## Abstract

Scholars have been showing great interest in revealing the mechanisms that govern the activities of the western Pacific subtropical high (WPSH). However, the problem currently remains unresolved. In this paper, a new model is developed to reveal the dynamical mechanism of the WPSH abnormal activities. Variables in the partial differential vorticity equations based on latent heat flux are separated in space and time using Galerkin methods. To overcome three deficiencies of the traditional highly truncated spectral method, spatial basis functions are reconstructed from observation field time series based on a combination of empirical orthogonal function decomposition and genetic algorithm. Based on the atmospheric vorticity equation, nonlinear ordinary differential equations of the WPSH can be obtained using an objective spatial basis function. Moreover, dynamical characteristics and actions, such as bifurcation and catastrophe induced by latent heat flux factors, are analyzed. Results show that the latent heat flux field in the Indian Ocean and the Bay of Bengal region determines the formation and rupture of the WPSH (such as the “double ridge” phenomenon). The enhancements of land–sea heating contrast and latent heat flux field in the South China Sea lead to WPSH abnormal activities and WPSH circulation anomalies, such as subtropical high northward jump and rapid westward extension. Linked with the real weather phenomena and the diagnostic analysis of the previous studies, the credibility of the bifurcation and catastrophe analysis is confirmed. This work documents new areas of research on the evolution mechanism of the WPSH under the action of latent heat flux from the view of chaotic dynamics.

## 1. Introduction

The western Pacific subtropical high (WPSH), which covers about 25% of the Northern Hemisphere’s surface, plays a very important role in the global circulation of the atmosphere and oceans (Huang et al. 2015). The WPSH has been identified as an important component of the East Asian summer monsoon (EASM) system (Miyasaka and Nakamura 2005; Sui et al. 2007; Mao et al. 2010; Bergman and Hendon 2000). It is, therefore, well known that the activity of the WPSH strongly influences the summer climate in East Asia. In particular, the WPSH plays a very important role in modulating the weather and climate in China, Korea, and Japan (Wu and Zhou 2008). In addition, the WPSH also has far-reaching effects on the summer climate of the United States (Wang 2001; Lau and Weng 2002). In recent years, repeated anomalies of the subtropical high (SH) have been observed and have resulted in frequent occurrences of severe meteorological disasters. For example, the catastrophic Yangtze River flood in August 1998 occurred because of the abnormal southward movement of the WPSH, and the South China and Jiangnan regions experienced 14 storms from May to July in 2010 as a result of WPSH abnormal activities (China National Climate Center 2011). These occurrences suggest that understanding the variations of the WPSH is essential for estimating the monsoon climatic changes over East Asia.

Owing to their dominance of the East Asian climate, previous studies have focused on investigating the laws of the abnormal activities and the dynamic mechanisms. Wu et al. (2002) derived many important findings in their comprehensive in-depth research of the formation and variation of the high. In the early stages, a low-order spectral method obtained multiequilibrium states that were often utilized for the high’s northward displacement. Subsequently, Fan and Miao (1996) explained the genesis by means of Rossby soliton theory. Many efforts have been made to establish the mechanisms that govern the WPSH activities. However, as the activities and variations of the WPSH are extremely complex, the mechanisms that underlie WPSH development and maintenance remain unresolved.

Ever since Charney and Devore (1979) studied multiple equilibria and stability, such as the bifurcation and catastrophe phenomenon, in the atmosphere using a highly truncated spectral method, many Chinese researchers have introduced this method to study the evolution mechanism of the WPSH (Li and Luo 1983; Miu and Ding 1985; Dong and Chou 1988). For example, Liu and Tao (1983) and Miu and Ding (1985) used a nonlinear forced dissipative highly truncated spectral model to simulate the WPSH seasonal northward movement, pointing out that various meridional heat forcings can lead north jumps of the WPSH. The spatial and temporal descriptions of WPSH activities could be more easily converted into the dynamical system in the proper spatial basis functions using a highly truncated spectral method, and then the impact of the dynamical characteristics and external forcing factors were discussed. So this method has been widely used for understanding the mutational mechanisms of the WPSH during the period 1980–2000 (Zhang and Yu 2000; Huang et al. 2001; Cao et al. 2003). However, as Cehelsky and Tung (1987) and Cao et al. (2003) have pointed out, studies of the multiple equilibria in the atmosphere with the highly truncated spectral method have at least three deficiencies: 1) the truncated trigonometric functions are artificially selected, 2) the different truncated trigonometric functions may lead to drastically different results, and 3) the truncated trigonometric functions are selected simply and ideally (e.g., the zonal and meridional 2–3 waves are selected), which do not accurately describe the real atmosphere. The three deficiencies create doubts whether the results obtained with this method can reveal the real atmospheric motion. These deficiencies also restrict detailed and in-depth analyses of the evolution mechanism of the WPSH, limiting the progress in the dynamical study of the WPSH over a long time period. These deficiencies require further optimization of the spatial basis functions to reveal the evolution mechanism of the WPSH deeply.

