Physical Mechanism of the Development and Extinction of the China Southwest Vortex

Abstract

In this paper, a typical vortex system based on quasi-linear thermal-dynamic equations to reflect the development and extinction of the China Southwest Vortex is established using the vortex motion stability method combined with the outer environmental field and cumulus convective latent heat release. The development and extinction of the China Southwest Vortex in catastrophic weather systems are studied from the aspects of stability and development mechanisms for the primary-stage China Southwest Vortex, the transition mechanism from the primary-stage China Southwest Vortex to the mature vortex, and stability and development mechanisms of the mature China Southwest Vortex. The results show the following: (1) the convergence and divergence of the surrounding flow field is the main factor influencing the development and extinction of the primary-stage China Southwest Vortex, while gravity wave disturbance is the main driving force for the maintenance and development of the primary vortex. Based on the convergence of the external flow field, the gravity wave disturbance must exceed the critical frequency, or the vortex will tend to die out. (2) The convergence and divergence of the surrounding flow field is also the main factor for the transition from the primary vortex to the mature vortex. Based on the convergence of the surrounding flow field, the primary vortex transforms into a mature vortex only when the gravity wave disturbance strongly exceeds the critical frequency and causes the vertical disturbance to become unstable. (3) The convergence and divergence of the external flow field is also the main factor for the development and extinction of the mature China Southwest Vortex. In the early stage, the vortex can be maintained and developed as long as the surrounding flow field converges. In the case of the divergence of the external flow field, the vortex may be maintained for a short time, but eventually dissipates when the gravity wave disturbance exceeds the critical frequency. In the later stage, under the convergence of the surrounding flow field, the vortex can be maintained when the gravity wave disturbance exceeds the critical frequency. However, with the divergence of the surrounding flow field, the vortex may be maintained for a short time, but it will eventually dissipate when the gravity wave disturbance is extremely strong. In addition, the observations of the evolution of China Southwest Vortexes and gravity wave activities under the influence of southwest airflow and atmospheric disturbance in the Western Sichuan Plateau–Sichuan Basin are explained by the above physical mechanism. It is also pointed out that the heating effect can be an obstacle to the development of the China Southwest Vortex by increasing the critical frequency of gravity waves during unstable layer formation, and the divergent environment flow field under the condition of stable layer formation. Therefore, this paper deepens the understanding of the evolution process and anomalous mechanisms of the China Southwest Vortex.

1. Introduction

The China Southwest Vortex is a cyclonic closed low-pressure weather system on the isobaric surface of 700 hPa or 850 hPa on the eastern side of the Qinghai–Tibet Plateau (mainly the Western Sichuan Plateau and Sichuan Basin). Its horizontal scale is about 300–500 km and it is a kind of β mesoscale weather system [1,2,3,4,5]. The China Southwest Vortex is an important weather system that causes rainstorms in China and is associated with many historical heavy rainstorms and waterlogging disasters. Not only is it an important precipitation system in southwest China, but the China Southwest Vortex and the disturbances it induces can also move eastward and develop in summer, leading to large-scale rainstorms and waterlogging in downstream areas [5,6,7]; for example, the rainstorms and waterlogging of the whole Yangtze River basin in the summer of 1998 and the large-scale rainstorms during 8–15 July 2012 and 10–16 July 2020. Therefore, research on the generation, development, and movement of the China Southwest Vortex has long attracted significant attention from the meteorological community, especially research on the development of the China Southwest Vortex, which is extremely significant in both theoretical research and practical forecasting.

