Cyclone Research*

Barra do Corda, Maranhao State (5<sup>o</sup>

vapor condensation, and, finally, fog formation.

with WDTW in Maceio (Alagoas State) [1, 5, 6, 11].

coastal region of the BNE [5].

fog/haze forecasting.

24 (1) and 4 (28) events/year, respectively. More frequently, fog was observed in

rare, only 4 days/year. Also, fog duration was very short, 1–2 h on average in the

Moreover, the physical mechanism of fog formation on the northern coast was atypical (typical radiation or advection fog does not occur typically) [6]. Summarizing the results of this chapter, it was possible to make the following conclusion regarding the processes of fog formation in the tropical region of the BNE. A weak confluence at the low-level trough (wave disturbances in trade winds—WDTW) contributes to weak pressure anomaly and creates the conditions for moisture convergence as well. A change in wind direction and air current from the river region contributes additional humidity for fog formation. Also, precipitation occurred before the fog events, contributing to humidification of the air through evaporation of the humid surface and raindrops. A warmer sea surface contributes to more evaporation and, as a consequence, increases the amount of water vapor in the surrounding air at the low levels near the coast. Positive latent heat flux shows a humidity increase and, therefore, moisture accumulation in the coastal region. Negative sensible heat flux results in air cooling, possible water

All this information about rare fog events with a short duration and atypical physical mechanisms of their formation demonstrates the numerous problems for

An intertropical convergence zone (ITCZ), a South Atlantic subtropical high (SASH), trade winds, and an upper tropospheric cyclonic vortex are typical synoptic scale systems associated with different weather conditions in the BNE [7].

For the first time, the influence of wave disturbances in trade winds (WDTW) on rain formation in the BNE was shown in Molion and Bernardo [8]. The relationship between different types of troughs and adverse meteorological phenomena (fog and thunderstorms) in Alagoas State was studied by Rodrigues et al. [9]. It was noted that 87% of the troughs were associated with wave disturbances in trade winds (WDTW) on the northwestern periphery of the subtropical South Atlantic High. Rare stratus cloud events with a duration of 4 days each on the northeast coast of Brazil also were formed in WDTW [10]. Fog formation is usually associated

The influence of the Brazilian Northeast Jet Stream (BNEJS) on weather in the BNE was described recently [12–14]. Particularly, the results of this study show that the period from April to October (a rainy period and the transition to a dry season in the coastal region) was characterized by a rather high number of fast BNEJSs, with high wind speed in the core, a predominant northwesterly direction, and the location of BNEJS between the upper tropospheric trough (UTT) and the SASH. Upper tropospheric cyclonic vortices (UTCVs) are important synoptic scale systems in this region; the greatest UTCV frequency occurs during the austral summer period, with the maximum frequency in January [15–18]. Intensive cloud development with precipitation was observed on the UTCV periphery and cloudlessness in their center as a result of a downward motion in its cold center and upward motion on the periphery. Rao and Bonatti [18] conjectured that barotropic instability of the regional mean basic winds at the upper levels could be one cause for UTCV formation. It was suggested that these air streams are associated with the subtropical jet stream and contribute to UTCV development. A strong positive shear zone was formed within the South Atlantic trough before the formation of the UTCV [19]. The same paper showed that the barotropic process dominates over the baroclinic

W), in 48 days/year, but haze was very

S, 45o

**28**

### **2.2 Analysis of meteorological phenomenon in the BNE**

Fog, mist, and weak rain events in the BNE were analyzed by SYNOP and METAR data. The BNE area and location of all meteorological stations are presented in **Figure 1**.

### **2.3 Synoptic and thermodynamic analysis**

Synoptic analysis was elaborated in the equatorial and tropical regions of both hemispheres between 30o N and 30o S and by latitude between 10o W and 60o W. This region includes the BNE, part of the South Atlantic and the TCs moving over the North Atlantic region.

The tropospheric structure for all TC events has been studied using data from the European Center for Medium-Range Weather Forecasts (ECMWF), with a resolution of 0.25°× 0.25°, 00, 12, 18 UTC (http://apps.ecmwf.int/datasets/ data/interim-full-daily/levtype=pl). Streamlines have been elaborated at the low (1000 hPa), middle (500 hPa), and high levels (300 and 200 hPa). Divergence maps were constructed only at the high levels (300 and 200 hPa). Both horizontal maps and vertical sections were used for vertical movement identification. These vertical sections passed through the TC center and the BNE region at the same longitude. Satellite infrared images from the GOES-13 and METEOSAT-10 (http://bancodedados.cptec.inpe.br) were used for synoptic system identification.

Thermodynamic analysis was elaborated for the days with fog, mist, and weak rain in the BNE. Vertical profiles were constructed using the same data from the ECMWF quoted above.

**31**

**Table 1.**

2013

2014

2015

*National Hurricane Center, 2017.*

*Information about time period and intensity of tropical cyclones (TC).*

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation…*

Information about tropical cyclones (TC) formed during 2013–2015 between

**3.2 Information about fog, mist, and rain formation by METAR and SYNOP data**

Information about fog, mist, and weak rain events in the BNE by METAR data

examples of maps with this information are presented in **Figure 2**. Fog, mist. and weak rain events were observed more frequently in the coastal region, e.g., **Figure 2**. Only one fog event was registered far from the coastal region, occurring in the Maranhao State (SBIZ station) (**Figure 2a**). Fog, mist, and weak rain events were observed more frequently in the southern region of the BNE, in the Bahia State. Six events were registered in the southern region of the Bahia (SBQV), not far from the coastal region (300 km approximately). Also, fog was detected in the coastal stations of this region (SBIL and SBPS). Only one event was recorded in the northern coastal region in the Pernambuco State (SBRF). An influence of seven TCs on fog rain formation in the BNE was confirmed (**Table 2**). During three TCs (Edouard, Fred, and Grace), METAR data did not record any fog events; at the same time, weak rain and mist were detected. A *Tropical disturbance and Tropical storm* were the predominant

**Name Data TC intensity**

Chantal 06 July 10 July 7–10 July *TS TS* Dorian 22 July 03 August 24–27 July *TS TS* Erin 15 August 20 August 16–20 August *TS TS-D*

Bertha 29 July 06 August 29–31 July *H TD* Edouard 10 September 23 September 10–14 September *H TD-H*

Danny 17 August 24 August 17–22 August *H TD-H* Erika 24 August 28 August 24–25 August *TS TS* Fred 30 August 06 September 02–05 September *H TS-TDep* Grace 05 September 09 September 06–09 September *TS TS-D* Ida 15 September 28 September 17–28 September *TS TD-D TD, tropical disturbance; TDep, tropical depression; TS, tropical storm; H, hurricane; D, dissipation. Source:* 

**Formation Dissipation BNE longitude Maximal BNE longitude**

W is presented in **Figure 2** and **Table 2**. Two

tropical cyclones were registered during this time period, and their influences have been analyzed. Five TCs were observed in 2015, three of them in 2013 and

W is presented in **Table 1**. Ten

*DOI: http://dx.doi.org/10.5772/intechopen.88804*

**3.1 Information about tropical cyclones**

N and the equator and passing through 35–50o

TC stages during its influence on the weather in the BNE.

**3. Results**

only two in 2014.

when TCs passed through 35–50o

20o

**Figure 1.** *The location of the BNE, States, and all meteorological stations (green points) with names.*

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation… DOI: http://dx.doi.org/10.5772/intechopen.88804*

## **3. Results**

*Current Topics in Tropical Cyclone Research*

**2.3 Synoptic and thermodynamic analysis**

in **Figure 1**.

hemispheres between 30o

North Atlantic region.

ECMWF quoted above.

**2.2 Analysis of meteorological phenomenon in the BNE**

N and 30o

dos.cptec.inpe.br) were used for synoptic system identification.

*The location of the BNE, States, and all meteorological stations (green points) with names.*

Fog, mist, and weak rain events in the BNE were analyzed by SYNOP and METAR data. The BNE area and location of all meteorological stations are presented

Synoptic analysis was elaborated in the equatorial and tropical regions of both

The tropospheric structure for all TC events has been studied using data from

Thermodynamic analysis was elaborated for the days with fog, mist, and weak rain in the BNE. Vertical profiles were constructed using the same data from the

region includes the BNE, part of the South Atlantic and the TCs moving over the

the European Center for Medium-Range Weather Forecasts (ECMWF), with a resolution of 0.25°× 0.25°, 00, 12, 18 UTC (http://apps.ecmwf.int/datasets/ data/interim-full-daily/levtype=pl). Streamlines have been elaborated at the low (1000 hPa), middle (500 hPa), and high levels (300 and 200 hPa). Divergence maps were constructed only at the high levels (300 and 200 hPa). Both horizontal maps and vertical sections were used for vertical movement identification. These vertical sections passed through the TC center and the BNE region at the same longitude. Satellite infrared images from the GOES-13 and METEOSAT-10 (http://bancodeda-

S and by latitude between 10o

W and 60o

W. This

**30**

**Figure 1.**

## **3.1 Information about tropical cyclones**

Information about tropical cyclones (TC) formed during 2013–2015 between 20o N and the equator and passing through 35–50o W is presented in **Table 1**. Ten tropical cyclones were registered during this time period, and their influences have been analyzed. Five TCs were observed in 2015, three of them in 2013 and only two in 2014.

## **3.2 Information about fog, mist, and rain formation by METAR and SYNOP data**

Information about fog, mist, and weak rain events in the BNE by METAR data when TCs passed through 35–50o W is presented in **Figure 2** and **Table 2**. Two examples of maps with this information are presented in **Figure 2**. Fog, mist. and weak rain events were observed more frequently in the coastal region, e.g., **Figure 2**. Only one fog event was registered far from the coastal region, occurring in the Maranhao State (SBIZ station) (**Figure 2a**). Fog, mist, and weak rain events were observed more frequently in the southern region of the BNE, in the Bahia State. Six events were registered in the southern region of the Bahia (SBQV), not far from the coastal region (300 km approximately). Also, fog was detected in the coastal stations of this region (SBIL and SBPS). Only one event was recorded in the northern coastal region in the Pernambuco State (SBRF). An influence of seven TCs on fog rain formation in the BNE was confirmed (**Table 2**). During three TCs (Edouard, Fred, and Grace), METAR data did not record any fog events; at the same time, weak rain and mist were detected. A *Tropical disturbance and Tropical storm* were the predominant TC stages during its influence on the weather in the BNE.


#### **Table 1.**

*Information about time period and intensity of tropical cyclones (TC).*

#### **Figure 2.**

*(a) Location of the meteorological stations detected fog (yellow points) in the BNE by METAR during all TC events. (b) and (c) Location of the meteorological stations detected mist events (yellow points) 29 July 2014 12 UTC in the BNE by METAR (c) and SYNOP (b). Green points mark all meteorological stations.*

#### **3.3 Synoptic systems associated with fog events**

#### *3.3.1 1000 hPa*

A synoptic situation at 1000 hPa was very similar during all fog, mist, and weak rain events (**Table 3**). Trade winds presented cyclonic curvature (*weak trough*) over the ocean close to the coastal area and anticyclonic curvature (*weak ridge*) over the

**33**

has action on trough formation.

*data (inclined) during the time period with TC passing through 35–50o*

*tropical storm. Source: DECEA, 2017.*

*3.3.2 500 hPa*

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation…*

**TC name,** *stage* **Data State Station Hours (local)**

25 July Paraiba SBKG 06, 09, 10 25 July Maranhao SBIZ 08 26 July Paraiba SBKG 03–08

18 August Pernambuco SBRF 15

31 July Bahia South SBQV 10, 11 **01** August Bahia South SBQV 09

20 August Bahia South SBQV 09, 10 21 August Paraiba SBKG 04, 05 21 August Bahia South SBQV 10, 11

*W.*

*Paraiba 06*

*Pernambuco 12*

Chantal *TS* 08 July Bahia South SBQV 08, 09 Dorian *TS* **22** July Bahia South SBQV 10, 11

Erin *TS* **15** August Bahia North SBSV 08–10

Bertha *TD 29 July Pernambuco 12*

Danny *TD 17 August Pernambuco 12*

Erika *TS* 24 August Bahia South SBIL 09, 10 Fred — — — — Grace — — — — Ida *TD* 18 September Bahia South SBPS 09 *Bold data show the information when a TC passes before or after the BNE longitudes. TD, tropical disturbance; TS,* 

Edouard — — —

northern region of the BNE (**Figure 3**). This weak trough was located close to the frontal extremity over the south of the Bahia State in five events. It is important to note that a frontal extremity does not have a direct influence in the fog region but

*Information (TC name, state, station, and hour) about fog events in the BNE by METAR data and SYNOP* 

Anticyclonic circulation over the BNE was the predominant process at the middle levels in days with fog, mist, and weak rain events (**Figure 4**, **Table 3**). Also, a convergence zone was observed in the north and middle regions of the BNE in all events. This convergence zone was formed between the currents from a tropical cyclone or tropical

N)

S, 54o

W).

depression in the Northern Hemisphere and this anticyclonic circulation in the Southern Hemisphere. The current from the NH was formed as a result of the divergence at the middle levels of the tropical cyclone or tropical depression. For example, **Figure 3** shows airflow from the tropical depression on the west coast of Africa (6o

converging with the air current of the high (with the center at 6o

*DOI: http://dx.doi.org/10.5772/intechopen.88804*

2013

2014

2015

**Table 2.**


*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation… DOI: http://dx.doi.org/10.5772/intechopen.88804*

*Bold data show the information when a TC passes before or after the BNE longitudes. TD, tropical disturbance; TS, tropical storm. Source: DECEA, 2017.*

#### **Table 2.**

*Current Topics in Tropical Cyclone Research*

**32**

*3.3.1 1000 hPa*

**Figure 2.**

**3.3 Synoptic systems associated with fog events**

A synoptic situation at 1000 hPa was very similar during all fog, mist, and weak rain events (**Table 3**). Trade winds presented cyclonic curvature (*weak trough*) over the ocean close to the coastal area and anticyclonic curvature (*weak ridge*) over the

*(a) Location of the meteorological stations detected fog (yellow points) in the BNE by METAR during all TC events. (b) and (c) Location of the meteorological stations detected mist events (yellow points) 29 July 2014 12* 

*UTC in the BNE by METAR (c) and SYNOP (b). Green points mark all meteorological stations.*

*Information (TC name, state, station, and hour) about fog events in the BNE by METAR data and SYNOP data (inclined) during the time period with TC passing through 35–50o W.*

northern region of the BNE (**Figure 3**). This weak trough was located close to the frontal extremity over the south of the Bahia State in five events. It is important to note that a frontal extremity does not have a direct influence in the fog region but has action on trough formation.

#### *3.3.2 500 hPa*

Anticyclonic circulation over the BNE was the predominant process at the middle levels in days with fog, mist, and weak rain events (**Figure 4**, **Table 3**). Also, a convergence zone was observed in the north and middle regions of the BNE in all events. This convergence zone was formed between the currents from a tropical cyclone or tropical depression in the Northern Hemisphere and this anticyclonic circulation in the Southern Hemisphere. The current from the NH was formed as a result of the divergence at the middle levels of the tropical cyclone or tropical depression. For example, **Figure 3** shows airflow from the tropical depression on the west coast of Africa (6o N) converging with the air current of the high (with the center at 6o S, 54o W).


*A, anticyclonic circulation; TO (trough over the ocean); RC (ridge over the continent); TW, trade winds nearly straight (no pronounced curvature); FZ (frontal zone), TCO (trough at the middle and high levels over the continent and ocean); SJSHS (subtropical jet stream of the Southern Hemisphere)—location of the northern boundary (o W); MTCVo and MTCVc—middle tropospheric cyclonic vortices over the oceanic and continental regions, respectively. Information inside brackets represents number of days.*

#### **Table 3.**

*Synoptic systems in the BNE at the low (1000 hPa), middle (500 hPa), and high (300 and 200 hPa) levels during tropical cyclones in the northern hemisphere.*

Middle tropospheric cyclonic vortices (MTCV) were observed in five studied events (**Table 3**). MTCVs were located over the adjacent ocean region in two of the five events, in the northeastern part and in the coastal region of the continent in one case each. Four MTCVs were typical in terms of the level of its location and duration, were short, and lasted 6–12 h. Only one MTCV lasted 36 h, and its center was located in the south of the BNE, and its circulation affected only southern part of the BNE. The MTCVs created the waves on the air current from the NH but did not change the principal influence of this northern current. Meanwhile one MTCV on 17 Ag 2013, 12UTC, was formed between west flow of the SH and

**35**

**Figure 4.**

*tropical cyclone; and -• -• -, convergence zone.*

**Figure 3.**

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation…*

*Streamlines at 1000 hPa on 22 July 2013, 18 UTC. Trough, dashed line; ridge, dotted line; and TC, tropical cyclone.*

*Streamlines at 500 hPa, 17 August 2013, 00UTC. H, anticyclonic circulation; TD, tropical disturbance; TC,* 

*DOI: http://dx.doi.org/10.5772/intechopen.88804*

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation… DOI: http://dx.doi.org/10.5772/intechopen.88804*

#### **Figure 3.**

*Current Topics in Tropical Cyclone Research*

Danny *TDis TO/RC* (17–19), *TO/RC/FZ* (20–21)

*TO/RC/FZ* (2–3)

Ida *TO/RC* (16–18) A (16–17)

*Information inside brackets represents number of days.*

*during tropical cyclones in the northern hemisphere.*

Fred *TO/RC* (1)

2013

**2014**

**2015**

**TC name,** *stage* **1000 hPa 500 hPa 200/300 hPa**

Chantal *TS TO/RC* (7), *TO/RC/FZ* (8,9) A (7–9) *TCO* (6, 7)

Erin *TO/RC/FZ* (16–17) MTCVc, A (16–18) A (16)

(23–25)

A, MTCVc (17–18), A

MTCVo A (16–17)

A (1–4) *TCO* (1–2)

(19–22)

Bertha *TDis TO/RC/FZ* (29–31–1) A (29–31–1) *TCO* (29–31–1), SJSHS

Edouard *TO/RC* (10–13) A (10), MTCVc (10, 11) *TCO* (10–11)

Erika *TO/RC/FZ* (24–26) A (24–26) *TCO* (24–26)

Grace *TW(6–9)* A (7–9) *TCO* (7)

*A, anticyclonic circulation; TO (trough over the ocean); RC (ridge over the continent); TW, trade winds nearly straight (no pronounced curvature); FZ (frontal zone), TCO (trough at the middle and high levels over the continent and ocean); SJSHS (subtropical jet stream of the Southern Hemisphere)—location of the northern boundary (o*

*MTCVo and MTCVc—middle tropospheric cyclonic vortices over the oceanic and continental regions, respectively.*

*Synoptic systems in the BNE at the low (1000 hPa), middle (500 hPa), and high (300 and 200 hPa) levels* 

Dorian *TS TO/RC/FZ* (22–26) A (22–25), MTCVo

A (8) *TCO* (9) SJSHS 10–15o

A (22–23) *TCO* (24, 25) A (26, 27) SJSHS 20o

*TCO* (17) A (18) SJSHS 20o

10–15o

A (12–13) SJSHS 10–15o

*TCO* (17–20) SJSHS 10–15o (17–20)

SJSHS 10–15o (24–26)

A (3–4) SJSHS 10–15o

A (8–9) SJSHS 10–15o

A (17–19)

*W);* 

**34**

**Table 3.**

Middle tropospheric cyclonic vortices (MTCV) were observed in five studied events (**Table 3**). MTCVs were located over the adjacent ocean region in two of the five events, in the northeastern part and in the coastal region of the continent in one case each. Four MTCVs were typical in terms of the level of its location and duration, were short, and lasted 6–12 h. Only one MTCV lasted 36 h, and its center was located in the south of the BNE, and its circulation affected only southern part of the BNE. The MTCVs created the waves on the air current from the NH but did not change the principal influence of this northern current. Meanwhile one MTCV on 17 Ag 2013, 12UTC, was formed between west flow of the SH and

*Streamlines at 1000 hPa on 22 July 2013, 18 UTC. Trough, dashed line; ridge, dotted line; and TC, tropical cyclone.*

#### **Figure 4.**

*Streamlines at 500 hPa, 17 August 2013, 00UTC. H, anticyclonic circulation; TD, tropical disturbance; TC, tropical cyclone; and -• -• -, convergence zone.*

northeast current from the tropical disturbance of the NH near Africa and lasted 6 h. Other four MTCVs were formed in the flow of the southern hemisphere without the TC influence.

#### *3.3.3 200/300 hPa*

The principal influence of the NH flow was observed at the high levels (**Figure 5**, **Table 3**). Divergent flow was observed at the high levels in the TC. The height of this divergent flow depended on the TC intensity and was registered at 300 and 200 hPa. This divergent flow at the high levels passed through the equator to the BNE region. This flow from the TC was connected with anticyclonic circulation over the equatorial and south Atlantic. Thus, a single current was formed in the Northern Hemisphere, from the north or northeast.

Also, the anticyclonic circulation was predominant over the central and south region of the BNE. The location of this high was dependent upon the subtropical jet stream of the Southern Hemisphere (SJSHS). The northern boundary of this jet stream reached 15 or 30o W in the study events and was not observed over the BNE region. Curvature of the SJSHS was cyclonic (*trough*, *upper tropospheric trough*) over the continental region and anticyclonic (*ridge*) over the ocean region.

These two currents from the Northern Hemisphere and anticyclonic circulation of the Southern Hemisphere created the convergence of air current which, in turn, produced air sinking.

Two examples of this flow at the high levels can be seen in **Figures 5** and **6**. The difference between these examples is in the location of the Highs over the Atlantic Ocean. Meanwhile, the location of the convergence zone over the BNE was very similar.

#### **Figure 5.**

*Streamlines at 300 hPa 18 September 2015, 12UTC. H*, *anticyclonic circulation; TD*, *tropical disturbance; and TC*, *tropical cyclone. -• -• -, convergence zone.*

**37**

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation…*

A slightly different situation was observed during the TC Dorian on 25 July 2013 (**Figure 7**). This difference was associated with jet stream formation from the Northern to Southern Hemisphere, denominated as the Brazilian Northeast Jet Stream—BNEJS [20]. The data from this paper shows that, generally, BNEJS from the north was observed very rarely. On this day, the BNEJS was formed between two highs: in the north (near Maranhao State, high 1) and northeast of Brazil (high 2). The high 1 location was more to the north than the typical high position described before. At the same time, the high 2 location was close to the continental region, with its center on the northeastern cape of the South American continent. These highs squeezed the flow from the TC, and so the BNEJS was created. Also, the

*Streamlines at 300 hPa 16 August 2013, 00UTC.* H*, anticyclonic circulation;* TD*, tropical disturbance;* TC*,* 

vergence zone between the currents from the south and north was detected in the

levels and anticyclonic circulation at the middle levels were formed.

**3.4 Mechanism of fog, mist, and weak rain formation**

A scheme of the synoptic systems, observed more frequently at all levels simultaneously during the TC and fog events, is presented in **Figure 8**. This figure shows the circulation pattern, which contributes to fog, mist, and weak rain formation in the BNE. As a result of this circulation, a convergence zone at the middle and high

The circulation described in Section 3.3 creates sinking over the BNE. Vertical movements were analyzed in all events, and an example of this vertical motion is presented in **Figure 9**. Lifting in TC Bertha can be seen in **Figure 9a** and **d**. The vertical movement reaches 0.3 m/s. Very intense diffluent current was observed at

W on this day. Therefore, the con-

*DOI: http://dx.doi.org/10.5772/intechopen.88804*

northern boundary of the SJSHS reached 15o

*tropical cyclone; and -• -• -, convergence zone.*

south BNE region.

**Figure 6.**

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation… DOI: http://dx.doi.org/10.5772/intechopen.88804*

#### **Figure 6.**

*Current Topics in Tropical Cyclone Research*

Northern Hemisphere, from the north or northeast.

without the TC influence.

stream reached 15 or 30o

produced air sinking.

similar.

*3.3.3 200/300 hPa*

northeast current from the tropical disturbance of the NH near Africa and lasted 6 h. Other four MTCVs were formed in the flow of the southern hemisphere

The principal influence of the NH flow was observed at the high levels (**Figure 5**, **Table 3**). Divergent flow was observed at the high levels in the TC. The height of this divergent flow depended on the TC intensity and was registered at 300 and 200 hPa. This divergent flow at the high levels passed through the equator to the BNE region. This flow from the TC was connected with anticyclonic circulation over the equatorial and south Atlantic. Thus, a single current was formed in the

Also, the anticyclonic circulation was predominant over the central and south region of the BNE. The location of this high was dependent upon the subtropical jet stream of the Southern Hemisphere (SJSHS). The northern boundary of this jet

region. Curvature of the SJSHS was cyclonic (*trough*, *upper tropospheric trough*) over

These two currents from the Northern Hemisphere and anticyclonic circulation of the Southern Hemisphere created the convergence of air current which, in turn,

Two examples of this flow at the high levels can be seen in **Figures 5** and **6**. The difference between these examples is in the location of the Highs over the Atlantic Ocean. Meanwhile, the location of the convergence zone over the BNE was very

*Streamlines at 300 hPa 18 September 2015, 12UTC. H*, *anticyclonic circulation; TD*, *tropical disturbance; and* 

the continental region and anticyclonic (*ridge*) over the ocean region.

