Physical Formation Mechanisms of the Southwest China Vortex

Abstract

On the basis of the Prandtl boundary layer theory and an improved perturbation method, the process of laminar flow bifurcating into the Southwest China vortex (SWV) in the Hengduan Mountains is studied. The results show that the formation of SWV is mainly determined by the speed of incoming airflow in the direction of the main axis of the Hengduan Mountains. The vortex is generated in the leeward area of the Hengduan Mountains when the speed of incoming airflow is greater than the critical velocity. Moreover, it means that the laminar flow bifurcates into a vortex. The formation position of the SWV is mainly determined by the relative position of the incoming airflow in the windward area of the Hengduan Mountains and the main axis of the Hengduan Mountains. The seasonal distribution of SWVs is determined by both the velocity of the incoming airflow and the relative position of the incoming airflow to the main axis of the Hengduan Mountains. These findings are consistent with the SWV observation facts, which not only adequately explain the physical formation mechanisms and processes of SWVs, but also present the formation location and seasonal distribution of SWVs. Meanwhile, a solution from laminar to vortex in circumflow motion is also presented.

1. Introduction

The Southwest China vortex (SWV) is a warm meso-α-scale cyclonic vortex at 700 hPa or 850 hPa in Southwest China, with a horizontal scale of 300–500 km. It forms under the special terrain and circulation conditions of the Tibetan Plateau (TP). The SWV occurs every month of the year, mostly from April to September, with the peak in summer, followed by spring, autumn, and winter [1,2,3,4,5]. The SWV is an important weather system that may cause a wide range of rainstorms in China. In terms of the intensity, frequency, and range of the caused rainstorms, the SWV is the second rainstorm weather system in China, with typhoon being the first [6]. Many extraordinary heavy rainstorms and floods in Chinese history are related to the SWV. Because the SWV and its induced disturbance can propagate eastward along the Meiyu front in summer, it is not only an important precipitation system in Southeast China but also a source of downpour in its downstream areas [7]. The SWV rainstorm is also a very complex and characteristic typical rainstorm phenomenon in China [5,8,9]. Therefore, the studies and forecast of SWVs and their impact on rainstorms have always been the focus and difficulty of synoptic meteorology.

The origin of the SWV is an important scientific issue. As early as in 1916, Zhu [10] pointed out that most of the storms in Chinese inland originate in Tibet or Sichuan. Most of their paths are along the Yangtze River Basin, and some are along the Yellow River Basin. This is the first time that the source, activity, and influence of low-level vortex systems such as the SWV and TP vortex were noted. Moreover, Ye et al. [11] also illustrated that the SWV is a dynamic vortex system caused by the special terrain of the TP, and the SWV is closely related to the shear line formed by the warm air on the east of the plateau and the southward-moving cold air. Wu et al. [12] pointed outthat the interaction between the terrain and the convergences caused by the northern branch of the mid-latitude westerly, the southern branch of the low-level monsoon jet, and the easterly of the western Pacific subtropical high (WPSH)plays an important role in the formation of the SWV. Xu et al. [13] illustrated that a stable laminar flow is formed between the humid-warm southwesterly flow in the low layer and the dry-cold westerly flow in the upper layer, and the interaction between the laminar flow and the terrain is most conducive to the formation of vortex disturbance. Gao [14] pointed out that the formation of SWV is a steady state related to the basin, valleys, and stratification of upper airflows, and different laminar flow has different effects on the SWV formation. Simulation studies also showed that the terrain plays an important dynamic role in the SWV formation [15,16,17,18].

There are three main source areas of SWVs—the Jiulongarea, the Sichuan Basin, and the Xiaojin area [5,19,20,21,22]. It was also found that, under the effect of local topography, the distribution, evolution, and influence of SWVs present some multiscale characteristics [23,24]. Simulation studies have shown that topography has an important dynamic effect on the formation of SWVs [18]. Recent studies revealed for the first time that the three SWV sources are not isolated but interconnected. The disturbance of the upstream vortex source has an important influence on the downstream vortex source [21].

Existing studies have fully demonstrated that the formation of SWV is the result of the interaction between the invariable topography and variable airflow [1,6,25,26], and the topography is the main reason [25,26,27]. In particular, Lu [1] explained the formation mechanism of SWV, and proposed that the westerly wind under topographic conditions may be the main factor of SWV formation, which provides a direction for theoretical research on the SWV formation mechanisms.

At present, it has been generally recognized that the formation of SWV is related to the complex and special topography of the TP and its surrounding areas, especially the TP, Hengduan Mountains, and Sichuan Basin. The SWV is the product of the land–atmosphere interaction [1,18,20]. Due to the invariance of topography, the variable airflow has become the most important factor affecting the SWV formation. Past studies have emphasized that, under favorable terrain conditions, only appropriate westerly wind can promote the formation of SWV. The intensity of the westerly wind affects the movement of SWV. Although the existing research results partially explain the formation and evolution of SWVs, they cannot fully reveal its formation processes. The reason for the difficulty in theoretical research is that the formation of SWV is closely related to the solid boundary (topography), and the detouring flow around the solid boundary is one of the classic topics of fluid mechanics, including complex flow phenomena and mechanisms [28]. The vortex is a “bridge” connecting laminar flow and turbulent flow, and its formation mechanism is the study focus of detouring flow. However, it has not been fully understood so far [28,29,30]. In order to deepen the understanding of the SWV formation, it is necessary to conductan in-depth analysis of the detouring flow that may generate the SWV.

Since the SWV is a shallow weather system in the nascent stage [1,2,19], the nascent SWV can be approximately regarded as a horizontal two-dimensional vortex. Therefore, the study of SWV formation mechanism only requires considering the detouring flow in the horizontal direction. Section 2 demonstrates that the main body of the detouring flow is the cross-section of the Hengduan Mountains. According to the near-cylinder characteristics of the Hengduan Mountains and the hydrodynamic similarity theory, the detouring flow around the Hengduan Mountains is similar to the motion of large-Reynolds-number fluid around a cylinder. For this kind of movement bypassing the cylinder, Prandtl [31] made a major assumption that the effect of viscosity only exists in the boundary layer near the solid wall, and considered that the fluid motion outside the boundary layer could be approximately regarded as ideal case. Thus, on the basis of the Prandtl boundary layer theory [28,30,31,32], this study adopts a kind of improved perturbation method [33] to approximate and analyze the structural change of the flow around the Hengduan Mountains from laminar flow to the vortex, so as to obtain the physical formation mechanisms of SWVs.