Therefore, further research on WPSH activities, especially on the dynamical behavior of the WPSH for abnormal activity, should be carried out. The selection of the spatial basis functions should be improved to make them more precise and targeted. How can the spatial structure information of the observation element field be objectively, reasonably, and effectively introduced into the spatial basis functions of a dynamical model? How can a nonlinear dynamical system of the WPSH close to the real atmosphere be obtained? Hong et al. (2015) introduced other parameters into the EASM to further refine the dynamical forecasting model of the WPSH area index (SI) and carried out forecast experiments. In addition, the dynamical activities of the WPSH were simply analyzed based on the model (Hong et al. 2014). However, the most important problem is that the model used in the study of Hong et al. (2014, 2015) did not have an obvious physical basis. To solve this problem, in the present paper, we propose empirical orthogonal function (EOF) and genetic algorithm (GA) to reconstruct the spatial basis functions of the atmospheric vorticity partial differential equations from the time series of the observation field, and then a dynamical model of the WPSH can be obtained. On this basis, the dynamical mechanism of abnormal WPSH activities affected by latent heat flux is also discussed.

The paper is organized as follows. We introduce the reanalysis data of the past 20 years in section 2. Spatial basis functions are reconstructed from observation field time series based on a combination of EOF decomposition and GA. Based on the atmospheric vorticity equation, the nonlinear ordinary differential equations of the WPSH are obtained using an objective spatial basis function, as described in sections 3. The dynamical equations of the WPSH under the action of latent heat flux are obtained in section 4. The equilibria of the dynamical WPSH model are investigated and the dynamical characteristics, such as bifurcation and catastrophe, are discussed. The evolution mechanism of the WPSH under the action of latent heat flux is fully analyzed in section 5. Finally, section 6 summarizes the results.

## 2. Research data and atmospheric vorticity equations

### a. Research data

We obtained daily data from May to October for a 20-yr period (1995–2014) from the National Centers for Environmental Prediction–National Center for Atmospheric Research reanalyses (Kalnay et al. 1996). These included the geopotential height field at 500 hPa and the latent heat flux in the Gaussian grid.

### b. Atmospheric vorticity equations

Atmospheric large-scale basic movements such as the WPSH can be described using a forced dissipative nonlinear vorticity equation (Zhang and Yu 2000; Cao et al. 2003):

where is the quasigeostrophic streamfunction; , where is the radius of the earth, is the angular speed of the earth, and is a central latitude; is the thermal forcing; and *k* is the eddy diffusivity parameter.

Because the large-scale movement of the atmosphere strictly conforms the geostrophic principle, the geopotential height fields and wind fields satisfy the following relationship (Lv et al. 1997): ; , , where represent the geopotential height, zonal wind, and meridional wind of the atmosphere, respectively; and is the geostrophic parameter.

Therefore, the above vorticity equation can be used to describe the potential field structure and abnormal changes of a large-scale system such as the WPSH. In addition, the equation can also be used to describe the corresponding distribution and variation of the flow field. Nondimensional treatment can be carried out on the above equation (Lv et al. 1997) to obtain the following dimensionless vorticity equation:

where , , , and represent the characteristics of the geostrophic parameter, the meridional gradient of , the horizontal scale, and the thermal forcing parameters, respectively.

Boundary conditions and are used as the solid wall condition, .

The WPSH research area is taken as

The Galerkin method can be used for time–space separation with the above partial differential vorticity equation to obtain an ordinary differential equation of the WPSH.

The variables and can be expanded to

where are the spatial basis functions of the streamfunction , which should satisfy the complete orthogonal character, and are the spatial structure distribution functions for describing the thermal field (these functions can be extracted from the observation latent heat flux field data).

In traditional dynamical analysis, the above spatial basis functions are often selected as relatively simple trigonometric functions of the zonal and meridional 2–3 waves to simulate the actual atmospheric circulation and the potential field structure. The structures have huge differences between the spatial basis functions and the actual weather systems, which indicate that the constructed weather model cannot objectively or accurately describe the structural characteristics and activity variability of weather systems.

