**4. Numerical analysis on the multiscale systems associated with the torrential rainfall**

The multiscale conception of helicity discussed above is fundamental for the understanding of energy cascade, which occurs either by Taylor's mechanism of stretching and spin-up of small-scale vortices due to large-scale strain or twisting of small-scale vortex filaments due to

The landfall process of Fung-Wong is simulated based on the experiments in Section 2. Statistical variables, including the errors of track and intensity, are calculated. The result indicates that the errors in the experiment VIRV are generally less than CTRL. For example, the errors of 24 h simulated track for the experiments CTRL and VIRV are 79 and 54 km, respectively. The improvement in TC intensity is about 27.6% in the simulation of maximum wind speed (MWS) and 16.4% in minimum sea level pressure (MSLP), respectively. Because of the "spin-up" process of numerical simulation, both the experiments exhibit smaller MWS and MSLP errors at 24 h than 12 h. However, the difference of MSLP between VIRV and CTRL increases with the model integration, indicating that the simulation of MSLP is sensitive to the initial condition. For the rainfall simulation (**Figure 2**), the rainfall pattern in VIRV agrees with the TRMM observation. In this regard, the results from VIRV simulations are used for the

**Figure 2.** The accumulated rainfall (mm) during 12UTC 27–12UTC 29, Jul 2009, from TRMM 3B42 (a) and numerical

**3. Numerical experiments on TC landfall process and evolution of** 

analysis of rainfall mechanism in the subsequent section.

282 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

a large-scale screw.

simulation from D1 (b).

**rainfall**

## **4.1. Quasi-frontal structure viewed from helicity, low-level jet (LLJ), and cold pool**

Numerous studies have highlighted the role of preexisting boundaries intersecting the primary convective system where cyclonic-only mesovortices were observed to form at the intersection point [40, 41]. More detailed analysis also indicated shearing instability [42] as the genesis mechanism for cyclonic-only low-level vortices formed along mesoscale boundaries such as gust fronts [43, 44]. These mesoscale boundaries are usually associated with frontal structures, which are required to be examined to clarify the mechanism of heavy rainfall. While it has long been recognized that the low-level jet (LLJ) is an efficient moisture transport mechanism [45] and a source of large-scale destabilization through warm advection [46, 47], the frontogenetical character of the boundary of LLJ can be important for the genesis of MCSs [48]. Therefore, the frontogenesis process similar to that of Augustine and Caracena [49] is investigated here to understand the characteristics of boundaries associated with MCSs. In particular, a timeaveraged composite vertical cross section at 6 h preceding the mature stage shows that the LLJ in the plane of the cross section ascending the northeastward sloping frontal surface. Trier et al. [50] argued that long-lived MCSs are aided by the frontogenetical lifting of air by the LLJ, which produces a zone of elevated conditional instability favorable for rainfall genesis.

Through the analysis on the evolution of *H*<sup>1</sup> during TC landfall, this study also identifies that relative warm and moist LLJs associated with TC inflow frequently appear and move toward the TC core region and finally meet with the strong cold convective downdrafts which was induced by the convective detrainment from the middle-to-upper troposphere (PBL, **Figure 3**). As a result, quasi-frontal structure is generated at the boundary between the warm LLJ and cold downdraft. The LLJ intensifies as the front gradually sharpens. The large curvature near the quasifront should serve to accelerate the buoyant air and the growth of convection. According to Eq. (4), the vertical shear of the angle between vector ⇀*v* and *u* component is equivalent to the ratio between helicity (*H*<sup>1</sup> ) and the square of total horizontal velocity. In this regard, positive helicity should be generated when the LLJ turns clockwise with height. It is interesting to find that before the occurrence of heavy rainfall, the shear vector of LLJ generally turns counterclockwise with height (\_\_\_ <sup>∂</sup>*<sup>α</sup>* ∂*z* is positive). This relation indicates the existence of negative helicity and cold advection. However, as heavy rainfall occurs, the shear vector turns clockwise with height, showing a positive helicity. It is also noticed that high potential vorticity (PV, >2.5 PVU) associated with the mesoscale disturbances mainly occupies 700–950 hPa before the genesis of heavy rainfall. These PV disturbances then grow rapidly and extend to the whole troposphere, in companion with the genesis of heavy rainfall. Moreover, to evaluate the influence of the frontogenetical forcing on the growth of heavy rainfall, the Sawyer-Eliassen equation is further calculated. It is found that evident frontogenetical forcing (stream function value is −10.1 hPa m s−1) is formed at the quasifrontal area to the southeastern part of heavy rainfall. The forcing is intensified (−12.2 hPa m s−1) and extended from 400 hPa to a lower level (700 hPa) at the peak time of rainfall (04UTC 28 Jul).