To this end, this study first establishes a boundary layer equation describing the airflow motion around the Hengduan Mountains. Then, the boundary layer equation is approximated using an improved perturbation method. Lastly, the necessary conditions for the formation of SWVs, as well as the geometric structure and initiation location of the SWVs, are analyzed by using the approximate velocity field.

2. Methods

In order to better answer how the SWV is formed, the Prandtl boundary layer detouring flow scheme is adopted in this study. First, we need to determine the main body of detouring flow, i.e., the TP or the Hengduan Mountains.

In fact, the SWV is mainly formed in the Jiulong area (located in the south of the Western Sichuan Plateau (WSP) on the east side of the TP), Xiaojin area (located in the middle of the WSP), and the downstream Sichuan Basin, i.e., concentrated in the WSP and the Sichuan Basin. When passing over plateaus or mountains on the edges of the plateau, westerly wind (including northwesterly and southwesterly wind) can form vortices. The anticyclonic (cyclonic) vortex forms on the northern (southern) edge of the plateau. Since the SWV is a cyclonic low-pressure system, it cannot be a vortex formed by westerly (or northwesterly) wind passing over the northern plateau or the northern part of the mountains. Therefore, it is only necessary to discuss the issue of the main body of the detouring flow, i.e., the southern margin of the TP or the Hengduan Mountains. Supposing the main body is the TP, at 700hPa, regardless of whether the incoming airflow is the westerly airflow from Central Asia or the southwesterly airflow from the Indian Ocean, a boundary layer is formed on the south side of the TP. It can be calculated that the thickness of the boundary layer ($\delta \approx 5\sqrt{\frac{\nu x}{U_{\infty }}}$) does not exceed 1 km ($U_{\infty }$ is the atmospheric characteristic velocity; $\nu$ is atmospheric viscosity coefficient), according to the boundary layer theory [28,34]. This shows that, even if a vortex is formed by the main body of the plateau, its location can only be in the west of the Hengduan Mountains, and the SWV cannot form. Therefore, the Hengduan Mountains should be the main body of the detouring flow that can generate SWVs.

In addition, the nascent SWV can be approximately regarded as a horizontal two-dimension vortex; hence, we only need to study the flow in the horizontal direction. Therefore, unlike the traditional vertical atmospheric boundary layer, the boundary layer in this study is horizontal. For this reason, this research takes the horizontal section of the Hengduan Mountains at 700 hPa as the main body of the detouring flow (see the red circle in Figure 1a).

The boundary layer equations describing the flow around the cylinder have strong nonlinear characteristics; hence, they need to be approximated. In the field of aerodynamics, many scholars [35,36,37,38,39] convert the detouring flow around plates, wedges, cylinders, and shrinkage troughs into ordinary differential equations through scale transformations to obtain simple “similar” solutions. These approximate solutions better explain the moving characteristics of the detouring flow under different main bodies. However, since the ordinary differential equations are the Blasius equation [40], they still have highly nonlinear characteristics. Therefore, such “similar” solutions still require numerical calculations, which is not conducive to analyzing the physical processes of SWV formation. Further analysis found that the boundary layer equations are a kind of generalized nonlinear evolution equation, which can be approximated by the perturbation method. Considering that the perturbation method can generally only deal with weak nonlinear differential equations, an improved perturbation method [33] is adopted to simplify the boundary layer equations and, thus, get the approximate solutions. On this basis, the physical mechanisms of SWV formation are studied.

3. Atmospheric Detouring-Flow Equations

The SWV is a shallow low-pressure system at its initiation stage [1,2]; hence, it can be regarded as a two-dimensional structure in this stage. At 700 hPa, the main body of the Hengduan Mountains is similar to a semicylinder, and its main axis is a straight line between the eastern edge of the Himalayas and the Sichuan Basin (about 30° to the latitude line). The southwesterly flow from the Bay of Bengal forms a detouring flow here. It should be pointed out that the actual cross-section of the Hengduan Mountains is an irregular surface and is idealized as a semicylinder (as shown in Figure 1b) to facilitate the calculation when establishing the surface coordinate system.

The equations of the flows around the Hengduan Mountains are established by the Prandtl boundary layer. Since the mechanical forcing of the Henduan Mountains plays a major role in the formation of SWV, the two-dimensional atmospheric motion equations and continuity equation are as expressed below.

Inner layer:

$$\begin{array}{l}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}-fv=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\nu \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right),\\ \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+fu=-\frac{1}{\rho }\frac{\partial p}{\partial y}+\nu \left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right),\\ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.\end{array}$$

(1)

Outer layer:

$$\frac{\partial U_e}{\partial t}+U_e\frac{\partial U_e}{\partial x}=-\frac{1}{\rho }\frac{\partial p}{\partial x}.$$

(2)

In the above equations, u, v, and p are the zonal velocity, meridional velocity, and pressure in the inner boundary layer of the Hengduan Mountains, respectively. $U_e$ is the zonal velocity of the outer boundary layer, $\rho$ the atmospheric density at 700 hPa (about $1.0\times {10}^3$ gm⁻³, $\nu$ is the atmospheric viscosity coefficient (about $1.5\times {10}^{-5}$ m²s⁻¹), and $f$ the geostrophic parameter (about ${10}^{-4}$ s⁻¹). Here, the inner layer is a two-dimensional rotating atmosphere motion equation without external forcing, and the outer layer is the potential equation of an ideal fluid. The inner and outer atmospheres are coupled through the pressure gradient force. Thus, a closed boundary layer equation system is established.

When the southwesterly airflow moves around the Hengduan Mountains, it is inevitable to consider the rigid boundary conditions on the edge of the mountains. Therefore, it is necessary to establish a coordinate system with the edge of the mountains as the axis. Due to the fact that the Hengduan Mountains can be regarded as a semicylinder, assuming that the atmosphere is parallel to the main axis of the Hengduan Mountains, a curved coordinate system can be established. The contact point (stagnation point) between the southwesterly airflow and the Hengduan Mountains is the coordinate origin, the edge of the TP is the x-axis, and the normal line of the TP is the y-axis (Figure 2).

Let $r(x)$ be the curvature radius of any point, and then the arc length of any two points in the coordinate system is calculated as follows:

$${(ds)}^2={(r+y)}^2{(d\theta )}^2+{(dy)}^2=h_1^2{(dx)}^2+h_2^2{(dy)}^2,$$

(3)

where $h_1=1+\frac{y}{r}=1+ky,h_2=1$.