W in the study events and was not observed over the BNE

**36**

**Figure 5.**

*TC*, *tropical cyclone. -• -• -, convergence zone.*

*Streamlines at 300 hPa 16 August 2013, 00UTC.* H*, anticyclonic circulation;* TD*, tropical disturbance;* TC*, tropical cyclone; and -• -• -, convergence zone.*

A slightly different situation was observed during the TC Dorian on 25 July 2013 (**Figure 7**). This difference was associated with jet stream formation from the Northern to Southern Hemisphere, denominated as the Brazilian Northeast Jet Stream—BNEJS [20]. The data from this paper shows that, generally, BNEJS from the north was observed very rarely. On this day, the BNEJS was formed between two highs: in the north (near Maranhao State, high 1) and northeast of Brazil (high 2). The high 1 location was more to the north than the typical high position described before. At the same time, the high 2 location was close to the continental region, with its center on the northeastern cape of the South American continent. These highs squeezed the flow from the TC, and so the BNEJS was created. Also, the northern boundary of the SJSHS reached 15o W on this day. Therefore, the convergence zone between the currents from the south and north was detected in the south BNE region.

A scheme of the synoptic systems, observed more frequently at all levels simultaneously during the TC and fog events, is presented in **Figure 8**. This figure shows the circulation pattern, which contributes to fog, mist, and weak rain formation in the BNE. As a result of this circulation, a convergence zone at the middle and high levels and anticyclonic circulation at the middle levels were formed.

#### **3.4 Mechanism of fog, mist, and weak rain formation**

The circulation described in Section 3.3 creates sinking over the BNE. Vertical movements were analyzed in all events, and an example of this vertical motion is presented in **Figure 9**. Lifting in TC Bertha can be seen in **Figure 9a** and **d**. The vertical movement reaches 0.3 m/s. Very intense diffluent current was observed at

#### **Figure 7.**

*Streamlines at 200 hPa, 25 July 2013, 06UTC. H, anticyclonic circulation; TC, tropical cyclone; and -• -• -, convergence zone. The black arrows show the BNEJS.*

#### **Figure 8.**

*Synoptic systems, observed more frequently at all levels simultaneously. TD, tropical disturbance; and TC, tropical cyclone.*

**39**

all TC days.

**Figure 9.**

*velocity (a, Pa s<sup>−</sup><sup>1</sup>*

**4. Conclusion**

ing its passage in this area*.*

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation…*

300 hPa in TC Bertha (**Figure 9b**). Airflow at the high levels from TC Bertha to the

*Circulation between TC, TD is*, *and fog formation region on 29 June 2014, 06 UTC, in different maps: Vertical* 

*), streamlines and wind velocity (c, m s<sup>−</sup><sup>1</sup>*

*W (longitude of TC Bertha).*

The impact of all studied (10) TCs on the weather conditions in the BNE was analyzed during July to September from 2013 to 2015. These TCs had a *tropical* 

Hemisphere. Only one, TC Bertha, had the stage of *tropical disturbance (TDis)* dur-

W and up to 20o

N in the Northern

*) all at 300 hPa and (d)* 

These circulations create sinking over the BNE and, as a result, humidity accumulation at the low levels and fog, mist, and weak rain formation. These phenomena were detected as the principal adverse phenomena in the BNE during

*) along 37.1o*

BNE is presented in **Figure 9a** and **c**.

*vertical section of vertical velocity (Pa s<sup>−</sup><sup>1</sup>*

*) and divergence (b, s<sup>−</sup><sup>1</sup>*

*storm* stage, when passing between 35 and 50o

*DOI: http://dx.doi.org/10.5772/intechopen.88804*

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation… DOI: http://dx.doi.org/10.5772/intechopen.88804*

**Figure 9.**

*Current Topics in Tropical Cyclone Research*

**38**

**Figure 8.**

*tropical cyclone.*

**Figure 7.**

*convergence zone. The black arrows show the BNEJS.*

*Streamlines at 200 hPa, 25 July 2013, 06UTC. H, anticyclonic circulation; TC, tropical cyclone; and -• -• -,* 

*Synoptic systems, observed more frequently at all levels simultaneously. TD, tropical disturbance; and TC,* 

*Circulation between TC, TD is*, *and fog formation region on 29 June 2014, 06 UTC, in different maps: Vertical velocity (a, Pa s<sup>−</sup><sup>1</sup> ) and divergence (b, s<sup>−</sup><sup>1</sup> ), streamlines and wind velocity (c, m s<sup>−</sup><sup>1</sup> ) all at 300 hPa and (d) vertical section of vertical velocity (Pa s<sup>−</sup><sup>1</sup> ) along 37.1o W (longitude of TC Bertha).*

300 hPa in TC Bertha (**Figure 9b**). Airflow at the high levels from TC Bertha to the BNE is presented in **Figure 9a** and **c**.

These circulations create sinking over the BNE and, as a result, humidity accumulation at the low levels and fog, mist, and weak rain formation. These phenomena were detected as the principal adverse phenomena in the BNE during all TC days.

#### **4. Conclusion**

The impact of all studied (10) TCs on the weather conditions in the BNE was analyzed during July to September from 2013 to 2015. These TCs had a *tropical storm* stage, when passing between 35 and 50o W and up to 20o N in the Northern Hemisphere. Only one, TC Bertha, had the stage of *tropical disturbance (TDis)* during its passage in this area*.*

The fog, mist, and weak rain in the BNE were the adverse phenomena associated more frequently with these TCs. Days with mist were also accompanied by weak rain. These phenomena were observed in all TC events and predominantly near the coastal region of the BNE.

The synoptic situation at 1000 hPa shows a wave disturbance near the BNE coastal area with a weak trough over the ocean close to the coastal area and a weak ridge over the northern region of the BNE. This situation was detected in all study events. Frontal extremity over the south of the Bahia State had an influence on trough intensification.

The convergence zone at the middle levels in the northern or middle region of the BNE was formed in all fog events between the currents from a TC in the Northern Hemisphere and anticyclonic circulation in the Southern Hemisphere. This anticyclonic circulation was predominant over the greater part of the BNE. Middle tropospheric cyclonic vortices were observed during half of the events but did not change the principal influence of the airflow from the TCs. One MTCV was formed between west flow of the Southern Hemisphere and northeast current from the tropical disturbance of the Northern Hemisphere.

The airflow at the high levels from the Northern Hemisphere is formed by the coupling (joining) of two currents: (1) from a TC and (2) from the high over the equatorial and south Atlantic. The confluence of these airflows with the current from the trough created by the subtropical jet stream of the Southern Hemisphere was the principal mechanism of the sinking in the BNE.

This sinking over the BNE formed the accumulation of humidity at the low levels and, as a result, fog, mist, and weak rain formation. Therefore, fog, mist, and weak rain were the principal adverse phenomena in the BNE during TC days.

This information can be used for short-term forecasting of adverse phenomena, such as fog and mist with weak rain in the BNE.

### **Acknowledgements**

The authors thank English editor Robert Dower for his careful revision of this manuscript.

#### **Author details**

Natalia Fedorova\*, Vladimir Levit and Lucas Carvalho Vieira Cavalcante Institute of Atmospheric Science, Federal University of Alagoas, Brazil

\*Address all correspondence to: nataliabras@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**41**

*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation…*

Atmospheric Science Letters. 2010. DOI:

[10] Gomes HB, Fedorova N, Levit V. Rare events of stratus clouds on the northeast coast of Brazil. Revista Brasileira de Meteorologia. 2011;**26**(1):9-18

[11] Fedorova N, Levit V, Silva AO, Santos DMB. Low visibility formation and forecasting on the northern coast of Brazil. Pure and Applied Geophysics. 2013;**170**(4):689-709. DOI: 10.1007/

[12] Fedorova N, Levit V, Campos AMV.

Frequency, wind speed and direction.

2018;**25**:254-260. DOI: 10.1002/met.1688

[13] Fedorova N, Levit V, Campos AMV.

[14] Fedorova N, Lyra MJA. Capítulo 1. Corrente de jato e fenômenos

[15] Gan MA, Kousky VE. Vórtices ciclônicos da Alta troposfera no Oceano Atlântico Sul (Cyclonic vortices in the upper troposphere the South Atlantic Ocean). Revista Brasileira de

associados. In: Meteorologia em tópicos. Vol. 5. Pelotas: UFPel; 2017. pp. 11-68.

Brazilian northeast jet stream:

Meteorological Applications.

Brazilian northeast jet stream: Association with synoptic scale systems. Meteorological Applications. 2018;**25**:261-268. DOI: 10.1002/met.1693

ISBN 978-85-68891-04-9

Meteorologia. 1986;**1**:19-28

1981;**33**(6):538-551

[16] Kousky VE, Gan MA. Upper tropospheric cyclonic vortices in the tropical South Atlantic. Tellus.

[17] Mishra SK, Rao VB, Franchito SH. Genesis of Northeast Brazil upper tropospheric cyclonic vortex: A

primitive equation barotropic instability study. Journal of the Atmospheric Sciences. 2007;**64**:1379-1392

10.1002/asl.273

s00024-012-0565-6

*DOI: http://dx.doi.org/10.5772/intechopen.88804*

[1] Fedorova N, Levit V. Fog in the Tropical Region. Fog Formation in the Tropical Region of the Northeast of Brazil. Saarbrücken, Germany: LAP LAMBERT Academic Publishing; 2016,

82 p. ISBN: 978-3-659-87098-9

Company; 1976. pp. 219-269

University of Alagoas; 2003

2008;**87**:268-278

s00024-014-1027-0

pp. 119-139

[2] Schwerdtfeger W. Climates of Central and South America. New York: Elsevier Scientific Publishing Company; 1976

[3] Ratisbona LR. The climate of Brazil. Chapter 5. In: Schwerdtfeger W, editor. Climates of Central and South America. Oxford: Elsevier Scientific Publishing

[4] Silveira PS. Analysis of the fog cases and stratus clouds in Airport of Maceio [MSc thesis]. Maceio, Brazil: Federal

[5] Fedorova N, Levit V, Fedorov D. Fog and stratus formation on the coast of Brazil. Atmospheric Research.

[6] Fedorova N, Levit V, Souza JL, Silva AO, Afonso JMS, Teodoro I. Fog events at Maceio airport on the northern coast of Brazil during 2002-2005 and 2007. Pure and Applied Geophysics. 2015;**172**:2727-2749. DOI: 10.1007/

[7] Satyamurty P, Nobre CA, Silva Dias PL. South America. In: Karoly DJ, Vincent DG, editors. Meteorology of Southern Hemisphere. Boston: American Meteorological Society; 1998.

[8] Molion LCB, Bernardo SO. Uma revisão da dinâmica das chuvas no Nordeste Brasileiro. Revista Brasileira de

[9] Rodrigues LRL, Fedorova N, Levit V. Adverse meteorological phenomena associated with low level baric troughs in the Alagoas State in 2003.

Meteorologia. 2002;**17**(1):1-10

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*Impacts of Tropical Cyclones in the Northern Atlantic on Adverse Phenomena Formation… DOI: http://dx.doi.org/10.5772/intechopen.88804*

## **References**

*Current Topics in Tropical Cyclone Research*

coastal region of the BNE.

trough intensification.

The fog, mist, and weak rain in the BNE were the adverse phenomena associated more frequently with these TCs. Days with mist were also accompanied by weak rain. These phenomena were observed in all TC events and predominantly near the

The synoptic situation at 1000 hPa shows a wave disturbance near the BNE coastal area with a weak trough over the ocean close to the coastal area and a weak ridge over the northern region of the BNE. This situation was detected in all study events. Frontal extremity over the south of the Bahia State had an influence on

The convergence zone at the middle levels in the northern or middle region of the BNE was formed in all fog events between the currents from a TC in the Northern Hemisphere and anticyclonic circulation in the Southern Hemisphere. This anticyclonic circulation was predominant over the greater part of the

BNE. Middle tropospheric cyclonic vortices were observed during half of the events but did not change the principal influence of the airflow from the TCs. One MTCV was formed between west flow of the Southern Hemisphere and northeast current

The airflow at the high levels from the Northern Hemisphere is formed by the coupling (joining) of two currents: (1) from a TC and (2) from the high over the equatorial and south Atlantic. The confluence of these airflows with the current from the trough created by the subtropical jet stream of the Southern Hemisphere

This sinking over the BNE formed the accumulation of humidity at the low levels and, as a result, fog, mist, and weak rain formation. Therefore, fog, mist, and weak rain were the principal adverse phenomena in the BNE during TC days.

This information can be used for short-term forecasting of adverse phenomena,

The authors thank English editor Robert Dower for his careful revision of this

from the tropical disturbance of the Northern Hemisphere.

was the principal mechanism of the sinking in the BNE.

such as fog and mist with weak rain in the BNE.

**40**

**Author details**

manuscript.

**Acknowledgements**

Natalia Fedorova\*, Vladimir Levit and Lucas Carvalho Vieira Cavalcante Institute of Atmospheric Science, Federal University of Alagoas, Brazil

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: nataliabras@gmail.com

provided the original work is properly cited.

[1] Fedorova N, Levit V. Fog in the Tropical Region. Fog Formation in the Tropical Region of the Northeast of Brazil. Saarbrücken, Germany: LAP LAMBERT Academic Publishing; 2016, 82 p. ISBN: 978-3-659-87098-9

[2] Schwerdtfeger W. Climates of Central and South America. New York: Elsevier Scientific Publishing Company; 1976

[3] Ratisbona LR. The climate of Brazil. Chapter 5. In: Schwerdtfeger W, editor. Climates of Central and South America. Oxford: Elsevier Scientific Publishing Company; 1976. pp. 219-269

[4] Silveira PS. Analysis of the fog cases and stratus clouds in Airport of Maceio [MSc thesis]. Maceio, Brazil: Federal University of Alagoas; 2003

[5] Fedorova N, Levit V, Fedorov D. Fog and stratus formation on the coast of Brazil. Atmospheric Research. 2008;**87**:268-278

[6] Fedorova N, Levit V, Souza JL, Silva AO, Afonso JMS, Teodoro I. Fog events at Maceio airport on the northern coast of Brazil during 2002-2005 and 2007. Pure and Applied Geophysics. 2015;**172**:2727-2749. DOI: 10.1007/ s00024-014-1027-0

[7] Satyamurty P, Nobre CA, Silva Dias PL. South America. In: Karoly DJ, Vincent DG, editors. Meteorology of Southern Hemisphere. Boston: American Meteorological Society; 1998. pp. 119-139

[8] Molion LCB, Bernardo SO. Uma revisão da dinâmica das chuvas no Nordeste Brasileiro. Revista Brasileira de Meteorologia. 2002;**17**(1):1-10

[9] Rodrigues LRL, Fedorova N, Levit V. Adverse meteorological phenomena associated with low level baric troughs in the Alagoas State in 2003.

Atmospheric Science Letters. 2010. DOI: 10.1002/asl.273

[10] Gomes HB, Fedorova N, Levit V. Rare events of stratus clouds on the northeast coast of Brazil. Revista Brasileira de Meteorologia. 2011;**26**(1):9-18

[11] Fedorova N, Levit V, Silva AO, Santos DMB. Low visibility formation and forecasting on the northern coast of Brazil. Pure and Applied Geophysics. 2013;**170**(4):689-709. DOI: 10.1007/ s00024-012-0565-6

[12] Fedorova N, Levit V, Campos AMV. Brazilian northeast jet stream: Frequency, wind speed and direction. Meteorological Applications. 2018;**25**:254-260. DOI: 10.1002/met.1688

[13] Fedorova N, Levit V, Campos AMV. Brazilian northeast jet stream: Association with synoptic scale systems. Meteorological Applications. 2018;**25**:261-268. DOI: 10.1002/met.1693

[14] Fedorova N, Lyra MJA. Capítulo 1. Corrente de jato e fenômenos associados. In: Meteorologia em tópicos. Vol. 5. Pelotas: UFPel; 2017. pp. 11-68. ISBN 978-85-68891-04-9

[15] Gan MA, Kousky VE. Vórtices ciclônicos da Alta troposfera no Oceano Atlântico Sul (Cyclonic vortices in the upper troposphere the South Atlantic Ocean). Revista Brasileira de Meteorologia. 1986;**1**:19-28

[16] Kousky VE, Gan MA. Upper tropospheric cyclonic vortices in the tropical South Atlantic. Tellus. 1981;**33**(6):538-551

[17] Mishra SK, Rao VB, Franchito SH. Genesis of Northeast Brazil upper tropospheric cyclonic vortex: A primitive equation barotropic instability study. Journal of the Atmospheric Sciences. 2007;**64**:1379-1392

[18] Rao VB, Bonatti JP. On the origin of upper tropospheric cyclonic vortex in the South Atlantic Ocean and adjoin Brazil during summer. Meteorology and Atmospheric Physics. 1987;**37**:11-16

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[21] Pontes da Silva BF, Fedorova N, Levit V, Peresetsky A. Sistemas sinóticos associados às precipitações intensas no Estado de Alagoas (Synoptic systems associated with the precipitations in the Alagoas State). Revista Brasileira de Meteorologia. 2011;**26**(3):295-310

[22] Fedorova N, Carvalho MH. Processos sinóticos em anos de La Niña e de El Niño. Parte II: Zonas Frontais. Revista Brasileira de Meteorologia. 2000;**15**(2):57-72

[23] Kousky VE. Frontal influences on Northeast Brazil. Monthly Weather Review. 1979;**107**(9):1140-1153

[24] Fedorova N, Levit V, Cruz CD. On frontal zone analysis in the tropical region of the Northeast Brazil. Pure and Applied Geophysics. 2016;**173**:1403-1421. DOI: 10.1007/s00024-015-1166-y

[25] Reeder MJ, Smith RK. Mesoscale meteorology. In: Meteorology of the Southern Hemisphere. Vol. 27(49). Boston: American Meteorological Society; 1998. pp. 201-241

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**Chapter 3**

**Abstract**

Summer 2006

*Tiffany Reyes and Bo-Wen Shen*

or decaying amplitudes of multiple AEWs.

**1. Introduction**

**43**

**Keywords:** recurrence plot, chaos, limit cycle, AEWs, Lorenz model

Recent studies suggest that accurate detection of recurrent, multiple, large-scale tropical waves, such as African easterly waves (AEWs), has the potential to help extend prediction lead times for tropical cyclone (TC) genesis. For example, using the NASA global mesoscale model (GMM, e.g., [1]), the formation, subsequent intensification, and movement of hurricane Helene (2006) were simulated to a degree of satisfaction from Day 22 to 30 within a 30-day model integration when multiple AEWs were realistically simulated (e.g., [2]). To date, the scalable parallel

ensemble empirical mode decomposition method (PEEMD; e.g., [3, 4]) has revealed the role of downscaling processes associated with environmental flows in determining the timing and location of hurricane Helene's formation (e.g., [5]), supporting the model's practical predictability over extended-range time scales. Prior to Helene's formation, observed AEWs had comparable but slightly different periods with both spatial and temporal variations. Thus, a local analysis of the

A Recurrence Analysis of Multiple

Accurate detection of large-scale atmospheric tropical waves, such as African easterly waves (AEWs), may help extend lead times for predicting tropical cyclone (TC) genesis. Since observed AEWs have comparable but slightly different periods showing spatial and temporal variations, local analysis of frequencies and amplitudes of AEWs is crucial for revealing the role of AEWs in the modulation of TC genesis. To achieve this goal, we investigate the recurrence plot (RP) method. A recurrence is defined when the trajectory of a state returns to the neighborhood of a previously visited state. To verify implementation of the RP method in Python and its capability for revealing a transition between different types of solutions, we apply the RP to analyze several idealized solutions, including periodic, quasiperiodic, chaotic and limit cycle solutions, and various types of solutions within the three- and five-dimensional Lorenz models. We then extend the RP analysis to two datasets from the European Centre for Medium-Range Weather Forecasts global reanalysis and global mesoscale model data in order to reveal the recurrence of multiple AEWs during summer 2006. Our results indicate that the RP analysis effectively displays the major features of time-varying oscillations and the growing

African Easterly Waves during

[27] Parmenter FC. A southern hemisphere cold-front passage near the equator. Bulletin of the American Meteorology Society. 1976;**57**:1435-1440

[28] Taljaard JJ. Synoptic meteorology of the Southern hemisphere. In: Meteorology of the Southern Hemisphere. Vol. 13(35). Boston: Press AMS; 1972. pp. 129-213

[29] Hastenrath S. Climate Dynamics of the Tropics. Dordrecht: Kluwer Academic Publishers; 1991. p. 488

[30] Vasquez T. Weather Forecasting Handbook. Garland, Texas: Weather Graphics Technologies; 2000. 98 p

**Chapter 3**

*Current Topics in Tropical Cyclone Research*

[18] Rao VB, Bonatti JP. On the origin of upper tropospheric cyclonic vortex in the South Atlantic Ocean and adjoin Brazil during summer. Meteorology and Atmospheric Physics. 1987;**37**:11-16

synoptic evolution. Monthly Weather

[28] Taljaard JJ. Synoptic meteorology of the Southern hemisphere. In: Meteorology of the Southern

Hemisphere. Vol. 13(35). Boston: Press

[29] Hastenrath S. Climate Dynamics of the Tropics. Dordrecht: Kluwer Academic Publishers; 1991. p. 488

[30] Vasquez T. Weather Forecasting Handbook. Garland, Texas: Weather Graphics Technologies; 2000. 98 p

Review. 1983;**111**:181-196

AMS; 1972. pp. 129-213

[27] Parmenter FC. A southern hemisphere cold-front passage near the equator. Bulletin of the American Meteorology Society. 1976;**57**:1435-1440

[19] Mishra SK, Rao VB, Gan MA. Structure and evolution of the largescale flow of an embedded upper tropospheric cyclonic vortex over Northeast Brazil. Monthly Weather

Review. 2001;**129**:1673-1688

s00024-016-1381-1

2000;**15**(2):57-72

[20] Fedorova N, Santos DMB, Lopes Segundo MM, Levit V. Middle tropospheric cyclonic vortex in northeastern Brazil and the tropical Atlantic. Pure and Applied Geophysics. 2017;**174**(1):397-411. DOI: 10.1007/

[21] Pontes da Silva BF, Fedorova N, Levit V, Peresetsky A. Sistemas sinóticos associados às precipitações intensas no Estado de Alagoas (Synoptic systems associated with the precipitations in the Alagoas State). Revista Brasileira de Meteorologia. 2011;**26**(3):295-310

[22] Fedorova N, Carvalho MH.

Processos sinóticos em anos de La Niña e de El Niño. Parte II: Zonas Frontais. Revista Brasileira de Meteorologia.

[23] Kousky VE. Frontal influences on Northeast Brazil. Monthly Weather Review. 1979;**107**(9):1140-1153

[24] Fedorova N, Levit V, Cruz CD. On frontal zone analysis in the tropical region of the Northeast Brazil. Pure and Applied Geophysics. 2016;**173**:1403-1421.

DOI: 10.1007/s00024-015-1166-y

Society; 1998. pp. 201-241

[25] Reeder MJ, Smith RK. Mesoscale meteorology. In: Meteorology of the Southern Hemisphere. Vol. 27(49). Boston: American Meteorological

[26] Fortune MA, Kousky VE. Two severe freezes in Brazil: Precursors and

**42**

## A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006

*Tiffany Reyes and Bo-Wen Shen*

## **Abstract**

Accurate detection of large-scale atmospheric tropical waves, such as African easterly waves (AEWs), may help extend lead times for predicting tropical cyclone (TC) genesis. Since observed AEWs have comparable but slightly different periods showing spatial and temporal variations, local analysis of frequencies and amplitudes of AEWs is crucial for revealing the role of AEWs in the modulation of TC genesis. To achieve this goal, we investigate the recurrence plot (RP) method. A recurrence is defined when the trajectory of a state returns to the neighborhood of a previously visited state. To verify implementation of the RP method in Python and its capability for revealing a transition between different types of solutions, we apply the RP to analyze several idealized solutions, including periodic, quasiperiodic, chaotic and limit cycle solutions, and various types of solutions within the three- and five-dimensional Lorenz models. We then extend the RP analysis to two datasets from the European Centre for Medium-Range Weather Forecasts global reanalysis and global mesoscale model data in order to reveal the recurrence of multiple AEWs during summer 2006. Our results indicate that the RP analysis effectively displays the major features of time-varying oscillations and the growing or decaying amplitudes of multiple AEWs.

**Keywords:** recurrence plot, chaos, limit cycle, AEWs, Lorenz model

## **1. Introduction**

Recent studies suggest that accurate detection of recurrent, multiple, large-scale tropical waves, such as African easterly waves (AEWs), has the potential to help extend prediction lead times for tropical cyclone (TC) genesis. For example, using the NASA global mesoscale model (GMM, e.g., [1]), the formation, subsequent intensification, and movement of hurricane Helene (2006) were simulated to a degree of satisfaction from Day 22 to 30 within a 30-day model integration when multiple AEWs were realistically simulated (e.g., [2]). To date, the scalable parallel ensemble empirical mode decomposition method (PEEMD; e.g., [3, 4]) has revealed the role of downscaling processes associated with environmental flows in determining the timing and location of hurricane Helene's formation (e.g., [5]), supporting the model's practical predictability over extended-range time scales. Prior to Helene's formation, observed AEWs had comparable but slightly different periods with both spatial and temporal variations. Thus, a local analysis of the

frequencies and amplitudes for multiple AEWs is desired in order to monitor the evolution of AEWs that may influence the timing and location of TC genesis.

To effectively reveal space-varying features and/or temporal oscillations in simulations and reanalysis data, recurrence plots (RPs) are employed (e.g., [6]). Recurrence is defined when the trajectory of a state returns back to the neighborhood of a previously visited state. Thus, recurrence may be viewed as a generalization of periodicity to brace quasiperiodicity [7] and chaos [8]. Measuring the patterns associated with recurrence provides valuable information regarding the oscillatory nature of a system. To verify implementation in Python, we test the RP method using several types of idealized solutions and three types of solutions within the three- and five-dimensional Lorenz model (3DLM and 5DLM). We then apply the method in order to analyze two global datasets.