The development of the China Southwest Vortex has always been an important topic for meteorologists and forecasters. Gao Shouting et al. [8] found that the Tibetan Plateau plays an important driving role in the development of a low vortex and cyclonic waves along the shear line through the trough experiment. Zou Bo et al. [9] found that non-equilibrium dynamic forcing in the lower atmosphere promoted the development of the China Southwest Vortex by stimulating airflow convergence and positive vorticity growth, and that positive vorticity advection forcing in the middle troposphere intensified the development of the vortex. Huang Fujun et al. [10] argued that an important factor in the evolution of the China Southwest Vortex into a baroclinic vortex is when a disturbance in the middle layer overlaps the low vortex and cold advection enters from the west of the vortex behind the disturbance. Based on the Hoskins potential vortex theory, Fan Ke et al. [11] studied the influence of upper tropospheric potential vortex disturbance on the occurrence and development of the China Southwest Vortex in the lower troposphere from the perspectives of conservation of the potential vortex on isobaric surfaces and conservation of the wet isentropic potential vortex. They found that upper tropospheric potential vortex disturbance is an important factor for the occurrence and development of the China Southwest Vortex in the lower troposphere; its influence on the vortex is shown in the characteristic slanting southward extension, with the descending height of the upper wet isentropic potential vortex surface and the convergence strength of the wind field corresponding well with the occurrence and development of the vortex. Yu Shuhua et al. [12] analyzed two vortex-accompanying activities of the Plateau Vortex and the China Southwest Vortex and pointed out that, for the two forms of the vortex—those induced by and coupled with the Plateau Vortex—the strengthening of the vortex is caused by the enhancement of baroclinic instability at 500 hPa and the southwesterly jet at 200 hPa influencing the plateau vortex and the China Southwest Vortex, respectively. Based on the WRF model, Li Xuesong et al. [13] demonstrated that surface heating over the Qinghai–Tibet Plateau results in the strong development and continuous eastward movement of the China Southwest Vortex, caused by significant changes in the atmospheric circulation in the middle and upper troposphere of the Qinghai–Tibet Plateau and downstream of it, which triggered continuous heavy precipitation in southern China in June 2010. Chen et al. [14] found that the condition of large relative helicity promotes the development and movement of the China Southwest Vortex, with the large-scale wind field continuing to deliver positive vorticity to the vortex.

In addition, it is generally recognized that the release of latent heat from condensation and the formation and maintenance of low-level jets are conducive to the development of the China Southwest Vortex. For example, the study of Kuo et al. [15] on heavy rain in the Sichuan Basin during 11–15 July 1981 showed that half of the heating caused by the convergence of convective vortex sensible heat and latent heat flux came from the release of latent heat in condensation, indicating that the release of latent heat from condensation, accompanied by cumulus convection, plays an important role in the development of the China Southwest Vortex. The study of Chen et al. [16] also shows that the release of latent heat from condensation is an important mechanism in the maintenance and development of the China Southwest Vortex. Chen Zhongming [17] found that the divergence of the outer environmental field and the release of latent heat from the secondary circulation cumulus caused by friction in the free boundary layer are the main factors contributing to the development of the China Southwest Vortex. According to numerical tests and diagnostic analysis on the formation process of the China Southwest Vortex, Zhao Ping et al. [18,19] found that latent heat does not affect the formation of the vortex, but only enhances it, and that latent heat can further develop the vortex by reducing the strength of the cyclonic convergence at the low level and the anticyclonic divergence at the high level. Through case analysis, Zhu He et al. [20] pointed out that abundant water vapor and unstable energy are favorable thermal conditions for the development of the China Southwest Vortex, and that the adjustment of upper-middle circulation accompanied by the fluctuation of the Qinghai–Tibet high is related to the development of convection and precipitation in the vortex. Wang Xiaofang et al. [21] found that the development of the China Southwest Vortex is always accompanied by a strong southwest low-level jet; the combined effect of positive vorticity advection at the upper level and warm advection at the lower level is a significant reason that the surrounding atmosphere maintains large non-thermal wind vorticity during movement of the vortex, leading to strong upward movement over the vortex.

As can be seen from the above, many significant advances have been made in previous studies on the thermal and dynamic structure of the China Southwest Vortex and its generation and development mechanisms. However, owing to the limitations of high-resolution observation data and numerical model technology, it is difficult to truly describe the overall structure and evolution characteristics of the China Southwest Vortex. In particular, the internal mechanism of the development and extinction of the vortex is not very clear [5]. It is thus necessary to strengthen the theoretical research. As a typical rotating fluid, the development and extinction of the China Southwest Vortex are essentially due to the instability of the vortex, which is a kind of centrifugal instability. Research on the stability of vortices began with the pioneering work of Rayleigh [22], but great progress was made by Howard and Gupta [23], who derived the governing equation for the analysis of the invisticity of the linear stability of concentrated vortices. Following these advances, the stability characteristics of various vortexes were analyzed and their stability criteria were summarized, including typical vortex flow, the Hamel–Oseen vortex, the Rankine vortex, the Burgers vortex, and so on [24,25,26,27,28,29,30,31]. Through stability analysis, the physical mechanisms of the development and rupture of various vortexes were studied. In the field of meteorology, Sullivan, Long, Martin, Yih, and Wu Jie Zhi et al. [32,33,34,35,36] simulated the atmospheric vortex structure of typhoons and tornadoes through vortex models and tried to find vortex solutions that satisfied the Navier–Stokes equation. However, this research mainly concerned the geometric structure of typhoons and tornadoes through vortex solutions. In addition, for the development and extinction of cyclones and the stability of vortices, Zhang Ming et al. [37] studied the instability of the barotropic rigid vortex base flow using the barotropic model in the column coordinate system to reveal the binocular structure of tropical cyclones and the unstable development of spiral waves. However, owing to the obvious vertical motion of tropical cyclones, the barotropic model cannot fully describe the development and extinction mechanisms. In conclusion, current research on the stability of vortex motion is not sufficient, mainly because the equation of the motion of vortexes is more complex than that of laminar flow and the analysis of the stability of vortex motion is more difficult than that of other fluid forms. Moreover, owing to the influence of the Earth’s rotation, the stability of atmospheric vortices is more complicated and their application in meteorology is rare. Therefore, it is necessary to strengthen the study of the dynamics of the China Southwest Vortex, establish a more general and accurate basic theory, and improve the quantitative understanding of the vortex.