middle-level air, (3) enhanced turbulent entrainment of dry air from above the top of PBL, and (4) loss of the oceanic heat and moisture source, causing air to cool dry adiabatically while flowing toward the center. In this study, the first mechanism seems the most plausible if one examines the temperature advection disclosed by large-scale helicity. In contrast, the second mechanism seems not considerable, as the cold air is not covered by the rainband as seen from the FY2C cloud image. Based on the analysis of the simulations, the other two mechanisms should have played an additional indirect role in inducing the cold flow in the

Numerical Analysis on the Simulated Heavy Rainfall Event of Tropical Cyclone Fung-Wong

http://dx.doi.org/10.5772/intechopen.72264

285

According to Romine and Wilhelmson [58], the Kelvin-Helmholtz instability is one of the principle mechanisms that result in the genesis of spiral rain band. Based on radar data, Weckwerth and Wakimoto [59] examined a mesoscale front in association with the outflow produced by the downdraft of a supercell. They found that the K-H instability and mesoscale convective cell develop at the top of the edge of front. Detailed investigation on the important features of the quasi-frontal structure, e.g., vertical wind shear, local Richardson number (less than 1/4), and the distribution of wave structure, which is parallel to the gustfront head and perpendicular to the low-level shear vector, shows high similarity with the characteristics of K-H waves from supercells [59] and hurricanes [58]. It is also noticed that the quasi-frontal region is characterized by vertical vorticity maxima. According to Atkins and Laurent [60], as the vorticity near the front can be amplified by updraft through stretching, the vorticity maxima provides favorable locations for the growth of instability and convection. These convective cells subsequently propagated back relative to the outflow boundary while they preferentially existed along the updraft side of the K-H waves. The spacing of the cells at the leading edge of the outflow boundary, particularly along the northern part, is approximately 5 km, which compares well with the wavelength of the K-H waves. It also indicates that the static instability increases and ultimately triggered upward motion at the front. As a result, horizontal shear of vertical motion across the front develops and contributes to the growth of helicity and helical disturbance. It is also recognized that the intersections between the vortex tubes at the gust-front head are characterized by vertical vorticity maxima. In addition to the K-H instability, this study also calculated the Brunt-Vaisala frequency atop the radial inflow layer at the spiral rainband and the decrease of Scorer number [61] with height below the cold flow, which indicates the presence of gravity wave with the wave period of 220 s. It thus may mix the dynamic-unstable sheared layer

The vertical wind profile averaged at the quasi-frontal region is further investigated. It is found that the profile is characterized with clockwise-turning and increased curvature of hodograph through a deep layer. According to Wu et al. [26], the increased curvature of hodograph is beneficial to the development of small-scale convection due to the energy cascading from the basic flow. In this regard, the process of energy cascading is further examined based

**4.2. K-H instability associated with the multiscale systems**

and contribute to the accumulation of rainfall [58].

*4.2.1. Numerical analysis on energy cascading*

on the numerical simulation results.

middle level.