The paper is organized as follows. Section 2.1 introduces the European Centre for Medium-Range Weather Forecasts (ECMWF) global reanalysis data and NASA GMM data (e.g., [1, 2, 4, 9]). We focus on a 30-day period in 2006 over which multiple AEWs were observed during the NASA African Monsoon Multidisciplinary Analyses (NAMMA) field campaign (e.g., [2, 10]). Section 2.2 introduces the 3DLM and 5DLM (e.g., [11–13]). Section 2.3 introduces RP methods for performing the local analysis. In Section 3.1, the method is applied in order to analyze four basic types of solutions, including (1) periodic, (2) quasiperiodic, (3) chaotic solutions, and (4) Gaussian white noise. While Section 3.2 presents an analysis of the idealized spiral sink, Section 3.3 extends the RP analysis for steadystate solutions of the 3DLM. Section 3.4 discusses limit cycle solutions of the 3DLM and 5DLM and their RP analysis. Section 3.5 presents an analysis of global reanalysis and GMM data in order to reveal the time-varying recurrent behavior of AEWs. Appendix A discusses the mathematical approach for the unique version of the 3DLM and 5DLM, referred to as the Version 2 (V2) system. V2 systems are used for either linear or nonlinear simulations. Additionally, the linearized V2 system is applied to perform an eigenvalue analysis to obtain the growth rates and oscillatory frequencies of solutions.

#### **2. Global data, the Lorenz model, and the recurrence analysis method**

As a result of higher spatial and temporal resolutions, GMM simulation data [1, 9] are also selected. The GMM that produced the dataset is composed of three major components: finite volume dynamics [14], National Center for Atmospheric Research (NCAR) Community Climate Model physics, and the NCAR Community Land Model. Simulations spanning the 35 day period began on August 22, 2006. However, our analysis only focuses on the first 30 days. The data set has a 15-min integration time step, 17 vertical levels, and a horizontal resolution of approximately 28 km. **Figure 1c** provides the time-altitude cross section of meridional

*A time-altitude cross section of meridional winds from the ERA-interim dataset (a) and global mesoscale model simulations (c). The color bar to the right of each figure represents the wind velocity in meters per second. Panel*

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006*

*DOI: http://dx.doi.org/10.5772/intechopen.86859*

In the real world, as a result of nonlinearity, system solutions may possess timevarying frequencies and amplitudes. To understand the performance of the selected method in analyzing nonlinear time series data, the 3DLM and 5DLM are used to provide three types of solutions at different ranges of parameters for verifications. The 5DLM, which extends the 3DLM and includes the 3DLM as a subset (e.g., [13]),

*<sup>d</sup><sup>τ</sup>* ¼ �*σ<sup>X</sup>* <sup>þ</sup> *<sup>σ</sup>Y,* (1)

*dX*

winds from the GMM dataset.

**Figure 1.**

**45**

is defined by Eqs. (1)–(5), as follows:

**2.2 The five-dimensional Lorenz model (5DLM)**

*(b) displays NAMMA observations for a comparison (e.g., [10]).*

#### **2.1 Global reanalysis and model data**

For this study, we select the latest European Centre for Medium-Range Weather Forecasts (ECMWF) global reanalysis (ERA-interim) dataset. The data spans the time period from January 1979 to the present day. The dataset has a sampling rate of 6 h and a horizontal grid spacing of 0*:*75°, yielding a spatial resolution of approximately 78 km. The following gridded products are available: 3-h surface fields, daily vertical integrations, 6-h upper air atmospheric fields for pressure, potential temperature, potential vorticity levels, and vertical coverage from the lower troposphere to the stratosphere on 60 model layers. Based on previous studies (e.g., [2, 5]), here, we focus on the 30-day period from August 22 to September 20, 2006. During this period, the NASA African Monsoon Multidisciplinary Analyses (NAMMA) field campaign documented multiple AEWs and provided a great opportunity to characterize the frequency of AEWs as well as their evolution and impact over continental western Africa. **Figure 1a** and **b** display multiple AEWs that were detected by a change in sign of meridional winds from the ERA-interim dataset and the NAMMA observations, respectively.

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006 DOI: http://dx.doi.org/10.5772/intechopen.86859*

#### **Figure 1.**

frequencies and amplitudes for multiple AEWs is desired in order to monitor the evolution of AEWs that may influence the timing and location of TC genesis.

ulations and reanalysis data, recurrence plots (RPs) are employed (e.g., [6]). Recurrence is defined when the trajectory of a state returns back to the neighborhood of a previously visited state. Thus, recurrence may be viewed as a generalization of periodicity to brace quasiperiodicity [7] and chaos [8]. Measuring the patterns associated with recurrence provides valuable information regarding the oscillatory nature of a system. To verify implementation in Python, we test the RP method using several types of idealized solutions and three types of solutions within the three- and five-dimensional Lorenz model (3DLM and 5DLM). We then apply

the method in order to analyze two global datasets.

*Current Topics in Tropical Cyclone Research*

frequencies of solutions.

**44**

**2.1 Global reanalysis and model data**

To effectively reveal space-varying features and/or temporal oscillations in sim-

The paper is organized as follows. Section 2.1 introduces the European Centre for Medium-Range Weather Forecasts (ECMWF) global reanalysis data and NASA GMM data (e.g., [1, 2, 4, 9]). We focus on a 30-day period in 2006 over which multiple AEWs were observed during the NASA African Monsoon Multidisciplinary Analyses (NAMMA) field campaign (e.g., [2, 10]). Section 2.2 introduces the 3DLM and 5DLM (e.g., [11–13]). Section 2.3 introduces RP methods for performing the local analysis. In Section 3.1, the method is applied in order to analyze four basic types of solutions, including (1) periodic, (2) quasiperiodic, (3) chaotic solutions, and (4) Gaussian white noise. While Section 3.2 presents an analysis of the idealized spiral sink, Section 3.3 extends the RP analysis for steadystate solutions of the 3DLM. Section 3.4 discusses limit cycle solutions of the 3DLM and 5DLM and their RP analysis. Section 3.5 presents an analysis of global reanalysis and GMM data in order to reveal the time-varying recurrent behavior of AEWs. Appendix A discusses the mathematical approach for the unique version of the 3DLM and 5DLM, referred to as the Version 2 (V2) system. V2 systems are used for either linear or nonlinear simulations. Additionally, the linearized V2 system is applied to perform an eigenvalue analysis to obtain the growth rates and oscillatory

**2. Global data, the Lorenz model, and the recurrence analysis method**

During this period, the NASA African Monsoon Multidisciplinary Analyses (NAMMA) field campaign documented multiple AEWs and provided a great opportunity to characterize the frequency of AEWs as well as their evolution and impact over continental western Africa. **Figure 1a** and **b** display multiple AEWs that were detected by a change in sign of meridional winds from the ERA-interim

dataset and the NAMMA observations, respectively.

For this study, we select the latest European Centre for Medium-Range Weather Forecasts (ECMWF) global reanalysis (ERA-interim) dataset. The data spans the time period from January 1979 to the present day. The dataset has a sampling rate of 6 h and a horizontal grid spacing of 0*:*75°, yielding a spatial resolution of approximately 78 km. The following gridded products are available: 3-h surface fields, daily vertical integrations, 6-h upper air atmospheric fields for pressure, potential temperature, potential vorticity levels, and vertical coverage from the lower troposphere to the stratosphere on 60 model layers. Based on previous studies (e.g., [2, 5]), here, we focus on the 30-day period from August 22 to September 20, 2006.

*A time-altitude cross section of meridional winds from the ERA-interim dataset (a) and global mesoscale model simulations (c). The color bar to the right of each figure represents the wind velocity in meters per second. Panel (b) displays NAMMA observations for a comparison (e.g., [10]).*

As a result of higher spatial and temporal resolutions, GMM simulation data [1, 9] are also selected. The GMM that produced the dataset is composed of three major components: finite volume dynamics [14], National Center for Atmospheric Research (NCAR) Community Climate Model physics, and the NCAR Community Land Model. Simulations spanning the 35 day period began on August 22, 2006. However, our analysis only focuses on the first 30 days. The data set has a 15-min integration time step, 17 vertical levels, and a horizontal resolution of approximately 28 km. **Figure 1c** provides the time-altitude cross section of meridional winds from the GMM dataset.

#### **2.2 The five-dimensional Lorenz model (5DLM)**

In the real world, as a result of nonlinearity, system solutions may possess timevarying frequencies and amplitudes. To understand the performance of the selected method in analyzing nonlinear time series data, the 3DLM and 5DLM are used to provide three types of solutions at different ranges of parameters for verifications. The 5DLM, which extends the 3DLM and includes the 3DLM as a subset (e.g., [13]), is defined by Eqs. (1)–(5), as follows:

$$\frac{dX}{d\tau} = -\sigma X + \sigma Y,\tag{1}$$

$$\frac{dY}{d\tau} = -XZ + rX - Y,\tag{2}$$

$$\frac{dZ}{d\tau} = XY - XY\_1 - bZ,\tag{3}$$

$$\frac{dY\_1}{d\tau} = \text{XZ} - 2\text{XZ}\_1 - d\_\circ Y\_1. \tag{4}$$

$$\frac{dZ\_1}{d\tau} = 2XY\_1 - 4bZ\_1. \tag{5}$$

the RP (e.g., [6]). As shown in **Figure 3a**, a typical RP for a simple periodic solution displays uniform diagonal lines. Here, it should be noted that the RP has a main black diagonal line with an angle of *π=*4, referred to as the line of identity (LOI).The threshold distance for recurrence (*ε*) is an important parameter to consider. If the value is too small, there may not be enough recurrences plotted, resulting in insufficient information required for conducting an accurate analysis. For a value that is too large, the appearance of many recurrent points may mask important structures that would otherwise be present within the RP. While there is much discussion regarding the proper selection of *ε*, a general rule is to find the smallest possible value that is capable of providing a sufficient number of recurrences for identifying recurrent structures within a dataset (e.g., [6]). Figure 2.10 of Reyes [24] demonstrates the impact with different values of *ε* on the RP for the

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006*

*DOI: http://dx.doi.org/10.5772/intechopen.86859*

*respectively. (d) Y*<sup>1</sup> *vs. Z*<sup>1</sup> *modes with r* ¼ 43*:*5 *(see details in [13]).*

*Phase space plots for the 3DLM (a) and 5DLM (b–d). All of the plots show Lorenz strange attractors. (a)* Y *vs.* Z *modes using the 3DLM with r* ¼ 25*. (b and c)* Y *vs. Z modes using the 5DLM with r* ¼ 25 *and r* ¼ 43*:*5*,*

In this section, we first discuss the RP analysis for idealized solutions, various types of solutions from the 3DLM and 5DLM. We then apply the RP to analyze the global reanalysis and the GMM 30-day simulation data in order to detect recurrent, multiple AEWs that play an important role in the modulation of TC genesis (e.g.,

3DLM.

**47**

**Figure 2.**

**3. Results**

Hurricane Helene (2006)).

Here, *τ* is the dimensionless time. In this study, the following parameters are kept constant: *b* ¼ 8*=*3, *d* ¼ 19*=*3, and *σ* ¼ 10. By comparison, we vary the value of the Rayleigh parameter, *r*, that represents a heating or forcing term. The term *XY*<sup>1</sup> in Eq. (3) plays a role in providing negative nonlinear feedback associated with small-scale dissipative processes (e.g., �*doY*<sup>1</sup> and �4*bZ*1) to stabilize the solutions (e.g., [13]). Equations. (1)–(3), without inclusion of the feedback term (*XY*1), are reduced to become the 3DLM. As discussed below, the 5DLM requires higher values of *r* for the onset of chaos (e.g., [13]). Recently, the 5DLM has been re-derived and analyzed in detail by Moon et al. [15] and Felicio and Rech [16], showing the model's robustness. Furthermore, the 5DLM has been extended to higherdimensional LMs (e.g., [17–20]) and a generalized LM (e.g., [21, 22]).

The 3DLM produces three types of solutions, including steady-state solutions, chaotic solutions, and nonlinear periodic solutions. We present the first two types of solutions below and discuss the third type of solution in Section 3.4. **Figure 2a** provides *Y* and *Z* modes of the solution for the strange attractor using the 3DLM with *r* ¼ 25. Irregular oscillations surrounding nontrivial critical points that are defined in Appendix A are indicative of chaotic solutions. In **Figure 2b**, the 5DLM with *r* ¼ 25 clearly produces a steady-state solution that spirals into the non-trivial critical point. **Figure 2c** from the 5DLM with *r* ¼ 43*:*5 appears more like **Figure 2a** and depicts an irregular oscillatory motion. Such behavior within the 5DLM only occurs when *r* ≥ *rc* ¼ 42*:*9 and when the other parameters are held at the aforementioned values. The corresponding solutions for Y1 and Z1 are shown in **Figure 2d**.

#### **2.3 Recurrence plot analysis**

The recurrent nature of a system can offer important insight into its dynamics, such as periodicities or other oscillatory behavior (e.g., growing or decaying oscillations). In the case of deterministic systems, the recurrence of states (i.e., a state returning to an arbitrarily close area at a point from a previous time within the phase space) is an important feature (e.g., [6]). Such recurrences can be visualized using a tool known as recurrence plots (RPs). RPs allow for the representation of multidimensional systems in a two-dimensional visualization; thus, they easily provide insight into high-dimensional systems. The RP plots a black point for each time ð Þ *i; j* in which there is a recurrence, more precisely given by the following equation:

$$R\_{i,j} = \Theta\left(\varepsilon - \|\overrightarrow{\mathbf{x}\_i} - \overrightarrow{\mathbf{x}\_j}\|\right) \quad i, j = 1, \dots, N,\tag{6}$$

where *N* is the number of states (*xi* !) considered, *ε* is the threshold distance for recurrence, ∥ � ∥ is the Euclidean norm, and Θð Þ� is the Heaviside function (e.g., [23]). The individual point ð Þ *i; j* does not offer any information regarding the states of the system at times *i* and *j*. However, the phase space trajectory can be reconstructed from the collection of all of the recurrence points present within

#### *A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006 DOI: http://dx.doi.org/10.5772/intechopen.86859*

#### **Figure 2.**

*dY*

*Current Topics in Tropical Cyclone Research*

*dZ*

*dZ*<sup>1</sup>

Here, *τ* is the dimensionless time. In this study, the following parameters are kept constant: *b* ¼ 8*=*3, *d* ¼ 19*=*3, and *σ* ¼ 10. By comparison, we vary the value of the Rayleigh parameter, *r*, that represents a heating or forcing term. The term *XY*<sup>1</sup> in Eq. (3) plays a role in providing negative nonlinear feedback associated with small-scale dissipative processes (e.g., �*doY*<sup>1</sup> and �4*bZ*1) to stabilize the solutions (e.g., [13]). Equations. (1)–(3), without inclusion of the feedback term (*XY*1), are reduced to become the 3DLM. As discussed below, the 5DLM requires higher values of *r* for the onset of chaos (e.g., [13]). Recently, the 5DLM has been re-derived and analyzed in detail by Moon et al. [15] and Felicio and Rech [16], showing the model's robustness. Furthermore, the 5DLM has been extended to higherdimensional LMs (e.g., [17–20]) and a generalized LM (e.g., [21, 22]).

The 3DLM produces three types of solutions, including steady-state solutions, chaotic solutions, and nonlinear periodic solutions. We present the first two types of solutions below and discuss the third type of solution in Section 3.4. **Figure 2a** provides *Y* and *Z* modes of the solution for the strange attractor using the 3DLM with *r* ¼ 25. Irregular oscillations surrounding nontrivial critical points that are defined in Appendix A are indicative of chaotic solutions. In **Figure 2b**, the 5DLM with *r* ¼ 25 clearly produces a steady-state solution that spirals into the non-trivial critical point. **Figure 2c** from the 5DLM with *r* ¼ 43*:*5 appears more like **Figure 2a** and depicts an irregular oscillatory motion. Such behavior within the 5DLM only occurs when *r* ≥ *rc* ¼ 42*:*9 and when the other parameters are held at the aforementioned values. The corresponding solutions for Y1 and Z1 are shown in **Figure 2d**.

The recurrent nature of a system can offer important insight into its dynamics, such as periodicities or other oscillatory behavior (e.g., growing or decaying oscillations). In the case of deterministic systems, the recurrence of states (i.e., a state returning to an arbitrarily close area at a point from a previous time within the phase space) is an important feature (e.g., [6]). Such recurrences can be visualized using a tool known as recurrence plots (RPs). RPs allow for the representation of multidimensional systems in a two-dimensional visualization; thus, they easily provide insight into high-dimensional systems. The RP plots a black point for each time ð Þ *i; j* in which there is a recurrence, more precisely given by the following equation:

> ! � *xj* !∥

recurrence, ∥ � ∥ is the Euclidean norm, and Θð Þ� is the Heaviside function (e.g., [23]). The individual point ð Þ *i; j* does not offer any information regarding the states of the system at times *i* and *j*. However, the phase space trajectory can be reconstructed from the collection of all of the recurrence points present within

*i, j* ¼ 1*,* …*,N,* (6)

!) considered, *ε* is the threshold distance for

*Ri,j* ¼ Θ *ε* � ∥*xi*

where *N* is the number of states (*xi*

**46**

*dY*<sup>1</sup>

**2.3 Recurrence plot analysis**

*<sup>d</sup><sup>τ</sup>* ¼ �*XZ* <sup>þ</sup> *rX* � *Y,* (2)

*<sup>d</sup><sup>τ</sup>* <sup>¼</sup> *XY* � *XY*<sup>1</sup> � *bZ,* (3)

*<sup>d</sup><sup>τ</sup>* <sup>¼</sup> *XZ* � <sup>2</sup>*XZ*<sup>1</sup> � *doY*1*,* (4)

*<sup>d</sup><sup>τ</sup>* <sup>¼</sup> <sup>2</sup>*XY*<sup>1</sup> � <sup>4</sup>*bZ*1*:* (5)

*Phase space plots for the 3DLM (a) and 5DLM (b–d). All of the plots show Lorenz strange attractors. (a)* Y *vs.* Z *modes using the 3DLM with r* ¼ 25*. (b and c)* Y *vs. Z modes using the 5DLM with r* ¼ 25 *and r* ¼ 43*:*5*, respectively. (d) Y*<sup>1</sup> *vs. Z*<sup>1</sup> *modes with r* ¼ 43*:*5 *(see details in [13]).*

the RP (e.g., [6]). As shown in **Figure 3a**, a typical RP for a simple periodic solution displays uniform diagonal lines. Here, it should be noted that the RP has a main black diagonal line with an angle of *π=*4, referred to as the line of identity (LOI).The threshold distance for recurrence (*ε*) is an important parameter to consider. If the value is too small, there may not be enough recurrences plotted, resulting in insufficient information required for conducting an accurate analysis. For a value that is too large, the appearance of many recurrent points may mask important structures that would otherwise be present within the RP. While there is much discussion regarding the proper selection of *ε*, a general rule is to find the smallest possible value that is capable of providing a sufficient number of recurrences for identifying recurrent structures within a dataset (e.g., [6]). Figure 2.10 of Reyes [24] demonstrates the impact with different values of *ε* on the RP for the 3DLM.

#### **3. Results**

In this section, we first discuss the RP analysis for idealized solutions, various types of solutions from the 3DLM and 5DLM. We then apply the RP to analyze the global reanalysis and the GMM 30-day simulation data in order to detect recurrent, multiple AEWs that play an important role in the modulation of TC genesis (e.g., Hurricane Helene (2006)).

#### **3.1 Analysis of four basic types of solutions**

**Figure 3** provides RPs for four types of solutions, including (a) periodic solutions that are characterized by uniform diagonal lines extending from edge to edge of the plot and spaced with an equal distance from one another; (b) quasiperiodic solutions with a similar structure to the periodic RP, with the exception that the distance between the diagonal lines varies; (c) chaotic solutions from the 3DLM with a Rayleigh parameter of *r* ¼ 28 whose plots have many short diagonal lines spaced various distances from one another; and (d) Gaussian white noise, characterized by a homogenous RP consisting of many individual points and little to no organized recurrent structures (e.g., [6]).

Other features of the RP analysis, as discussed below, include (1) vertical or horizontal lines or clusters, which indicate small or no change in solution amplitudes over a certain period of time, and (2) white bands, which suggest transitions from one type of dynamics to another. More than one distinct line or dot structure may also appear within a single RP. Line structures and the length and number of lines, as well as their positioning, describe the dynamics for each system (e.g., [25]). When applying this method to real data, the overall structures should still be visible, although they may or may not be as neatly depicted.

> Previously, performance of the RP was demonstrated using idealized data. In practice, most real data has some amount of random noise. Therefore, understanding the capabilities of the method in analyzing noisy data, before applying them to real or non-idealized datasets, is important. Here, we present an analysis of composite data consisting of the raw data and additional noise. **Figure 4a** displays the RP for idealized periodic solutions generated from *y* ¼ sin 2ð Þ *πt=*16 , 0 ≤ *t* ≤ 128, and Δ*t* ¼ 0*:*2. **Figure 4b** provides the RP for the same dataset superimposed with Gaussian white noise with an amplitude of 0.1. While the overall diagonal line structure is still present, some visual differences are apparent. The most prominent differences include thicker diagonal lines and a fluctuation in thickness. The differences are due to so-called false recurrence resulting from the noise itself and the

*Recurrence plots of raw data with a periodic solution given by y* ¼ *sin* ð Þ 2*πt=*16 *(a) and the raw data*

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006*

*DOI: http://dx.doi.org/10.5772/intechopen.86859*

*superimposed with Gaussian white noise that has a finite amplitude of 0.1 (b).*

In addition to the four fundamental solutions in **Figure 3**, real-world model data

may include growing or decaying oscillatory solutions. Mathematically, such a solution can be represented as *<sup>y</sup>* <sup>¼</sup> *<sup>e</sup><sup>α</sup><sup>t</sup>* sin ð Þ *<sup>β</sup><sup>t</sup>* <sup>þ</sup> *<sup>θ</sup>* , where *<sup>θ</sup>* is a phase constant. This type of solution can be obtained from a linearized system with a complex eigenvalue, *λ* ¼ *α* þ *iβ*, where *α* and *β* are real numbers. Next, we discuss the conditions under which a recurrence plot can be defined for the general oscillatory solution. To facilitate discussions, we further assume *β* . 0 and consider three cases with (i) *α* ¼ 0 for a simple oscillation, (ii) *α* . 0 for a growing oscillation (i.e., a spiral source), and (iii) *α* , 0 for a decaying oscillation (i.e., a spiral sink), respectively. RPs for spiral sinks or sources display a different structure as compared to RPs for periodic and quasiperiodic solutions, as discussed below. **Figure 5a** displays an oscillatory solution, *<sup>y</sup>* <sup>¼</sup> *<sup>e</sup>*�0*:*1*<sup>t</sup>* sin ð Þ*<sup>t</sup>* (i.e., ð Þ¼ � *<sup>α</sup>; <sup>β</sup>; <sup>θ</sup>* ð Þ <sup>0</sup>*:*1*;* <sup>1</sup>*;* <sup>0</sup> ). The solution decays with time over the time interval from 0 to 128. When the solutions produce time-varying amplitudes, the RPs display vertical and horizontal lines. As the solution decays toward zero, the lines become denser (the upper right-hand corner of **Figure 5c**). Namely, the density of the lines in these RPs increases when the solution is closer to the mean value (i.e., zero for this idealized solution). The appearance of

slightly higher value of *ε* required in order to obtain a reasonable plot.

**3.2 Analysis of a spiral sink and source**

**Figure 4.**

horizontal and vertical lines is explained below.

**49**

#### **Figure 3.**

*Recurrence plots of (a) periodic solutions produced using y* ¼ *sin* ð Þ 2*πt=*4 *, (b) quasiperiodic solutions produced using <sup>y</sup>* <sup>¼</sup> *sin* ð Þþ <sup>16</sup>*π<sup>t</sup> sin* <sup>8</sup> ffiffiffiffiffiffi ð Þ<sup>2</sup> <sup>p</sup> *<sup>π</sup><sup>t</sup>* � �*, (c) chaotic solutions produced by the 3DLM, and (d) Gaussian white noise.*

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006 DOI: http://dx.doi.org/10.5772/intechopen.86859*

**Figure 4.** *Recurrence plots of raw data with a periodic solution given by y* ¼ *sin* ð Þ 2*πt=*16 *(a) and the raw data superimposed with Gaussian white noise that has a finite amplitude of 0.1 (b).*

Previously, performance of the RP was demonstrated using idealized data. In practice, most real data has some amount of random noise. Therefore, understanding the capabilities of the method in analyzing noisy data, before applying them to real or non-idealized datasets, is important. Here, we present an analysis of composite data consisting of the raw data and additional noise. **Figure 4a** displays the RP for idealized periodic solutions generated from *y* ¼ sin 2ð Þ *πt=*16 , 0 ≤ *t* ≤ 128, and Δ*t* ¼ 0*:*2. **Figure 4b** provides the RP for the same dataset superimposed with Gaussian white noise with an amplitude of 0.1. While the overall diagonal line structure is still present, some visual differences are apparent. The most prominent differences include thicker diagonal lines and a fluctuation in thickness. The differences are due to so-called false recurrence resulting from the noise itself and the slightly higher value of *ε* required in order to obtain a reasonable plot.

