3. Results and discussion

relationship; if ϕ<sup>x</sup>1xj

222 Wavelet Theory and Its Applications

2.3.6. Significance tests

other parameters.

Figure 2. Phase difference circle.

dividing the phase difference, ϕ1j:qj

2.3.7. Wavelet packages and parameters used

Eð Þ 90; 180 , then xj leads x1; if ϕ<sup>x</sup>1xj

Phase difference can be converted into instantaneous time lag between two time series by

It is important to assess the statistical significance of the wavelet, cross-wavelet power and the wavelet coherence. The assessment of the statistical significance levels and confidence intervals

Ng and Kwok provided the software for CWT, WTC, and XWT, which is available at http:// www.cityu.edu.hk/gcacic/wavelet. It is used in collaboration with the software provided by Grinsted, which is available at http://www.glaciology.net/wavelet-coherence. The software for PWC was provided by Aguiar-Conraria and Soares and is available at http://sites.google.com/ site/aguiarconraria/joanasoares-wavelets. Some of the parameters used in the analysis are as follows: the mother wavelet function—Morlet; the sampling time (dt)—1 h; the spacing between discrete scales (dj)—0.5 h; the level of significance—5%; and the number of Monte Carlo simulations used to assess statistical significance—1000. Default values were taken for

against red noise backgrounds was done using direct Monte Carlo simulations.

Eð Þ �180; �90 , then x<sup>1</sup> leads xj (see Figure 2).

by the angular frequency corresponding to the scale s, ωð Þs .

## 3.1. Analysis of period characteristics

#### 3.1.1. Variability of seawater flux

Figure 3 shows the seawater flux per unit width at Nakaura Watergate and its CWT coefficient chart. Positive values of the time series indicate seawater flux toward the Japan Sea and negative toward Lake Nakaumi. The CWT coefficient chart for seawater flux has stable period characteristics, with high power oscillations in the 12-h and 1-day period band throughout the analysis period. Both the red color and the black contour indicate that cycles are strong and statistically significant at 95% confidence level (hereinafter statistically significant).

#### 3.1.2. Variability of tide level

Figure 4 shows time series plot and CWT coefficient chart for the tide level. The high-power tide level oscillations have statistically significant periods of 12 h and 1 day. This implies considerable power spreads throughout the semi-diurnal and diurnal bands throughout the analysis period. The oscillations indicate spring-neap tidal variations since they appear twice a month. Also observed is a relatively strong statistically significant, though not regular, 2–6-day period cycle that occurs mainly in winter (December to March). Tide level and atmospheric pressure are negatively correlated. The time series of both shows that as the atmospheric pressure increases, the tide level decreases and vice versa (Figures 4 and 5).

#### 3.1.3. Variability of sea level atmospheric pressure

The CWT coefficient chart for the atmospheric pressure (Figure 5) shows continuous statistically significant high power 64-day period cycles from April 2002. There are also 128-day to 1-year period cycles throughout the analysis period. There are some discontinuous and irregular high power oscillations in the 1-day and 2–32-day period cycles.

Figure 3. (a) The time series of salinity flux (salt flux (kg/m/s)) and (b) its wavelet power spectrum for the Jan 2001 to Oct 2003 period. Period is in days. The red color designates high power oscillations whilst blue is low power oscillations. The black contour designates 95% confidence level, using red noise as the background spectrum. White regions on either end indicate the "cone of influence" where edge effects become important.

#### 3.1.4. Variability of river discharge

Figure 6 shows the time series plot and CWT coefficient chart for the total river discharge in Hii River. Wavelet coefficients acutely vary from the highest to the lowest, indicating an unstable river discharge. The chart also shows a distinct character that has long vertical peaks like a raindrop, which indicate that the period of oscillation varies from high to low almost instantaneously. Each peak of energy corresponds to a high river discharge. From the time series plot and CWT coefficient chart, it is evident that the highest river flow rate occurs in June and July. However, there were little rains in the summer of 2002 compared to that of 2001 and 2003. The CWT coefficient chart did show the river discharge in the summer of 2002 as significant; however, the wavelet analysis of the periods June–September 2002 indicated high energy in June and July [16]. The river discharge during June–July 2002 was dwarfed by that of June–July 2001 and 2003 and hence the absence of high energy on the CWT coefficient chart.