Undoubtedly, an effective way to reveal the physical mechanism of the development and extinction of the China Southwest Vortex is to analyze the stability of its structure and explore its extinction. As the China Southwest Vortex is a shallow weather system at its primary stage and a deep warm-wet and low-pressure system at its mature stage [38], the vortex can be approximated as a pure vortex structure at its primary stage (a two-dimensional steady axisymmetrical basic flow without vertical motion), while it can be approximated as a spiral vortex structure at its maturity stage (a two-dimensional axisymmetrical basic flow with vertical motion) [39]. Therefore, the development and extinction of the China Southwest Vortex will be studied in two stages. Firstly, the stability and development mechanisms of the primary-stage China Southwest Vortex are analyzed. Secondly, the transition mechanisms from the primary stage to the mature stage are analyzed. Finally, the stability and development mechanisms of the mature vortex are analyzed.

2. Research Ideas and Methods

The physical mechanism causing the instability of vortex motion is centrifugal force. When a particle of fluid is disturbed and moves to a larger radius in a rotating motion, a larger centrifugal force will be generated if the fluid rotates faster in the new environment. Then, the pressure difference of the original basic flow is not sufficient to balance the centrifugal force, causing the fluid element to move outward continuously and the disturbance to increase. Eventually, the motion becomes unstable. For inviscose fluid, let the initial radius of the fluid particle be $r_1$ and the angular velocity be ${\mathsf{\Omega }}_1$; it moves to the radius $r_2$ ($r_2>r_1$) after being disturbed. Because it is an inviscose fluid, according to the conservation of angular momentum, the angular velocity of the fluid element is ${\mathsf{\Omega }}_2{}^{\prime }={\mathsf{\Omega }}_1{r}_1^2/r_2^2$. If $\left|{\mathsf{\Omega }}_2{}^{\prime }\right|>\left|{\mathsf{\Omega }}_2\right|$, and the new centrifugal force exceeds the centrifugal force of the undisturbed fluid. If $\left|{\mathsf{\Omega }}_1{r}_1^2\right|>\left|{\mathsf{\Omega }}_2{r}_2^2\right|$, the rotating motion will be unstable. This means that the rotating fluid motion is unstable when $\frac{d}{dr}({\mathsf{\Omega }}_1{r}^2)<0$. This is known as the Rayleigh criterion of rotating Couette flow [22]. Considering that the China Southwest Vortex is a typical vortex structure, in this study, the development and extinction of the vortex will be discussed from the perspective of vortex stability according to the Rayleigh criterion.

3. Thermal-Dynamic Equations of the China Southwest Vortex

Considering that the China Southwest Vortex is a quasi-symmetric system, it can be described by the atmospheric thermal-dynamics equations of the Boussinesq approximation in the symmetric cylindrical coordinate system $(r,\theta ,z)$. Its dimensionless equations are as follows [40]:

$$\begin{array}{l}\frac{\partial v_r}{\partial t}+v_r\frac{\partial v_r}{\partial r}+v_z\frac{\partial v_r}{\partial z}-\frac{v_{\theta }^2}{r}-\frac{1}{R_o}{v}_{\theta }=-\frac{\partial p}{\partial r},\\ \frac{\partial v_{\theta }}{\partial t}+v_r\frac{\partial v_{\theta }}{\partial r}+v_z\frac{\partial v_{\theta }}{\partial z}+\frac{v_r{v}_{\theta }}{r}+\frac{1}{R_o}{v}_r=0,\\ \frac{\partial v_z}{\partial t}+v_r\frac{\partial v_z}{\partial r}+v_z\frac{\partial v_z}{\partial z}=\frac{1}{{\beta }^2}\left(-\frac{\partial p}{\partial z}+\vartheta \right),\\ \frac{\partial \vartheta }{\partial t}+v_r\frac{\partial \vartheta }{\partial r}+v_z\frac{\partial \vartheta }{\partial z}=-R_i{v}_z+Q,\\ \frac{1}{r}\frac{\partial (r{v}_r)}{\partial r}+\frac{\partial v_z}{\partial z}=0,\end{array}$$

(1)

where $v_r$, $v_{\theta }$, $v_z$, $\vartheta$, and $p$ represent radial velocity, tangential velocity, vertical velocity, potential temperature, and atmospheric pressure, respectively. $\beta =\frac{H}{R}$. $H$ is the characteristic height of the primary southwest vortex. $R$ is the characteristic scale of the southwest vortex. $R_o$ is the Rossby number, representing the ratio of horizontal inertia force to Coriolis force. $R_i$ is the Richardson number, representing the ratio of buoyancy to shear force.

For heating conditions, large-scale rising motion and latent heat are considered two kinds of condensation heating. Then, the heating form can be set as follows [41,42,43]:

$$Q=R_s{v}_z+\eta v_{zB}$$

(2)

where $R_s$ is the dimensionless latent heat condensation parameter and $R_s{v}_z$ represents latent heat (cumulus convective heating). $v_{zB}$ is the vertical velocity at the top of the boundary layer. $\eta$ is the dimensionless large-scale upward motion condensation parameter. $\eta v_{zB}$ represents large-scale upward motion condensation heating under Ekman suction.

According to the characteristics of the China Southwest Vortex, the motion of the vortex can be decomposed into $\beta$ mesoscale basic flow with radial shear and $\gamma$ mesoscale disturbance flow, which is decomposed as follows:

$$v_r=U(r)+\epsilon v_r{}^{\prime }, v_{\theta }=V(r)+\epsilon v_{\theta }{}^{\prime }, v_z=W(r)+\epsilon v_z{}^{\prime }, p=P(r)+\epsilon p^{\prime }, \vartheta =\mathsf{\Theta }(r)+\epsilon {\vartheta }^{\prime }$$

(3)

Considering that the $\gamma$ mesoscale is approximately $\frac{1}{10}$ of the $\beta$ mesoscale, $\epsilon \approx {10}^{-1}$ is a small parameter. By inserting Formula (3) into Equation (1), one obtains

$$\begin{array}{l}U\frac{\mathrm{d}U}{\mathrm{d}r}-\frac{V^2}{r}-\frac{1}{R_o}V+\frac{\partial P}{\partial r}+\epsilon \left\{\frac{\partial v_r^{’}}{\partial t}+\frac{\mathrm{d}U}{\mathrm{d}r}{v}_r^{’}+U\frac{\partial v_r^{’}}{\partial r}+W\frac{\partial v_r^{’}}{\partial z}-\frac{2V{v}_{\theta }^{’}}{r}-\frac{v_{\theta }^{’}}{R_o}+\frac{\partial p^{\prime }}{\partial r}\right\}+o\left(\epsilon \right)=0,\\ U\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{UV}{r}+\frac{1}{R_o}U+\epsilon \left\{\frac{\partial v_{\theta }^{’}}{\partial t}+\frac{\mathrm{d}V}{\mathrm{d}r}{v}_r^{’}+U\frac{\partial v_{\theta }^{’}}{\partial r}+W\frac{\partial v_{\theta }^{’}}{\partial z}+\frac{U{v}_{\theta }^{’}}{r}+\frac{V{v}_r^{’}}{r}+\frac{1}{R_o}{v}_r^{’}\right\}+o\left(\epsilon \right)=0,\\ U\frac{\mathrm{d}W}{\mathrm{d}r}-\frac{\Theta }{{\beta }^2}+\epsilon \left\{\frac{\partial v_z^{’}}{\partial t}+v_r^{’}\frac{\mathrm{d}W}{\mathrm{d}r}+U\frac{\mathrm{d}{v}_z^{’}}{\mathrm{d}r}+W\frac{\partial v_z^{’}}{\partial z}+\frac{1}{{\beta }^2}\left(\frac{\partial p^{\prime }}{\partial z}-{\vartheta }^{\prime }\right)\right\}+o\left(\epsilon \right)=0,\\ \left.U\frac{\mathrm{d}\Theta }{\mathrm{d}r}+R_{i}W-\eta v_{vB}+\epsilon \{\frac{\partial {\vartheta }^{\prime }}{\partial t}+U\frac{\mathrm{d}{\vartheta }^{\prime }}{\mathrm{d}r}+v_r^{’}\frac{\partial \Theta }{\partial r}+W\frac{\partial {\vartheta }^{\prime }}{\partial z}+\left(R_i-R_s\right)v_z^{’}\right\}+o\left(\epsilon \right)=0,\\ \frac{1}{r}\frac{\mathrm{d}\left(rU\right)}{\mathrm{d}r}+\epsilon \left\{\frac{\partial \left(r{v}_r^{’}\right)}{\partial r}+\frac{\partial v_z^{’}}{\partial z}\right\}=0.\end{array}$$