**Figure 3.** Schematic diagram of the multiscale quasi-frontal structure that results in the heavy rainfall.

Studies on tropical convection [51–53] suggested that convective rainfall rate is directly related to cold pool intensity. A number of idealized numerical simulations of tropical [54] and midlatitude [55, 56] convection also have shown that cold pool may intensify when the low- to midlevel moisture is drier. Here, this study explores such relationship based on the simulation for Fung-Wong's inner rainbands where the most intensive rainfall appears. It is found that maximum Δθ deficits ranged from 1 to 5 K with a mean value of 2.7 K, while maximum Δθ<sup>e</sup> deficits ranged from 1 to 14 K with a mean of 6.2 K. Thus, Fung-Wong's cold pools which lead to the heavy rainfall was equally intense as cold pools observed in other tropical storms. Moreover, in the cross-band direction, convergence of strong storm-relative inflow along the cold pool leading edge was coincident with a modest meso-high pressure anomaly, while inflow divergence prevailed through the collocated rainfall and cold pool maxima (figure is not given). Such structure is qualitatively consistent with many prior studies of TC outer rainbands and their associated cold pools [11, 12, 53]. In the along-band direction, several cold pools show signs of upband expansion while being advected downband by the prevailing cyclonic flow.

Meanwhile, the simulated wind profiles at the site of the most intensive cold pool exhibited easterly surface winds that veered with height, with 0–6 km shear vectors oriented toward the north at ~13 m s−1. The strength and orientation (primarily crossband) of the 0–3 km shear vectors were marginally consistent with expectations for intense, long-lived rainbands [57]. Simulations further indicate a shallow moist layer near the surface (below ~1 km), drier air at mid-levels (~2–6 km), and moister air aloft. The midlevel RH minima approached 30–35% near 3–5 km AGL.

According to previous studies, the physical mechanisms for the genesis of this kind of cold pool include (1) advection of cooler and drier air from over land, (2) enhancement of rainband convection over land leading to mesoscale saturated downdrafts of cool and dry middle-level air, (3) enhanced turbulent entrainment of dry air from above the top of PBL, and (4) loss of the oceanic heat and moisture source, causing air to cool dry adiabatically while flowing toward the center. In this study, the first mechanism seems the most plausible if one examines the temperature advection disclosed by large-scale helicity. In contrast, the second mechanism seems not considerable, as the cold air is not covered by the rainband as seen from the FY2C cloud image. Based on the analysis of the simulations, the other two mechanisms should have played an additional indirect role in inducing the cold flow in the middle level.

### **4.2. K-H instability associated with the multiscale systems**

According to Romine and Wilhelmson [58], the Kelvin-Helmholtz instability is one of the principle mechanisms that result in the genesis of spiral rain band. Based on radar data, Weckwerth and Wakimoto [59] examined a mesoscale front in association with the outflow produced by the downdraft of a supercell. They found that the K-H instability and mesoscale convective cell develop at the top of the edge of front. Detailed investigation on the important features of the quasi-frontal structure, e.g., vertical wind shear, local Richardson number (less than 1/4), and the distribution of wave structure, which is parallel to the gustfront head and perpendicular to the low-level shear vector, shows high similarity with the characteristics of K-H waves from supercells [59] and hurricanes [58]. It is also noticed that the quasi-frontal region is characterized by vertical vorticity maxima. According to Atkins and Laurent [60], as the vorticity near the front can be amplified by updraft through stretching, the vorticity maxima provides favorable locations for the growth of instability and convection. These convective cells subsequently propagated back relative to the outflow boundary while they preferentially existed along the updraft side of the K-H waves. The spacing of the cells at the leading edge of the outflow boundary, particularly along the northern part, is approximately 5 km, which compares well with the wavelength of the K-H waves. It also indicates that the static instability increases and ultimately triggered upward motion at the front. As a result, horizontal shear of vertical motion across the front develops and contributes to the growth of helicity and helical disturbance. It is also recognized that the intersections between the vortex tubes at the gust-front head are characterized by vertical vorticity maxima. In addition to the K-H instability, this study also calculated the Brunt-Vaisala frequency atop the radial inflow layer at the spiral rainband and the decrease of Scorer number [61] with height below the cold flow, which indicates the presence of gravity wave with the wave period of 220 s. It thus may mix the dynamic-unstable sheared layer and contribute to the accumulation of rainfall [58].