According to the arc length defined in Equation (3), the coordinate transformation is defined as follows:

$$\begin{array}{l}\tilde{x}=\left(1+\frac{y}{r}\right)x=\left(1+ky\right)x,\\ \tilde{y}=y.\end{array}$$

(4)

In these equations, $k(x)=\frac{1}{r(x)}$ is the curvature of the Hengduan Mountains. Under the assumption that the Hengduan Mountains are regarded as the semicylinder, $k(x)$ is taken as a constant $k$. Through the coordinate transformation defined in Equation (4), the two-dimensional atmospheric motion equations and continuity equation (Equation (1)) can be transformed. Equation (1) of the inner layer of detouring flow is transformed as follows ($\tilde{x}$ and $\tilde{y}$ are respectively denoted as $x$ and $y$):

$$\begin{array}{l}\frac{\partial u}{\partial t}+\frac{1}{1+ky}u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}-\frac{k}{1+ky}uv-fv\\ =-\frac{1}{\rho }\frac{1}{1+ky}\frac{\partial p}{\partial x}+\nu [\frac{1}{{(1+ky)}^2}\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}-\frac{y}{{(1+ky)}^3}\frac{\partial k}{\partial x}\frac{\partial u}{\partial x}\\ +\frac{1}{1+ky}\frac{\partial u}{\partial y}-\frac{k^2}{{(1+ky)}^2}u+\frac{1}{{(1+ky)}^3}\frac{\partial k}{\partial x}v+\frac{2k}{{(1+ky)}^2}\frac{\partial v}{\partial x}],\\ \frac{\partial v}{\partial t}+\frac{1}{1+ky}u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}-\frac{k}{1+ky}{u}^2+\frac{f}{1+ky}u\\ =-\frac{1}{\rho }\frac{\partial p}{\partial y}+\nu [\frac{1}{{(1+ky)}^2}\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}-\frac{y}{{(1+ky)}^3}\frac{\partial k}{\partial x}\frac{\partial v}{\partial x}\\ +\frac{1}{1+ky}\frac{\partial v}{\partial y}-\frac{k^2}{{(1+ky)}^2}v-\frac{1}{{(1+ky)}^3}\frac{\partial k}{\partial x}u-\frac{2k}{{(1+ky)}^2}\frac{\partial u}{\partial x}],\\ \frac{\partial u}{\partial x}+\frac{\partial }{\partial y}[(1+ky)v]=0.\end{array}$$

(5)

Let $R$ be the characteristic radius of the Hengduan Mountains ($R\sim {10}^5\mathrm{m}$), and then $k\sim {10}^{-5}{\mathrm{m}}^{-1}$. Let $\delta$ be the thickness of the boundary layer, and then $\delta \approx 5\sqrt{\frac{\nu x}{U_{\infty }}}\sim 7\mathrm{m}$. Let $U_{\infty }$ be the characteristic velocity of the outer layer of detouring flow, and then $U_{\infty }\sim 10\mathrm{m}{\mathrm{s}}^{-1}$. Therefore, it can be assumed that $\delta k\le 1$ and ${\delta }^2\frac{dk}{dx}\le 1$. $U_{\infty },R$ is selected as the standard quantity. Then, if $\frac{u}{U_{\infty }}$, $\frac{p}{\rho U_{\infty }^2}$, $\frac{\partial }{\partial t}$, $\frac{\partial }{\partial x}$ are the order of magnitude of 1 and $\frac{\partial }{\partial y}$ is the order of magnitude of ${\delta }^{-1}$,

$$\frac{\delta }{L}\approx o\left({\mathrm{Re}}^{-\frac{1}{2}}\right).$$

(6)

After ignoring small quantities of the magnitude of $\delta$ and higher, Equation (5) can be simplified to

$$\begin{array}{l}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\nu \frac{{\partial }^{2}u}{\partial y^2},\\ -k{u}^2+\frac{f}{1+ky}u=-\frac{1}{\rho }\frac{\partial p}{\partial y},\\ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.\end{array}$$

(7)

Integrating the second form of Equation (7) and considering $k\sim {10}^{-5}{\mathrm{m}}^{-1}$ and $f\sim {10}^{-4}{\mathrm{s}}^{-1}$, we can get

$$p(x,\delta ,t)-p(x,0,t)=\rho {\displaystyle {\int }_0^{\delta }\left(k{u}^2-\frac{fu}{1+ky}\right)dy}=o(\delta ).$$

(8)

Ignoring small quantities $\delta$, then

$$p(x,\delta ,t)=p(x,0,t).$$

(9)

This means that the pressure in the boundary layer is constant in the y-axis direction. Accordingly, the equations of the inner layer of detouring flow are as follows:

$$\begin{array}{l}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\nu \frac{{\partial }^{2}u}{\partial y^2},\\ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.\end{array}$$

(10)

For the outer layer of detouring flow, $V_e\ll U_e$. Then, ignoring the normal velocity $V_e$, we get

$$\begin{array}{l}\frac{\partial U_e}{\partial t}+U_e\frac{\partial U_e}{\partial x}=-\frac{1}{\rho }\frac{\partial p}{\partial x},\\ -k{U}_e{}^2+\frac{f{U}_e}{1+ky}=-\frac{1}{\rho }\frac{\partial p}{\partial y}.\end{array}$$

(11)

Take any point $y_0$ in the y-axis direction near the boundary layer, and integrate the form of Equation (11) in $[\delta ,y_0]$, considering $k\sim {10}^{-5}{\mathrm{m}}^{-1}$, $f\sim {10}^{-4}{\mathrm{s}}^{-1}$, and $U_{\infty }\sim 10\mathrm{m}{\mathrm{s}}^{-1}$, we get

$$p(x,y_0,t)-p(x,\delta ,t)=\rho {\displaystyle {\int }_{\delta }^{y_0}\left(k{U}_e{}^2-\frac{f{U}_e}{1+ky}\right)dy}\approx 0.$$

(12)

Then, near the boundary layer, there is

$$p(x,y,t)=p(x,\delta ,t)=p(x,0,t),$$

(13)

i.e., the pressure in the outer layer of detouring flow is constant in the y-axis direction. Therefore, in the inner layer of detouring flow,

$$\frac{\partial U_e}{\partial t}+U_e\frac{\partial U_e}{\partial x}=-\frac{1}{\rho }\frac{\partial p}{\partial x},$$

(14)

where $U_e$ is the speed of the outer layer flow along the edge of the Hengduan Mountains, and $V_e$ is the speed of the outer layer flow along the normal line to the edge of the Hengduan Mountains.