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Figure 7 shows the CWT coefficient chart for the wind velocity vectors. There are observations of fluctuating medium power in the 0.5-day period band for the North–South (wy) wind velocity component throughout the analysis period. The strong and statistically significant oscillations are in the 2–14-day period band throughout the analysis period. The East–West (wx) wind velocity component has discontinuous and irregular high power oscillations in the 2–16-day period band. There are continuous medium-power oscillations in the 32-day period

The WTC and XWT of the tide level and the seawater flux are shown in Figure 8, which displays that significant power sections appear continuously throughout the analysis period. This indicates

Figure 7. Wind velocity (wx (a & c) and wy (b & d))—time-series plot (a & b) and time-series wavelet power spectrum (c &

3.1.5. Variability of wind velocity

band and 64-day to 1-year period band.

3.2. Analysis of dynamic relationships

d) for the Jan 2001 to Oct 2003 period.

3.2.1. Correlation between the tide level and the seawater flux

Figure 4. Tide level—time-series plot (a) and time-series wavelet power spectrum (b) for the Jan 2001 to Oct 2003 period.

Figure 5. Atmospheric pressure–time-series plot (a) and time-series wavelet power spectrum (b) for the Jan 2001 to Oct 2003 period.

Figure 6. River discharge—time-series plot (a) and time-series wavelet power spectrum (b) for the Jan 2001 to Oct 2003 period.

instantaneously. Each peak of energy corresponds to a high river discharge. From the time series plot and CWT coefficient chart, it is evident that the highest river flow rate occurs in June and July. However, there were little rains in the summer of 2002 compared to that of 2001 and 2003. The CWT coefficient chart did show the river discharge in the summer of 2002 as significant; however, the wavelet analysis of the periods June–September 2002 indicated high energy in June and July [16]. The river discharge during June–July 2002 was dwarfed by that of June–July 2001 and 2003 and hence the absence of high energy on the CWT coefficient chart.

## 3.1.5. Variability of wind velocity

3.1.4. Variability of river discharge

224 Wavelet Theory and Its Applications

2003 period.

period.

Figure 6 shows the time series plot and CWT coefficient chart for the total river discharge in Hii River. Wavelet coefficients acutely vary from the highest to the lowest, indicating an unstable river discharge. The chart also shows a distinct character that has long vertical peaks like a raindrop, which indicate that the period of oscillation varies from high to low almost

Figure 4. Tide level—time-series plot (a) and time-series wavelet power spectrum (b) for the Jan 2001 to Oct 2003 period.

Figure 5. Atmospheric pressure–time-series plot (a) and time-series wavelet power spectrum (b) for the Jan 2001 to Oct

Figure 6. River discharge—time-series plot (a) and time-series wavelet power spectrum (b) for the Jan 2001 to Oct 2003

Figure 7 shows the CWT coefficient chart for the wind velocity vectors. There are observations of fluctuating medium power in the 0.5-day period band for the North–South (wy) wind velocity component throughout the analysis period. The strong and statistically significant oscillations are in the 2–14-day period band throughout the analysis period. The East–West (wx) wind velocity component has discontinuous and irregular high power oscillations in the 2–16-day period band. There are continuous medium-power oscillations in the 32-day period band and 64-day to 1-year period band.

#### 3.2. Analysis of dynamic relationships

#### 3.2.1. Correlation between the tide level and the seawater flux

The WTC and XWT of the tide level and the seawater flux are shown in Figure 8, which displays that significant power sections appear continuously throughout the analysis period. This indicates

Figure 7. Wind velocity (wx (a & c) and wy (b & d))—time-series plot (a & b) and time-series wavelet power spectrum (c & d) for the Jan 2001 to Oct 2003 period.

Figure 8. The WTC (a) and XWT (b) of the tide level and the salinity flux for the Jan 2001 to Oct 2003 period. The arrows represent phase difference. The arrow pointing up and to the right—in-phase relationship with the tide level leading. The arrow pointing down and to the left—out-of-phase relationship with the tide level leading.

coherence in the 2–16 period band. However, the XWT shows the occasional appearance of the significant power sections with irregular intervals especially during the summer period. The power section appears around June, July, September, and January. The influence of the river discharge on the seawater flux failed to pass the significance test at 5% level in other months

Figure 10. The WTC (a) and XWT (b) of the atmospheric pressure and the salinity flux for the Jan 2001 to Oct 2003 period.