Therefore, we propose an idea to reconstruct the spatial basis functions from the time series of the observation potential field. First, sequence EOF decomposition is used on the 20-yr average of the WPSH potential field reanalysis data to extract the main modes of the WPSH spatial structure, which are the objective fitting reconstruction targets of the spatial basis functions in the WPSH dynamical system. Then, the least errors and complete orthogonality are constructed as the dual constraint function. The GA is used to globally optimize the coefficients of the spatial basis functions. Finally, the spatial basis functions, which are accurate approximations of the actual weather, are obtained. On this basis, a nonlinear ordinary differential dynamical model of the WPSH can be objectively and reasonably constructed.

## 3. Spatial basis function reconstruction

To describe the basic features of WPSH activities in summer and the dynamical response to thermal forcing, we use 20 yr of the daily 500-hPa potential field (the original field) in the region 0°–90°N, 90°E–180° to reconstruct the spatial basis functions of the dynamical WPSH model.

### a. EOF decomposition

The method of EOF analysis widely used in earth sciences is a field analytical method. It is a decomposition of a signal or dataset in terms of orthogonal basis functions, which are determined from the data (Dommenget and Latif 2002). After the EOF expansion, the long series of the field observation elements can be transformed into a linear superposition of the typical spatial field and time weight coefficient series.

Based on the above EOF decomposition approach, first, the EOF method is used to decompose the above potential field data. Three typical spatial fields are shown as blue lines in Fig. 1. The cumulative variance contribution of the first three typical spatial fields reaches 91.77%, which indicates that the first three typical spatial fields can basically represent the spatial structure of the potential field. As shown in Fig. 1, the first space mode (81.65% of the total variance) shows that there is a negative center at high latitudes (Inner Mongolia Autonomous Region and Heilongjiang Province). The East Asian trough is deepened, and the northwest flow after the trough is strengthened, which cause a polar cold air southward outbreak. These are typical latitudinal distribution differences. The second space mode (7.25% of the total variance) shows that there is an obvious positive center in the WPSH region of the mid- and high latitudes. And the third space mode (2.87% of the total variance) also indicates that there is a strong positive high center in the WPSH region and a strong negative center in the high-latitude continental region, which are more typical of meridional distribution differences.

Therefore, the structure of the first three typical spatial fields can actually behave as the basic characteristics of large-scale weather systems, such as the WPSH. The least error and complete orthogonality are constructed as a dual constraint function. The GA and the surface fitting method are used to globally reconstruct the spatial basis functions based on the first three typical spatial fields.

### b. Spatial basis function reconstruction

Using the first three typical spatial fields after the EOF decomposition as a goal, the trigonometric functions are built as the generalized spatial basis functions:

The reconstruction of the above generalized basis functions can be attributed to optimization of the trigonometric parameters under two constraints of the least error and complete orthogonality. Therefore, the constraint functions can be constructed as follows:

The error between the calculation of the spatial basis functions and the corresponding EOF typical field is minimum.

The spatial basis functions must be met for complete orthogonality; ; , , where

*D*is the integration interval of the model, .

Conventional parametric optimization methods (such as the hill-climbing method and the gradient descent method) often yield local optima and are sensitive to the initial guess. Therefore, we use the GA to optimize the parameters of the spatial basis functions. In recent years, GAs have been extensively used for global optimization. They are excellent for global searching and parallel computing; the error convergence rate can be greatly improved by GAs and they are, therefore, very helpful in optimizing parameters (Taherdangkoo et al. 2012).

Using the least error and complete orthogonality as the two constraints, the GA is used to search for the optimal parameters in the parameter space.

Set the population parameters as and take the minimum square error as the objective function. The complete orthogonal conditions should also be met: ; , . The specific steps of GA parameter optimization include coding and creating the initial population, a calculation of fitness, choosing the male parent, crossover, and variation. The termination condition is taken as the optimal objective function value (). The theory of calculation and a detailed explanation are reported by Taherdangkoo et al. (2012). After about the fourth round of the GA and optimization searching, the spatial basis functions can be reconstructed as follows:

The reconstructed spatial basis functions are shown as black lines in Fig. 1. Comparing the black lines and the blue lines, we can see that the spatial structural features of the reconstruction results are very close to those of the observation spatial field, which basically reflects the spatial distribution of the WPSH potential field. The correlation coefficients between the first three observation spatial modes and the first three reconstruction spatial modes in Fig. 1 reach 0.9243, 0.9027, and 0.8980, respectively.