(4)

The above are the $\epsilon$ problems of motion and the thermal field. As Equation (4) is a strong nonlinear equation, there are some difficulties in mathematical processing. Therefore, the quasi-linear model, which omits the strong nonlinear term and retains the weak nonlinear term, is used to analyze the development and extinction of the China Southwest Vortex.

The ${\epsilon }^0$-order quasilinear problem is as follows:

$$\begin{array}{l}-\frac{V^2}{r}-\frac{V}{R_o}=-\frac{\mathrm{d}P}{\mathrm{d}r},\\ \frac{V}{r}-\frac{1}{R_o}=0,\\ -R_{i}W+\eta v_{zB}=0.\end{array}$$

(5)

The first line of Equation (5) indicates that the basic field satisfies the gradient wind balance. As the China Southwest Vortex is a cyclone, $V>0$. The China Southwest Vortex has a low-pressure center structure. The third line of Equation (5) indicates that the vertical motion of the basic field is driven by condensation heating of large-scale ascending motion.

The ${\epsilon }^1$-order quasilinear problem is as follows:

$$\begin{array}{l}\frac{\partial v_r^{’}}{\partial t}+\frac{\mathrm{d}U}{\mathrm{d}r}{v}_r^{’}+U\frac{\partial v_r^{’}}{\partial r}+W\frac{\partial v_r^{’}}{\partial z}-\frac{2V{v}_{\theta }^{’}}{r}-\frac{v_{\theta }^{’}}{R_o}+\frac{\partial p^{\prime }}{\partial r}=0,\\ \frac{\partial v_{\theta }^{’}}{\partial t}+\frac{\mathrm{d}V}{\mathrm{d}r}{v}_r^{’}+U\frac{\partial v_{\theta }^{’}}{\partial r}+W\frac{\partial v_{\theta }^{’}}{\partial z}+\frac{U{v}_{\theta }^{’}}{r}+\frac{V{v}_r^{’}}{r}+\frac{1}{R_o}{v}_r^{’}=0,\\ \frac{\partial v_z^{’}}{\partial t}+v_r^{’}\frac{\mathrm{d}W}{\mathrm{d}r}+U\frac{\mathrm{d}{v}_z^{’}}{\mathrm{d}r}+W\frac{\partial v_z^{’}}{\partial z}+\frac{1}{{\beta }^2}(\frac{\partial p^{\prime }}{\partial z}-{\vartheta }^{\prime })=0,\\ \frac{\partial {\vartheta }^{\prime }}{\partial t}+U\frac{\mathrm{d}{\vartheta }^{\prime }}{\mathrm{d}r}+v_r^{’}\frac{\partial \Theta }{\partial r}+W\frac{\partial {\vartheta }^{\prime }}{\partial z}+\left(R_i-R_s\right)v_z^{’}=0,\\ \frac{\partial \left(r{v}_r^{’}\right)}{\partial r}+\frac{\partial v_z^{’}}{\partial z}=0.\end{array}$$

(6)

Equation (6) presents the disturbance flow equations. Combined with the basic flow in Equation (5), the physical mechanism of the development and extinction of the China Southwest Vortex can be discussed by analyzing the centrifugal stability of Equation (6).