Let

$$t=\frac{R}{U_{\infty }}{t}^{\prime },u=U_{\infty }{u}^{\prime },v=\frac{\delta U_{\infty }}{R}{v}^{\prime },x=R{x}^{\prime },y=\delta y^{\prime },p=\rho U_{\infty }^2,\nu =\frac{{\delta }^2{U}_{\infty }}{R}{\nu }^{\prime },$$

(15)

yielding the following dimensionless equations (ignoring the apostrophe):

Inner layer:

$$\begin{array}{l}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\nu \frac{{\partial }^{2}u}{\partial y^2},\\ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.\end{array}$$

(16)

Outer layer:

$$\frac{\partial U_e}{\partial t}+U_e\frac{\partial U_e}{\partial x}=-\frac{1}{\rho }\frac{\partial p}{\partial x}.$$

(17)

From the scale analysis above, it can be seen that in the flow around the Hengduan Mountains, the Coriolis force in the normal direction of the edge of the Hengduan Mountains has a negligible effect on the pressure, whether inside or outside the boundary layer. Therefore, the motion Equations (16) and (17) of the flow around the Hengduan Mountains in this study are consistent with those of the nonrotating circulation. Although Equations (16) and (17) are simplified equations, they can fully reflect the nature of the detouring flow around the Hengduan Mountains. It is difficult to directly solve the flow Equations (16) and (17) due to the strong nonlinearity of the advection term in the equation. Therefore, we canget the vortex structure solutions by the perturbation method.

4. Approximate Solutions of the Flow around the Hengduan Mountains

Under the curved coordinate system in Section 3, the flow around the Hengduan Mountains can be regarded as the flow around a cylinder. Thus, the outer layer flow can be regarded as steady, and Equation (17) can be simplified as

$$U_e\frac{\partial U_e}{\partial x}=-\frac{1}{\rho }\frac{\partial p}{\partial x}.$$

(18)

Substituting Equation (16) into Equation (18), we can get

$$\begin{array}{l}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=U_e\frac{\mathrm{d}{U}_e}{\mathrm{d}x}+\nu \frac{{\partial }^{2}u}{\partial y^2},\\ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.\end{array}$$

(19)

The corresponding initial conditions of Equation (19) are as follows:

$$u(0,x,y)=U_e,v(0,x,y)=0.$$

(20)

The corresponding boundary conditions of Equation (19) are as follows:

$$\begin{array}{l}u(\tau ,x,0)=0,v(\tau ,x,0)=0,\\ u(\tau ,x,1)=U_e,v(\tau ,x,0)=0.\end{array}$$

(21)

From the continuity equation (the second form of Equation (19)), we introduce the stream function $\psi (x,y)$. Then,

$$u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x}.$$

(22)

Therefore, the equation satisfied by the stream function is as follows:

$$\frac{\partial }{\partial t}\left(\frac{\partial \psi }{\partial y}\right)+\frac{\partial \psi }{\partial y}\frac{{\partial }^2\psi }{\partial x\partial y}-\frac{\partial \psi }{\partial x}\frac{{\partial }^2\psi }{\partial y^2}=U_e\frac{\mathrm{d}{U}_e}{\mathrm{d}x}+\nu \frac{{\partial }^3\psi }{\partial y^3}.$$

(23)

The corresponding initial conditions are as follows:

$$\frac{\partial \psi }{\partial y}(0,x,y)=\phi (x,y),\frac{\partial \psi }{\partial x}(0,x,y)=0.$$

(24)

The corresponding boundary conditions are as follows:

$$\begin{array}{l}\frac{\partial \psi }{\partial y}(\tau ,x,0)=0,\frac{\partial \psi }{\partial x}(\tau ,x,0)=0,\\ \frac{\partial \psi }{\partial y}(\tau ,x,1)=U_e,\frac{\partial \psi }{\partial x}(\tau ,x,1)=0.\end{array}$$

(25)

According to the theory of inviscid and non-rotating flow around a cylinder [28], the velocity distribution of the outer layer flow is (dimensionless system) as follows:

$$U_e=2\sin \theta =2\sin x,$$

(26)

where $\theta$ is the angle from the stagnation point, which can be transformed as follows:

$$\tau =vt,\epsilon =\frac{1}{\nu },$$

(27)

and then Equation (23) is transformed into

$$\frac{\partial }{\partial \tau }\left(\frac{\partial \psi }{\partial y}\right)+\epsilon \frac{\partial \psi }{\partial y}\frac{{\partial }^2\psi }{\partial x\partial y}-\epsilon \frac{\partial \psi }{\partial x}\frac{{\partial }^2\psi }{\partial y^2}=\epsilon U_e\frac{\mathrm{d}{U}_e}{\mathrm{d}x}+\frac{{\partial }^3\psi }{\partial y^3}.$$

(28)

According to the transformation of Equation (15), it can be seen that $\epsilon =\frac{1}{\nu }>1$. Nonlinear Equation (28) cannot be expanded by $\epsilon$. the following transformation can be introduced:

$$\gamma =\frac{\epsilon }{{\sigma }^2+\epsilon },$$

(29)

where $\sigma >1$ (convergence parameter) and $\gamma \approx \frac{1}{2}<1$. Instituting Equation (29) into Equation (28), we get

$$(1-\gamma )\frac{\partial }{\partial \tau }\left(\frac{\partial \psi }{\partial y}\right)+\gamma r^2\frac{\partial \psi }{\partial y}\frac{{\partial }^2\psi }{\partial x\partial y}-\gamma r^2\frac{\partial \psi }{\partial x}\frac{{\partial }^2\psi }{\partial y^2}=\gamma r^2{U}_e\frac{\mathrm{d}{U}_e}{\mathrm{d}x}+(1-\gamma )\frac{{\partial }^3\psi }{\partial y^3}.$$

(30)

The stream function $\psi$ can be expanded into a series of $\gamma$.

$$\psi ={\psi }_0+\gamma {\psi }_1+{\gamma }^2{\psi }_2+\cdots .$$

(31)

The corresponding initial conditions are as follows:

$$\begin{array}{l}\frac{\partial {\psi }_0}{\partial y}(0,x,y)=\phi (x,y),\frac{\partial {\psi }_0}{\partial x}(0,x,y)=0,\\ \frac{\partial {\psi }_i}{\partial y}(0,x,y)=0,\frac{\partial {\psi }_i}{\partial x}(0,x,y)=0, (i=1,2,3\cdots ).\end{array}$$

(32)

The corresponding boundary conditions are as follows:

$$\begin{array}{l}\frac{\partial {\psi }_i}{\partial y}(\tau ,x,0)=0,\frac{\partial {\psi }_i}{\partial x}(\tau ,x,0)=0,(i=0,1,2\cdots )\\ \frac{\partial {\psi }_0}{\partial y}(\tau ,x,1)=U_e,\frac{\partial {\psi }_0}{\partial x}(\tau ,x,1)=0, \\ \frac{\partial {\psi }_i}{\partial y}(\tau ,x,1)=0,\frac{\partial {\psi }_i}{\partial x}(\tau ,x,1)=0,(i=1,2\cdots ). \end{array}$$

(33)

Instituting Equation (31) into Equation (30), we get the $i$th-order problem about $\psi$.