### *4.2.1. Numerical analysis on energy cascading*

Studies on tropical convection [51–53] suggested that convective rainfall rate is directly related to cold pool intensity. A number of idealized numerical simulations of tropical [54] and midlatitude [55, 56] convection also have shown that cold pool may intensify when the low- to midlevel moisture is drier. Here, this study explores such relationship based on the simulation for Fung-Wong's inner rainbands where the most intensive rainfall appears. It is found that maxi-

ranged from 1 to 14 K with a mean of 6.2 K. Thus, Fung-Wong's cold pools which lead to the heavy rainfall was equally intense as cold pools observed in other tropical storms. Moreover, in the cross-band direction, convergence of strong storm-relative inflow along the cold pool leading edge was coincident with a modest meso-high pressure anomaly, while inflow divergence prevailed through the collocated rainfall and cold pool maxima (figure is not given). Such structure is qualitatively consistent with many prior studies of TC outer rainbands and their associated cold pools [11, 12, 53]. In the along-band direction, several cold pools show signs of upband expansion while being advected downband by the prevailing cyclonic flow.

Meanwhile, the simulated wind profiles at the site of the most intensive cold pool exhibited easterly surface winds that veered with height, with 0–6 km shear vectors oriented toward the north at ~13 m s−1. The strength and orientation (primarily crossband) of the 0–3 km shear vectors were marginally consistent with expectations for intense, long-lived rainbands [57]. Simulations further indicate a shallow moist layer near the surface (below ~1 km), drier air at mid-levels (~2–6 km), and moister air aloft. The midlevel RH minima approached 30–35% near

According to previous studies, the physical mechanisms for the genesis of this kind of cold pool include (1) advection of cooler and drier air from over land, (2) enhancement of rainband convection over land leading to mesoscale saturated downdrafts of cool and dry

3–5 km AGL.

deficits

mum Δθ deficits ranged from 1 to 5 K with a mean value of 2.7 K, while maximum Δθ<sup>e</sup>

**Figure 3.** Schematic diagram of the multiscale quasi-frontal structure that results in the heavy rainfall.

284 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

The vertical wind profile averaged at the quasi-frontal region is further investigated. It is found that the profile is characterized with clockwise-turning and increased curvature of hodograph through a deep layer. According to Wu et al. [26], the increased curvature of hodograph is beneficial to the development of small-scale convection due to the energy cascading from the basic flow. In this regard, the process of energy cascading is further examined based on the numerical simulation results.

Using the conception of difference in total energy (DTE), Tan et al. (hereafter T04) [62] examined the impacts of initial small-scale disturbance on a "surprise" snowstorm through the analysis of energy cascading. Similar to the definition of DTE by T04, the total energy (*TE*) is defined here by considering kinetic and internal components:

$$TE = \frac{1}{2} \left( L\_{l, \downarrow k}{}^2 + V\_{l, \downarrow k}{}^2 + kT\_{l \downarrow k}{}^2 \right) \tag{5}$$

The nonlinear multiscale transition of energy from basic flow to the process of disturbance

represents the weakening phase of environmental vertical wind shear by convection, during which larger-scale instability is eliminated by the small-scale motion. Similar to WL92, this

by D1 is generally higher than that of D2, with the difference more evident after landfall, and

Detailed analysis also shows that the largest value of the generation of available potential energy (CAPE) and its conversion to kinetic energy occurs at the region of the heaviest rain in the simulation of TC Fung-Wong. The values decrease rapidly from TC core region to the outer radial belts. The cloud scales essentially extract energy from the TC scale (azimuthally averaged wavenumber 0) system. It implies that the TC scale is barotropically unstable to the cloud scales (wavenumbers 1 and 2). This is essentially a cascading process where energy is conveyed from the larger to the smaller scales. The generation of available potential energy and its transformation to kinetic energy takes place directly on the larger scales of TC Fung-Wong. Using a spectral closure calculation, Andre and Lesieur [66] showed that transport of energy through the inertial range is sensitive to the presence of helicity. In the calculations of this study, the time evolution of the energy spectrum toward the k−5/3 form is slowed down when helicity is injected at small wavenumbers. It thus supports the argument by Tsinober and Levich [67] that helical structures might be an inherent part of the turbulent energy cas-