For the boundary conditions of perturbation in Equation (6), the following considerations can be made:

(1)

In the near field, i.e., the center of the southwest vortex, $r=0$ is a mathematical singularity in Equation (6), not a physical singularity. Then, the physical quantity in $r=0$ should be single-valued, smooth, and bounded:

$$\begin{array}{l}\underset{r\to 0}{\lim }\frac{\partial \overrightarrow{v}}{\partial \theta }=0,\underset{r\to 0}{\lim }\frac{\partial p}{\partial \theta }=0,\\ \underset{r\to 0}{\lim }\frac{\partial \overrightarrow{v}}{\partial \theta }=0,\underset{r\to 0}{\lim }\frac{\partial p}{\partial \theta }=0,\end{array}$$

(7)

where $\overrightarrow{v}$ is the velocity vector. Based on Equation (6) and the smoothness condition (7), the boundary conditions in the near field $r=0$ can be deduced as follows:

$${v^{\prime }}_r(0,z)=0,{v^{\prime }}_{\theta }(0,z)=0,\frac{\partial {v^{\prime }}_z}{\partial r}\left|{}_{(0,z)}\right.=0,\frac{\partial p^{\prime }}{\partial r}\left|{}_{(0,z)}\right.=0,{\vartheta }^{\prime }(0,z)=0.$$

(8)

(2)

In the far field, all perturbations disappear:

$${v^{\prime }}_r(R,z)=0,{v^{\prime }}_{\theta }(R,z)=0,{v^{\prime }}_z(R,z)=0,p^{\prime }(R,z)=0,{\vartheta }^{\prime }(R,z)=0.$$

(9)

Next, considering that the structure of the China Southwest Vortex is different at the primary stage and the mature stage, the physical mechanism of the development and extinction of the China Southwest Vortex is discussed in terms of these two stages.

4. Physical Mechanism of the Development and Extinction of the China Southwest Vortex in the Primary Stage

Because the $\beta$ mesoscale basic flow of the China Southwest Vortex represents the circulation field outside the vortex, the influence of the external circulation field on the vortex can be discussed by analyzing the effect of the basic flow on the disturbance flow. According to the research results of Liu Chun et al. [39], the primary southwest vortex is a pure vortex structure (a steady, two-dimensional, axisymmetric basic flow without vertical motion). Then, let the basic flow be pure vortex motion:

$$U\equiv 0,V=V(r),W\equiv 0,P=P(r),\mathsf{\Theta }\equiv 0$$

(10)

As the upward motion of the China Southwest Vortex is not obvious at the primary stage, the vertical velocity $v_{zB}=0$, which indicates the ${\epsilon }^0$ order quasilinear problem (5) of the top of the boundary layer, is still valid. By substituting Equation (10) into Equation (6), the perturbation flow field satisfies the following quasilinear equation:

$$\begin{array}{l}\frac{\partial {v^{\prime }}_r}{\partial t}-\left(\frac{2V}{r}+\frac{1}{R_o}\right)v_{\theta }{}^{\prime }=-\frac{\partial p^{\prime }}{\partial r},\\ \frac{\partial v_{\theta }{}^{\prime }}{\partial t}+\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right){v^{\prime }}_r=0,\\ \frac{\partial {v^{\prime }}_z}{\partial t}=\frac{1}{{\beta }^2}\left(-\frac{\partial p^{\prime }}{\partial z}+{\vartheta }^{\prime }\right),\\ \frac{\partial {\vartheta }^{\prime }}{\partial t}=(R_s-R_i){v^{\prime }}_z,\\ \frac{1}{r}\frac{\partial (r{v^{\prime }}_r)}{\partial r}+\frac{\partial {v^{\prime }}_z}{\partial z}=0.\end{array}$$

(11)

Next, based on Equation (11) and boundary conditions (8) and (9), the physical mechanism of the development and extinction of the primary China Southwest Vortex can be studied. These mechanisms include the following: (1) the mechanism of the development and extinction of the primary China Southwest Vortex (two-dimensional pure vortex structure) and (2) the mechanism of the transition from the primary-stage China Southwest Vortex (two-dimensional pure vortex structure) to the mature-stage China Southwest Vortex (three-dimensional twisting vortex structure).

As Equation (11) describes, the primary China Southwest Vortex does not consist of linear equations, but quasilinear equations, so the physical mechanisms of the development and extinction of the primary China Southwest Vortex cannot be discussed using a direct solution. Therefore, the Sturm–Liouville eigenvalue theory is used in this study to analyze the stability of Equation (11), after which the physical mechanism of the development and extinction of the China Southwest Vortex can be discussed.