#### **3.2 Analysis of a spiral sink and source**

In addition to the four fundamental solutions in **Figure 3**, real-world model data may include growing or decaying oscillatory solutions. Mathematically, such a solution can be represented as *<sup>y</sup>* <sup>¼</sup> *<sup>e</sup><sup>α</sup><sup>t</sup>* sin ð Þ *<sup>β</sup><sup>t</sup>* <sup>þ</sup> *<sup>θ</sup>* , where *<sup>θ</sup>* is a phase constant. This type of solution can be obtained from a linearized system with a complex eigenvalue, *λ* ¼ *α* þ *iβ*, where *α* and *β* are real numbers. Next, we discuss the conditions under which a recurrence plot can be defined for the general oscillatory solution. To facilitate discussions, we further assume *β* . 0 and consider three cases with (i) *α* ¼ 0 for a simple oscillation, (ii) *α* . 0 for a growing oscillation (i.e., a spiral source), and (iii) *α* , 0 for a decaying oscillation (i.e., a spiral sink), respectively.

RPs for spiral sinks or sources display a different structure as compared to RPs for periodic and quasiperiodic solutions, as discussed below. **Figure 5a** displays an oscillatory solution, *<sup>y</sup>* <sup>¼</sup> *<sup>e</sup>*�0*:*1*<sup>t</sup>* sin ð Þ*<sup>t</sup>* (i.e., ð Þ¼ � *<sup>α</sup>; <sup>β</sup>; <sup>θ</sup>* ð Þ <sup>0</sup>*:*1*;* <sup>1</sup>*;* <sup>0</sup> ). The solution decays with time over the time interval from 0 to 128. When the solutions produce time-varying amplitudes, the RPs display vertical and horizontal lines. As the solution decays toward zero, the lines become denser (the upper right-hand corner of **Figure 5c**). Namely, the density of the lines in these RPs increases when the solution is closer to the mean value (i.e., zero for this idealized solution). The appearance of horizontal and vertical lines is explained below.

**3.1 Analysis of four basic types of solutions**

*Current Topics in Tropical Cyclone Research*

organized recurrent structures (e.g., [6]).

**Figure 3.**

*noise.*

**48**

*using <sup>y</sup>* <sup>¼</sup> *sin* ð Þþ <sup>16</sup>*π<sup>t</sup> sin* <sup>8</sup> ffiffiffiffiffiffi

although they may or may not be as neatly depicted.

**Figure 3** provides RPs for four types of solutions, including (a) periodic solutions that are characterized by uniform diagonal lines extending from edge to edge of the plot and spaced with an equal distance from one another; (b) quasiperiodic solutions with a similar structure to the periodic RP, with the exception that the distance between the diagonal lines varies; (c) chaotic solutions from the 3DLM with a Rayleigh parameter of *r* ¼ 28 whose plots have many short diagonal lines spaced various distances from one another; and (d) Gaussian white noise, characterized by a homogenous RP consisting of many individual points and little to no

Other features of the RP analysis, as discussed below, include (1) vertical or horizontal lines or clusters, which indicate small or no change in solution amplitudes over a certain period of time, and (2) white bands, which suggest transitions from one type of dynamics to another. More than one distinct line or dot structure may also appear within a single RP. Line structures and the length and number of lines, as well as their positioning, describe the dynamics for each system (e.g., [25]). When applying this method to real data, the overall structures should still be visible,

*Recurrence plots of (a) periodic solutions produced using y* ¼ *sin* ð Þ 2*πt=*4 *, (b) quasiperiodic solutions produced*

ð Þ<sup>2</sup> <sup>p</sup> *<sup>π</sup><sup>t</sup>* � �*, (c) chaotic solutions produced by the 3DLM, and (d) Gaussian white*

Here, we provide an additional analysis for the pattern of the RP in **Figure 5**. As shown in **Figure 6**, a distance (*D*) between two points *y t*ð Þ<sup>1</sup> and *y t*ð Þ<sup>2</sup> with a time lag of *<sup>T</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>β</sup>* is given by *D* ¼ ∣*y t*ð Þ� <sup>2</sup> *y t*ð Þ<sup>1</sup> ∣, where *t*<sup>2</sup> ¼ *t*<sup>1</sup> þ *T*. We can obtain the following equation:

$$D = |e^{at\_1} \sin\left(\beta t\_2 + \theta\right) - e^{at\_1} \sin\left(\beta t\_1 + \theta\right)|\tag{7}$$

$$= \left| e^{a\left(t\_1 + \frac{2\pi}{\beta}\right)} \sin\left(\beta \left(t\_1 + \frac{2\pi}{\beta}\right) + \theta\right) - e^{at\_1} \sin\left(\beta t\_1 + \theta\right)\right|$$

$$= \left| e^{\frac{2\pi}{\beta}} e^{at\_1} \sin\left(\beta t\_1 + \theta\right) - e^{at\_1} \sin\left(\beta t\_1 + \theta\right)\right|\tag{8}$$

$$= \left| e^{\frac{2\pi}{\beta}} e^{at\_1} \sin\left(\beta t\_1 + \theta\right) - e^{at\_1} \sin\left(\beta t\_1 + \theta\right)\right|$$

$$= \left| e^{at\_1} \sin\left(\beta t\_1 + \theta\right) \left(e^{\frac{2\pi}{\beta}} - 1\right)\right|.$$

Recurrence appears when the distance, *D*, is small (i.e., *D* , *ε*, where *ε* represents a threshold), requiring

$$\left| e^{at\_1} \sin \left( \beta t\_1 + \theta \right) \left( e^{\frac{2\pi}{\beta}} - 1 \right) \right| < \varepsilon. \tag{9}$$

Therefore, (i) for a simple oscillation, *α* ¼ 0, Eq. (8) is always valid, suggesting that two points with a time lag of *T* define a recurrent point within the recurrence

*M*

 

*ε:* (10)

*e* 2*απ <sup>β</sup>* � 1 

The value of *αt*<sup>1</sup> roughly determines if the above inequality is valid. As a result, Eq. (10) suggests that (ii) for a spiral source, *α* . 0, a recurrence point appears when *t*<sup>1</sup> is small and that (iii) for a spiral sink, *α* , 0, a recurrence point appears when *t*<sup>1</sup> is large. This type of RP appears in the red box in **Figure 5c**. Additionally, (iv) when sin ð Þ¼ *βt*<sup>1</sup> þ *θ* 0 in Eq. (7a), the point at *t* ¼ *t*<sup>1</sup> can produce recurrence with all of the points at *t*<sup>2</sup> that have very small amplitudes (i.e., the amplitudes are close to zero). Thus, as shown in **Figure 5c**, the appearance of continuous horizontal lines indicates a transition from the third type of solution (i.e., a spiral sink) into the 4th type of solution with a small or zero amplitude. Note that the distance of two horizontal lines determined in (iv) yields an estimate of a half of the period (i.e., *T=*2.) In general, Eq. (9) suggests that the aforementioned recurrence threshold should be selected by taking the rate of growth or decay of the oscillatory solutions

While the 3DLM with *r* . 24*:*74 produces chaotic solutions, it simulates steadystate solutions when *r* , 24*:*74. Since the steady-state solution displays a decaying oscillatory mode, which may resemble a weakening AEW, its RP is analyzed below. **Figure 7a** displays numerical solutions of the Y mode of the 3DLM with *r* ¼ 10. At approximately *t* ¼ 7*:*0, only small fluctuations in the solution remain (i.e., the solution has a small amplitude over this interval). **Figure 7b** displays the

corresponding recurrence plot with *ε* ¼ 0*:*05. Initially, horizontal and vertical lines appear, consistent with the previous analysis using **Figures 5** and **6**. As the solution

begins to converge to a steady state, the plot rapidly becomes denser.

 

plot. For a specific *t*<sup>1</sup> where sin ð Þ¼ *βt*<sup>1</sup> þ *θ* 1*=M* 6¼ 0, Eq. (8) leads to:

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006*

*DOI: http://dx.doi.org/10.5772/intechopen.86859*

∣*e <sup>α</sup>t*<sup>1</sup> ∣ ,

*An oscillatory mode and two points at t*<sup>1</sup> *and t*2*, where t*<sup>2</sup> ¼ *t*<sup>1</sup> þ *T.*

**Figure 6.**

**51**

into account, which will be the subject of a future study.

**3.3 Analysis of a steady-state solution within the 3DLM**

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006 DOI: http://dx.doi.org/10.5772/intechopen.86859*

**Figure 6.** *An oscillatory mode and two points at t*<sup>1</sup> *and t*2*, where t*<sup>2</sup> ¼ *t*<sup>1</sup> þ *T.*

Therefore, (i) for a simple oscillation, *α* ¼ 0, Eq. (8) is always valid, suggesting that two points with a time lag of *T* define a recurrent point within the recurrence plot. For a specific *t*<sup>1</sup> where sin ð Þ¼ *βt*<sup>1</sup> þ *θ* 1*=M* 6¼ 0, Eq. (8) leads to:

$$|e^{at\_1}| \le \left| \frac{M}{\left(e^{\frac{2\pi}{\beta}} - 1\right)}\right| e. \tag{10}$$

The value of *αt*<sup>1</sup> roughly determines if the above inequality is valid. As a result, Eq. (10) suggests that (ii) for a spiral source, *α* . 0, a recurrence point appears when *t*<sup>1</sup> is small and that (iii) for a spiral sink, *α* , 0, a recurrence point appears when *t*<sup>1</sup> is large. This type of RP appears in the red box in **Figure 5c**. Additionally, (iv) when sin ð Þ¼ *βt*<sup>1</sup> þ *θ* 0 in Eq. (7a), the point at *t* ¼ *t*<sup>1</sup> can produce recurrence with all of the points at *t*<sup>2</sup> that have very small amplitudes (i.e., the amplitudes are close to zero). Thus, as shown in **Figure 5c**, the appearance of continuous horizontal lines indicates a transition from the third type of solution (i.e., a spiral sink) into the 4th type of solution with a small or zero amplitude. Note that the distance of two horizontal lines determined in (iv) yields an estimate of a half of the period (i.e., *T=*2.) In general, Eq. (9) suggests that the aforementioned recurrence threshold should be selected by taking the rate of growth or decay of the oscillatory solutions into account, which will be the subject of a future study.

#### **3.3 Analysis of a steady-state solution within the 3DLM**

While the 3DLM with *r* . 24*:*74 produces chaotic solutions, it simulates steadystate solutions when *r* , 24*:*74. Since the steady-state solution displays a decaying oscillatory mode, which may resemble a weakening AEW, its RP is analyzed below. **Figure 7a** displays numerical solutions of the Y mode of the 3DLM with *r* ¼ 10. At approximately *t* ¼ 7*:*0, only small fluctuations in the solution remain (i.e., the solution has a small amplitude over this interval). **Figure 7b** displays the corresponding recurrence plot with *ε* ¼ 0*:*05. Initially, horizontal and vertical lines appear, consistent with the previous analysis using **Figures 5** and **6**. As the solution begins to converge to a steady state, the plot rapidly becomes denser.

Here, we provide an additional analysis for the pattern of the RP in **Figure 5**. As shown in **Figure 6**, a distance (*D*) between two points *y t*ð Þ<sup>1</sup> and *y t*ð Þ<sup>2</sup> with a time

*<sup>α</sup>t*<sup>2</sup> sin ð Þ� *<sup>β</sup>t*<sup>2</sup> <sup>þ</sup> *<sup>θ</sup> <sup>e</sup>*

*<sup>β</sup> <sup>e</sup><sup>α</sup>t*<sup>1</sup> sin ð Þ� *<sup>β</sup>t*<sup>1</sup> <sup>þ</sup> *<sup>θ</sup> <sup>e</sup><sup>α</sup>t*<sup>1</sup> sin ð Þ *<sup>β</sup>t*<sup>1</sup> <sup>þ</sup> *<sup>θ</sup>*

*<sup>β</sup> <sup>e</sup><sup>α</sup>t*<sup>1</sup> sin ð Þ� *<sup>β</sup>t*<sup>1</sup> <sup>þ</sup> *<sup>θ</sup> <sup>e</sup><sup>α</sup>t*<sup>1</sup> sin ð Þ *<sup>β</sup>t*<sup>1</sup> <sup>þ</sup> *<sup>θ</sup>*

2*απ <sup>β</sup>* � 1 

*<sup>α</sup>t*<sup>1</sup> sin ð Þ *<sup>β</sup>t*<sup>1</sup> <sup>þ</sup> *<sup>θ</sup> <sup>e</sup>*

*A time evolution of the solution <sup>y</sup>* <sup>¼</sup> *<sup>e</sup>*�01*:<sup>t</sup> sin t*ð Þ *(a), its power spectrum (b), and RP (c) for <sup>τ</sup>* <sup>∈</sup>½ � *<sup>0</sup>; <sup>128</sup> .*

2*π β* 

*<sup>β</sup>* is given by *D* ¼ ∣*y t*ð Þ� <sup>2</sup> *y t*ð Þ<sup>1</sup> ∣, where *t*<sup>2</sup> ¼ *t*<sup>1</sup> þ *T*. We can obtain the

þ *θ*

 *:*

2*απ <sup>β</sup>* � 1 

 

Recurrence appears when the distance, *D*, is small (i.e., *D* , *ε*, where *ε* repre-

*<sup>α</sup>t*<sup>1</sup> sin ð Þ *<sup>β</sup>t*<sup>1</sup> <sup>þ</sup> *<sup>θ</sup>* <sup>∣</sup> (7)

 

, *ε:* (9)

(8)

� *<sup>e</sup><sup>α</sup>t*<sup>1</sup> sin ð Þ *<sup>β</sup>t*<sup>1</sup> <sup>þ</sup> *<sup>θ</sup>*

 

 

lag of *<sup>T</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup>*

following equation:

¼ *e*

*Current Topics in Tropical Cyclone Research*

¼ *e* 2*απ*

¼ *e* 2*απ*

¼ 

sents a threshold), requiring

**Figure 5.**

**50**

*D* ¼ ∣*e*

*<sup>α</sup> <sup>t</sup>*1þ2*<sup>π</sup>* ð Þ*<sup>β</sup>* sin *<sup>β</sup> <sup>t</sup>*<sup>1</sup> <sup>þ</sup>

*<sup>e</sup>αt*<sup>1</sup> sin ð Þ *<sup>β</sup>t*<sup>1</sup> <sup>þ</sup> *<sup>θ</sup> <sup>e</sup>*

 *e*

**Figure 7.** *Solutions for the* Y *mode of the 3DLM with r* ¼ *10 (a) and the corresponding recurrence plot (b). Power spectra for Y mode solutions of the 3DLM V2, FN* ¼ 0 *(c) and FN* ¼ *1 (d).*

By comparison, **Figure 7c** displays the power spectrum for the linear 3DLM V2 (*FN* ¼ 0) with the same parameter values as the solution in **Figure 7a**. Eigenvalues for the 3DLM V2 *FN* ¼ 0 are approximately �0*:*59550 � 6*:*17416*i,* � 12*:*47567, indicating that the frequency of a linear solution near the nontrivial critical point is 6*:*174161*=*2*π* ¼ 0*:*9826, as confirmed by the power spectrum. However, such a spectral analysis does not reveal a local transition from a decaying oscillation to a constant solution. **Figure 7d** provides the power spectrum for the corresponding nonlinear simulations with *FN* ¼ 1 within the 3DLM V2, which produces a result similar to panel (c). Comparable spectra for linear and nonlinear steady-state solutions are due to the fact that perturbations (which measure departures from the nontrivial critical point) become smaller as time proceeds when solutions move closer to the critical point.

In **Figure 8** for the 3DLM, we plot the time evolution of the X mode in panel (a). As shown, the solution begins by an oscillation with small time-varying amplitudes and gradually increases to become regularly oscillatory at approximately *τ* ¼ 2*:*1.

*Time evolution of X mode solutions (a) and X vs. Y mode solutions (b), (c) of the 3DLM V2 (FN* ¼ 1*) and the original 3DLM with r* ¼ 800*. Panels (a) and (b) display solutions over the time interval [0, 5], while panel (c) shows the solution over the interval [2.5, 5] during which the orbit becomes regularly oscillatory in (a).*

corresponding nontrivial critical point and then becomes a closed curve, leading to a limit cycle that encloses trivial and nontrivial critical points. The closed and isolated features of the limit cycle are clearly shown in panel (c) over the time interval

For comparison, a limit cycle solution of the 5DLM is plotted in **Figure 9**. Just as in plots for the 3DLM, the X mode of the 5DLM grows until it becomes oscillatory with a nearly constant amplitude after *τ* ¼ 3*:*2. **Figure 9b** plots solutions of the X and Y modes over *τ* ∈ ½ � 0*;* 8 . The solution spirals outward from its initial position and gradually forms a limit cycle. Panel (c) displays the solution in the *X* � *Y* space over *τ* ∈ ½ � 4*;* 8 , showing a closed curve. The above spiral orbit and limit cycle and their

Recurrence plots for limit cycle solutions exemplify the method's capacity for

*r* ¼ 800. In both of these RPs, four distinct line patterns can be sequentially identified: (1) a dark block or a cluster of points (bottom left), (2) vertical and horizontal lines, (3) a large white band or a lack of points (middle), and (4) uniformly spaced diagonal lines (upper right). As shown in **Figures 8** and **9**, these patterns correspond to a transition from a spiral solution near a nontrivial critical point to a (nonlinear) periodic solution that encloses trivial and nontrivial critical points.

local analysis. **Figure 10a** and **b** provides RPs for the 3DLM and 5DLM with

In the *X* � *Y* phase space in panel (b), the solution spirals away from the

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006*

*DOI: http://dx.doi.org/10.5772/intechopen.86859*

*τ* ∈ ½ � 2*:*5*;* 5 , displaying only one trajectory.

**Figure 8.**

**53**

transition are analyzed below using the RP.

#### **3.4 Analysis of limit cycle solutions**

At moderate heating parameters, both the 3DLM and 5DLM models produce chaotic solutions when other parameters are held constant. By comparison, for very large values of *r*, the 3DLM and 5DLM produce limit cycle solutions. A limit cycle is defined as a closed and isolated orbit to which nearby trajectories converge (e.g., [26, 27]). In **Figures 8** and **9**, limit cycle solutions are simulated in order to compare the 3DLM V2 and 5DLM V2 with the original 3DLM and 5DLM with *r* ¼ 800. In all three panels for both plots, only the trajectory in red is seen, indicating that the V2 with *FN* ¼ 1 and the original versions produce the same solutions over the given time intervals.

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006 DOI: http://dx.doi.org/10.5772/intechopen.86859*

**Figure 8.**

By comparison, **Figure 7c** displays the power spectrum for the linear 3DLM V2 (*FN* ¼ 0) with the same parameter values as the solution in **Figure 7a**. Eigenvalues for the 3DLM V2 *FN* ¼ 0 are approximately �0*:*59550 � 6*:*17416*i,* � 12*:*47567, indicating that the frequency of a linear solution near the nontrivial critical point is 6*:*174161*=*2*π* ¼ 0*:*9826, as confirmed by the power spectrum. However, such a spectral analysis does not reveal a local transition from a decaying oscillation to a constant solution. **Figure 7d** provides the power spectrum for the corresponding nonlinear simulations with *FN* ¼ 1 within the 3DLM V2, which produces a result similar to panel (c). Comparable spectra for linear and nonlinear steady-state solutions are due to the fact that perturbations (which measure departures from the nontrivial critical point) become smaller as time proceeds when solutions move

*Solutions for the* Y *mode of the 3DLM with r* ¼ *10 (a) and the corresponding recurrence plot (b). Power spectra*

At moderate heating parameters, both the 3DLM and 5DLM models produce chaotic solutions when other parameters are held constant. By comparison, for very large values of *r*, the 3DLM and 5DLM produce limit cycle solutions. A limit cycle is defined as a closed and isolated orbit to which nearby trajectories converge (e.g., [26, 27]). In **Figures 8** and **9**, limit cycle solutions are simulated in order to compare the 3DLM V2 and 5DLM V2 with the original 3DLM and 5DLM with *r* ¼ 800. In all three panels for both plots, only the trajectory in red is seen, indicating that the V2 with *FN* ¼ 1 and the original versions produce the same solutions over the given

closer to the critical point.

**Figure 7.**

time intervals.

**52**

**3.4 Analysis of limit cycle solutions**

*for Y mode solutions of the 3DLM V2, FN* ¼ 0 *(c) and FN* ¼ *1 (d).*

*Current Topics in Tropical Cyclone Research*

*Time evolution of X mode solutions (a) and X vs. Y mode solutions (b), (c) of the 3DLM V2 (FN* ¼ 1*) and the original 3DLM with r* ¼ 800*. Panels (a) and (b) display solutions over the time interval [0, 5], while panel (c) shows the solution over the interval [2.5, 5] during which the orbit becomes regularly oscillatory in (a).*

In **Figure 8** for the 3DLM, we plot the time evolution of the X mode in panel (a). As shown, the solution begins by an oscillation with small time-varying amplitudes and gradually increases to become regularly oscillatory at approximately *τ* ¼ 2*:*1. In the *X* � *Y* phase space in panel (b), the solution spirals away from the corresponding nontrivial critical point and then becomes a closed curve, leading to a limit cycle that encloses trivial and nontrivial critical points. The closed and isolated features of the limit cycle are clearly shown in panel (c) over the time interval *τ* ∈ ½ � 2*:*5*;* 5 , displaying only one trajectory.

For comparison, a limit cycle solution of the 5DLM is plotted in **Figure 9**. Just as in plots for the 3DLM, the X mode of the 5DLM grows until it becomes oscillatory with a nearly constant amplitude after *τ* ¼ 3*:*2. **Figure 9b** plots solutions of the X and Y modes over *τ* ∈ ½ � 0*;* 8 . The solution spirals outward from its initial position and gradually forms a limit cycle. Panel (c) displays the solution in the *X* � *Y* space over *τ* ∈ ½ � 4*;* 8 , showing a closed curve. The above spiral orbit and limit cycle and their transition are analyzed below using the RP.

Recurrence plots for limit cycle solutions exemplify the method's capacity for local analysis. **Figure 10a** and **b** provides RPs for the 3DLM and 5DLM with *r* ¼ 800. In both of these RPs, four distinct line patterns can be sequentially identified: (1) a dark block or a cluster of points (bottom left), (2) vertical and horizontal lines, (3) a large white band or a lack of points (middle), and (4) uniformly spaced diagonal lines (upper right). As shown in **Figures 8** and **9**, these patterns correspond to a transition from a spiral solution near a nontrivial critical point to a (nonlinear) periodic solution that encloses trivial and nontrivial critical points.

#### **Figure 9.**

*Time evolution of X mode solutions (a) and X vs. Y mode solutions (b), (c) of the 5DLM V2 (FN* ¼ 1*) and the original 5DLM with r* ¼ 800*. Panels (a) and (b) show solutions over the time interval [0, 8], while panel (c) occurs over the interval [4, 8] during which the solution becomes a limit cycle.*

Specifically, the solutions initially oscillate with very small amplitudes, progress into oscillations with larger amplitudes, and finally turn into regular oscillations. Since the patterns of the RPs directly correspond to a change in solution types, the RPs are capable of clearly identifying local dynamics and transition. By comparison, a spectral analysis cannot reveal such a local transition.

#### **3.5 Analysis of global reanalysis and model data**

Here, we apply the RP to analyze the case with multiple AEWs. **Figure 11** plots the time series for wind velocity at the 850 hPa level for the ERA-interim data (a) and GMM simulations (b). In panel (a), the amplitude of the signal is nearly constant over the entire duration. In panel (b), the signal's amplitude generally decays first and then grows with time. In comparison to the observations from the NAMMA field campaign (Figure 1b or Figure 4a of [2]), previous studies (e.g., [2, 28]) have indicated that the GMM simulations more accurately represent actual observations as compared to the reanalysis data.

**Figure 11** displays the AEWs from the ERA-interim and GMM data, to be analyzed below. We first create two sets of idealized oscillatory solutions that mimic the "observed" and simulated AEWs. **Figure 12a** displays an idealized solution consisting of a periodic solution with a frequency of 1 and a periodic solution with a frequency of 2, separated by periodic solutions with negligible amplitude, similar to the ERA-interim data. In this simplified case, the oscillatory solutions both have the same amplitude. The corresponding power spectrum indicates peaks

at 1 and 2 but does not readily offer information regarding the wave transitions (not shown). The RP of the signal, as seen in **Figure 12b**, displays uniform diagonal lines in each corner and a solid block of recurrent points in the center. These features

*A recurrence plot of a transition from a spiral orbit (the bottom left corner) to a limit cycle (the top right*

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006*

*DOI: http://dx.doi.org/10.5772/intechopen.86859*

*corner) within the 3DLM (top) and 5DLM (bottom) using r* ¼ *800.*

**Figure 10.**

**55**

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006 DOI: http://dx.doi.org/10.5772/intechopen.86859*

#### **Figure 10.**

Specifically, the solutions initially oscillate with very small amplitudes, progress into oscillations with larger amplitudes, and finally turn into regular oscillations. Since the patterns of the RPs directly correspond to a change in solution types, the RPs are capable of clearly identifying local dynamics and transition. By comparison,

*Time evolution of X mode solutions (a) and X vs. Y mode solutions (b), (c) of the 5DLM V2 (FN* ¼ 1*) and the original 5DLM with r* ¼ 800*. Panels (a) and (b) show solutions over the time interval [0, 8], while panel (c)*

Here, we apply the RP to analyze the case with multiple AEWs. **Figure 11** plots

the time series for wind velocity at the 850 hPa level for the ERA-interim data (a) and GMM simulations (b). In panel (a), the amplitude of the signal is nearly constant over the entire duration. In panel (b), the signal's amplitude generally decays first and then grows with time. In comparison to the observations from the NAMMA field campaign (Figure 1b or Figure 4a of [2]), previous studies (e.g., [2, 28]) have indicated that the GMM simulations more accurately represent actual

**Figure 11** displays the AEWs from the ERA-interim and GMM data, to be analyzed below. We first create two sets of idealized oscillatory solutions that mimic the "observed" and simulated AEWs. **Figure 12a** displays an idealized solution consisting of a periodic solution with a frequency of 1 and a periodic solution with a frequency of 2, separated by periodic solutions with negligible amplitude, similar to the ERA-interim data. In this simplified case, the oscillatory solutions both have the same amplitude. The corresponding power spectrum indicates peaks

a spectral analysis cannot reveal such a local transition.