High-resolution simulations are performed with nonhydrostatic WRF mesoscale numerical model to clarify the multiscale mechanisms leading to the heavy rainfall of TC Fung-Wong during landfall on southwestern coast of China. Numerical analysis shows that quasi-frontal structures are frequently generated at the boundary of warm LLJs associated with TC inflow and cold convective downdrafts, which favor the genesis of intensive rainfall. Some important features of the quasi-frontal structures, e.g., intensive vertical wind shear and small local Richardson number, exhibit similarity with that of K-H waves from supercell. The hodograph of LLJ which turns clockwise with height tends to produce positive helicity and favors the

For the future study, the structure, organization, and impact of convective rainfall systems in TCs during landfall remain a fruitful area for research. Convective cells are known to be favored downshear in TCs due to the shear-induced increase in convergence and upward motion downshear [68]. The variations of helicity and CAPE described in this paper should

cade and thereby suppress the nonlinear terms responsible for the cascade.

genesis of convection. An evident antiphase relationship between *H*<sup>1</sup>

rainfall suggests the energy transition from CAPE to kinematic energy.

 of D1 rapidly decreases. The amplitude of energy transition in D2 increases persistently as the typhoon makes landfall, which might relate to the development of small-scale convec-

merged together, in concert with the rainfall intensification.

∂*u*¯/∂*z* − *v*′ *w*′

produced by D1 and D2. It shows that the *Et*

Numerical Analysis on the Simulated Heavy Rainfall Event of Tropical Cyclone Fung-Wong

∂*v*¯/∂*z* (hereafter WL92) [26]. Here *Et*

http://dx.doi.org/10.5772/intechopen.72264

evolution is nearly in positive correlation with *H*<sup>2</sup>

is examined before and after the heavy rainfall, during

produced

287

and CAPE during heavy

(*Et*

the *Et*

(instead of *H*<sup>1</sup>

which the maximas of *Et*

) can be also represented by *Et* = −*u*′ *w*′

tion. It is also noticed that the tendency of *Et*

). The distribution of *Et*

study examined the difference of *Et*

**5. Summary and discussion**

where *U*, *V*, and *T* are horizontal u-wind, v-wind components, and temperature, respectively. *k* = *Cp* /*Tr* (the reference temperature *Tr* =287 K). I, j, and k are the numbers of x, y, and σ grid points, respectively.

A power spectrum of *TE*, averaged in the region of heavy rainfall, is analyzed. Wavenumbers 0, 1, and 2 are the TC scales following Krishnamurti et al. [63]. Meanwhile, the scales of the individual deep convective clouds reside around the azimuthal wave numbers 20–30. According to Saltzman [64], the TC scale is about several hundreds of kilometers, whereas the scale of convection, including updrafts and adjacent downdrafts, is only a few kilometers. It shows that a sizeable portion of the variance of *TE* is contributed by the first few harmonics (0–4) in the innermost region. The contribution from wavenumbers 3 to 55 (associated with medium- to small-scale processes) accounts for less than 10% of total *TE*, which agrees with the results from quasi-geostrophic models [65].

To better understand the energy transition during rainfall, this section further examines the relation between helicity, the magnitude of which is associated with kinematic energy, and CAPE, an indicator of potential energy. Krishnamurti et al. [63], by examining the scale interaction of hurricane inferred from the decomposition of the liquid water mixing ratio fields, found that nonlinear interaction of kinetic energy and available potential energy among cloud scales and the hurricane scale provide the energy to drive the hurricane. The generation of available potential energy and its transformation to kinetic energy takes place directly on the larger scales of the hurricane. Their results among hurricane scales and smaller scales show largely a cascade of energy, that is, hurricane scales lose energy when they interact with other scales.