To adopt the Sturm–Liouville eigenvalue theory, Equation (11) needs to be further processed. According to the fifth line of Equation (11), namely, the incompressible property under axial symmetry, the disturbance flow function can be introduced to represent the disturbance radial velocity and disturbance vertical velocity.

$${v^{\prime }}_r=-\frac{1}{r}\frac{\partial {\psi }^{\prime }}{\partial z},{v^{\prime }}_z=\frac{1}{r}\frac{\partial {\psi }^{\prime }}{\partial r}.$$

(12)

Substituting (12) into Equation (11), the equation of the perturbed flow function ${\psi }^{\prime }$ is obtained:

$$\begin{array}{l}\left\{{\beta }^2\frac{{\partial }^2}{\partial t^2}+(R_i-R_s)\right\}\left(\frac{1}{r}\frac{{\partial }^2{\psi }^{\prime }}{\partial r^2}-\frac{1}{r^2}\frac{\partial {\psi }^{\prime }}{\partial r}\right)\\ =\left\{\frac{{\partial }^2}{\partial t^2}+\left(\frac{1}{R_o}+\frac{2V}{r}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right\}\left(-\frac{1}{r}\frac{{\partial }^2{\psi }^{\prime }}{\partial z^2}\right).\end{array}$$

(13)

According to the research results of Ma Huiyang [29], the disturbance wave propagated by vortex motion is a dispersive wave, while the study of Chen Wei et al. [44] shows that the evolution process of the vortex is accompanied by gravity wave activity. Therefore, for the China Southwest Vortex, this disturbance wave is a gravity wave. Considering that the horizontal scale of the China Southwest Vortex is much larger than the vertical scale, the horizontal wave number of its gravity wave is much smaller than the vertical wave number. Therefore, the disturbance flow can be assumed to take the following form:

$${\psi }^{\prime }=\widehat{\psi }(r)\exp [i(kz-\omega t)]$$

(14)

where $\widehat{\psi }$ is the amplitude of the disturbance component, $k$ is the wave number along the direction $z$, and $\omega$ is the frequency.

First, the physical mechanisms of the development and extinction of two-dimensional pure vortex structures are discussed. Therefore, only the stability of the perturbation radial velocity ${v^{\prime }}_r$ needs to be discussed, while the stability of the $\widehat{\psi }$ needs to be discussed according to Equations (12) and (14). Then, by substituting Equation (14) into Equation (13), the equation of the disturbance amplitude $\widehat{\psi }$ is obtained:

$$\frac{\mathrm{d}}{\mathrm{d}r}\left\{f(r)\frac{\mathrm{d}\widehat{\psi }}{\mathrm{d}r}\right\}+\left\{\lambda g(r)-h(r)\right\}\widehat{\psi }=0$$

(15)

where

$$\begin{array}{l}\lambda =k^2/{\omega }^2,\\ f(r)=\frac{1}{r},\\ g(r)={\omega }^2\left(\frac{1}{R_o}+\frac{2V}{r}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)/\left[{\beta }^2{\omega }^2-(R_i-R_s)\right]r,\\ h(r)=k^2{\omega }^2/\left[{\beta }^2{\omega }^2-(R_i-R_s)\right]r.\end{array}$$

According to the boundary conditions (8) and (9) and Equations (13) and (14), the boundary conditions of the amplitude in Equation (15) of the perturbation flow function can be obtained:

(1)

Boundary conditions of the near field $r=0$:

$$\widehat{\psi }(0)=0.$$

(16)

(2)

Boundary conditions of the far field $r=R$:

$$\widehat{\psi }(R)=0.$$

(17)

Equation (15) and boundary conditions (16) and (17) form a typical Sturm–Liouville eigenvalue problem. According to the Sturm–Liouville eigenvalue theory, if $g(r)$ is always greater than zero in $[0,R]$, the eigenvalues $\lambda$ are all positive and $\omega =\pm \sqrt{\frac{k^2}{\lambda }}$ are real, which indicates that the vortex flow is stable. If $g(r)$ is not always greater than zero in $[0,R]$, the eigenvalues $\lambda$ are all negative, or negative in some regions, and $\omega =\pm \sqrt{\frac{k^2}{\lambda }}$ are pure imaginary numbers at least in some regions, which indicates that the vortex flow is unstable. Then, the value of $g(r)$ inside $[0,R]$ can be analyzed further.