*occurs over the interval [4, 8] during which the solution becomes a limit cycle.*

*Current Topics in Tropical Cyclone Research*

**3.5 Analysis of global reanalysis and model data**

**Figure 9.**

**54**

observations as compared to the reanalysis data.

*A recurrence plot of a transition from a spiral orbit (the bottom left corner) to a limit cycle (the top right corner) within the 3DLM (top) and 5DLM (bottom) using r* ¼ *800.*

at 1 and 2 but does not readily offer information regarding the wave transitions (not shown). The RP of the signal, as seen in **Figure 12b**, displays uniform diagonal lines in each corner and a solid block of recurrent points in the center. These features

**Figure 12c** presents an idealized solution with the following three components: decaying oscillations, small variations with small amplitudes, and growing oscillations. The idealized data resemble the GMM data. The first and third components correspond to a spiral sink and a spiral source, respectively. **Figure 12d** displays the solution's RP with horizontal and vertical lines. Such a RP pattern resembles a combined feature of the RP in **Figure 7b** for a spiral sink and **Figure 10b** for a spiral

The RPs for the above idealized data, shown in **Figure 12b** and **d**, are used for a comparison with the RPs of the ERA-interim and GMM data, as shown in **Figure 13a**

**Figure 13b** provides the RP for normalized data from the 850 hPa level gener-

**Figure 12d**, with a darker area in the middle and a distinct line pattern surrounding the center. Compared to **Figure 13a**, one major difference is the appearance of horizontal and vertical lines instead of diagonal lines. As suggested by the idealized solution in **Figure 12c** and **d**, this type of structure indicates decaying and growing oscillations, which is also similar to the combined feature in **Figures 7b** and **10b**. The distance between the vertical and horizontal lines can be used to estimate a period for GMM data of approximately 5 days, consistent with observations as well

*Recurrence plots for the normalized wind velocity data at the 850 hPa level for the ERA-interim data (a) and global mesoscale model data (b). Panels (a) and (b) produce the RP analyses that are consistent with those in*

ated by the GMM. The line structure present displays some similarities to

and **b**, to reveal the features of observed and simulated AEWs in **Figure 11**. **Figure 13a** presents the RP for normalized ERA-interm data at the 850 hPa level. A solid black region is present in the middle, indicating slow variations in states across this region. In areas surrounding the center, the familiar diagonal line structure associated with oscillatory behavior, and possessing some degree of periodicity, can be identified. The data results in a diagonal line structure in RPs that is comparable to **Figure 12b**, consisting of periodic solutions. Based on visual inspection, most of the recurrence points fall into diagonal lines, while a few stand alone. Finding an estimated period for the recurrence of AEWs through the RP is feasible. By calculating the distance between them, we obtain a good measure of the time interval between AEW occurrences. Using this approach, the distance between diagonal lines

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006*

*DOI: http://dx.doi.org/10.5772/intechopen.86859*

ranges from 3.5 to 5 days, consistent with the observed AEWs.

as the RP analysis of the ERA-interim data.

**Figure 13.**

**57**

*Figure 12b and d, respectively.*

outward solution.

#### **Figure 11.**

*Time series of the wind velocity at the 850 hPa level for the ERA-interim dataset (a) and the global mesoscale model (b).*

#### **Figure 12.**

*Idealized oscillatory data used to mimic multiple AEWs with two different periods (a) and decaying and growing amplitudes (c). The corresponding RPs are seen in (b) and (d), respectively.*

indicate transitions between all three types of solution components. The first component with a frequency of 1 is shown by the distance between the diagonal lines. The black section in the center corresponds to no or slow variations of the states in time. For the third component (i.e., the second periodic component), uniform diagonal lines are present and are separated by a length of 0.5, consistent with the frequency of 2.

#### *A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006 DOI: http://dx.doi.org/10.5772/intechopen.86859*

**Figure 12c** presents an idealized solution with the following three components: decaying oscillations, small variations with small amplitudes, and growing oscillations. The idealized data resemble the GMM data. The first and third components correspond to a spiral sink and a spiral source, respectively. **Figure 12d** displays the solution's RP with horizontal and vertical lines. Such a RP pattern resembles a combined feature of the RP in **Figure 7b** for a spiral sink and **Figure 10b** for a spiral outward solution.

The RPs for the above idealized data, shown in **Figure 12b** and **d**, are used for a comparison with the RPs of the ERA-interim and GMM data, as shown in **Figure 13a** and **b**, to reveal the features of observed and simulated AEWs in **Figure 11**. **Figure 13a** presents the RP for normalized ERA-interm data at the 850 hPa level. A solid black region is present in the middle, indicating slow variations in states across this region. In areas surrounding the center, the familiar diagonal line structure associated with oscillatory behavior, and possessing some degree of periodicity, can be identified. The data results in a diagonal line structure in RPs that is comparable to **Figure 12b**, consisting of periodic solutions. Based on visual inspection, most of the recurrence points fall into diagonal lines, while a few stand alone. Finding an estimated period for the recurrence of AEWs through the RP is feasible. By calculating the distance between them, we obtain a good measure of the time interval between AEW occurrences. Using this approach, the distance between diagonal lines ranges from 3.5 to 5 days, consistent with the observed AEWs.

**Figure 13b** provides the RP for normalized data from the 850 hPa level generated by the GMM. The line structure present displays some similarities to **Figure 12d**, with a darker area in the middle and a distinct line pattern surrounding the center. Compared to **Figure 13a**, one major difference is the appearance of horizontal and vertical lines instead of diagonal lines. As suggested by the idealized solution in **Figure 12c** and **d**, this type of structure indicates decaying and growing oscillations, which is also similar to the combined feature in **Figures 7b** and **10b**. The distance between the vertical and horizontal lines can be used to estimate a period for GMM data of approximately 5 days, consistent with observations as well as the RP analysis of the ERA-interim data.

#### **Figure 13.**

*Recurrence plots for the normalized wind velocity data at the 850 hPa level for the ERA-interim data (a) and global mesoscale model data (b). Panels (a) and (b) produce the RP analyses that are consistent with those in Figure 12b and d, respectively.*

indicate transitions between all three types of solution components. The first component with a frequency of 1 is shown by the distance between the diagonal lines. The black section in the center corresponds to no or slow variations of the states in time. For the third component (i.e., the second periodic component), uniform diagonal lines are present and are separated by a length of 0.5, consistent with the

*Idealized oscillatory data used to mimic multiple AEWs with two different periods (a) and decaying and*

*growing amplitudes (c). The corresponding RPs are seen in (b) and (d), respectively.*

*Time series of the wind velocity at the 850 hPa level for the ERA-interim dataset (a) and the global mesoscale*

frequency of 2.

**56**

**Figure 12.**

**Figure 11.**

*Current Topics in Tropical Cyclone Research*

*model (b).*

## **4. Concluding remarks**

Accurate detection of recurrent, multiple AEWs and their evolution (e.g., intensification) may provide a good indicator for determining the timing and location of tropical cyclone formation and initial intensification. Since large-scale AEWs have better predictability, such an indicator may help extend the lead time of TC formation prediction. To achieve this goal, in this study, we first deployed the recurrence plot method in Python and verified our implementation using several types of idealized solutions and three types of solutions obtained in the classical 3DLM and 5DLM. We then applied the RP method to analyze two, real-world datasets, ERA-interim data and global mesoscale model data, in order to reveal the time-varying amplitudes and frequencies of multiple AEWs over a 30-day period between late August and September 2006. Compared to traditional global analysis methods (e.g., spectral analysis), the RP method is effective in showing temporal growing or decaying oscillations and the transition between these two types of solutions. As a result, the RP analysis of global data not only produces a good estimate of the period for AEWs but also displays the weakening or intensifying trend of AEW intensities, as shown in **Figure 13**.

determining the "recurrence rate" and "determinism" in order to quantitatively measure the recurrence and determinism (or "predictability") of the recurrent

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006*

We thank Dr. Y.-L. Wu, Dr. J. Chern, and Ms. S. Faghih-Naini for their scientific discussions and technical help. Thanks are also extended to Dr. B. Bailey, Dr. F. De Sales, and anonymous reviewers for their comments on the original manuscript. We are grateful for the support from the College of Science of

The perturbation method was applied to derive the so-called V2 system in order to analyze solutions of the 3DLM, as well as the 5DLM, with initial conditions near the nontrivial critical point. The 5DLM V2 and its non-dissipative V2 were previously documented in [13] and [7], respectively. This appendix simply discusses the 3DLM V2. For the 3DLM, in Eqs. (1)–(3) without the feedback term *XY*1, the total field is decomposed into the basic state (*Xc, Yc, Zc*) and the perturbation (*X*<sup>0</sup>

with respect to the basic state in the form of *X* ¼ *Xc* þ *X*<sup>0</sup> for each state variable,

*<sup>d</sup><sup>τ</sup>* <sup>¼</sup> *XcYc* <sup>þ</sup> *XcY*<sup>0</sup> <sup>þ</sup> *YcX*<sup>0</sup> <sup>þ</sup> *<sup>X</sup>*<sup>0</sup>

independent, nontrivial critical point solutions, defined as (e.g., [11])

*dX*<sup>0</sup>

Note that the above basic state solutions are determined using the time-

*Xc* <sup>¼</sup> *Yc* ¼ � ffiffiffiffiffiffiffi

*<sup>d</sup><sup>τ</sup>* <sup>¼</sup> ð Þ *<sup>r</sup>* � *Zc <sup>X</sup>*<sup>0</sup> � *XcZ*<sup>0</sup> � *<sup>Y</sup>*<sup>0</sup> � **FN** *<sup>X</sup>*<sup>0</sup>

*<sup>d</sup><sup>τ</sup>* <sup>¼</sup> *YcX*<sup>0</sup> <sup>þ</sup> *XcY*<sup>0</sup> � *bZ*<sup>0</sup> <sup>þ</sup> **FN** *<sup>X</sup>*<sup>0</sup>

which describe the time evolution of the perturbations. *FN* is a flag identifying whether the system is fully nonlinear (*FN* ¼ 1) or not (*FN* ¼ 0). The above system

Since the basic state solutions are time independent, Eqs. (A1)–(A3) are reduced

*<sup>d</sup><sup>τ</sup>* ¼ �*<sup>σ</sup> Xc* <sup>þ</sup> *<sup>X</sup>*<sup>0</sup> ð Þþ *<sup>σ</sup> Yc* <sup>þ</sup> *<sup>Y</sup>*<sup>0</sup> ð Þ*,* (A1)

*Y*<sup>0</sup> ð Þ � *b Zc* þ *Z*<sup>0</sup> ð Þ*:* (A3)

*Zc* ¼ *r* � 1*,* (A4)

*<sup>d</sup><sup>τ</sup>* ¼ �*<sup>σ</sup> <sup>X</sup>*<sup>0</sup> � *<sup>Y</sup>*<sup>0</sup> ð Þ*,* (A6)

p *:* (A5)

*Z*<sup>0</sup> ð Þ*,* (A7)

*Y*<sup>0</sup> ð Þ*,* (A8)

*Z*<sup>0</sup> ð Þ þ *r Xc* þ *X*<sup>0</sup> ð Þ� *Yc* þ *Y*<sup>0</sup> ð Þ*,* (A2)

*bZc*

*, Y*0 *, Z*0 )

solutions.

**Acknowledgements**

San Diego State University.

yielding a new set of equations:

*dZc dτ* þ

*dY*0

*dYc dτ* þ

to become

**59**

**A. Appendix A: the 3DLM Version 2**

*DOI: http://dx.doi.org/10.5772/intechopen.86859*

*dXc dτ* þ

*dZ*0

*dY*<sup>0</sup>

*dZ*<sup>0</sup>

*dX*0

*<sup>d</sup><sup>τ</sup>* ¼ � *XcZc* <sup>þ</sup> *XcZ*<sup>0</sup> <sup>þ</sup> *ZcX*<sup>0</sup> <sup>þ</sup> *<sup>X</sup>*<sup>0</sup>

A summary on the performance of the RP in analyzing various types of solutions is provided below. We first performed the analysis for (1) four basic types of data, including periodic, quasiperiodic, and chaotic solutions as well as Gaussian white noise, (2) idealized decaying/growing oscillations, (3) steady-state solutions using the 3DLM, (4) limit cycle solutions using the 3DLM and 5DLM with a focus on the transition from a spiral solution near a nontrivial critical point to a nonlinear periodic solution that encloses trivial and nontrivial critical points, and (5) idealized solutions that mimic the features of selected AEW data with different periods and different amplitudes. For basic types of solutions, the RP produces a consistent analysis as compared to previous studies. For time-varying oscillations, we discussed how recurrence may lead to horizontal and vertical lines in RPs. The distance between two consecutive horizontal (or vertical) lines gives an estimate of half of a period (i.e., *T=*2). For the fourth and fifth types of datasets, the corresponding RP analyses clearly display local transitions between solution components with various periods or amplitudes.

Based on its promising performance in revealing the features of idealized solutions, the selected RP method was applied to analyze ERA-interim data and GMM data, yielding a RP analysis consistent with the analysis of the idealized AEWs from the fifth dataset. While the ERA-interim data resemble the idealized data in **Figure 12a** with comparable amplitudes but different periods, the GMM data resemble the idealized data in **Figure 12c** with decaying and growing oscillations. The RP analysis of the ERA-interim data in **Figure 13a** produces an estimated period of 3.5–5 days, and a spectral analysis of the same data yields a dominant period of 4.7 days. Both estimated periods are consistent with the observed results. However, only the RP analysis can reveal local variations or transitions. By comparison, the RP pattern of the global model data in **Figure 13b** that displays horizontal and vertical lines is similar to the RP plot of the idealized data in **Figure 12d**. While the RP produces a comparable period of approximately 5 days, it additionally reveals realistic information regarding the weakening and intensifying trend of AEWs.

Recent studies within simplified and generalized Lorenz models [22, 29–32] suggest the importance of detecting oscillatory components for extending the lead time of weather prediction. The analysis with RP is encouraging but largely produces qualitative information. Our future work includes an application of recurrence quantification analysis (RQA) methods (e.g., [33–35]) for quantitatively

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006 DOI: http://dx.doi.org/10.5772/intechopen.86859*

determining the "recurrence rate" and "determinism" in order to quantitatively measure the recurrence and determinism (or "predictability") of the recurrent solutions.

### **Acknowledgements**

**4. Concluding remarks**

*Current Topics in Tropical Cyclone Research*

trend of AEW intensities, as shown in **Figure 13**.

ponents with various periods or amplitudes.

trend of AEWs.

**58**

Accurate detection of recurrent, multiple AEWs and their evolution (e.g., intensification) may provide a good indicator for determining the timing and location of tropical cyclone formation and initial intensification. Since large-scale AEWs have better predictability, such an indicator may help extend the lead time of TC formation prediction. To achieve this goal, in this study, we first deployed the recurrence plot method in Python and verified our implementation using several types of idealized solutions and three types of solutions obtained in the classical 3DLM and 5DLM. We then applied the RP method to analyze two, real-world datasets, ERA-interim data and global mesoscale model data, in order to reveal the time-varying amplitudes and frequencies of multiple AEWs over a 30-day period between late August and September 2006. Compared to traditional global analysis methods (e.g., spectral analysis), the RP method is effective in showing temporal growing or decaying oscillations and the transition between these two types of solutions. As a result, the RP analysis of global data not only produces a good estimate of the period for AEWs but also displays the weakening or intensifying

A summary on the performance of the RP in analyzing various types of solutions is provided below. We first performed the analysis for (1) four basic types of data, including periodic, quasiperiodic, and chaotic solutions as well as Gaussian white noise, (2) idealized decaying/growing oscillations, (3) steady-state solutions using the 3DLM, (4) limit cycle solutions using the 3DLM and 5DLM with a focus on the transition from a spiral solution near a nontrivial critical point to a nonlinear periodic solution that encloses trivial and nontrivial critical points, and (5) idealized solutions that mimic the features of selected AEW data with different periods and different amplitudes. For basic types of solutions, the RP produces a consistent analysis as compared to previous studies. For time-varying oscillations, we discussed how recurrence may lead to horizontal and vertical lines in RPs. The distance between two consecutive horizontal (or vertical) lines gives an estimate of

half of a period (i.e., *T=*2). For the fourth and fifth types of datasets, the

the fifth dataset. While the ERA-interim data resemble the idealized data in **Figure 12a** with comparable amplitudes but different periods, the GMM data resemble the idealized data in **Figure 12c** with decaying and growing oscillations. The RP analysis of the ERA-interim data in **Figure 13a** produces an estimated period of 3.5–5 days, and a spectral analysis of the same data yields a dominant period of 4.7 days. Both estimated periods are consistent with the observed results.

However, only the RP analysis can reveal local variations or transitions. By comparison, the RP pattern of the global model data in **Figure 13b** that displays horizontal and vertical lines is similar to the RP plot of the idealized data in

**Figure 12d**. While the RP produces a comparable period of approximately 5 days, it additionally reveals realistic information regarding the weakening and intensifying

Recent studies within simplified and generalized Lorenz models [22, 29–32] suggest the importance of detecting oscillatory components for extending the lead time of weather prediction. The analysis with RP is encouraging but largely produces qualitative information. Our future work includes an application of recurrence quantification analysis (RQA) methods (e.g., [33–35]) for quantitatively

corresponding RP analyses clearly display local transitions between solution com-

Based on its promising performance in revealing the features of idealized solutions, the selected RP method was applied to analyze ERA-interim data and GMM data, yielding a RP analysis consistent with the analysis of the idealized AEWs from

We thank Dr. Y.-L. Wu, Dr. J. Chern, and Ms. S. Faghih-Naini for their scientific discussions and technical help. Thanks are also extended to Dr. B. Bailey, Dr. F. De Sales, and anonymous reviewers for their comments on the original manuscript. We are grateful for the support from the College of Science of San Diego State University.

## **A. Appendix A: the 3DLM Version 2**

The perturbation method was applied to derive the so-called V2 system in order to analyze solutions of the 3DLM, as well as the 5DLM, with initial conditions near the nontrivial critical point. The 5DLM V2 and its non-dissipative V2 were previously documented in [13] and [7], respectively. This appendix simply discusses the 3DLM V2. For the 3DLM, in Eqs. (1)–(3) without the feedback term *XY*1, the total field is decomposed into the basic state (*Xc, Yc, Zc*) and the perturbation (*X*<sup>0</sup> *, Y*0 *, Z*0 ) with respect to the basic state in the form of *X* ¼ *Xc* þ *X*<sup>0</sup> for each state variable, yielding a new set of equations:

$$\frac{dX\_{\epsilon}}{d\tau} + \frac{dX'}{d\tau} = -\sigma(X\_{\epsilon} + X') + \sigma(Y\_{\epsilon} + Y'),\tag{A1}$$

$$\frac{dY\_{\varepsilon}}{d\tau} + \frac{dY'}{d\tau} = -(X\_{\varepsilon}Z\_{\varepsilon} + X\_{\varepsilon}Z' + Z\_{\varepsilon}X' + X'Z') + r(X\_{\varepsilon} + X') - (Y\_{\varepsilon} + Y'), \tag{A2}$$

$$\frac{d\mathbf{Z}\_c}{d\tau} + \frac{d\mathbf{Z}'}{d\tau} = (\mathbf{X}\_c \mathbf{Y}\_c + \mathbf{X}\_c \mathbf{Y}' + \mathbf{Y}\_c \mathbf{X}' + \mathbf{X}' \mathbf{Y}') - b(\mathbf{Z}\_c + \mathbf{Z}').\tag{A3}$$

Note that the above basic state solutions are determined using the timeindependent, nontrivial critical point solutions, defined as (e.g., [11])

$$Z\_{\mathfrak{c}} = r - \mathfrak{1},\tag{A4}$$

$$X\_{\mathfrak{c}} = Y\_{\mathfrak{c}} = \pm \sqrt{bZ\_{\mathfrak{c}}}.\tag{A5}$$

Since the basic state solutions are time independent, Eqs. (A1)–(A3) are reduced to become

$$\frac{dX'}{d\tau} = -\sigma(X'-Y'),\tag{A6}$$

$$\frac{dY'}{d\tau} = (r - Z\_{\epsilon})X' - X\_{\epsilon}Z' - Y' - \mathbf{FN}(X'Z'),\tag{A7}$$

$$\frac{dZ'}{d\tau} = Y\_c X' + X\_c Y' - bZ' + \mathbf{FN}(X'Y'),\tag{A8}$$

which describe the time evolution of the perturbations. *FN* is a flag identifying whether the system is fully nonlinear (*FN* ¼ 1) or not (*FN* ¼ 0). The above system

#### **Figure A1.**

*Time evolution of the X mode solutions of the original 3DLM and 3DLM V2 with FN* ¼ 1 *over the time interval 0* ≤ *τ* ≤ *50 with a time step Δτ* ¼ *0:001.*

#### **Figure A2.**

*Time evolution of Y*<sup>0</sup> *linear (blue) and nonlinear (red) solutions using the 3DLM V2 with 0* ≤ *τ* ≤ *110 and* Δ*τ* ¼ *0:001.*

**Author details**

CA, USA

**61**

Tiffany Reyes and Bo-Wen Shen\*

\*Address all correspondence to: bshen@sdsu.edu

provided the original work is properly cited.

Department of Mathematics and Statistics, San Diego State University, San Diego,

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006*

*DOI: http://dx.doi.org/10.5772/intechopen.86859*

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

with *FN* ¼ 0 or *FN* ¼ 1 is referred to as a Version 2 system, denoted by V2. In the V2 system, there are no nonlinear terms that involve the product of two basic state variables. The system only allows interactions between one basic state variable and one perturbation (i.e., *XcY*<sup>0</sup> ) or between two perturbation variables (e.g., *X*<sup>0</sup> *Y*0 ). By setting *FN* ¼ 0, the system is linear with respect to the perturbation. When the initial perturbations are small, nonlinear terms with two perturbation variables are smaller. Thus, such a linear system is capable of depicting the initial time evolution of the full 3DLM when the perturbation is small. In comparison to the full 3DLM, the nonlinear V2 model (*FN* ¼ 1) produces the same solutions, even at the onset of chaotic irregular oscillations, as shown in **Figure A1**. The time evolutions of the *Y*<sup>0</sup> solutions for both *FN* ¼ 0 and *FN* ¼ 1 are shown in **Figure A2**. The solutions are initially the same and begin to diverge as time further progresses, indicating the impact of nonlinearity. An eigenvalue analysis of the 3DLM V2 is provided in Section 2.1.3 of Reyes [24].

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006 DOI: http://dx.doi.org/10.5772/intechopen.86859*

## **Author details**

with *FN* ¼ 0 or *FN* ¼ 1 is referred to as a Version 2 system, denoted by V2. In the V2 system, there are no nonlinear terms that involve the product of two basic state variables. The system only allows interactions between one basic state variable and

*Time evolution of Y*<sup>0</sup> *linear (blue) and nonlinear (red) solutions using the 3DLM V2 with 0* ≤ *τ* ≤ *110 and*

*Time evolution of the X mode solutions of the original 3DLM and 3DLM V2 with FN* ¼ 1 *over the time interval*

setting *FN* ¼ 0, the system is linear with respect to the perturbation. When the initial perturbations are small, nonlinear terms with two perturbation variables are smaller. Thus, such a linear system is capable of depicting the initial time evolution of the full 3DLM when the perturbation is small. In comparison to the full 3DLM, the nonlinear V2 model (*FN* ¼ 1) produces the same solutions, even at the onset of chaotic irregular oscillations, as shown in **Figure A1**. The time evolutions of the *Y*<sup>0</sup> solutions for both *FN* ¼ 0 and *FN* ¼ 1 are shown in **Figure A2**. The solutions are initially the same and begin to diverge as time further progresses, indicating the impact of nonlinearity. An eigenvalue analysis of the 3DLM V2 is provided in

) or between two perturbation variables (e.g., *X*<sup>0</sup>

*Y*0 ). By

one perturbation (i.e., *XcY*<sup>0</sup>

**Figure A1.**

**Figure A2.**

Δ*τ* ¼ *0:001.*

*0* ≤ *τ* ≤ *50 with a time step Δτ* ¼ *0:001.*

*Current Topics in Tropical Cyclone Research*

Section 2.1.3 of Reyes [24].