The above reconstruction spatial basis function in (6) obtained from the observation can be put into the streamfunction equation in (3). Combined with the spectrum expansion of the thermal forcing , it can be an entry into the nondimensional partial differential vorticity equations to conduct a Galerkin decomposition. Then, the vorticity equations are multiplied by and integrated along the research areas : . Finally, the partial differential vorticity equations in (2) can be converted to the following ordinary differential equations:

## 4. Dynamical equations of the WPSH under the action of latent heat flux

The in forced heat can reflect the basic spatial structure modes of the heat latent flux. Previous studies have revealed that latent heat flux is one of the important factors determining the position and strength of the WPSH (Wu and Liu 2003; Miyasaka and Nakamura 2005; Seager et al. 2003). Therefore, this paper mainly discusses the dynamical equations of the WPSH under the action of latent heat flux.

To make the theoretical model approach as similar as possible to the actual situation, 20 yr (1995–2014) of the daily latent heat flux field (the original field) is taken for EOF decomposition. The cumulative variance contribution of the first two modes reaches 76.3%, which indicates that these two modes can basically reflect the actual situation of the latent heat flux field. The first two spatial fields are shown in Figs. 2a and 2c.

Figure 2a shows that the first space mode of the latent heat flux field is positive in most parts of the East Asian continent and that the positive center is located to the west of the Baikal Lake region. Negative values appear in the east of Japan. Both negative and positive values exhibit an antiphase distribution. Because the daily latent heat flux fields taken for EOF decomposition are from 1995 to 2014, the figure shows that after the 1970s, when the latent heat flux in the tropical western Pacific reduces, the summer 500-hPa potential height high in East Asia increases, resulting in the summer monsoon being weakened in East Asia and the abnormal southward transport of land–sea heating contrast, which is consistent with the results of Miyasaka and Nakamura (2005) and Park et al. (2010). There is also a high-value center in the Kuroshio and its extension region and a low-value center in the Pacific warm pool area, indicating that the land–sea heating contrast of the Kuroshio and its extension region exchange more frequently, which is also consistent with previous research (Qi et al. 2008; Wang et al. 2012). Figure 2c also shows a high-value center in the Kuroshio and its extension region.

Similar to the reconstruction approach of the spatial basis functions , the EOF spatial mode functions of the latent heat flux field *Q* can be reconstructed as follows:

and

The spatial structures of the reconstructed functions are shown in Figs. 2b and 2d. The correlation coefficients between the first two reconstruction spatial modes and the first two observation spatial modes in Fig. 2 reach 0.815 and 0.877, respectively. As shown in Fig. 2, although some details are not reconstructed, the basic form and the centers of high values and low values are accurately reconstructed.

The reconstructions of are substituted into the above ordinary differential equations in (7) and are integrated. The following dynamical equations, which consider the spatial structure information of the latent heat flux field, can be obtained:

Based on the equations, the dynamical mechanism of morphological variation and activities of the WPSH under the action of latent heat flux can be analyzed and discussed.

## 5. Bifurcation and catastrophe of multiple equilibria

For large-scale weather systems, such as the WPSH, when they are in a relatively stable state, the time-varying items of the dynamical model have small values, and the items on the left-hand side of the equations are zero. Therefore, a singular spot can be achieved by solving the roots of the equilibrium equations of the constant differential coefficient system. Then, the stability can be analyzed. Taking the above dynamical equations in (9) of the WPSH as an example, the items on the left-hand side of the equations are zero. Finally, the equations that give the morphology and activity of the WPSH in the quasi–steady case can be obtained as follows:

### a. Discrimination theory of equilibrium stability

To solve the equilibrium, the equilibrium stability can be analyzed. The equilibrium of the general dynamical system can be obtained from the linearized system near the equilibrium (Kang and Tsuda 2009):

Suppose the above system has an equilibrium , that is . We examine the stability of its nearby orbit. Suppose

Because is an equilibrium, . Then, ignoring higher-order terms , the following equation can be obtained:

The above system (14) is linear. Its solution evolution of a fixed point in most cases determines the stability of the fixed point solution in the system in (11). The general solution is , where the initial value is and is the *N*th eigenvalue of . Specifically, the eigenvalue equation of this system is as follows:

Taking the dynamical equations in (9) of the WPSH as an example, the eigenvalue equation is as in (15). The parameters satisfy the next Jacobian matrix:

The equation of the eigenvalues can be converted as follows:

The solution of this three-order equation consists of the eigenvalues of . We can judge whether the equilibrium is stable at this time according to the following three theorems.

Theorem 1: When all of the eigenvalues of have negative real parts, the equilibrium of the system in (9) is asymptotically stable.