The $O({\gamma }^0)$ problem satisfies the linear partial differential equation as follows:

$$\frac{{\partial }^2{\psi }_0}{\partial y\partial \tau }-\frac{{\partial }^3{\psi }_0}{\partial y^3}=0.$$

(34)

Let

$$u_i=\frac{\partial {\psi }_i}{\partial y},(i=0,1,2\cdots ),$$

(35)

and Equation (30) is transformed into

$$\frac{\partial u_0}{\partial \tau }-\frac{{\partial }^2{u}_0}{\partial y^2}=0.$$

(36)

The boundary conditions of Equation (36) are inhomogeneous. Then, we need to homogenize its boundaries. Considering that Equation (36) is linear and its solutions can be superimposed, we let

$$u_0(y,t)=w_0(y,t)+{\vartheta }_0(y,t),$$

(37)

where

$$w_0(y,t)=U_e\left[1-\cos \left(\frac{\pi y}{2}\right)\right],$$

and ${\vartheta }_0(y,t)$ meets the following relation:

$$\begin{array}{l}\frac{\partial {\vartheta }_0}{\partial \tau }-\frac{{\partial }^2\vartheta }{\partial y^2}=0,\\ {\vartheta }_0(0,\tau )=0,{\vartheta }_0(1,\tau )=0,\\ {\vartheta }_0(y,0)=U_e\cos \left(\frac{\pi y}{2}\right).\end{array}$$

(38)

Solving Equation (38), we get

$${\vartheta }_0=U_e\cos \left(\frac{\pi y}{2}\right)e^{-{\left(\frac{\pi }{2}\right)}^2\tau }.$$

(39)

According to Equations (37) and (39), we get

$$u_0(\tau ,x,y)=U_e\left[1-\cos \left(\frac{\pi y}{2}\right)\right]+U_e\cos \left(\frac{\pi y}{2}\right)e^{-{\left(\frac{\pi }{2}\right)}^2\tau }.$$

(40)

Integrating Equation (40) over $[0,y]$, we get the expression of ${\psi }_0$ as follows:

$${\psi }_0(\tau ,x,y)=U_e\left[y-\frac{2}{\pi }\sin \left(\frac{\pi y}{2}\right)+\frac{2}{\pi }\sin \left(\frac{\pi y}{2}\right)e^{-{\left(\frac{\pi }{2}\right)}^2\tau }\right].$$

(41)

The $O({\gamma }^1)$ problem is as follows, which satisfies the nonlinear partial differential equation.

$$\frac{\partial }{\partial \tau }\left(\frac{\partial {\psi }_1}{\partial y}\right)-\frac{{\partial }^2}{\partial y^2}\left(\frac{\partial {\psi }_1}{\partial y}\right)={\sigma }^2\left(\frac{\partial {\psi }_0}{\partial x}\frac{{\partial }^2{\psi }_0}{\partial y^2}-\frac{\partial {\psi }_0}{\partial y}\frac{{\partial }^2{\psi }_0}{\partial y\partial x}+U_e\frac{\mathrm{d}{U}_e}{\mathrm{d}x}\right).$$

(42)

By institute Equation (41) into Equation (42), we get

$$\begin{array}{l}\frac{\partial }{\partial \tau }\left(\frac{\partial {\psi }_1}{\partial y}\right)-\frac{{\partial }^2}{\partial y^2}\left(\frac{\partial {\psi }_1}{\partial y}\right)\\ ={\sigma }^2\sin x\cos x-{\sigma }^2\sin x\cos x{\left[1-\cos \left(\frac{\pi y}{2}\right)+\cos \left(\frac{\pi y}{2}\right)e^{-{\left(\frac{\pi }{2}\right)}^2\tau }\right]}^2\\ +{\sigma }^2\sin x\cos x\left[y-\frac{2}{\pi }\sin \left(\frac{\pi y}{2}\right)+\frac{2}{\pi }\sin \left(\frac{\pi y}{2}\right)e^{-{\left(\frac{\pi }{2}\right)}^2\tau }\right]\left[\frac{\pi }{2}\sin \left(\frac{\pi y}{2}\right)-\frac{\pi }{2}\sin \left(\frac{\pi y}{2}\right)e^{-{\left(\frac{\pi }{2}\right)}^2\tau }\right].\end{array}$$

(43)

From the initial conditions in Equation (32) and the boundary conditions in Equation (33), we can solve Equation (43) and get

$$u_1(\tau ,x,y)=-\frac{{\sigma }^2\sin x\cos x}{2{\pi }^2}\left[{\pi }^2{y}^2+4\pi y\sin \left(\frac{\pi y}{2}\right)+32\cos \left(\frac{\pi y}{2}\right)-32\right]+\zeta (\tau ,x,y),$$

(44)

where $\zeta (\tau ,x,y)$ is the function including the time $\tau$. According to Equation (44) and the transformation of Equation (29), we can get the first-order approximation of $u$ as follows:

$$\begin{array}{l}u(t,x,y)=\sin x\left[1-\cos \left(\frac{\pi y}{2}\right)\right]\\ -\frac{\gamma {\sigma }^2\sin x\cos x}{2{\pi }^2}\left[{\pi }^2{y}^2+4\pi y\sin \left(\frac{\pi y}{2}\right)+32\cos \left(\frac{\pi y}{2}\right)-32\right]+\xi (t,x,y),\end{array}$$

(45)

where $\xi (t,x,y)$ is the function including the time $\tau$ with the form of $e^{-{\chi }^{2}t}$, which is dissipative. This means that, affected by the blocking of the Hengduan Mountains, the velocity of the flow along the curved surface decreases with time in the boundary layer.

5. The Physical Formation Mechanisms of the Southwest China Vortex

In the previous section, the first-order approximate solution of detouring flow around the Hengduan Mountains was obtained through an improved perturbation method. In this section, we discuss the formation conditions of SWVs through the approximate solution in Equation (45).