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The XWT and WTC of the atmospheric pressure and the seawater flux shown in Figure 10 occasionally display extensive significant power sections in the 0.5-, 1–, and 256-day to 1-year period band, which stands out throughout analysis period, testing the existence of the correlation between the atmospheric pressure and the seawater flux. The WTC shows occasional

Figure 11. The WTC (a) and XWT (b) for the wx wind velocity component and salinity flux for the Jan 2001 to Oct 2003

3.2.3. Correlation between the atmospheric pressure and the seawater flux

and the summer of 2002.

period.

correlation in the 2–16-day period band.

Figure 9. The WTC (a) and XWT (b) of river discharge and salinity flux for the Jan 2001 to Oct 2003 period.

that the influence of the tide level on the seawater flux is strong. Both WTC and XWT show significant power sections in the semi-diurnal and diurnal periods. WTC also shows almost continuous coherence between the tide level and the seawater flux in the 2–16-day period band. XWT does not show much common power in the 2–16-day period band. This indicates that the tide level influences the seawater flux mainly in the 0.5-day and 1-day period band. Both WTC and XWT show that in the 0.5-day and 1-day period band, the tide level and the seawater flux have an antiphase relationship with tides leading (the arrow pointing down and to the left). That is, a rise in the tide level leads to an increase in the negative seawater flux (seawater flux into Lake Nakaumi is denoted as the negative flux in this study).

#### 3.2.2. Correlation between the river discharge and the seawater flux

Extensive significant power sections show the influence of river discharge on salinity, Figure 9. The WTC shows continuous coherence in the 16–128-day period band and discontinuous Use of Wavelet Techniques in the Study of Seawater Flux Dynamics in Coastal Lakes http://dx.doi.org/10.5772/intechopen.75177 227

Figure 10. The WTC (a) and XWT (b) of the atmospheric pressure and the salinity flux for the Jan 2001 to Oct 2003 period.

coherence in the 2–16 period band. However, the XWT shows the occasional appearance of the significant power sections with irregular intervals especially during the summer period. The power section appears around June, July, September, and January. The influence of the river discharge on the seawater flux failed to pass the significance test at 5% level in other months and the summer of 2002.

#### 3.2.3. Correlation between the atmospheric pressure and the seawater flux

that the influence of the tide level on the seawater flux is strong. Both WTC and XWT show significant power sections in the semi-diurnal and diurnal periods. WTC also shows almost continuous coherence between the tide level and the seawater flux in the 2–16-day period band. XWT does not show much common power in the 2–16-day period band. This indicates that the tide level influences the seawater flux mainly in the 0.5-day and 1-day period band. Both WTC and XWT show that in the 0.5-day and 1-day period band, the tide level and the seawater flux have an antiphase relationship with tides leading (the arrow pointing down and to the left). That is, a rise in the tide level leads to an increase in the negative seawater flux (seawater flux into Lake

Figure 9. The WTC (a) and XWT (b) of river discharge and salinity flux for the Jan 2001 to Oct 2003 period.

Figure 8. The WTC (a) and XWT (b) of the tide level and the salinity flux for the Jan 2001 to Oct 2003 period. The arrows represent phase difference. The arrow pointing up and to the right—in-phase relationship with the tide level leading. The

arrow pointing down and to the left—out-of-phase relationship with the tide level leading.

226 Wavelet Theory and Its Applications

Extensive significant power sections show the influence of river discharge on salinity, Figure 9. The WTC shows continuous coherence in the 16–128-day period band and discontinuous

Nakaumi is denoted as the negative flux in this study).

3.2.2. Correlation between the river discharge and the seawater flux

The XWT and WTC of the atmospheric pressure and the seawater flux shown in Figure 10 occasionally display extensive significant power sections in the 0.5-, 1–, and 256-day to 1-year period band, which stands out throughout analysis period, testing the existence of the correlation between the atmospheric pressure and the seawater flux. The WTC shows occasional correlation in the 2–16-day period band.

Figure 11. The WTC (a) and XWT (b) for the wx wind velocity component and salinity flux for the Jan 2001 to Oct 2003 period.