The evolution of CAPE and helicity of TC circulation during landfall (12UTC 27–12UTC 29 July) is investigated. It shows that an approximated negative correlation exists between *H*<sup>1</sup> and CAPE before the occurrence of rainfall, which is mainly featured by the decrease (increase) of CAPE (*H*<sup>1</sup> ). However after rainfall, the original negative relation is replaced by an approximate positive correlation, which decreases simultaneously. The decrease of CAPE should be associated with the significantly reduced heat flux from land surface and the large consumption of CAPE during the rainfall process. Scatter plot also shows that intensive *H*<sup>1</sup> corresponds to low CAPE (e.g., CAPE of 3500 J kg−1 vs. *H*<sup>1</sup> of 50–150 m<sup>2</sup> s−2) during the growth of convection. In other words, kinematic energy increases as potential energy is consumed. However, there is no clear correlation between CAPE and *H*<sup>2</sup> , indicating that the energy from CAPE might not directly fuel the growth of small-scale convection. Moreover, over the land, *H*<sup>1</sup> is positive in more than 66.6% of the rainfall region. The maximum of CAPE is 3500 J kg−1 with *H*<sup>1</sup> of 50–150 m<sup>2</sup> s−2. Five percent of *H*<sup>1</sup> are greater than 400 m<sup>2</sup> s−2, with the biggest of 800 m<sup>2</sup> s−2. Over land, more than 95% of *H*<sup>1</sup> is positive, with *H*<sup>1</sup> decreasing with CAPE. Most of the intensive *H*<sup>1</sup> corresponds to CAPE less than 500 J kg−1, while over the ocean, negative *H*<sup>1</sup> corresponds to the CAPE as low as 1000 J kg−1.

The nonlinear multiscale transition of energy from basic flow to the process of disturbance (*Et* ) can be also represented by *Et* = −*u*′ *w*′ ∂*u*¯/∂*z* − *v*′ *w*′ ∂*v*¯/∂*z* (hereafter WL92) [26]. Here *Et* represents the weakening phase of environmental vertical wind shear by convection, during which larger-scale instability is eliminated by the small-scale motion. Similar to WL92, this study examined the difference of *Et* produced by D1 and D2. It shows that the *Et* produced by D1 is generally higher than that of D2, with the difference more evident after landfall, and the *Et* of D1 rapidly decreases. The amplitude of energy transition in D2 increases persistently as the typhoon makes landfall, which might relate to the development of small-scale convection. It is also noticed that the tendency of *Et* evolution is nearly in positive correlation with *H*<sup>2</sup> (instead of *H*<sup>1</sup> ). The distribution of *Et* is examined before and after the heavy rainfall, during which the maximas of *Et* merged together, in concert with the rainfall intensification.

Detailed analysis also shows that the largest value of the generation of available potential energy (CAPE) and its conversion to kinetic energy occurs at the region of the heaviest rain in the simulation of TC Fung-Wong. The values decrease rapidly from TC core region to the outer radial belts. The cloud scales essentially extract energy from the TC scale (azimuthally averaged wavenumber 0) system. It implies that the TC scale is barotropically unstable to the cloud scales (wavenumbers 1 and 2). This is essentially a cascading process where energy is conveyed from the larger to the smaller scales. The generation of available potential energy and its transformation to kinetic energy takes place directly on the larger scales of TC Fung-Wong. Using a spectral closure calculation, Andre and Lesieur [66] showed that transport of energy through the inertial range is sensitive to the presence of helicity. In the calculations of this study, the time evolution of the energy spectrum toward the k−5/3 form is slowed down when helicity is injected at small wavenumbers. It thus supports the argument by Tsinober and Levich [67] that helical structures might be an inherent part of the turbulent energy cascade and thereby suppress the nonlinear terms responsible for the cascade.