In Equation (45), the approximate solution consists of a dissipative term (with time variation) and a non-dissipative term (without time variation). In order to discuss the formation of the vortex, considering that the SWV is the final state of flow around the Hengduan Mountains, then, when $t\to +\infty$, there is

$$\begin{array}{l}u(x,y)=\sin x\left[1-\cos \left(\frac{\pi y}{2}\right)\right]\\ -\frac{\gamma {\sigma }^2\sin 2x}{4{\pi }^2}\left[{\pi }^2{y}^2+4\pi y\sin \left(\frac{\pi y}{2}\right)+32\cos \left(\frac{\pi y}{2}\right)-32\right].\end{array}$$

(46)

In Equation (46), $u$ consists of three parts. The first is the outer boundary compensation term, the second is the pressure gradient force term, and the third is the nonlinear term. Considering the Equation (46) is in the dimensionless form, we can institute Equation (46) into Equation (15) and get

$$\begin{array}{l}u(x,y)=U_{\infty }\sin \left(\frac{x}{R}\right)\left[1-\cos \left(\frac{\pi y}{2\delta }\right)\right]\\ -\frac{{\sigma }^2{U}_{\infty }^2}{4{\pi }^2(1+\nu {\sigma }^2)}\sin \left(\frac{2x}{R}\right)\left[\frac{{\pi }^2{y}^2}{{\delta }^2}+\frac{4\pi y}{\delta }\sin \left(\frac{\pi y}{2\delta }\right)+32\cos \left(\frac{\pi y}{2\delta }\right)-32\right].\end{array}$$

(47)

According to Schlichting [28], backflow is a necessary condition for the formation of vortex, and $u$ of the backflow meets $\frac{\partial u}{\partial y}=0$. Therefore, it is necessary to determine whether a backflow can appear; that is, we need to analyze whether the above equation has real roots in the interval of $(0,\delta ]$. In order to make the convergence domain of the series expansion in Equation (31) large enough, r should be a small enough value; thus, we take $r=10\pi$. Setting the main axis of the Hengduan Mountains $R\approx 3.5\times {10}^5\mathrm{m}$ and $\delta \approx 5\mathrm{m}$ at 700 hPa, the equation $\frac{\partial u}{\partial y}=0$ has a solution in the interval $(0,\delta ]$ when $U_{\infty }>11.3\mathrm{m}{\mathrm{s}}^{-1}$. According to the velocity gradient, it can be known that the gradient of the outer boundary compensation term is always positive when $\theta \in [0,\pi ]$, and the gradient of the pressure gradient term is negative when $\delta$ is sufficiently large, which mainly suppresses the outer boundary compensation term. This means that, if the velocity is too low, the gradient of the pressure gradient term is small, and the gradient of the outer boundary compensation term cannot be suppressed to zero.

From a physical point of view, in the deceleration area, the kinetic energy of the atmosphere is continuously consumed, and it also flows downstream under the reaction of pressure. According to the geopotential theory, if the incoming airflow is faster, the pressure gradient force is correspondingly larger and continues to increase in the downstream direction. When the kinetic energy is consumed to a certain extent, the atmosphere near the surface of the Hengduan Mountains can no longer overcome the action of pressure and continue to flow. Because of the continuity requirement, once the atmosphere stops moving forward, the fluid in downstream areas must flow backward (the point at which backflow occurs is called the separation point). On the contrary, if the incoming airflow is slow, the pressure gradient force is correspondingly too small to consume kinetic energy, and it cannot stop the forward movement of the atmosphere in the boundary layer. The results are consistent with the boundary layer theory. If the characteristic scale of the cylinder is constant, when the incoming flow velocity is low, the movement of the rear of the cylinder can be regarded as constant; when the incoming flow velocity exceeds a critical value, the movement becomes periodic [21,27]. The fact that the nascent SWV is periodic can be seen in the analysis below.

Although the backflow is the key to the formation of SWVs, whether the SWV can form requires further analysis. The SWV is a system with a typical geometric structure; hence, the issue of SWV formation is studied with the geometric structure of a two-dimensional flow field. The spatial structure of atmospheric motion in two-dimensional space can be described as follows:

$$\begin{array}{l}\frac{dx}{dt}=u(x,y),\\ \frac{dy}{dt}=v(x,y).\end{array}$$

(48)

If the SWV can form, Equation (46) must meet two conditions: (1) there are equilibrium points within the boundary layer; (2) the equilibrium points meet the geometric structure of the vortex. Thus, we analyze the formation conditions of the SWV from the two points.

According to the above analysis, at first, we should consider the existence of the equilibrium points in Equation (48). These equilibrium points $(x^{\ast },y^{\ast })$ should meet the following equations:

$$\begin{array}{l}u(x^{\ast },y^{\ast })=U_{\infty }\sin \left(\frac{x^{\ast }}{R}\right)\left[1-\cos \left(\frac{\pi y^{\ast }}{2\delta }\right)\right]\\ -\frac{{\sigma }^2{U}_{\infty }^2}{4{\pi }^2(1+\nu {\sigma }^2)}\sin \left(\frac{2x^{\ast }}{R}\right)\left[\frac{{\pi }^2{y}^{\ast }{}^2}{{\delta }^2}+\frac{4\pi y^{\ast }}{\delta }\sin \left(\frac{\pi y^{\ast }}{2\delta }\right)+32\cos \left(\frac{\pi y^{\ast }}{2\delta }\right)-32\right]=0,\\ v(x^{\ast },y^{\ast })=-R{U}_{\infty }\cos \left(\frac{x^{\ast }}{R}\right)\left[y^{\ast }+\frac{2\delta }{\pi }\sin \left(\frac{\pi y^{\ast }}{2\delta }\right)\right]\\ +\frac{{\sigma }^{2}R{U}_{\infty }^2}{8{\pi }^2(1+\nu {\sigma }^2)}\cos \left(\frac{2x^{\ast }}{R}\right)\left[\frac{{\pi }^2{y}^{\ast }{}^3}{3{\delta }^2}+2y^{\ast }\cos \left(\frac{\pi y^{\ast }}{2\delta }\right)+\frac{60\delta }{\pi }\sin \left(\frac{\pi y^{\ast }}{2\delta }\right)-32y^{\ast }\right]=0.\end{array}$$

(49)

By solving Equation (49), it can be known that, when $U_{\infty }>18.9\mathrm{m}{\mathrm{s}}^{-1}$, Equation (49) always has a nonzero solution less than $\delta$. For example, when $\overline{U}=25\mathrm{m}{\mathrm{s}}^{-1}$, the solution of Equation (49) is $(x^{*},y^{*})=(9.45\times {10}^5\mathrm{m},4.1\mathrm{m})$; correspondingly, $\theta \approx {154}^{°}$(Figure 3d).