3.2.4. Correlation between the wind velocity and the seawater flux

analysis period.

The XWT and WTC for the wind velocity vectors and the seawater flux are shown in Figures 11 and 12. East–West (wx) wind velocity vector also influences seawater flux. The WTC indicates significant discontinuous and irregular power sections on the periods 2–32 days throughout the statistical intervals as shown in Figure 11. The XWT shows some discontinuous and irregular significant power sections in the 0.5- and 1-day period band, testifying that sometimes a correlation exists between East–West wind velocity component and seawater flux.

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The XWT and WTC for North–South (wy) wind velocity vector display continuous extensive significant power sections, and their center focuses on the period of 0.5 day (Figure 12). The significant power sections also appear at irregular intervals with varying periods in the 2–16-day period band. Powerful influence of wind speed is consistent in the 0.5 day throughout the

Figures 13–17 shows PWC, the relationship, in the time-frequency domain, between seawater flux and each of the forcing variables, after eliminating the effect of other variables. Two summer

Figure 14. Partial wavelet coherence of river discharge versus salinity flux for the period (a) Jun-Sep 2001, (b) Dec 2001 -

Mar 2002, (c) Dec 2002 - Mar 2003, (d) Apr-Jul 2003.

3.3. Analysis of dynamic characteristics using partial wavelet coherence

Figure 12. The WTC (a) and XWT (b) for the wy wind velocity component and salinity flux for the Jan 2001 to Oct 2003 period.

Figure 13. Partial wavelet coherence of observed tides versus salinity flux for the period (a) Jun-Sep 2001, (b) Dec 2001 - Mar 2002, (c) Dec 2002 - Mar 2003, (d) Apr-Jul 2003. The thick black line indicates the cone of influence that delimits the region not influenced by edge effects.

#### 3.2.4. Correlation between the wind velocity and the seawater flux

The XWT and WTC for the wind velocity vectors and the seawater flux are shown in Figures 11 and 12. East–West (wx) wind velocity vector also influences seawater flux. The WTC indicates significant discontinuous and irregular power sections on the periods 2–32 days throughout the statistical intervals as shown in Figure 11. The XWT shows some discontinuous and irregular significant power sections in the 0.5- and 1-day period band, testifying that sometimes a correlation exists between East–West wind velocity component and seawater flux.

The XWT and WTC for North–South (wy) wind velocity vector display continuous extensive significant power sections, and their center focuses on the period of 0.5 day (Figure 12). The significant power sections also appear at irregular intervals with varying periods in the 2–16-day period band. Powerful influence of wind speed is consistent in the 0.5 day throughout the analysis period.

## 3.3. Analysis of dynamic characteristics using partial wavelet coherence

Figure 12. The WTC (a) and XWT (b) for the wy wind velocity component and salinity flux for the Jan 2001 to Oct 2003

Figure 13. Partial wavelet coherence of observed tides versus salinity flux for the period (a) Jun-Sep 2001, (b) Dec 2001 - Mar 2002, (c) Dec 2002 - Mar 2003, (d) Apr-Jul 2003. The thick black line indicates the cone of influence that delimits the

period.

228 Wavelet Theory and Its Applications

region not influenced by edge effects.

Figures 13–17 shows PWC, the relationship, in the time-frequency domain, between seawater flux and each of the forcing variables, after eliminating the effect of other variables. Two summer

Figure 14. Partial wavelet coherence of river discharge versus salinity flux for the period (a) Jun-Sep 2001, (b) Dec 2001 - Mar 2002, (c) Dec 2002 - Mar 2003, (d) Apr-Jul 2003.