**60**

Tiffany Reyes and Bo-Wen Shen\* Department of Mathematics and Statistics, San Diego State University, San Diego, CA, USA

\*Address all correspondence to: bshen@sdsu.edu

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

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[2] Shen B-W, Tao W-K, Wu M-L. African easterly waves in 30-day highresolution global simulations: A case study during the 2006 NAMMA period. Geophysical Research Letters. 2010;**37**: L18803. DOI: 10.1029/2010GL044355

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[6] Marwan N, Webber CL Jr. Recurrence Quantification Analysis: Theory and Best Practices. Switzerland: Springer International Publishing; 2015. p. 421

[7] Faghih-Naini S, Shen B-W. Quasi-periodic in the five-dimensional non-dissipative Lorenz model: The role of the extended nonlinear feedback loop. International Journal of Bifurcation and Chaos. 2018;**28**(6): 1850072. DOI: 10.1142/ S0218127418500724

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[16] Felicio CC, Rech PC. On the dynamics of five- and six-dimensional Lorenz models. Journal of Physics Communications. 2018;**2**:025028

[17] Shen B-W. Nonlinear feedback in a six-dimensional Lorenz model. Impact of an additional heating term. Nonlinear Processes in Geophysics. 2015;**22**: 749-764. DOI: 10.5194/npg-22-749-2015

*DOI: http://dx.doi.org/10.5772/intechopen.86859*

*A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006*

[24] Reyes TAL. Applying recurrence quantification analysis methods for the analysis of global reanalysis and model data to reveal the local oscillations of multiple African easterly waves during 2006 [Master thesis]. San Diego State

University; 2018. p. 64

[25] Webber CL Jr. Recurrence quantification of fractal structures. Front Physiotherapy. 2012;**3**:382. DOI:

[26] Jordan DW, Smith P. Nonlinear Ordinary Differential Equations. An Introduction for Scientists and Engineers. 4th ed. New York: Oxford University Press; 2007. p. 560

[27] Nagel RK, Saff E, Snider A. Fundamentals of Differential

10.1029/2009JD013140

2008. p. 912

Equations. 7th ed. New York: Pearson;

[28] Shen B-W, Tao W-K, Lau W, Atlas R. Predicting tropical cyclogenesis with a global mesoscale model: Hierarchical multiscale interactions during the formation of tropical cyclone Nargis (2008). Journal of Geophysical Research. 2010;**115**:D14102. DOI:

[29] Shen B-W. On the predictability of 30-day global mesoscale simulations of multiple african easterly waves during summer 2006: A view with a generalized Lorenz model. Geosciences

[30] Shen B-W. Homoclinic orbits and

[31] Shen B-W, Pielke RA Sr, Zeng X, Baik J-J, Faghih-Naini S, Cui J, Atlas R. Is weather chaotic? Coexistence of chaos and order within a generalized Lorenz

2019b;**9**(7):281. DOI: 10.3390/

solitary waves within the nondissipative Lorenz model and KdV equation. International Journal of Bifurcation and Chaos. DOI: 10.1142/ S0218127420502570. (in press)

geosciences9070281

10.3389/fphys.2012.00382

[18] Shen B-W. Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model. Nonlinear Processes in Geophysics. 2016;**23**:189-203. DOI:

[19] Shen B-W. On an extension of the nonlinear feedback loop in a ninedimensional Lorenz model. Chaotic Modeling and Simulation (CMSIM).

[20] Shen BW, Reyes T, Faghih-Naini S. Coexistence of chaotic and non-chaotic orbits in a new nine-dimensional Lorenz model. In: Skiadas CH, Lubashevsky I, editors. The 11th Chaos International Conference; Springer Proceedings in Complexity; Cham: Springer; 2019. DOI:

[21] Reyes TAL, Shen B-W. A recurrence analysis of chaotic and non-chaotic solutions within a generalized ninedimensional Lorenz model. Chaos, Solitons & Fractals. 2019:**125**(2019): 1-12. DOI: 10.1016/j.chaos. 2019.05.003

[22] Shen B-W. Aggregated negative feedback in a generalized Lorenz model. International Journal of Bifurcation and Chaos. 2019;**29**(3):1950037. DOI: 10.1142/S0218127419500378

[23] Marwan N, Romano MC, Thiel M, Kurths J. Recurrence plots for the analysis of complex systems. Physics

Reports. 2007;**438**:237-329

**63**

10.1007/978-3-030-15297-0\_22

10.5194/npg-23-189-2016

2017;**2**:147157

S021812741 7501760

[8] Thompson JMT, Stewart HB. Nonlinear Dynamics and Chaos. 2nd ed. Chichester, United Kingdom: John Wiley & Sons, Ltd.; 2002. p. 437

[9] Shen B-W, Atlas R, Chern J-D, Reale O, Lin S-J, Lee T, et al. The 0.125 degree finite-volume general circulation model on the NASA Columbia supercomputer: Preliminary simulations of mesoscale vortices. Geophysical Research Letters. 2006;**33**:L05801. DOI: 10.1029/ 2005GL024594

[10] Zipser EJ et al. The Saharan air layer and the fate of African easterly waves— NASA AMMA field study of tropical cyclogenesis. Bulletin of the American Meteorological Society. 2009;**90**: 1137-1156

[11] Lorenz E. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences. 1963;**20**:130-141

[12] Lorenz EN. The Essence of Chaos. Seattle: University of Washington Press; 1993. p. 227

[13] Shen B-W. Nonlinear feedback in a five-dimensional Lorenz model. Journal of the Atmospheric Sciences. 2014;**71**: 1701-1723. DOI: 10.1175/JAS-D-13-0223.1

[14] Lin SJ. A vertically Lagrangian finite volume dynamical core for global models. Monthly Weather Review. 2004; **132**:2293-2307. DOI: 10.1175/1520-0493

[15] Moon S, Han B-S, Park J, Seo JM, Baik J-J. Periodicity and chaos of highorder Lorenz systems. International Journal of Bifurcation and Chaos. 2017; *A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006 DOI: http://dx.doi.org/10.5772/intechopen.86859*

**27**(11):1750176. DOI: 10.1142/ S021812741 7501760

**References**

[1] Shen B-W, Atlas R, Reale O, Lin S-J, Chern J-D, Chang J, et al. Hurricane forecasts with a global mesoscaleresolving model: Preliminary results with hurricane Katrina (2005).

*Current Topics in Tropical Cyclone Research*

non-dissipative Lorenz model: The role of the extended nonlinear feedback loop. International Journal of Bifurcation and Chaos. 2018;**28**(6):

1850072. DOI: 10.1142/ S0218127418500724

[8] Thompson JMT, Stewart HB.

2006;**33**:L05801. DOI: 10.1029/

2005GL024594

1137-1156

1993. p. 227

Nonlinear Dynamics and Chaos. 2nd ed. Chichester, United Kingdom: John Wiley & Sons, Ltd.; 2002. p. 437

[9] Shen B-W, Atlas R, Chern J-D, Reale O, Lin S-J, Lee T, et al. The 0.125 degree finite-volume general circulation model on the NASA Columbia supercomputer: Preliminary simulations of mesoscale vortices. Geophysical Research Letters.

[10] Zipser EJ et al. The Saharan air layer and the fate of African easterly waves— NASA AMMA field study of tropical cyclogenesis. Bulletin of the American Meteorological Society. 2009;**90**:

[11] Lorenz E. Deterministic nonperiodic

[12] Lorenz EN. The Essence of Chaos. Seattle: University of Washington Press;

[13] Shen B-W. Nonlinear feedback in a five-dimensional Lorenz model. Journal of the Atmospheric Sciences. 2014;**71**: 1701-1723. DOI: 10.1175/JAS-D-13-0223.1

[14] Lin SJ. A vertically Lagrangian finite volume dynamical core for global models. Monthly Weather Review. 2004; **132**:2293-2307. DOI: 10.1175/1520-0493

[15] Moon S, Han B-S, Park J, Seo JM, Baik J-J. Periodicity and chaos of highorder Lorenz systems. International Journal of Bifurcation and Chaos. 2017;

flow. Journal of the Atmospheric Sciences. 1963;**20**:130-141

Geophysical Research Letters. 2006;**33**: L13813. DOI: 10.1029/2006GL026143

[3] Shen B-W, Cheung S, Li J-LF,Wu Y-L,

Shen SS. Multiscale processes of Hurricane Sandy (2012) as revealed by the parallel ensemble empirical mode

decomposition and advanced

S2424922X16500054

ensemble empirical mode decomposition (PEEMD) and its application for earth science data analysis. Computing in Science & Engineering. 2017;**19**(5):49-57. DOI:

10.1109/MCSE.2017.3421555

15-0257.1

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visualization technology. Advances in Data Science and Adaptive Analysis. 2016;**08**:1650005. DOI: 10.1142/

[4] Shen B-W, Cheung S, Wu Y, Li F, Kao D. Parallel implementation of the

[5] Wu Y-L, Shen B-W. An evaluation of the parallel ensemble empirical mode decomposition method in revealing the role of downscaling processes associated with African easterly waves in tropical cyclone genesis. Journal of Atmospheric and Oceanic Technology. 2016;**33**: 1611-1628. DOI: 10.1175/JTECH-D-

[6] Marwan N, Webber CL Jr. Recurrence Quantification Analysis: Theory and Best

Quasi-periodic in the five-dimensional

Practices. Switzerland: Springer International Publishing; 2015. p. 421

[7] Faghih-Naini S, Shen B-W.

[2] Shen B-W, Tao W-K, Wu M-L. African easterly waves in 30-day highresolution global simulations: A case study during the 2006 NAMMA period. Geophysical Research Letters. 2010;**37**: L18803. DOI: 10.1029/2010GL044355

[16] Felicio CC, Rech PC. On the dynamics of five- and six-dimensional Lorenz models. Journal of Physics Communications. 2018;**2**:025028

[17] Shen B-W. Nonlinear feedback in a six-dimensional Lorenz model. Impact of an additional heating term. Nonlinear Processes in Geophysics. 2015;**22**: 749-764. DOI: 10.5194/npg-22-749-2015

[18] Shen B-W. Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model. Nonlinear Processes in Geophysics. 2016;**23**:189-203. DOI: 10.5194/npg-23-189-2016

[19] Shen B-W. On an extension of the nonlinear feedback loop in a ninedimensional Lorenz model. Chaotic Modeling and Simulation (CMSIM). 2017;**2**:147157

[20] Shen BW, Reyes T, Faghih-Naini S. Coexistence of chaotic and non-chaotic orbits in a new nine-dimensional Lorenz model. In: Skiadas CH, Lubashevsky I, editors. The 11th Chaos International Conference; Springer Proceedings in Complexity; Cham: Springer; 2019. DOI: 10.1007/978-3-030-15297-0\_22

[21] Reyes TAL, Shen B-W. A recurrence analysis of chaotic and non-chaotic solutions within a generalized ninedimensional Lorenz model. Chaos, Solitons & Fractals. 2019:**125**(2019): 1-12. DOI: 10.1016/j.chaos. 2019.05.003

[22] Shen B-W. Aggregated negative feedback in a generalized Lorenz model. International Journal of Bifurcation and Chaos. 2019;**29**(3):1950037. DOI: 10.1142/S0218127419500378

[23] Marwan N, Romano MC, Thiel M, Kurths J. Recurrence plots for the analysis of complex systems. Physics Reports. 2007;**438**:237-329

[24] Reyes TAL. Applying recurrence quantification analysis methods for the analysis of global reanalysis and model data to reveal the local oscillations of multiple African easterly waves during 2006 [Master thesis]. San Diego State University; 2018. p. 64

[25] Webber CL Jr. Recurrence quantification of fractal structures. Front Physiotherapy. 2012;**3**:382. DOI: 10.3389/fphys.2012.00382

[26] Jordan DW, Smith P. Nonlinear Ordinary Differential Equations. An Introduction for Scientists and Engineers. 4th ed. New York: Oxford University Press; 2007. p. 560

[27] Nagel RK, Saff E, Snider A. Fundamentals of Differential Equations. 7th ed. New York: Pearson; 2008. p. 912

[28] Shen B-W, Tao W-K, Lau W, Atlas R. Predicting tropical cyclogenesis with a global mesoscale model: Hierarchical multiscale interactions during the formation of tropical cyclone Nargis (2008). Journal of Geophysical Research. 2010;**115**:D14102. DOI: 10.1029/2009JD013140

[29] Shen B-W. On the predictability of 30-day global mesoscale simulations of multiple african easterly waves during summer 2006: A view with a generalized Lorenz model. Geosciences 2019b;**9**(7):281. DOI: 10.3390/ geosciences9070281

[30] Shen B-W. Homoclinic orbits and solitary waves within the nondissipative Lorenz model and KdV equation. International Journal of Bifurcation and Chaos. DOI: 10.1142/ S0218127420502570. (in press)

[31] Shen B-W, Pielke RA Sr, Zeng X, Baik J-J, Faghih-Naini S, Cui J, Atlas R. Is weather chaotic? Coexistence of chaos and order within a generalized Lorenz

model. Bulletin of American Meteorological Society. 2020: 1-28. DOI: 10.1175/BAMS-D-19-0165.1

[32] Shen B-W, Pielke RA Sr, Zeng X, Baik J-J, Faghih-Naini S, Cui J, Atlas R, Reyes TA. Is Weather chaotic? Coexisting chaotic and non-chaotic attractors within Lorenz models. The 13th Chaos International Conference (CHAOS2020); 9-12 June 2020. (virtual conference)

[33] Webber CL Jr, Zbilut JP. Dynamical assessment of physiological systems and states using recurrence plot strategies. Journal of Applied Physiology. 1994; **76**(2):965-973

[34] Zbilut JP, Webber CL Jr. Embeddings and delays as derived from quantification of recurrence plots. Physics Letters A. 1992;**171**(34):199-203. DOI: 10.1016/0375-9601(92)90426-M

[35] Zbilut JP, Webber CL Jr. Recurrence quantification analysis: Introduction and historical context. International Journal of Bifurcation and Chaos. 2007; **17**(10):3477-3481. DOI: 10.1142/ S0218127407019238

**65**

**Chapter 4**

**Abstract**

**1. Introduction**

Tropical Cyclones

*Zhiyuan Wu and Mack Conde*

rainfall under the influence of tropical cyclones.

formation of a typhoon disaster chain [4–6].

temperature, sea salinity, extreme rainfall; air-sea interaction

Response of the Coastal Ocean to

The Northwest Pacific and the South China Sea region are the birthplaces of most monsoon disturbances and tropical cyclones and are an important channel for the generation and transmission of water vapor. The Northwest Pacific plays a major role in regulating interdecadal and long-term changes in climate. China experiences the largest number of typhoon landfalls and the most destructive power affected by typhoons in the world. The hidden dangers of typhoon disasters are accelerating with the acceleration of urbanization, the rapid development of economic construction and global warming. The coastal cities are the most dynamic and affluent areas of China's economic development. They are the strong magnetic field that attracts international capital in China, and are also the most densely populated areas and important port groups in China. Although these regions are highly developed, they are vulnerable to disasters. When typhoons hit, the economic losses and casualties caused by gale, heavy rain and storm surges were particularly serious. This chapter reviews the response of coastal ocean to tropical cyclones, included sea surface temperature, sea surface salinity, storm surge simulation and extreme

**Keywords:** tropical cyclones, typhoons, coastal ocean dynamics, sea surface

Tropical cyclones are some of the most destructive natural disasters, which often bring huge losses to people's life and property. The Northwest Pacific and the South China Sea regions are the birthplaces of most monsoons and typhoons and are important channels for the generation and transmission of water vapor [1–3]. The influence of a typhoon on a region is often not only a heavy wind disaster. At the same time, the heavy rain, extreme waves, storm surges, and coastal inundation that are produced will also have a huge impact on the region, which will result in the

There are more than 20 typhoons in the Pacific Northwest each year, which is the region with the most frequent typhoon activity in the world. China, which has a long coastline on the west coast of the Pacific Ocean, is the country with the most frequent typhoon attacks in the world, with an average of 9.3 per year, resulting in a direct economy every year. The losses exceeded 100 billion yuan and the number of casualties reached thousands. Therefore, a comprehensive understanding and in-depth study of the typhoon process, especially the improvement of typhoon

## **Chapter 4**

model. Bulletin of American

10.1175/BAMS-D-19-0165.1

Reyes TA. Is Weather chaotic? Coexisting chaotic and non-chaotic attractors within Lorenz models. The 13th Chaos International Conference (CHAOS2020); 9-12 June 2020. (virtual

[34] Zbilut JP, Webber CL Jr.

conference)

**76**(2):965-973

S0218127407019238

**64**

Meteorological Society. 2020: 1-28. DOI:

*Current Topics in Tropical Cyclone Research*

[32] Shen B-W, Pielke RA Sr, Zeng X, Baik J-J, Faghih-Naini S, Cui J, Atlas R,

[33] Webber CL Jr, Zbilut JP. Dynamical assessment of physiological systems and states using recurrence plot strategies. Journal of Applied Physiology. 1994;

Embeddings and delays as derived from quantification of recurrence plots. Physics Letters A. 1992;**171**(34):199-203. DOI: 10.1016/0375-9601(92)90426-M

[35] Zbilut JP, Webber CL Jr. Recurrence quantification analysis: Introduction and historical context. International Journal of Bifurcation and Chaos. 2007; **17**(10):3477-3481. DOI: 10.1142/

## Response of the Coastal Ocean to Tropical Cyclones

*Zhiyuan Wu and Mack Conde*

## **Abstract**

The Northwest Pacific and the South China Sea region are the birthplaces of most monsoon disturbances and tropical cyclones and are an important channel for the generation and transmission of water vapor. The Northwest Pacific plays a major role in regulating interdecadal and long-term changes in climate. China experiences the largest number of typhoon landfalls and the most destructive power affected by typhoons in the world. The hidden dangers of typhoon disasters are accelerating with the acceleration of urbanization, the rapid development of economic construction and global warming. The coastal cities are the most dynamic and affluent areas of China's economic development. They are the strong magnetic field that attracts international capital in China, and are also the most densely populated areas and important port groups in China. Although these regions are highly developed, they are vulnerable to disasters. When typhoons hit, the economic losses and casualties caused by gale, heavy rain and storm surges were particularly serious. This chapter reviews the response of coastal ocean to tropical cyclones, included sea surface temperature, sea surface salinity, storm surge simulation and extreme rainfall under the influence of tropical cyclones.

**Keywords:** tropical cyclones, typhoons, coastal ocean dynamics, sea surface temperature, sea salinity, extreme rainfall; air-sea interaction

### **1. Introduction**

Tropical cyclones are some of the most destructive natural disasters, which often bring huge losses to people's life and property. The Northwest Pacific and the South China Sea regions are the birthplaces of most monsoons and typhoons and are important channels for the generation and transmission of water vapor [1–3]. The influence of a typhoon on a region is often not only a heavy wind disaster. At the same time, the heavy rain, extreme waves, storm surges, and coastal inundation that are produced will also have a huge impact on the region, which will result in the formation of a typhoon disaster chain [4–6].

There are more than 20 typhoons in the Pacific Northwest each year, which is the region with the most frequent typhoon activity in the world. China, which has a long coastline on the west coast of the Pacific Ocean, is the country with the most frequent typhoon attacks in the world, with an average of 9.3 per year, resulting in a direct economy every year. The losses exceeded 100 billion yuan and the number of casualties reached thousands. Therefore, a comprehensive understanding and in-depth study of the typhoon process, especially the improvement of typhoon

monitoring and early warning capabilities, is an inevitable requirement for the disaster reduction work of our national defense platform.

In the past few decades, from the perspective of atmospheric science, the research on the mechanisms of typhoon development, numerical simulation and forecasting has made great progress. However, as a strong atmospheric process, typhoons have violent disturbances, the cumulative effects of many typhoons will also have a significant impact on the ocean's thermo-salt structure and ocean circulation and global ocean heat transport. These effects will counteract typhoons, affect not only the intensity and path of specific typhoons, but also the global the low-frequency variation characteristics of the typhoon. But so far, the lack of on-site observation data during the typhoon has made the study of multi-scale response and feedback mechanism of typhoon not deep enough. People cannot simulate the interaction process between the ocean and typhoon well. The reliable initial ocean field required for typhoon forecasting has greatly limited the further improvement of typhoon research and forecasting capabilities.

The typhoon is a devastating natural disaster that has long been a focus of attention in the field of atmospheric and oceanic research [7, 8]. With the rapid development of computers, the numerical simulation of typhoons is becoming increasingly developed, and the model resolution is getting higher and higher [9]. The Pacific Northwest is the most concentrated area of global tropical cyclones (also known as typhoons in the Pacific Northwest). China is located on the west coast of the Pacific Ocean, with a long coastline and a special geographical position on the southeast coast. It has been hit by typhoons frequently, with an average annual rate of 9.3, ranking first in the world. The typhoon is one of the most serious natural disasters in China [10–12]. The annual direct economic losses caused by the typhoon are nearly 100 billion yuan, and the number of casualties is thousands. On the one hand, the strong winds and heavy rains that landed in the typhoon brought huge meteorological disasters to the vast areas of China, posing a huge threat to the people's lives, property and production activities. On the other hand, huge waves and storm surges caused by typhoons have also caused serious marine disasters, which have caused major safety hazards and economic losses to offshore operations and transportation, coastal protection projects, marine fisheries and marine aquaculture. Coastal areas are one of China's most economically developed regions, which are vulnerable to the effects of marine disasters [13–16].

Therefore, studying the movement mechanism of typhoon, accurately forecasting the influence of typhoon and reducing storm surge disasters have important social value for the protection of national economic development and people's lives and property safety.

From the perspective of practical application, improving the monitoring and forecasting ability of typhoon is the fundamental goal of typhoon research. Due to the multi-scale characteristics of the interaction between ocean and typhoon, the ocean data assimilation for the typhoon process should also be multi-scale. Due to extreme sea conditions under typhoon conditions Harsh, satellite remote sensing data has become an important data source for assimilation. However, remote sensing can only provide sea surface information. At present, people usually use the projection mapping method and multiple dynamic constraints to map surface information to the ocean subsurface and assimilate. Some assimilation methods still lack universality. How to establish a sea surface data assimilation method that considers more dynamic constraints and is more suitable for typhoon conditions is an important part of the future sea-air coupled data assimilation research. For the actual operational forecast of the typhoon, International or domestic still rely mainly on numerical weather patterns. After long-term exploration and improvement, the main forecasting modes are for atmospheric processes (such as atmospheric

**67**

*Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

with the traditional atmospheric model.

forecasting capability.

ated sea surface height changes.

boundary layer physical properties, cloud physical processes, atmospheric turbulent energy calculations, cumulus convective parameterization schemes, etc.). There has been considerable progress in simulation and forecasting capabilities. The understanding of ocean feedback is insufficient. The current typhoon numerical (weather) forecasting model still has significant errors, especially for typhoon intensity and wind and rain distribution forecast. The United States is the first to develop a sea-air coupled hurricane (the Atlantic called hurricane, the Pacific Ocean). After years of operational operation, the improved air-sea coupled model has improved both the intensity of the hurricane and the path prediction compared

In summary, the interaction between the ocean and typhoon has obvious scientific significance and important practical value. At various time and space scales, people's understanding of the mechanism of the ocean response to typhoon and modulation is obviously insufficient. At present, air-sea coupling The model's ability to simulate, assimilate and forecast the typhoon process is still very limited, which has become a bottleneck problem to further improve the typhoon

With an increase in sea surface temperature (SST), the total number of tropical cyclones in the North Pacific, Indian Ocean, and southwest Pacific Ocean decreases, and the cyclone development period shortens, but the number and proportion of

In the study of air-sea interaction, the response of the upper ocean to typhoons is a hot topic [18]. Typhoon transit can cause ocean mixing and upwelling, and sea surface cooling is the main feature [19]. The cooling caused by a typhoon is mainly related to the intensity, propagation speed of the typhoon, and the ocean condition before typhoon arrival, such as the location of cold vortices, the thermodynamic

Cyclonic wind stress results in the upwelling of sea water in the center of the path, the decrease of sea surface temperature, and the heat transfer from the surface to the atmosphere. Strong winds cause turbulent mixing of the ocean, entraining cold water from the lower layer into the mixing layer, resulting in cooling of the upper sea water and deepening of the mixing layer [10–21]. Inclusive mixing disturbances cause 85% of irreversible ocean heat to enter the atmosphere; direct air-sea interaction plays a minor role in surface cooling. Mixed layer plays an important role in sea surface cooling [22–24]. After typhoon transit, the ocean response is mostly the internal nonlocal baroclinic process caused by wind stress. In baroclinic driving stage, the flow of mixing layer 1 m/s is induced by vertical mixing, and the flow of near-inertial oscillation frequency wave into thermocline, which lasts for 5–10 days. The barotropic driving process usually results in geostrophic currents and associ-

Using the observed data to study the ocean response to typhoons is a common research method. However, due to the severe weather conditions during the transit of tropical cyclones, it is very difficult to obtain fixed-point observation data. There is a strong mass transport, energy exchange, and interaction between the

The response mechanism of the ocean to typhoon can be considered from two levels. First, the typhoon-driven mesoscale three-dimensional ocean circulation will have a significant impact on the local dynamics and thermal processes. The resulting near-inertial internal waves and vortices can input a large amount of mechanical

atmosphere and the ocean during a typhoon process [25, 26] (**Figure 1**).

**2. Sea surface temperature response to tropical cyclones**

tropical cyclones reaching super typhoon intensity increases greatly [17].

structure of the upper ocean, the position of the 26°C isotherm, etc.

*Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

*Current Topics in Tropical Cyclone Research*

monitoring and early warning capabilities, is an inevitable requirement for the

In the past few decades, from the perspective of atmospheric science, the research on the mechanisms of typhoon development, numerical simulation and forecasting has made great progress. However, as a strong atmospheric process, typhoons have violent disturbances, the cumulative effects of many typhoons will also have a significant impact on the ocean's thermo-salt structure and ocean circulation and global ocean heat transport. These effects will counteract typhoons, affect not only the intensity and path of specific typhoons, but also the global the low-frequency variation characteristics of the typhoon. But so far, the lack of on-site observation data during the typhoon has made the study of multi-scale response and feedback mechanism of typhoon not deep enough. People cannot simulate the interaction process between the ocean and typhoon well. The reliable initial ocean field required for typhoon forecasting has greatly limited the further

The typhoon is a devastating natural disaster that has long been a focus of attention in the field of atmospheric and oceanic research [7, 8]. With the rapid development of computers, the numerical simulation of typhoons is becoming increasingly developed, and the model resolution is getting higher and higher [9]. The Pacific Northwest is the most concentrated area of global tropical cyclones (also known as typhoons in the Pacific Northwest). China is located on the west coast of the Pacific Ocean, with a long coastline and a special geographical position on the southeast coast. It has been hit by typhoons frequently, with an average annual rate of 9.3, ranking first in the world. The typhoon is one of the most serious natural disasters in China [10–12]. The annual direct economic losses caused by the typhoon are nearly 100 billion yuan, and the number of casualties is thousands. On the one hand, the strong winds and heavy rains that landed in the typhoon brought huge meteorological disasters to the vast areas of China, posing a huge threat to the people's lives, property and production activities. On the other hand, huge waves and storm surges caused by typhoons have also caused serious marine disasters, which have caused major safety hazards and economic losses to offshore operations and transportation, coastal protection projects, marine fisheries and marine aquaculture. Coastal areas are one of China's most economically developed regions,

disaster reduction work of our national defense platform.

improvement of typhoon research and forecasting capabilities.

which are vulnerable to the effects of marine disasters [13–16].