Theorem 2: When at least one of the eigenvalues of has a positive real part, the equilibrium of the system in (9) is unstable in a Lyapunov sense.

Theorem 3: When all of the eigenvalues of have zero real parts, the stability of equilibrium in the system in (9) depends on the higher-order terms of the Taylor series.

The situation of theorem 3 is called the critical case, which often appears in conservative systems. Based on the above discussion, we can easily judge the stability of the equilibrium states.

### b. Dynamical characteristic mechanisms of the WPSH

Based on the above equilibrium theory, we attempt to explain the abnormal mechanism of the WPSH. The distributions and changes of the equilibrium in the system can roughly represent the performance and condition of the WPSH in quasi–steady form with heat latent flux evolution.

Thus, changing the and parameter values of the equation via numerical calculation results in a subsequent distribution change, which may be used to approximately reflect the evolution of the WPSH system with external parameters.

If the number and/or style of the equilibria vary with external parameters, this equilibrium change is known as bifurcation. Conversely, a jumping phenomenon from one stable equilibrium to another stable equilibrium with external parameters, is known as catastrophe.

From the formula , we know that are the first and the second time series of the latent heat flux field based on EOF decomposition. Through EOF decomposition of the latent heat flux field data of 20 yr (1995–2014), time series of (3680 data) can be obtained. The maximum is 174.133 and the minimum is 28.405. Time series of (3680 data) can also be obtained. The maximum is 686.720 and the minimum is 411.289. The ranges of and values can be expanded a little. So the ranges of and values are selected as [20, 180] and [400, 700].

Then, mechanisms of dynamical characteristics of WPSH can be discussed with the change of the latent heat flux parameters and .

#### 1) Bifurcation of the equilibrium induced by latent heat flux parameter variance and aberrance of the WPSH

Calculating the range of and , we found that when , increased from 400 to 700, hence, there will be a bifurcation phenomenon. The change of with is shown in Fig. 3.

As demonstrated in Fig. 3, the model equilibria of , and change with the latent heat flux parameter . The equilibria of , and appear to vary obviously with an increase in parameter . When , the equilibrium values of , and are observed to remain relatively small (i.e., stable state of equilibrium). Conversely, when , with an increasing value of , the equilibrium values of , and are shown to increase both abruptly and significantly. In particular, two diverse equilibria of , and are observed, and are characterized as high-value and low-value equilibria, respectively. In this case, reducing is shown to significantly reduce the difference between the two equilibria. This situation indicates that there are two forms of the WPSH in certain latent heat flux; this is the “double ridge” phenomenon. When , the two equilibria of , and merge into one stable equilibrium, before which, stability remains constant. The change of shifting equilibrium of , and (Fig. 3) is found to reflect the change in the actual weather patterns.

For large-scale weather systems such as the WPSH, the wind field and potential field basically meet a stringent geostrophic approximation (Lv et al. 1997), which can be described as , , and . Therefore, with the evolution of multiple equilibria, the corresponding potential field and wind field (the synthesis field of *u* and *υ* winds) can be discussed.

When = 80 and , which corresponds to the situation before bifurcation, we select 71 cases. The values of these cases range from 400 to 542 and each value interval is 2. The values of and of each case can be substituted into , and the corresponding latent heat flux field of each case can be obtained. So we can draw the composite map of the latent heat flux field, shown as Fig. 4a. This latent heat flux field only has a high-value center of 250 maximum values the western Pacific warm pool area. The composite 850-hPa wind field and 500-hPa potential field are shown in Figs. 4b and 4c.

As shown in Figs. 4b and 4c, it can be observed that there is an anticyclonic circulation at 10°N and 150°E, and that there is an SH in the region 5°–20°N, 130°–160°E in the potential field, which indicates that with the increase of the latent heat flux in the western Pacific warm pool area, the SH begins to appear over the tropical oceans in East Asia. The above idealized atmosphere model can broadly reflect the basic weather situation that the latent heat flux is weaker and that the WPSH ridge line is single in the Northern Hemisphere.

When = 80 and , which corresponds to the situation after bifurcation, we select 79 cases. The values of these cases range from 542 to 700 and each value interval is 2. The values of and of each case can be substituted into , and the corresponding latent heat flux field of each case can be obtained. So we can draw the composite map of the latent heat flux field, shown as Fig. 5a. This latent heat flux field shows an obvious high-value center in the Indian Ocean and the Bay of Bengal and the maximum value of the center can reach 350. The latent heat flux in the Bay of Bengal increases, then the Indian monsoon breaks, and an Indian monsoon trough forms. The composite 850-hPa wind field and 500-hPa potential field are shown in Figs. 5b and 5c.