Figure 15. Partial wavelet coherence of atmospheric pressure versus salinity flux for the period (a) Jun-Sep 2001, (b) Dec 2001 - Mar 2002, (c) Dec 2002 - Mar 2003, (d) Apr-Jul 2003.

is high, it is not statistically significant. The winter periods also showed some significant correlation between the river discharge and the seawater flux. 2001–2002 winters have coherences, which are discontinuous and occur at irregular intervals. March 2002 exhibits a significant continuous coherence in the 4–24-day period band. 2002–2003 winters have statistically significant coherences that occur continuously in the 16–32-day period band. This shows the existence of the relationship between the river discharge and the seawater flux, which generally coincides with an increased river discharge. The effect of increased river flows due to rain,

Figure 16. Partial wavelet coherence of East–West (wx) wind velocity component versus salinity flux for the period (a)

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During summer, a discontinuous and irregular statistically significant relationship between the atmospheric pressure and the seawater flux exists in the 4–16-day period band, indicating the existence of an on-and-off correlation between the tide level and the seawater flux (Figure 15). During winter, a continuous relationship exists in the 16–32-day period band. WTC and XWT show that the atmospheric pressure's influence on the flux of seawater is not stable, implying that it is short-lived and has a weak influence on seawater flux (Figure 10). However, PWC

Figures 16 and 17 show partial wavelet coherence between the wind velocity and the seawater flux. An unstable relationship between wind vectors and seawater flux is exhibited. The East– West (wx) wind velocity component and seawater flux have a discontinuous and irregular

typhoon events, and water releases upstream is clearly shown.

Jun-Sep 2001, (b) Dec 2001 - Mar 2002, (c) Dec 2002 - Mar 2003, (d) Apr-Jul 2003.

3.3.3. PWC between the atmospheric pressure and the seawater flux

3.3.4. PWC between the wind velocity and the seawater flux

shows that the atmospheric pressure sometimes influences the seawater flux.

seasons and two winter seasons were analyzed separately in order to visualize the relationships that might otherwise be lost in a long-term analysis. The analysis of 2002 summer season was done before and will be compared with the current analysis [16].

#### 3.3.1. PWC between the tide level and the seawater flux

Figure 13 shows extensive statistically significant coherence at the 5% level in all the seasons, indicating the existence of relationship between observed tides and salinity transport. The center of power sections focuses on periods 0.5 and 1 day. Tides have a positive impact on seawater flux over the periods 0.5 and 1 day throughout the year. This study reinforces previous conclusion that short-term salinity transport is highly influenced by tides [16].

#### 3.3.2. PWC between the river discharge and the seawater flux

PWC between the river discharge and the seawater flux, after controlling for other forcing variables, shows statistically significant in-phase relationship in the 3–16-day period in all the seasons analyzed (Figure 14). The 2001 summer (June and July) and spring/summer 2003 (April–June) show a significant continuous coherence between the river discharge and the seawater flux in the 16-day period band. In July 2003, though the coherence in the 16-day band Use of Wavelet Techniques in the Study of Seawater Flux Dynamics in Coastal Lakes http://dx.doi.org/10.5772/intechopen.75177 231

Figure 16. Partial wavelet coherence of East–West (wx) wind velocity component versus salinity flux for the period (a) Jun-Sep 2001, (b) Dec 2001 - Mar 2002, (c) Dec 2002 - Mar 2003, (d) Apr-Jul 2003.

is high, it is not statistically significant. The winter periods also showed some significant correlation between the river discharge and the seawater flux. 2001–2002 winters have coherences, which are discontinuous and occur at irregular intervals. March 2002 exhibits a significant continuous coherence in the 4–24-day period band. 2002–2003 winters have statistically significant coherences that occur continuously in the 16–32-day period band. This shows the existence of the relationship between the river discharge and the seawater flux, which generally coincides with an increased river discharge. The effect of increased river flows due to rain, typhoon events, and water releases upstream is clearly shown.

#### 3.3.3. PWC between the atmospheric pressure and the seawater flux

seasons and two winter seasons were analyzed separately in order to visualize the relationships that might otherwise be lost in a long-term analysis. The analysis of 2002 summer season was

Figure 15. Partial wavelet coherence of atmospheric pressure versus salinity flux for the period (a) Jun-Sep 2001, (b) Dec

Figure 13 shows extensive statistically significant coherence at the 5% level in all the seasons, indicating the existence of relationship between observed tides and salinity transport. The center of power sections focuses on periods 0.5 and 1 day. Tides have a positive impact on seawater flux over the periods 0.5 and 1 day throughout the year. This study reinforces previous conclusion that short-term salinity transport is highly influenced by tides [16].