Therefore, studying the movement mechanism of typhoon, accurately forecasting the influence of typhoon and reducing storm surge disasters have important social value for the protection of national economic development and people's lives

From the perspective of practical application, improving the monitoring and forecasting ability of typhoon is the fundamental goal of typhoon research. Due to the multi-scale characteristics of the interaction between ocean and typhoon, the ocean data assimilation for the typhoon process should also be multi-scale. Due to extreme sea conditions under typhoon conditions Harsh, satellite remote sensing data has become an important data source for assimilation. However, remote sensing can only provide sea surface information. At present, people usually use the projection mapping method and multiple dynamic constraints to map surface information to the ocean subsurface and assimilate. Some assimilation methods still lack universality. How to establish a sea surface data assimilation method that considers more dynamic constraints and is more suitable for typhoon conditions is an important part of the future sea-air coupled data assimilation research. For the actual operational forecast of the typhoon, International or domestic still rely mainly on numerical weather patterns. After long-term exploration and improvement, the main forecasting modes are for atmospheric processes (such as atmospheric

**66**

and property safety.

boundary layer physical properties, cloud physical processes, atmospheric turbulent energy calculations, cumulus convective parameterization schemes, etc.). There has been considerable progress in simulation and forecasting capabilities. The understanding of ocean feedback is insufficient. The current typhoon numerical (weather) forecasting model still has significant errors, especially for typhoon intensity and wind and rain distribution forecast. The United States is the first to develop a sea-air coupled hurricane (the Atlantic called hurricane, the Pacific Ocean). After years of operational operation, the improved air-sea coupled model has improved both the intensity of the hurricane and the path prediction compared with the traditional atmospheric model.

In summary, the interaction between the ocean and typhoon has obvious scientific significance and important practical value. At various time and space scales, people's understanding of the mechanism of the ocean response to typhoon and modulation is obviously insufficient. At present, air-sea coupling The model's ability to simulate, assimilate and forecast the typhoon process is still very limited, which has become a bottleneck problem to further improve the typhoon forecasting capability.

## **2. Sea surface temperature response to tropical cyclones**

With an increase in sea surface temperature (SST), the total number of tropical cyclones in the North Pacific, Indian Ocean, and southwest Pacific Ocean decreases, and the cyclone development period shortens, but the number and proportion of tropical cyclones reaching super typhoon intensity increases greatly [17].

In the study of air-sea interaction, the response of the upper ocean to typhoons is a hot topic [18]. Typhoon transit can cause ocean mixing and upwelling, and sea surface cooling is the main feature [19]. The cooling caused by a typhoon is mainly related to the intensity, propagation speed of the typhoon, and the ocean condition before typhoon arrival, such as the location of cold vortices, the thermodynamic structure of the upper ocean, the position of the 26°C isotherm, etc.

Cyclonic wind stress results in the upwelling of sea water in the center of the path, the decrease of sea surface temperature, and the heat transfer from the surface to the atmosphere. Strong winds cause turbulent mixing of the ocean, entraining cold water from the lower layer into the mixing layer, resulting in cooling of the upper sea water and deepening of the mixing layer [10–21]. Inclusive mixing disturbances cause 85% of irreversible ocean heat to enter the atmosphere; direct air-sea interaction plays a minor role in surface cooling. Mixed layer plays an important role in sea surface cooling [22–24]. After typhoon transit, the ocean response is mostly the internal nonlocal baroclinic process caused by wind stress. In baroclinic driving stage, the flow of mixing layer 1 m/s is induced by vertical mixing, and the flow of near-inertial oscillation frequency wave into thermocline, which lasts for 5–10 days. The barotropic driving process usually results in geostrophic currents and associated sea surface height changes.

Using the observed data to study the ocean response to typhoons is a common research method. However, due to the severe weather conditions during the transit of tropical cyclones, it is very difficult to obtain fixed-point observation data. There is a strong mass transport, energy exchange, and interaction between the atmosphere and the ocean during a typhoon process [25, 26] (**Figure 1**).

The response mechanism of the ocean to typhoon can be considered from two levels. First, the typhoon-driven mesoscale three-dimensional ocean circulation will have a significant impact on the local dynamics and thermal processes. The resulting near-inertial internal waves and vortices can input a large amount of mechanical

#### **Figure 1.**

*The spatial distribution of change in sea surface temperature (ΔSST) in the northern area of the South China Sea under the influence of typhoon Kai-tak (2012).*

energy into the ocean, thus significantly enhancing the local The ocean mixes and changes the warm salt structure of the upper ocean. Second, in the interior of the ocean, the energy input into the ocean by typhoons in the form of near-inertial internal waves travels along the oceanic thermocline to distant places, such as the entire tropical Pacific. During the propagation process, they interact nonlinearly with the original internal waves and near-inertial oscillations inside the ocean, which affects the ocean basin scale and even the global energy distribution, and leads to an increase in ocean mixing rate in some specific regions. The modulation of the typhoon by the ocean can also be considered from two scales. On the weather scale, the ocean plays a very important role in the movement and action of typhoons.

The maturity stage is mainly characterized by negative feedback that reduces the sea surface temperature. However, when the upper ocean warm water is thicker, the typhoon transit will not cause obvious sea temperature anomaly, and the lack of negative ocean feedback can cause the typhoon to strengthen. The interaction between the ocean mesoscale process and the typhoon is currently a focus of typhoon research. Usually, the warm vortex can quickly strengthen the typhoon, and the cold vortex can quickly weaken the typhoon. At the climate scale, global warming and interannual and interdecadal variations of climate can cause changes in ocean circulation and thermal conditions, resulting in low-frequency modulation of the intensity and frequency of typhoons.

#### **3. Sea surface salinity response to tropical cyclones**

The typhoon is one of the most serious natural disasters that affects the coastal ocean environment in China [27, 28], especially in the eastern and southern estuaries, such as the Yangtze River Estuary [29] and the Pearl River Estuary [30–33]. During a typhoon, the coupling of various dynamic factors, such as wind, waves, storm surges, and river runoff, greatly enhances the mass and energy exchange of various interfaces in the ocean and is accompanied by heavy rain and storm runoff on the surface [34–37]. Scouring can transport a large amount of minerals

**69**

**Figure 2.**

*ROMS model (beginning on 2012-08-15 00:00:00 UTC).*

*Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

of ecosystems in this region on a long-term scale [8, 51].

environment [38–40].

from the land to an estuary offshore, causing sudden changes in the water quality of the estuary, which may have an important impact on the marine ecological

On the one hand, typhoon transit strengthens the mixing process of offshore water [41–43]. On the other hand, the heavy rainfall brought by a typhoon rapidly increases river runoff into the sea, and a large amount of land-based materials are washed away and brought into the estuary offshore area [44–47]. These changes due to the influence of a typhoon significantly affect the physical, chemical and biological processes of estuarine offshore waters, which in turn have an impact on the structure and function of the ecosystem [48–50]. Studying the changes of the estuarine nearshore environment under the influence of a typhoon and its ecological effects are of great importance for further understanding the evolution process

Field observations show that the salinity of the surface water of an estuary usually shows a sharp change during a typhoon and the resulting rain, which gradually rises after entering the recovery period [8, 52–54]. During typhoon crossing, the disturbance caused by strong winds strengthens the mixing process of the estuary and its adjacent waters. However, this process has a passing impact on the water environment, and the runoff diluting water expansion and the external seawater intrusion play a greater role in changing the water environment after a typhoon. Among these, the strengthening of a typhoon after the expansion of fresh water greatly affects the upper water, the upper salinity decreases after the typhoon, and the nutrient salt concentration increases significantly. External seawater intrusion substantially changes the bottom water environment. The salinity of the bottom

*Changes in stratifications salinity influenced by typhoon Kai-tak based on the fully coupled WRF-SWAN-*

#### *Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

*Current Topics in Tropical Cyclone Research*

*Sea under the influence of typhoon Kai-tak (2012).*

of the intensity and frequency of typhoons.

**3. Sea surface salinity response to tropical cyclones**

The typhoon is one of the most serious natural disasters that affects the coastal ocean environment in China [27, 28], especially in the eastern and southern estuaries, such as the Yangtze River Estuary [29] and the Pearl River Estuary [30–33]. During a typhoon, the coupling of various dynamic factors, such as wind, waves, storm surges, and river runoff, greatly enhances the mass and energy exchange of various interfaces in the ocean and is accompanied by heavy rain and storm runoff on the surface [34–37]. Scouring can transport a large amount of minerals

energy into the ocean, thus significantly enhancing the local The ocean mixes and changes the warm salt structure of the upper ocean. Second, in the interior of the ocean, the energy input into the ocean by typhoons in the form of near-inertial internal waves travels along the oceanic thermocline to distant places, such as the entire tropical Pacific. During the propagation process, they interact nonlinearly with the original internal waves and near-inertial oscillations inside the ocean, which affects the ocean basin scale and even the global energy distribution, and leads to an increase in ocean mixing rate in some specific regions. The modulation of the typhoon by the ocean can also be considered from two scales. On the weather scale, the ocean plays a very important role in the movement and action of typhoons. The maturity stage is mainly characterized by negative feedback that reduces the sea surface temperature. However, when the upper ocean warm water is thicker, the typhoon transit will not cause obvious sea temperature anomaly, and the lack of negative ocean feedback can cause the typhoon to strengthen. The interaction between the ocean mesoscale process and the typhoon is currently a focus of typhoon research. Usually, the warm vortex can quickly strengthen the typhoon, and the cold vortex can quickly weaken the typhoon. At the climate scale, global warming and interannual and interdecadal variations of climate can cause changes in ocean circulation and thermal conditions, resulting in low-frequency modulation

*The spatial distribution of change in sea surface temperature (ΔSST) in the northern area of the South China* 

**68**

**Figure 1.**

from the land to an estuary offshore, causing sudden changes in the water quality of the estuary, which may have an important impact on the marine ecological environment [38–40].

On the one hand, typhoon transit strengthens the mixing process of offshore water [41–43]. On the other hand, the heavy rainfall brought by a typhoon rapidly increases river runoff into the sea, and a large amount of land-based materials are washed away and brought into the estuary offshore area [44–47]. These changes due to the influence of a typhoon significantly affect the physical, chemical and biological processes of estuarine offshore waters, which in turn have an impact on the structure and function of the ecosystem [48–50]. Studying the changes of the estuarine nearshore environment under the influence of a typhoon and its ecological effects are of great importance for further understanding the evolution process of ecosystems in this region on a long-term scale [8, 51].

Field observations show that the salinity of the surface water of an estuary usually shows a sharp change during a typhoon and the resulting rain, which gradually rises after entering the recovery period [8, 52–54]. During typhoon crossing, the disturbance caused by strong winds strengthens the mixing process of the estuary and its adjacent waters. However, this process has a passing impact on the water environment, and the runoff diluting water expansion and the external seawater intrusion play a greater role in changing the water environment after a typhoon. Among these, the strengthening of a typhoon after the expansion of fresh water greatly affects the upper water, the upper salinity decreases after the typhoon, and the nutrient salt concentration increases significantly. External seawater intrusion substantially changes the bottom water environment. The salinity of the bottom

#### **Figure 2.**

*Changes in stratifications salinity influenced by typhoon Kai-tak based on the fully coupled WRF-SWAN-ROMS model (beginning on 2012-08-15 00:00:00 UTC).*

layer increases after a typhoon, and the nutrient concentration of nitrogen and silicon decreases.

Typhoons or tropical cyclones are strong wind events in the climate system and are a strong form of air-sea interaction. The strong vertical mixing and wind field generated by a typhoon has a major impact on the upper ocean dynamics and ecosystem [55]. Due to typhoons, there is a decrease in sea level, a decrease in sea surface temperature, an increase in phytoplankton blooms and a decrease in primary productivity, which also affect marine fisheries [56–58]. Typhoons mainly affect the marine ecological environment through two physical mechanisms: (1) after a typhoon, a cold vortex is formed, causing seawater to upwell and the lower layer of cold nutrient water is transported to the upper layer [59, 60]; and (2) the typhoon intensifies the vertical mixing of the upper ocean by a strong wind process [61–64].

At present, most research on the sea surface salinity (SSS) response to typhoons is limited to the estuary area. According to the physical and biochemical environmental conditions of the estuary, SSS may show an upward or downward trend after typhoon transit [2, 22, 65–75]. However, studies on marine ecological factors, especially SSS and the response to typhoon transit, are limited and have not been discussed in detail [76–79]. The South China Sea (CSC) is the largest marginal sea in the Pacific Northwest, and is also a frequent typhoon zone, but it is difficult to obtain measured data during typhoons.

Due to the harsh meteorological conditions during typhoon transit, the use of on-site observation methods in an estuary to study the changes in the marine environment before and after a typhoon is very limited. The numerical simulation method is an effective way to study the distribution characteristics of fresh and salt water in an estuary under the influence of a typhoon (**Figure 2**).

#### **4. Storm surge modeling during tropical cyclones**

The Northwest Pacific and the South China Sea region are the birthplaces of most monsoons and typhoons and are an important channel for the generation and transmission of water vapor [18, 80–84]. The Northwest Pacific plays a major role in regulating interdecadal and long-term changes in climate [46, 85, 86]. China is the region with the largest number of typhoons and the most destructive power affected by typhoons in the world [87, 88].

Compared with large-scale phenomena such as global climate change, small- and medium-scale phenomena such as typhoons and thunderstorms have an even greater impact on people's production and life [6, 89, 90]. Typhoons and hurricanes present some of the greatest threats to life and damage to property [91]. The influence of a typhoon on a region is often not only a heavy wind disaster. At the same time, the heavy rain, extreme waves, storm surges and beach erosion [24] that are produced will also have a huge impact on the region, which will result in the formation of a typhoon disaster chain [19, 92–94]. Therefore, studying the movement mechanism of typhoon, accurately forecasting the influence of typhoon and reducing storm surge disasters have important social value for the protection of national economic development and human and property safety.

Tropical cyclones (TCs) present some of the greatest threats to life [25, 95–98] and damage to property [99]. The SLOSH model was widely used in storm surge simulation in seas, lakes, and on land. Blumberg and Mellor (1987) developed the POM model to simulate large-scale ocean and coastal water levels, and flow field changes. Many ocean models have been developed and used for the simulation of storm surges, such as the ECOM model, ROMS model, CH3D-IMS model, CEST model, SELFE model, Delft3D model, ADCIRC model and FVCOM model. They

**71**

**Figure 3.**

*Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

effective method to solve this problem (**Figure 3**).

have achieved very good results and laid the foundation for understanding the dynamic mechanism of storm surges. The development of a coupled atmospheric and ocean model had significant effects on improving the accuracy of numerical prediction. The establishment of a coupled atmosphere and ocean model is an

The typhoon numerical model is the focus of typhoon research and the key to typhoon forecasting. The modern model has a certain forecasting ability for the typhoon path, but the forecast of typhoon intensity is still a recognized problem in the international meteorological community. The reason is that, besides the understanding and simulation of the atmospheric environment and the structure of the typhoon itself is not accurate enough, it is also one of the important reasons for the lack of understanding of the complexity and feedback of related ocean dynamics and thermal processes. When the typhoon transits, it exerts a great shearing force on the sea surface. The related wave breaking and the interaction between the wind field and the Stokes drifting can generate a large amount of turbulent kinetic energy, which produces a wave below the sea surface. The turbulent enhancement zone enhances the rate of turbulence dissipation in the upper ocean. Therefore, the establishment of a relatively complete marine hybrid scheme is an important way to improve the maritime-coupled typhoon model. In addition, improving the sea surface flux parameterization scheme under strong wind conditions is also an urgent need to improve the model prediction capability. With the rapid increase of computing power and technology, the air-sea coupled typhoon model has broken through the limitations of the early axisymmetric typhoon model and the mixed-layer ocean model, and replaced it with a complete fully coupled ocean

*Spatial distribution of storm surge level influenced by typhoon Kai-tak (start at 2012-08-15 00:00:00 UTC).*

#### *Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

*Current Topics in Tropical Cyclone Research*

obtain measured data during typhoons.

affected by typhoons in the world [87, 88].

development and human and property safety.

silicon decreases.

layer increases after a typhoon, and the nutrient concentration of nitrogen and

Typhoons or tropical cyclones are strong wind events in the climate system and are a strong form of air-sea interaction. The strong vertical mixing and wind field generated by a typhoon has a major impact on the upper ocean dynamics and ecosystem [55]. Due to typhoons, there is a decrease in sea level, a decrease in sea surface temperature, an increase in phytoplankton blooms and a decrease in primary productivity, which also affect marine fisheries [56–58]. Typhoons mainly affect the marine ecological environment through two physical mechanisms: (1) after a typhoon, a cold vortex is formed, causing seawater to upwell and the lower layer of cold nutrient water is transported to the upper layer [59, 60]; and (2) the typhoon intensifies the vertical mixing of the upper ocean by a strong wind process [61–64]. At present, most research on the sea surface salinity (SSS) response to typhoons is limited to the estuary area. According to the physical and biochemical environmental conditions of the estuary, SSS may show an upward or downward trend after typhoon transit [2, 22, 65–75]. However, studies on marine ecological factors, especially SSS and the response to typhoon transit, are limited and have not been discussed in detail [76–79]. The South China Sea (CSC) is the largest marginal sea in the Pacific Northwest, and is also a frequent typhoon zone, but it is difficult to

Due to the harsh meteorological conditions during typhoon transit, the use of on-site observation methods in an estuary to study the changes in the marine environment before and after a typhoon is very limited. The numerical simulation method is an effective way to study the distribution characteristics of fresh and salt

The Northwest Pacific and the South China Sea region are the birthplaces of most monsoons and typhoons and are an important channel for the generation and transmission of water vapor [18, 80–84]. The Northwest Pacific plays a major role in regulating interdecadal and long-term changes in climate [46, 85, 86]. China is the region with the largest number of typhoons and the most destructive power

Compared with large-scale phenomena such as global climate change, small- and medium-scale phenomena such as typhoons and thunderstorms have an even greater impact on people's production and life [6, 89, 90]. Typhoons and hurricanes present some of the greatest threats to life and damage to property [91]. The influence of a typhoon on a region is often not only a heavy wind disaster. At the same time, the heavy rain, extreme waves, storm surges and beach erosion [24] that are produced will also have a huge impact on the region, which will result in the formation of a typhoon disaster chain [19, 92–94]. Therefore, studying the movement mechanism of typhoon, accurately forecasting the influence of typhoon and reducing storm surge disasters have important social value for the protection of national economic

Tropical cyclones (TCs) present some of the greatest threats to life [25, 95–98] and damage to property [99]. The SLOSH model was widely used in storm surge simulation in seas, lakes, and on land. Blumberg and Mellor (1987) developed the POM model to simulate large-scale ocean and coastal water levels, and flow field changes. Many ocean models have been developed and used for the simulation of storm surges, such as the ECOM model, ROMS model, CH3D-IMS model, CEST model, SELFE model, Delft3D model, ADCIRC model and FVCOM model. They

water in an estuary under the influence of a typhoon (**Figure 2**).

**4. Storm surge modeling during tropical cyclones**

**70**

have achieved very good results and laid the foundation for understanding the dynamic mechanism of storm surges. The development of a coupled atmospheric and ocean model had significant effects on improving the accuracy of numerical prediction. The establishment of a coupled atmosphere and ocean model is an effective method to solve this problem (**Figure 3**).

The typhoon numerical model is the focus of typhoon research and the key to typhoon forecasting. The modern model has a certain forecasting ability for the typhoon path, but the forecast of typhoon intensity is still a recognized problem in the international meteorological community. The reason is that, besides the understanding and simulation of the atmospheric environment and the structure of the typhoon itself is not accurate enough, it is also one of the important reasons for the lack of understanding of the complexity and feedback of related ocean dynamics and thermal processes. When the typhoon transits, it exerts a great shearing force on the sea surface. The related wave breaking and the interaction between the wind field and the Stokes drifting can generate a large amount of turbulent kinetic energy, which produces a wave below the sea surface. The turbulent enhancement zone enhances the rate of turbulence dissipation in the upper ocean. Therefore, the establishment of a relatively complete marine hybrid scheme is an important way to improve the maritime-coupled typhoon model. In addition, improving the sea surface flux parameterization scheme under strong wind conditions is also an urgent need to improve the model prediction capability. With the rapid increase of computing power and technology, the air-sea coupled typhoon model has broken through the limitations of the early axisymmetric typhoon model and the mixed-layer ocean model, and replaced it with a complete fully coupled ocean

**Figure 3.** *Spatial distribution of storm surge level influenced by typhoon Kai-tak (start at 2012-08-15 00:00:00 UTC).*

and atmosphere model. At present, the world's 1/32 to 1/900 degree resolution ocean model is being developed, which will provide strong support for the study of small-scale processes in the ocean and the multi-scale interaction between ocean and typhoon.

## **5. Future plans**

The interaction between the ocean and the typhoon is a major scientific issue with significant scientific significance and important practical value. In recent years, with the support of national major scientific research projects, China has comprehensively utilized on-site observations from the perspective of air-sea interaction. Research methods such as theoretical analysis, data assimilation, and model prediction systematically study the response and modulation mechanism of the upper ocean to the typhoon, the interaction between the ocean and atmospheric observation system for the typhoon, the ocean mesoscale process and the typhoon, and the ocean to the typhoon. A series of innovations have been achieved in lowfrequency response and modulation, physical mechanisms and parameterization of typhoons affecting the upper oceans, ocean multi-source data assimilation and parameter estimation during the typhoon, and ocean-air coupled prediction technology and applications in typhoons and marine environments. These research results will provide a solid theoretical foundation and technical support for further improving the forecast level of typhoon business in China, and make substantial contributions to the major national needs of disaster prevention and reduction.

However, it must also be recognized that China is still very lacking in the research field of interaction between ocean and typhoon, and there is still a big gap with the international advanced level. Compared with the national demand for disaster prevention, there are still obvious deficiencies. Based on the research results, we believe that the major scientific problems and major challenges in the interaction between oceans and typhoons are mainly reflected in the following points.


**73**

**6. Summary**

*Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

> Sea, and the small-scale circulation and vortex superimposed on these largescale circulations. Typhoons can not only affect and even drive small- and medium-scale ocean circulation and vortex on the weather scale. The rotation can also affect the large-scale ocean circulation of the climatic state by changing the thermal salt structure of the upper ocean. Therefore, the response of the multi-scale circulation system of the upper ocean to the typhoon includes various dynamic processes, thermal processes, and nonlinearities between them. Interactions, these are major challenges in the study of the interaction between ocean and typhoon. Reveal the propagation, transfer and dissipation mechanisms of near-inertial energy input into the ocean by typhoons, and understand the mesoscale processes such as ocean vortex and internal waves during typhoon transit. Response characteristics and excitation mechanism to determine the "heat pump" and "cold suction" of the typhoon. The different

effects on ocean stratification are the core of solving these problems.

solved by this question. The key to the question.

in the interaction between ocean and typhoon.

c.A quantitative study on the modulation of typhoon intensity by the dynamic and thermal structures of the upper ocean. The dynamic and thermal structure of the upper ocean determines the magnitude of sensible heat and latent heat flux at the air-sea interface during typhoon transit. The maintenance and development of typhoons, especially the changes in typhoon intensity, depending on the energy and water vapor provided by these fluxes. Therefore, the dynamic and thermal structures of the upper oceans can play an important role in modulating the intensity of typhoons. The path and intensity are closely related, but since the typhoon intensity is directly affected by the energy provided by the ocean and is the weak link of the current typhoon forecast, we should pay special attention to the modulation of the typhoon intensity by the ocean. If the marine environment does not change, this modulation can be easily estimated from the upper maritime structure of the climatic state. But the problem is that the dynamic and thermal structures of the upper ocean are constantly changing at various spatial and temporal scales. Understand the feedback mechanism of the maritime mesoscale process on the typhoon on the weather scale, reveal the climate. The low-frequency variation of the upper ocean circulation and heat content under changing background should be

In short, based on the existing research foundation and experience, we suggest that in the future research on the interaction between ocean and typhoon. On the basis of the mechanism, the typhoon intensity and the forecasting ability of the marine environment are improved, and the predictability of typhoon low-frequency variability is evaluated, making China one of the world's leading researchers

Observations over the past few decades have shown that the frequency, intensity, and duration of tropical cyclones vary over the interannual, interdecadal, and even longer timescales. Global warming caused by human activities and low-frequency natural oscillations in the Earth's climate system may have an impact on typhoons, but the relative importance of the two is still controversial. Whatever the case, the role of the ocean is unquestionable. Because on a long-term scale, the memory of the climate system is mainly stored in the ocean, any low-frequency variation must be related to the ocean. Previous studies on the low-frequency modulation

*Current Topics in Tropical Cyclone Research*

and typhoon.