As shown in Fig. 5b, it can be observed that there is an anticyclonic circulation in the region 20°–30°N, 120°–140°E. There is also an anticyclone in the region 25°–40°N, 150°E–180°. There are two SHs at the north and south positions in the potential field (points A and B in Fig. 5c). The SH in the north position has a strong center of 5900 gpm and its ridge is in the northwest–southeast direction, located near 20°–35°N. The SH in the south position is relatively weak and its ridge line is in the northeast–southwest direction, located near 10°–30°N. Comparing the structures of Figs. 4c and 5c, after bifurcation, the main body of the SH disrupts, and then divides into a strong, north, west SH (point A in Fig. 5c) and a weak, south, east SH (point B in Fig. 5c), which shows the coexistence of two SHs and two ridge lines.

We can give some statistics for the frequency of the double-ridge phenomenon for and , respectively. When = 80 and , among 71 cases, the 500-hPa potential fields of 13 cases have the double-ridge phenomenon. So the frequency of double-ridge phenomenon for can be calculated, which is 18.31%. The frequency of double-ridge phenomenon for is 82.55%, which is much more than those of , indicating that the double-ridge phenomenon is easy to appear after bifurcation.

The above analysis shows that when the latent heat flux parameter is increased to a critical value, the equilibrium solutions of the corresponding potential streamfunctions appear as bifurcations. The weather situation may represent two SHs and their ridge lines in the tropical and subtropical regions, which coincides with the double-ridge phenomenon of the WPSH (Qi et al. 2008; Zhan et al. 2004). The latent heat flux parameter behaves as the bifurcation factor, which indicates that the increase of the latent heat flux field in the Indian Ocean and the Bay of Bengal will cause the break of the Indian monsoon and finally lead to complex morphology and structure of the flow and potential field in the subtropical region. The formation of the SH is mainly due to the effect of the latent heat flux caused by the sea–land thermal contrast (Xu et al. 2001; Wang and Shi 2005). The latent heat flux field in the Indian Ocean and the Bay of Bengal region determines the formation and rupture of the WPSH.

#### 2) Catastrophe of the equilibrium induced by latent heat flux parameter variance and aberrance of the WPSH

Calculating the range of and , we found when , increased from 20 to 180, there is a catastrophe phenomenon. The change of with is shown in Fig. 6.

As shown in Fig. 6, for equilibrium, there is the phenomenon of catastrophe. From to , one stable equilibrium state suddenly becomes the other stable equilibrium state. Moreover, a large difference of can be found when there is a jump from a stable low equilibrium to a stable high equilibrium. Taking for example, where , can be observed to be within a low-value equilibrium ranging from 0.07 to 0.2, while at , the equilibrium of increases rapidly to higher value states between 0.3 and 0.34 (Fig. 6a). Such catastrophes of equilibrium will cause a sudden change in the SH morphology. For example, it will cause the discontinuous north jumping phenomenon of the SH ridge line in the rainy season.

When and = 420, which corresponds to the situation before the catastrophe, we select 51 cases. The values of these cases range from 20 to 122 and each value interval is 2. The values of and of each case can be substituted into , and the corresponding latent heat flux field of each case can be obtained. So we can draw the composite map of the latent heat flux field, shown as Fig. 7a. This latent heat flux field shows no obvious high-value center and the maximum value is just 240, which indicates that the latent heat flux field is weak at this time. The composite 850-hPa wind field and 500-hPa potential field are shown in Figs. 7b and 7c.

As shown in Figs. 7b and 7c, it can be observed that there is an anticyclonic circulation at 15°N and 155°E, and that there is an SH in the region 5°–20°N, 145°–170°E in the potential field, which indicates that the SH begins to stretch eastward to the tropical oceans in East Asia. The above idealized atmosphere model can broadly reflect the basic weather situation that the latent heat flux is weaker and that the WPSH does not jump northward to the East Asian continent in the Northern Hemisphere.

When and = 420, which corresponds to the situation after the catastrophe, we select 29 cases. The values of these cases range from 122 to 180 and each value interval is 2. The values of and of each case can be substituted into , and the corresponding latent heat flux field of each case can be obtained. So we can draw the composite map of the latent heat flux field, shown as Fig. 8a. This latent heat flux field shows an obvious high-value center in the South China Sea and the Kuroshio area. The maximum value of the center can reach 450. This indicates that when the latent heat flux in the South China Sea is enhanced, the South China Sea monsoon trough moves eastward in the summer season and the South China Sea monsoon break outs. The composite 850-hPa wind field and 500-hPa potential field are shown in Figs. 8b and 8c.