The equilibrium points have different topological structures such as knot, saddle point, focus, and center. However, only the focus and center correspond to the vortex structure. Hence, it is necessary to further analyze the properties of the obtained equilibrium points. To this end, the Jacobi matrix corresponding to the equilibrium points of Equation (48) is discussed.

$${\left[\begin{array}{cc}\frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}\end{array}\right]|}_{(x^{\ast },y^{\ast })}.$$

(50)

Considering the incompressible nature of the atmosphere, the corresponding characteristic equation of the Jacobi matrix defined by Equation (50) is

$${\lambda }^2-\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)\lambda +\left(\frac{\partial u}{\partial x}\frac{\partial v}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}\right)={\lambda }^2+\left(\frac{\partial u}{\partial x}\frac{\partial v}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}\right)=0.$$

(51)

It can be seen from the calculation that, when $U_{\infty }>18.9\mathrm{m}{\mathrm{s}}^{-1}$, $\left(\frac{\partial u}{\partial x}\frac{\partial v}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}\right)|{}_{(x^{\ast },y^{\ast })}>0$. Correspondingly, the characteristic Equation (51) has a pair of conjugate pure imaginary roots. Then, the equilibrium point $(x^{\ast },y^{\ast })$ is the center.

To sum up, there are three situations for the detouring flow around the Hengduan Mountains.

(1)

When incoming flow velocity in the windward area of the Hengduan Mountains is small, there is no backflow around the Hengduan Mountains (Figure 3a).

(2)

When the flow velocity $U_{\infty }>11.3\mathrm{m}{\mathrm{s}}^{-1}$ in the windward area of the Hengduan Mountains, the flow around the Hengduan Mountains will form a backflow. At this time, the backflow will break away from the laminar flow and continue to move downstream (Figure 3b).

(3)

When the incoming flow velocity $U_{\infty }>18.9\mathrm{m}{\mathrm{s}}^{-1}$ in the windward area of the Hengduan Mountains, the backflow will wind up in the process of moving downstream, and finally form a vortex with a central feature (Figure 3c,d).

Figure 3 is a schematic diagram of the flow field in the curved coordinate system, which is offset by about 90° from the traditional longitude–latitude coordinate system. Therefore, the vortex in Figure 4d is an anticyclone limit cycle structure, while it is a cyclone-limit cycle structure in the latitude–longitude coordinate system. The above analysis shows that there are two necessary conditions for SWV formation. The first one is that, at the edge of the Hengduan Mountains, the incoming airflow diverges, i.e., backflow occurs, and the laminar flow undergoes a fundamental change. The second is that the backflow winds up and form a cyclone-limit cycle structure in the longitude–latitude coordinate system.

In addition, the central structure of the nascent vortex is determined by the incompressible nature of the atmosphere, which is common to all two-dimensional incompressible fluid vortices. However, the actually observed SWV is often a focal structure, which also means that the nascent vortex is two-dimensional, while the mature vortex is a three-dimensional structure. The results are consistent with the Chen [19]. The SWV in the initial stage is a shallow weather system below 500 hPa. The SWV in the mature stage is a circular asymmetric mesoscale system and a deep warm-humid low-pressure system. Its positive vorticity region can extend up to 100 hPa [41]. From the perspective of the three-dimensional structure, if the Hengduan Mountains is an ideal cylinder and the vertical distribution of the wind speed and direction of the incoming airflow are consistent, the nascent SWV should be a regular cylinder with a central structure (consistent in the vertical direction). However, the Hengduan Mountains are not an ideal cylinder, and the vertical distributions of the incoming wind speed and direction are inconsistent; thus, the nascent SWV should be an irregular cylinder, but each section has a central structure.

6. Physical Mechanisms of the Initiation Location and Seasonal Distribution of the Southwest China Vortex

Statistical results show that there are three SWV source areas, namely, the Jiulong area, the Sichuan Basin, and the Xiaojin area [5,19,20,21,22]. In terms of seasonal distribution, the SWV occurs most in summer, followed by spring, autumn, and winter [1,2,3,4,5]. At present, the explanation of these statistical results remains inconclusive and needs further research.

The Prandtl boundary layer theory and the perturbation method have been used to study the conditions for SWV formation under ideal conditions (the southwest airflow is about 30° with the latitude line). However, the direction of the incoming airflow in the windward area of the Hengduan Mountains is variable; therefore, different directions of incoming airflow will affect the initial velocity in the main-axis direction and the pressure gradient force in the boundary layer, which may further affect the formation and location of the SWV. Simulation studies also showed that different airflow directions and speeds have different effects on SWV formation, and the westerly wind is especially important for the formation of SWV [22]. To this end, it is necessary to further analyze how different directions and speeds of the airflow impact the initiation location and seasonal distribution of SWVs. Note that the SWV discussed in this study is in the nascent stage, and the actually observed SWV is an atmospheric vortex developed on this basis. It can be seen from Section 2 that the formation of the nascent vortex is mainly affected by the westerly wind or southwesterly wind. Considering the pressure distribution of the flow around the Hengduan Mountains, we use the main axis direction of the Hengduan Mountains (about 30° angle with the latitude line) as the criterion to analyze the effect of airflow direction on the initiation location and seasonal distribution of SWVs.

According to the geopotential theory, the pressure gradient force in the boundary layer is the key that affects the position of the separation point and, thus, the position of the nascent vortex. The pressure gradient force is related to the direction of the incoming airflow in the windward area of the Hengduan Mountains. Therefore, the issue of the SWV source is discussed from the direction of the incoming airflow in the windward area of the Hengduan Mountains. From the results of Section 5, it can be seen that, when the velocity of incoming airflow is large enough and is at the angle of 25°–35° with the latitude line, the center of the nascent vortex satisfies $\theta \in ({150}^{°},{160}^{°})$, which moves downstream from the boundary layer to the western edge of the Sichuan Basin. Thus, that area becomes the Sichuan Basin vortex source. If the angle between the incoming airflow and the latitude line is greater than 35°, the pressure gradient in the boundary layer will be enhanced, and the reaction of pressure will increase in the deceleration zone. The kinetic energy consumption of the atmosphere is accelerated, which will result in a more westerly location of separation point. Then, the position of the vortex center will be $\theta <{150}^{°}$. The nascent vortex will move downstream of the boundary layer to the western edge of the Sichuan Basin, where the Jiulong vortex source is. If the angle between the incoming airflow and the latitude line is less than 25°, the pressure gradient in the boundary layer will be reduced, and the reaction of the pressure will decrease in the deceleration zone. Thus, the kinetic energy consumption of the atmosphere slows down, resulting in a more easterly location of the separation point. Then, the position of the vortex center meets $\theta >{160}^{°}$. The nascent vortex will move downstream of the boundary layer to the center of the western Sichuan Plateau, where the Xiaojin vortex source is. Since the separation point of the detouring flow around the Hengduan Mountains is only related to the pressure gradient, when the velocity of the incoming airflow is large enough to form a nascent vortex, the SWV source is only related to the direction of the incoming airflow, but not to itsspeed.