**5. Future plans**

and atmosphere model. At present, the world's 1/32 to 1/900 degree resolution ocean model is being developed, which will provide strong support for the study of small-scale processes in the ocean and the multi-scale interaction between ocean

The interaction between the ocean and the typhoon is a major scientific issue with significant scientific significance and important practical value. In recent years, with the support of national major scientific research projects, China has comprehensively utilized on-site observations from the perspective of air-sea interaction. Research methods such as theoretical analysis, data assimilation, and model prediction systematically study the response and modulation mechanism of the upper ocean to the typhoon, the interaction between the ocean and atmospheric observation system for the typhoon, the ocean mesoscale process and the typhoon, and the ocean to the typhoon. A series of innovations have been achieved in lowfrequency response and modulation, physical mechanisms and parameterization of typhoons affecting the upper oceans, ocean multi-source data assimilation and parameter estimation during the typhoon, and ocean-air coupled prediction technology and applications in typhoons and marine environments. These research results will provide a solid theoretical foundation and technical support for further improving the forecast level of typhoon business in China, and make substantial contributions to the major national needs of disaster prevention and reduction. However, it must also be recognized that China is still very lacking in the research field of interaction between ocean and typhoon, and there is still a big gap with the international advanced level. Compared with the national demand for disaster prevention, there are still obvious deficiencies. Based on the research results, we believe that the major scientific problems and major challenges in the interaction between oceans and typhoons are mainly reflected in the following

a.On-site observations are still very scarce. As described in this paper, China has already made important practices in ocean monitoring of typhoon processes and has obtained valuable on-site observations. Especially in the field of sea-air coordinated observation, China has launched A useful attempt. After the technology and security conditions are more mature, the typhoon observations coordinated by the sea-air will provide the necessary information for deepening the typhoon research. In addition, due to the harsh sea conditions during the typhoon, the long-term monitoring system for the typhoon process is still missing. The Pacific region and the northern part of the South China Sea are the regions with the highest typhoon in the world, and are almost the only way for typhoons that cause major disasters in our country. Therefore, long-term observation networks are built and maintained in the region (for example, the cross buoy/potential system) An array of observations for the basic structure is an effective means of

enhancing ocean and atmospheric monitoring during the typhoon.

b.The response mechanism of the multi-scale circulation system of the upper ocean to the typhoon needs to be deepened. The circulation system of the upper ocean is very complicated. The typhoon prevailing in the northwestern Pacific includes the North Pacific subtropical circulation and tropical circulation driven by the trade wind, by buoyancy flux. The shallow transfected circulation of the North Pacific, the monsoon-driven circulation of the South China

**72**

points.

Sea, and the small-scale circulation and vortex superimposed on these largescale circulations. Typhoons can not only affect and even drive small- and medium-scale ocean circulation and vortex on the weather scale. The rotation can also affect the large-scale ocean circulation of the climatic state by changing the thermal salt structure of the upper ocean. Therefore, the response of the multi-scale circulation system of the upper ocean to the typhoon includes various dynamic processes, thermal processes, and nonlinearities between them. Interactions, these are major challenges in the study of the interaction between ocean and typhoon. Reveal the propagation, transfer and dissipation mechanisms of near-inertial energy input into the ocean by typhoons, and understand the mesoscale processes such as ocean vortex and internal waves during typhoon transit. Response characteristics and excitation mechanism to determine the "heat pump" and "cold suction" of the typhoon. The different effects on ocean stratification are the core of solving these problems.

c.A quantitative study on the modulation of typhoon intensity by the dynamic and thermal structures of the upper ocean. The dynamic and thermal structure of the upper ocean determines the magnitude of sensible heat and latent heat flux at the air-sea interface during typhoon transit. The maintenance and development of typhoons, especially the changes in typhoon intensity, depending on the energy and water vapor provided by these fluxes. Therefore, the dynamic and thermal structures of the upper oceans can play an important role in modulating the intensity of typhoons. The path and intensity are closely related, but since the typhoon intensity is directly affected by the energy provided by the ocean and is the weak link of the current typhoon forecast, we should pay special attention to the modulation of the typhoon intensity by the ocean. If the marine environment does not change, this modulation can be easily estimated from the upper maritime structure of the climatic state. But the problem is that the dynamic and thermal structures of the upper ocean are constantly changing at various spatial and temporal scales. Understand the feedback mechanism of the maritime mesoscale process on the typhoon on the weather scale, reveal the climate. The low-frequency variation of the upper ocean circulation and heat content under changing background should be solved by this question. The key to the question.

In short, based on the existing research foundation and experience, we suggest that in the future research on the interaction between ocean and typhoon. On the basis of the mechanism, the typhoon intensity and the forecasting ability of the marine environment are improved, and the predictability of typhoon low-frequency variability is evaluated, making China one of the world's leading researchers in the interaction between ocean and typhoon.

## **6. Summary**

Observations over the past few decades have shown that the frequency, intensity, and duration of tropical cyclones vary over the interannual, interdecadal, and even longer timescales. Global warming caused by human activities and low-frequency natural oscillations in the Earth's climate system may have an impact on typhoons, but the relative importance of the two is still controversial. Whatever the case, the role of the ocean is unquestionable. Because on a long-term scale, the memory of the climate system is mainly stored in the ocean, any low-frequency variation must be related to the ocean. Previous studies on the low-frequency modulation

of tropical cyclones in the ocean have focused on the correlation analysis between tropical sea surface temperature and typhoon parameters, but such analysis has its limitations. For example, the variation of the total power consumption of the Atlantic tropical cyclone has a good correlation with the variation of the sea surface temperature. If this empirical relationship is brought into the climate model, the total power consumption of the Atlantic tropical cyclone will increase by 3 times by the end of the 21st century. However, if a similar empirical relationship is established by subtracting the global tropical average from the tropical Atlantic sea surface temperature variation, the total tropical Atlantic cyclone power consumption predicted by the climate model remains essentially unchanged. This shows that the Atlantic tropical cyclone has been mainly modulated by natural low-frequency oscillations for the past 30 years.

In addition to high-resolution models, advanced data assimilation techniques are also essential to improve the simulation and forecasting capabilities of the typhoon model. Data assimilation can assimilate data from different sources, different time and space, and different elements into the dynamic model, and obtain an analysis field that is more detailed than the observation data and more realistic than the model results. For the assimilation of ocean data in the typhoon process, the most important problem is how to achieve multi-scale, multi-variable assimilation, extract the information reflecting the multi-scale interaction between ocean and typhoon in the observation system, and ensure the consistency of the model state field correction; The determination of the dependent background field error covariance matrix is also a problem.

In summary, the response and modulation mechanism of the ocean to typhoons is an international frontier proposition for marine and atmospheric science research. It is extremely challenging in terms of theoretical methods, observation techniques, model development and data assimilation. Taking this as an entry point, it is expected to achieve breakthrough basic research results, develop and improve marine science theories, and promote the interdisciplinary and common development of marine and atmospheric sciences while meeting the major needs of the country.

## **Acknowledgements**

The study was supported by the National Natural Science Foundation of China (Grant Nos. 51809023, 51839002, and 51879015). The partial support also comes from the Open Research Foundation of Key Laboratory of the Pearl River Estuarine Dynamics and Associated Process Regulation, Ministry of Water Resources ([2018] KJ03), and the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 19C0092).

**75**

**Author details**

Changsha, China

Durham, NH, USA

Zhiyuan Wu1,2,3\* and Mack Conde4

Hunan Province, Changsha, China

Dartmouth, New Bedford, MA, USA

\*Address all correspondence to: zwu@csust.edu.cn

provided the original work is properly cited.

1 School of Hydraulic Engineering, Changsha University of Science and Technology,

2 Key Laboratory of Water-Sediment Sciences and Water Disaster Prevention of

4 School of Marine Science and Ocean Engineering, University of New Hampshire,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

3 School for Marine Science and Technology, University of Massachusetts

*Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

*Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

*Current Topics in Tropical Cyclone Research*

oscillations for the past 30 years.

covariance matrix is also a problem.

the country.

**Acknowledgements**

(Grant No. 19C0092).

of tropical cyclones in the ocean have focused on the correlation analysis between tropical sea surface temperature and typhoon parameters, but such analysis has its limitations. For example, the variation of the total power consumption of the Atlantic tropical cyclone has a good correlation with the variation of the sea surface temperature. If this empirical relationship is brought into the climate model, the total power consumption of the Atlantic tropical cyclone will increase by 3 times by the end of the 21st century. However, if a similar empirical relationship is established by subtracting the global tropical average from the tropical Atlantic sea surface temperature variation, the total tropical Atlantic cyclone power consumption predicted by the climate model remains essentially unchanged. This shows that the Atlantic tropical cyclone has been mainly modulated by natural low-frequency

In addition to high-resolution models, advanced data assimilation techniques are also essential to improve the simulation and forecasting capabilities of the typhoon model. Data assimilation can assimilate data from different sources, different time and space, and different elements into the dynamic model, and obtain an analysis field that is more detailed than the observation data and more realistic than the model results. For the assimilation of ocean data in the typhoon process, the most important problem is how to achieve multi-scale, multi-variable assimilation, extract the information reflecting the multi-scale interaction between ocean and typhoon in the observation system, and ensure the consistency of the model state field correction; The determination of the dependent background field error

In summary, the response and modulation mechanism of the ocean to typhoons

The study was supported by the National Natural Science Foundation of China (Grant Nos. 51809023, 51839002, and 51879015). The partial support also comes from the Open Research Foundation of Key Laboratory of the Pearl River Estuarine Dynamics and Associated Process Regulation, Ministry of Water Resources ([2018] KJ03), and the Research Foundation of Education Bureau of Hunan Province, China

is an international frontier proposition for marine and atmospheric science research. It is extremely challenging in terms of theoretical methods, observation techniques, model development and data assimilation. Taking this as an entry point, it is expected to achieve breakthrough basic research results, develop and improve marine science theories, and promote the interdisciplinary and common development of marine and atmospheric sciences while meeting the major needs of

**74**

## **Author details**

Zhiyuan Wu1,2,3\* and Mack Conde4

1 School of Hydraulic Engineering, Changsha University of Science and Technology, Changsha, China

2 Key Laboratory of Water-Sediment Sciences and Water Disaster Prevention of Hunan Province, Changsha, China

3 School for Marine Science and Technology, University of Massachusetts Dartmouth, New Bedford, MA, USA

4 School of Marine Science and Ocean Engineering, University of New Hampshire, Durham, NH, USA

\*Address all correspondence to: zwu@csust.edu.cn

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[69] Nakada S, Kobayashi S, Hayashi M, Ishizaka J, Akiyama S, Fuchi M, et al. High-resolution surface salinity maps in coastal oceans based on geostationary ocean color images: Quantitative analysis of river plume dynamics. Journal of Oceanography. 2018;**74**:287-304

[70] Rudzin J, Shay L, Johns W. The influence of the barrier layer on SST response during tropical cyclone wind forcing using idealized experiments. Journal of Physical Oceanography. 2018;**48**:1951-1968

[71] Lengaigne M, Neetu S, Samson G, Vialard J, Krishnamohan K, Masson S, et al. Influence of air–sea coupling on Indian ocean tropical cyclones. Climate Dynamics. 2019;**52**:577-598

[72] Steffen J, Bourassa M. Barrier layer development local to tropical cyclones based on Argo float observations. Journal of Physical Oceanography. 2018;**48**:1951-1968

[73] Wu Z, Chen J, Jiang C, et al. Numerical investigation of Typhoon Kai-tak (1213) using a mesoscale coupled WRF-ROMS model—Part II: Wave effects[J]. Ocean Engineering; 2020;**196**:106805

[74] Blumberg AF, Georgas N, Yin L, et al. Street-scale modeling of storm surge inundation along the New Jersey Hudson river waterfront. Journal of Atmospheric and Oceanic Technology. 2015;**32**(8):1486-1497

[75] Blumberg AF, Mellor GL. A description of a three-dimensional coastal ocean circulation model. In: Three-Dimensional Coastal Ocean Models. Washington, USA; 1987. pp. 1-16 [76] Chen T, Zhang Q, Wu Y, et al. Development of a wave-current model through coupling of FVCOM and SWAN. Ocean Engineering. 2018;**164**:443-454

[77] Chen S, Qian YK, Peng S. Effects of various combinations of boundary layer schemes and microphysics schemes on the track forecasts of tropical cyclones over the South China Sea. Natural Hazards. 2015;**78**(1):61-74

[78] Craig AP, Jacob R, Kauffman B, et al. CPL6: The new extensible, high performance parallel coupler for the community climate system model. The International Journal of High Performance Computing Applications. 2005;**19**(3):309-327

[79] Di Liberto T, Colle BA, Georgas N, et al. Verification of a multimodel storm surge ensemble around new York City and Long Island for the cool season. Weather and Forecasting. 2011;**26**(6):922-939

[80] Forbes C, Rhome J, Mattocks C, et al. Predicting the storm surge threat of hurricane Sandy with the National weather service SLOSH model. Journal of Marine Science and Engineering. 2014;**2**(2):437-476

[81] Ge Z, Dai Z, Pang W, et al. LIDARbased detection of the post-typhoon recovery of a meso-macro-tidal beach in the Beibu gulf, China. Marine Geology. 2017;**391**:127-143

[82] Haidvogel DB, Arango H, Budgell WP, et al. Ocean forecasting in terrain-following coordinates: Formulation and skill assessment of the regional ocean modeling system. Journal of Computational Physics. 2008;**227**(7):3595-3624

[83] Hu D, Wu L, Cai W, et al. Pacific western boundary currents and their roles in climate. Nature. 2015;**522**(7556):299

**81**

*Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

> [93] Liu D, Pang L, Xie B. Typhoon disaster in China: Prediction, prevention, and mitigation. Natural Hazards. 2009;**49**(3):421-436

[94] Liu N, Ling T, Wang H, et al. Numerical simulation of typhoon Muifa (2011) using a coupled oceanatmosphere-wave-sediment transport

(COAWST) modeling system.

2015;**14**(2):199-209

Journal of Ocean University of China.

[95] Mei X, Dai Z, Darby SE, et al. Modulation of extreme flood levels by impoundment significantly offset by floodplain loss downstream of the three gorges dam. Geophysical Research

Letters. 2018;**45**(7):3147-3155

[96] Mooney PA, Mulligan FJ, Bruyère CL, et al. Investigating the performance of coupled WRF-ROMS simulations of hurricane Irene (2011) in a regional climate modeling framework. Atmospheric Research. 2019;**215**:57-74

[97] Neumann JE, Emanuel K, Ravela S, et al. Joint effects of storm surge and sealevel rise on US coasts: New economic estimates of impacts, adaptation, and benefits of mitigation policy. Climatic Change. 2015;**129**(1-2):337-349

[98] Pattanayak S, Mohanty UC, Rao AD. Simulation of storm surges in the bay of Bengal using one-way coupling between NMM-WRF and IITD storm surge model. Marine Geodesy.

[99] Rego JL, Li C. Nonlinear terms in storm surge predictions: Effect of tide and shelf geometry with case study from hurricane Rita. Journal of Geophysical Research, Oceans. 2010;**115**(C6). DOI:

2016;**39**(5):376-400

10.1029/2009JC005285

[84] Hu K, Chen Q, Wang H. A numerical study of vegetation impact on reducing storm surge by wetlands in a semi-enclosed estuary. Coastal

Engineering. 2015;**95**:66-76

[86] Jelesnianski CP, Chen J, Shaffer WA. SLOSH: Sea, lake, and overland surges from hurricanes. In: National Weather Service. USA; 1992

[88] Lakshmi DD, Murty PLN, Bhaskaran PK, et al. Performance of WRF-ARW winds on computed storm surge using hydodynamic model for Phailin and Hudhud cyclones. Ocean Engineering. 2017;**131**:135-148

[89] Laprise R. The Euler equations of motion with hydrostatic pressure as an independent variable. Monthly Weather

Review. 1992;**120**(1):197-207

Letters. 2006;**33**(2):L02604

[91] Li Y, Peng S, Yan J, et al. On improving storm surge forecasting using an adjoint optimal technique. Ocean Modelling. 2013;**72**:185-197

[92] Lorbacher K, Dommenget D, Niiler PP. Ocean mixed layer depth: A subsurface proxy of ocean-atmosphere variability. Journal of Geophysical Research, Oceans. 2006;**111**:520-522

[90] Li M, Zhong L, Boicourt WC, et al. Hurricane-induced storm surges, currents and destratification in a semienclosed bay. Geophysical Research

SCRIP/

[87] Jones PW. A Users Guide for SCRIP: A Spherical Coordinate Remapping and Interpolation Package. Los Alamos National Laboratory; 1998. Available from: http://climate.lanl.gov/Software/

[85] Islam T, Srivastava PK, Rico-Ramirez MA, et al. Tracking a tropical cyclone through WRF–ARW simulation and sensitivity of model physics. Natural Hazards. 2015;**76**(3):1473-1495

*Response of the Coastal Ocean to Tropical Cyclones DOI: http://dx.doi.org/10.5772/intechopen.90620*

[84] Hu K, Chen Q, Wang H. A numerical study of vegetation impact on reducing storm surge by wetlands in a semi-enclosed estuary. Coastal Engineering. 2015;**95**:66-76

*Current Topics in Tropical Cyclone Research*

[68] Lee J, Moon I, Moon J, Kim S, Jeong Y, Koo J. Impact of typhoons on the Changjiang plume extension in the yellow and East China seas. Journal of Geophysical Research, Oceans.

[76] Chen T, Zhang Q, Wu Y, et al. Development of a wave-current model through coupling of FVCOM and SWAN. Ocean Engineering.

[77] Chen S, Qian YK, Peng S. Effects of various combinations of boundary layer schemes and microphysics schemes on the track forecasts of tropical cyclones over the South China Sea. Natural

[78] Craig AP, Jacob R, Kauffman B, et al. CPL6: The new extensible, high performance parallel coupler for the community climate system model. The International Journal of High Performance Computing Applications.

[79] Di Liberto T, Colle BA, Georgas N, et al. Verification of a multimodel storm surge ensemble around new York City and Long Island for the cool season. Weather and Forecasting.

[80] Forbes C, Rhome J, Mattocks C, et al. Predicting the storm surge threat of hurricane Sandy with the National weather service SLOSH model. Journal of Marine Science and Engineering.

[81] Ge Z, Dai Z, Pang W, et al. LIDARbased detection of the post-typhoon recovery of a meso-macro-tidal beach in the Beibu gulf, China. Marine Geology.

2018;**164**:443-454

Hazards. 2015;**78**(1):61-74

2005;**19**(3):309-327

2011;**26**(6):922-939

2014;**2**(2):437-476

2017;**391**:127-143

[82] Haidvogel DB, Arango H, Budgell WP, et al. Ocean forecasting in terrain-following coordinates: Formulation and skill assessment of the regional ocean modeling system. Journal of Computational Physics.

2008;**227**(7):3595-3624

2015;**522**(7556):299

[83] Hu D, Wu L, Cai W, et al. Pacific western boundary currents and their roles in climate. Nature.

[69] Nakada S, Kobayashi S, Hayashi M, Ishizaka J, Akiyama S, Fuchi M, et al. High-resolution surface salinity maps in coastal oceans based on geostationary ocean color images: Quantitative analysis of river plume dynamics. Journal of Oceanography. 2018;**74**:287-304

[70] Rudzin J, Shay L, Johns W. The influence of the barrier layer on SST response during tropical cyclone wind forcing using idealized experiments. Journal of Physical Oceanography.

[71] Lengaigne M, Neetu S, Samson G, Vialard J, Krishnamohan K, Masson S, et al. Influence of air–sea coupling on Indian ocean tropical cyclones. Climate

[72] Steffen J, Bourassa M. Barrier layer development local to tropical cyclones based on Argo float observations. Journal of Physical Oceanography.

[73] Wu Z, Chen J, Jiang C, et al. Numerical investigation of Typhoon Kai-tak (1213) using a mesoscale coupled WRF-ROMS model—Part II: Wave effects[J]. Ocean Engineering;

[74] Blumberg AF, Georgas N, Yin L, et al. Street-scale modeling of storm surge inundation along the New Jersey Hudson river waterfront. Journal of Atmospheric and Oceanic Technology.

2017;**122**:4962-4973

2018;**48**:1951-1968

2018;**48**:1951-1968

2020;**196**:106805

2015;**32**(8):1486-1497

[75] Blumberg AF, Mellor GL. A description of a three-dimensional coastal ocean circulation model. In: Three-Dimensional Coastal Ocean Models. Washington, USA; 1987. pp. 1-16

Dynamics. 2019;**52**:577-598

**80**

[85] Islam T, Srivastava PK, Rico-Ramirez MA, et al. Tracking a tropical cyclone through WRF–ARW simulation and sensitivity of model physics. Natural Hazards. 2015;**76**(3):1473-1495

[86] Jelesnianski CP, Chen J, Shaffer WA. SLOSH: Sea, lake, and overland surges from hurricanes. In: National Weather Service. USA; 1992

[87] Jones PW. A Users Guide for SCRIP: A Spherical Coordinate Remapping and Interpolation Package. Los Alamos National Laboratory; 1998. Available from: http://climate.lanl.gov/Software/ SCRIP/

[88] Lakshmi DD, Murty PLN, Bhaskaran PK, et al. Performance of WRF-ARW winds on computed storm surge using hydodynamic model for Phailin and Hudhud cyclones. Ocean Engineering. 2017;**131**:135-148

[89] Laprise R. The Euler equations of motion with hydrostatic pressure as an independent variable. Monthly Weather Review. 1992;**120**(1):197-207

[90] Li M, Zhong L, Boicourt WC, et al. Hurricane-induced storm surges, currents and destratification in a semienclosed bay. Geophysical Research Letters. 2006;**33**(2):L02604

[91] Li Y, Peng S, Yan J, et al. On improving storm surge forecasting using an adjoint optimal technique. Ocean Modelling. 2013;**72**:185-197

[92] Lorbacher K, Dommenget D, Niiler PP. Ocean mixed layer depth: A subsurface proxy of ocean-atmosphere variability. Journal of Geophysical Research, Oceans. 2006;**111**:520-522

[93] Liu D, Pang L, Xie B. Typhoon disaster in China: Prediction, prevention, and mitigation. Natural Hazards. 2009;**49**(3):421-436

[94] Liu N, Ling T, Wang H, et al. Numerical simulation of typhoon Muifa (2011) using a coupled oceanatmosphere-wave-sediment transport (COAWST) modeling system. Journal of Ocean University of China. 2015;**14**(2):199-209

[95] Mei X, Dai Z, Darby SE, et al. Modulation of extreme flood levels by impoundment significantly offset by floodplain loss downstream of the three gorges dam. Geophysical Research Letters. 2018;**45**(7):3147-3155

[96] Mooney PA, Mulligan FJ, Bruyère CL, et al. Investigating the performance of coupled WRF-ROMS simulations of hurricane Irene (2011) in a regional climate modeling framework. Atmospheric Research. 2019;**215**:57-74

[97] Neumann JE, Emanuel K, Ravela S, et al. Joint effects of storm surge and sealevel rise on US coasts: New economic estimates of impacts, adaptation, and benefits of mitigation policy. Climatic Change. 2015;**129**(1-2):337-349

[98] Pattanayak S, Mohanty UC, Rao AD. Simulation of storm surges in the bay of Bengal using one-way coupling between NMM-WRF and IITD storm surge model. Marine Geodesy. 2016;**39**(5):376-400

[99] Rego JL, Li C. Nonlinear terms in storm surge predictions: Effect of tide and shelf geometry with case study from hurricane Rita. Journal of Geophysical Research, Oceans. 2010;**115**(C6). DOI: 10.1029/2009JC005285

**Chapter 5**

Intensity

*Boris Yurchak*

**Abstract**

The Use of a Spiral Band Model to

Spiral cloud-rain bands (SCRBs) are some of the most distinguishing features inherent in satellite and radar images of tropical cyclones (TC). The subject of the proposed research is the finding of a physically substantiated method for estimation of the TC's intensity using SCRBs' configuration parameters. To connect a rainband pattern to a physical process that conditions the spiraling feature of a rainband, it is assumed that the rainband's configuration near the core of a TC is governed primarily by a streamline. In turn, based on the distribution of primarily forces in a TC, an analytical expression as a combination of hyperbolic and logarithmic spirals (HLS) for the description of TC spiral streamline (rainband) is retrieved. Parameters of the HLS are determined by the physical parameters of a TC, particularly, by the maximal wind speed (MWS). To apply this theoretical finding to practical estimation of the TC's intensity, several approximation techniques are developed to "convert" rainband configuration to the estimation of the MWS. The developed techniques have been tested by exploring satellite infrared imageries and airborne and coastal radar data, and the outcomes were compared with in situ measurements

**Keywords:** hyperbolic-logarithmic spiral, tropical cyclone, spiral cloud-rain bands,

The issue addressed in this chapter relates to methods for estimating the intensity of a tropical cyclone (TC) from the characteristics of its cloud-rain field (CRF) structure. In general, these methods are empirical and semiempirical, i.e., they are based on the correlation of the structural features of the CRF with the intensity of the TC found from observations from satellites and radars. The most widely used method in operational practice is the Dvorak method [1, 2]. This chapter relates to the exploring of one of the most pronounced structural elements of the CRF, which are spiral cloud-rain bands (SCRBs). Attention was first attracted to these bands by Wexler [3] based on aircraft observations. It was suggested that SCRBs indicate the mature cyclone and its organization follows the streamlines. The authors of [4] described the SCRBs observed on the radar and suggested to use a modified logarithmic spiral to express its configuration in mathematic form. Although SCRBs have been studied for a long time, there is currently no consensus about their origin

Estimate Tropical Cyclone

of wind speeds and the best track data of tropical cyclones.

maximum wind speed, approximation

**1. Introduction**

**83**

## **Chapter 5**