As shown in Figs. 8b and 8c, it can be observed that there is an obvious anticyclonic circulation at 35°N and 145°E, and that there is an SH in the region 25°–40°N, 130°–160°E in the potential field. Compared with Fig. 7c, the strength and scope of the WPSH appears to be significantly enhanced, with a north jump and westward extension. The SH ridge line quickly jumps north from 15° to 30°N and it also quickly extends west from the region 145°–170°E to the region 130°–160°E.

We can give some statistics for the frequency of WPSH ridge farther north than 25°N for and , respectively. When and = 420, among 51 cases, WPSH ridges on the 500-hPa potential fields of 6 cases are farther north than 25°N. So the frequency of WPSH ridge farther north than 25°N for can be calculated, which is 11.76%. While the frequency of WPSH ridge farther north than 25°N for is 86.71%, which is much more than those of , indicating that the WPSH ridge is easy to jump farther north after the catastrophe.

The above results show when the latent heat flux parameter is increased to a critical value, the equilibrium solutions of the corresponding potential streamfunctions appear as catastrophes. The latent heat flux parameter behaves as the catastrophe factor, which shows that the active land–sea heating contrast and the enhanced latent heat flux in the South China Sea and the Kuroshio area will lead to WPSH abnormal activities and WPSH circulation anomalies, such as SH north jumping and rapid westward extension.

### c. Discussion

The bifurcation and catastrophe analysis from the view of chaotic dynamics in section 5b reveals the important role of the latent heat flux in the Indian monsoon region and the South China Sea monsoon region for SH circulation anomalies. This is consistent with the real weather diagnostic results of previous studies (Zhou et al. 2008; Subrahmanyam and Wang 2011; Chen et al. 2012; Wang et al. 2012; Wang et al. 2014; Wu et al. 2013). The important role of wind-induced variations of latent heat transfer in warming of the southern Indian Ocean has been noted in observational study by Wu and Liu (1999). There is increasing evidence that the change of the East Asian summer monsoon is closely related with the land–sea thermal contrast (latent heat fluxes) over the Indian Ocean (Zhou et al. 2008, 2009; Li et al. 2010), resulting in the WPSH abnormal activities (Zhou et al. 2008). Subrahmanyam and Wang (2011) have found that the enhanced latent heat flux over the Indian Ocean in the summer is responsible for stronger atmospheric circulation, leading to WPSH abnormal activities. Zhang et al.(2005) and Wang et al. (2014) also analyzed the relationship between latent heat net flux over the Bay of Bengal and WPSH based on the singular value decomposition (SVD) method. They found that in the summer and autumn season, when there is an abnormal enhancement of the latent heat flux over the Bay of Bengal, the height field near the Ural Mountain region is positive, easily forming a blocking high. When the height field between the Balkhash Lake and the Baikal Lake is negative, a broad trough is easily formed, which will help high-latitude cold air move southeastward. So this will cause an abnormal positive height field in eastern China and the northwest Pacific Ocean, indicating that at this time WPSH is unusually enhanced and northward, and prone to fracture. Moreover, Chen et al. (2012) pointed out that the variation of air–sea heat fluxes over the Indian Ocean influences the onset of South China Sea summer monsoon, resulting in the abnormal activities of WPSH. Xu et al. (2001) and Wang and Shi (2005) found when the latent heat flux over the Bay of Bengal increases unusually, the values of the outgoing longwave radiation (OLR) in the Sea of Japan, East China Sea, and the northwest Pacific Ocean are bigger than usual, indicating that the downward flows in these areas are strong, which will cause formation and rupture of WPSH. At this situation, WPSH is also unusually strong.

Wang et al. (2012) analyzed the variability of the South China Sea monsoon and its relationship with latent heat flux in the Kuroshio area. The latent heat flux variability in the Kuroshio area had positive correlation with variability of the South China Sea monsoon, which results the abnormal north jump of WPSH. Sui et al. (2012) also pointed out that the latent heat flux in the South China Sea is one of the important factors determining if the South China Sea summer monsoon (SCSSM) onsets sooner or later; also, it is proven that the outbreak of the SCSSM is closely related with the activities of the WPSH. Wang et al. (2014) found that the strength of the sea–land thermal contrast in the South China Sea is related to the SCSSM. In the summer, the strong sea–land thermal contrast (strong latent heat flux) in the South China Sea will cause the outbreak and enhancement of the SCSSM and further lead to the unusual enhancement and north jump of the WPSH. In addition, Wu et al. (2013) pointed out that the latent heat flux in the South China Sea affects the WPSH through influencing the upper monsoon circulation and the Walker circulation, and further affects the strength of SCSSM.