PWC between the river discharge and the seawater flux, after controlling for other forcing variables, shows statistically significant in-phase relationship in the 3–16-day period in all the seasons analyzed (Figure 14). The 2001 summer (June and July) and spring/summer 2003 (April–June) show a significant continuous coherence between the river discharge and the seawater flux in the 16-day period band. In July 2003, though the coherence in the 16-day band

done before and will be compared with the current analysis [16].

3.3.1. PWC between the tide level and the seawater flux

2001 - Mar 2002, (c) Dec 2002 - Mar 2003, (d) Apr-Jul 2003.

230 Wavelet Theory and Its Applications

3.3.2. PWC between the river discharge and the seawater flux

During summer, a discontinuous and irregular statistically significant relationship between the atmospheric pressure and the seawater flux exists in the 4–16-day period band, indicating the existence of an on-and-off correlation between the tide level and the seawater flux (Figure 15). During winter, a continuous relationship exists in the 16–32-day period band. WTC and XWT show that the atmospheric pressure's influence on the flux of seawater is not stable, implying that it is short-lived and has a weak influence on seawater flux (Figure 10). However, PWC shows that the atmospheric pressure sometimes influences the seawater flux.

#### 3.3.4. PWC between the wind velocity and the seawater flux

Figures 16 and 17 show partial wavelet coherence between the wind velocity and the seawater flux. An unstable relationship between wind vectors and seawater flux is exhibited. The East– West (wx) wind velocity component and seawater flux have a discontinuous and irregular

revealed fundamental characteristics in the variation of forcing parameters and seawater flux, as well as their interactions. The only constraint in this study was a high computation time due to

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The CWT results show that the seawater flux and the tide level have regular oscillations in the 12-h and 1-day period band, indicating that the influence of astronomical tides is dominant. River discharge from the Hii River does not exhibit any periodical variations due to the irregularity of precipitation and the controlled release from upstream reservoirs. Atmospheric pressure exhibits a continuous high power (lasting over a month) with a period range from 16 day to 1-year. East–West (wx) and North–South (wy) wind velocity components show irregular

WTC, XWT, and PWC revealed the influence of tide level, river discharge, atmospheric pressure, and wind velocity on seawater flux. WTC, XWT, and PWC showed that tides are consistently influential on the seawater flux in the 0.5- and 1-day period band. River discharge influenced seawater flux after heavy rains or water releases from upstream reservoirs. Atmospheric pressure and wind velocity occasionally influence seawater flux at Nakaura Watergate and may have an indirect influence on salinity transport through their effect on sea surface elevation. High drops of atmospheric pressure occasionally resulted in an increased tide level. This study reiterated the importance of tides in the transport of seawater in and out of Lakes

To conclude, the wavelet analysis of seawater intrusion studies proved useful. Wavelet coherence is helpful in the study of relationships between two time series. Partial wavelet coherence reveals the relationship between two time series after removing the effect of other time series. This is very useful when a dependent variable is under the influence of two or more variables. Wavelet analysis performs spectral analysis in frequency-time domain, revealing time-varying

Shimane Prefectural Institute of Public Health and Environmental Science provided the salin-

\*, Yoshiyuki Nakamura<sup>1</sup> and Hiroshi Kamiya2

1 Graduate School of Urban Innovation, Yokohama National University, Yokohama, Japan 2 Research Centre for Coastal Lagoon Environments, Shimane University, Matsue, Japan

ity, water temperature, and current velocity data used in this study.

\*Address all correspondence to: muchebve-edwin-tg@ynu.jp

1000 Monte Carlo simulation runs.

Shinji and Nakaumi.

relationships across frequencies.

Acknowledgements

Author details

Edwin Muchebve1

oscillations with periods between 2 and 16 days.

Figure 17. Partial wavelet coherence of North–South (wy) wind velocity component versus salinity flux for the period (a) Jun-Sep 2001, (b) Dec 2001 - Mar 2002, (c) Dec 2002 - Mar 2003, (d) Apr-Jul 2003).

relationship, mainly in the periods between 2 and 8 days. A continuous month-long relationship exists in the 8–16-day period band. The statistically significant month-long power sections for the North–South (wy) wind velocity component exist for 2001–2002 winters, 2002–2003 winters, and 2003 summer. The statistically significant correlation between the North–South (wy) wind velocity component and seawater flux exists mainly in the 2–16-day period band. Short-term oscillations are irregular and short-lived.