According to the analysis in Section 5, the formation of the nascent vortex is only related to the velocity of the incoming airflow in the direction of the main axis of the Hengduan Mountains. In terms of velocity decomposition, the velocity of the incoming airflow in the main-axis direction is affected by both the speed and the direction of the actual incoming airflow. Therefore, below, we discuss the seasonal distribution of the SWV from both the speed and the direction of the incoming airflow, as well as the climatic background. The climatic background shows that the SWV initiation area is mainly controlled by the westerlies of mid–high latitudes and the subtropical and tropical southerly circulation of mid–low latitudes. Thus, the north of the SWV initiation area is generally affected by westerly and northwesterly wind, especially in the winter half-year; the south of the SWV initiation area is mostly affected by southwesterly wind and southeasterly wind, especially in the summer half-year. As for the Heng duan Mountains, it is mainly affected by westerly, southwesterly, and southeasterly wind. Specifically, there is more westerly wind in the winter half-year, and more southwesterly and southeasterly wind in the summer half-year. Considering that the southeasterly wind is not conducive to the formation of SWV, it is only necessary to study the influence of the westerly wind in winter and southwesterly wind in summer. For all the atmospheric circulation passing through the Hengduan Mountains, it can be decomposed into the main axis and its vertical direction components. The main-axis component mainly produces the boundary layer, while the vertical component mainly changes the pressure gradient of the outer layer, which will transmit to the inner boundary layer. For the westerly wind, the angle between its direction and the main axis of the Hengduan Mountains is 30°, and the direction deviation is relatively large. For the southwesterly wind (the angle with the latitudinal line is 45°), the angle between its direction and the main axis of the Hengduan Mountains is 15°, and the direction deviation is small. For the same incoming flow velocity, if the deviation of the wind direction is smaller, the velocity of the main axis component will be larger, and the possibility of exceeding the critical velocity will be higher. This is why the probability of SWV formation in summer is greater, but the probability in winter is smaller, and this is also the reason why the Jiulong vortex source is more concentrated than the Sichuan Basin vortex source and the Xiaojin vortex source [41,42].

In general, the SWV source is mainly affected by the direction of the incoming airflow and has nothing to do with its speed. The seasonal distribution of the SWV is related to both the direction and the speed of the incoming airflow. The results of this study successfully explain the observed fact that the three main vortex sources are inJiulong, Xiaojin, and Sichuan basin, and confirm the reasons for the variation of SWV seasonal distribution.

7. Conclusions and Discussion

It is generally recognized that the friction effect of the terrain boundary is the main influence factor for the SWV formation, but it involves the complex mechanism of the flow arounda solid boundary.The existing research on the terrain effect is qualitative analysis, and does not reflect the vortex structure. In order to deepen the understanding on the SWV formation, it is necessary to conductan in-depth analysis of the detouring flow that may generate SWV. From the perspective of fluid dynamics, a vortex is a “bridge” connecting laminar flow and turbulent flow, and its formation mechanism is the study focus of detouring flow. So far, it has not been fully understood, and new solutions are urgently needed.

On the basis of past studies, this study further investigated the physical mechanisms of SWV formation using the Prandtl boundary layer theory, the improved perturbation method, and the boundary layer equations with strong nonlinearity. The main conclusions are that the fluid structure changes, and the laminar flow bifurcates into a vortex.

The formation of SWV is mainly determined by the incoming airflow in the main-axis direction of the Hengduan Mountains. When the speed of the incoming airflow is greater than the critical speed, i.e., when the Reynolds number is greater than the critical value, the vortex structure is formed in the leeward region of the Hengduan Mountains. Furthermore, the nascent vortex presents a two-dimensional central structure.

The formation location of SWV is mainly determined by the relative position of the incoming airflow in the windward area of the Hengduan Mountains and the main axis of the Hengduan Mountains. When the angle between the incoming airflow in the windward area of the Hengduan Mountains and the latitude line is 25°–35°, less than 25°,andgreater than 35°, the SWV source corresponds to the Sichuan Basin, Xiaojinarea, and Jiulong area, respectively.

The seasonal distribution of SWV is mainly determined by the velocity of the incoming airflow in the windward area of the Hengduan Mountains and the relative position of the incoming airflow and the main axis of the Hengduan Mountains. The angle between the southwesterly wind and the main-axis direction is smaller than that between the westerly wind and the main-axis direction. Therefore, for the two types of wind with the same incoming speed, the main axis component speed of the southwesterly wind will be greater, resulting in a greater possibility of exceeding the critical speed. This means that the southwesterly wind is more likely to generate SWV than the westerly wind. Therefore, the SWV is more likely to appear in summer when the southwesterly airflow prevails than in winter when the westerly airflow prevails, and the Jiulong vortex source is more concentrated than the Sichuan Basin vortex source and the Xiaojin vortex source.

The study revealed the necessary conditions for the SWV formation, the geometric structure of the SWV and other physical mechanisms, and better explained the initiation location and seasonal distribution of SWVs. The results of this studyare consistent with the actual situation. SWV is the result of variable airflow and constant terrain condition. This study deepens the understanding of SWV formation.

Note that there were inevitably some shortcomings in this study. First, the edges of the Hengduan Mountains were smoothed in this study, but the actual edges of the Hengduan Mountains at 700 hPa are uneven. This produced a “golf” effect, resulting in a sharper turbulence effect of atmospheric motion downstream of the stagnation point. Second, the outer pressure field and flow field in the east and north of the Hengduan Mountains were not considered, both of which can more or less affect the SWV formation and the distribution of vortex source. In addition, this study only gave the physical processes of SWV formation. In the future, we will conduct a comprehensive analysis and research on the development, movement, and dissipation of SWVs from the perspective of vortex dynamic stability.