The above diagnostic analyses of the previous studies are consistent with our previous conclusions in section 5b, verifying the credibility of the bifurcation and catastrophe analysis. However, previous research was restricted to diagnostic analyses based on traditional analysis methods, such as SVD, principal component analysis (PCA), etc. From the viewpoint of the bifurcation and catastrophe, we reasonably and carefully reveal that the anomaly of the WPSH is mainly caused by the latent heat fluxes, and specifically distinguish the different influences of the latent heat flux in the two regions on WPSH anomalies. The latent heat flux field in the Indian Ocean and the Bay of Bengal region determines the formation and rupture of the WPSH (i.e., the double-ridge phenomenon). The enhancement of the latent heat flux field in the South China Sea and the Kuroshio area is one of the great reasons leading to WPSH abnormal activities and WPSH circulation anomalies, such as SH north jumping and rapid westward extension.

Previous studies have debated the reasons for the occurrence and maintenance of the SH double ridge. Zhan et al. (2004) analyzed the phenomenon of WPSH double ridges in 1998, proposing that the formation of double ridges may be related to the latent heat flux of the Indian monsoon abnormal outbreak. Qi et al. (2008) proposed that the formation and development of double ridges are related to the outbreak of the Somali low-level jet. The mechanisms of the double-ridge phenomenon have been extensively researched over the years, but few studies have given reasonable explanations from a dynamical system perspective. In our study, based on the dynamical system theory, we reveal that the sudden increase in the latent heat flux of the Indian monsoon is the important reason for the occurrence and maintenance of the SH double ridge, which extends the comments of Zhan et al. (2004).

## 6. Conclusions

The WPSH is an important member of the EASM system, being in a nonlinear monsoon circulation system together with the other members. The WPSH also plays a major role in the East Asian climate. Studies focusing on the laws of the WPSH abnormal activities have been investigated. In this paper, to further reveal the mechanisms of WPSH development and maintenance, the dynamical equations of the WPSH under the action of latent heat flux have been obtained and the dynamical characteristics of the WPSH are studied. The evolution mechanism between the WPSH and the summer monsoon system has been analyzed. The major results can be summarized as follows:

The research idea for optimizing spatial basis functions from observation field time series based on a combination of EOF decomposition and GA has been proposed. Based on the atmospheric vorticity equation, the nonlinear ordinary differential equations of the WPSH could be obtained using an objective spatial basis function.

We have investigated the mechanism of WPSH aberrance and the equilibrium stability of the WPSH system under the action of latent heat flux, and analyzed the dynamical characteristics, such as bifurcation and catastrophe induced by external forcings. Changes of latent heat flux parameters and in this model could lead to instability and bifurcation of the equilibrium of the WPSH system.

Bifurcation and catastrophe of equilibria in the dynamical equations of WPSH under the action of the latent heat flux have been studied. Our paper further confirms the view proposed by previous research (Wang et al. 2012; Sui et al. 2012) that latent heat flux is one of the important factors determining the position and strength of the WPSH. Also, some new ideas are put forward. The latent heat flux field in the Indian Ocean and the Bay of Bengal region determines the formation and rupture of the WPSH (i.e., the “double ridge” phenomenon). The land–sea heating contrast in the South China Sea is active, and the latent heat flux in the South China Sea is enhanced, which are important reasons for WPSH circulation anomalies, such as the SH northward jump and rapid westward extension. Linked with the real weather phenomena and the diagnostic analysis of the previous studies, the credibility of the bifurcation and catastrophe analysis from the viewpoint of chaotic dynamics has been confirmed.

This paper mainly discusses the dynamical equations of the WPSH under the action of latent heat flux. How the sensible heat affects the WPSH activities will be our next work. And although we have overcome three deficiencies of traditional highly truncated spectral method and revealed some new views, this model is relatively simple and can only achieve qualitative analysis. Detailed descriptions and further research still need to be introduced by the comprehensive means of numerical simulation, which will also be the subject of our next work.

## Acknowledgments

We thank the two anonymous reviewers and the editor, Paul E. Roundy, for their thoughtful comments and suggestion. This study was supported by the Chinese National Natural Science Fund for young scholars (41005025/D0505) and the Chinese National Natural Science Fund (41375002, 41075045, 41276088, and 41306010).

## REFERENCES

*China Meteorological News*, 16 January 2011, 5th ed.