Before discussing the stability, the distribution of $V(r)$, that is, the fundamental field of the China Southwest Vortex, will be discussed. As the pressure gradient is forced, the pressure gradient force in the vortex is conducted by the external circulation. Therefore, the pressure gradient force $-\frac{\mathrm{d}P}{\mathrm{d}r}$ in the basic field can be regarded as the effect of the external circulation on the motion of the vortex. From the vorticity formula $\zeta =\frac{1}{r}\frac{\partial (r{v}_{\theta })}{\partial r}-\frac{1}{r}\frac{\partial v_r}{\partial \theta }$, the vorticity formula $\zeta =\frac{1}{r}\frac{\partial (rV)}{\partial r}=\frac{V}{r}+\frac{\mathrm{d}V}{\mathrm{d}r}$ of the external circulation can be obtained. Therefore, ${\zeta }_{R_o}=\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}$ can be denoted as the geostrophic vorticity of the external circulation.

As the motion of the China Southwest Vortex evolves with time, only the time mode is considered in this paper, where the wave number along the direction $z$ is real. For the primary-stage China Southwest Vortex, $V_r\approx {10}^0 {\mathrm{ms}}^{-1}$, $V_z\approx {10}^0{ \mathrm{ms}}^{-1}$, $R\approx {10}^2 \mathrm{m}$, and $H\approx {10}^2 \mathrm{m}$; thus, $R_o\approx {10}^2$, $R_e\approx {10}^7$, $P_r\approx {10}^0$, $\beta \approx 1$, and $R_s\approx {10}^{-1}$. Considering that the vertical motion of the China Southwest Vortex is weak at the primary stage, and the stratification of the atmosphere is relatively stable, $R_i\approx {10}^0$. Thus, $(R_s-R_i)<0$. Assuming $\mathsf{\Gamma }={\beta }^2{\omega }^2+(R_s-R_i)$, the following two situations can be known based on Sturm–Liouville eigenvalue theory:

(1)

If $\left|\omega \right|<\frac{\sqrt{-(R_s-R_i)}}{\beta }$, then $\mathsf{\Gamma }<0$ and $h(r)<0$. In this case, Equation (15) only has a zero solution, which means that, when the stratification of the atmosphere is stable and the gravity wave disturbance frequency is too low, the primary China Southwest Vortex will disperse and die out.

(2)

If $\left|\omega \right|>\frac{\sqrt{-(R_s-R_i)}}{\beta }$, then $\mathsf{\Gamma }>0$. When the pressure gradient force $\frac{\mathrm{d}P}{\mathrm{d}r}$ increases, the geostrophic vorticity ${\zeta }_{R_o}>0$ of the external circulation and the external circulation is convergent based on gradient wind equilibrium. In this case, $g(r)$ is always greater than zero. According to the eigenvalue theory, the amplitude $\widehat{\psi }(r)$ of the disturbance and the radial velocity $v_r{}^{\prime }$ of the corresponding disturbance is stable. Thus, the primary China Southwest Vortex is maintained and developed correspondingly. When the pressure gradient force $\frac{\mathrm{d}P}{\mathrm{d}r}$ decreases, the geostrophic vorticity ${\zeta }_{R_o}<0$ of the external circulation and the external circulation is divergent on the basis of gradient wind equilibrium. In this case, for $g(r)$, negative numbers exist in the interval $[0,R]$. According to the eigenvalue theory, the amplitude $\widehat{\psi }(r)$ and the radial velocity $v_r{}^{\prime }$ of the corresponding disturbance are unstable. Therefore, the primary China Southwest Vortex will gradually die out.

The above analysis shows that the convergence and divergence of the external circulation is the main factor for the development and extinction of the China Southwest Vortex. Accordingly, based on the convergence of the external circulation, the gravity wave disturbance must meet certain conditions to maintain and develop the China Southwest Vortex. As the Richardson number is one order of magnitude larger than the latent heat condensation parameter under stable stratification of the atmosphere, the latent heating effect of the primary-stage China Southwest Vortex is small. Therefore, the main driving force for the maintenance and development of the primary-stage China Southwest Vortex is gravity wave disturbance. Only when the gravity wave disturbance frequency exceeds the critical frequency can the external circulation overcome the inertial centrifugal force to cause the primary China Southwest Vortex to develop; otherwise, it will tend to die out.

According to existing studies [45,46], when the vertical shear of the average airflow speed is too large, resulting in unstable shear, the fluctuation can obtain energy from the dynamic unstable flow and develop, and trigger gravity waves. For the southwest vortex, with the complex and steep terrain of the West Sichuan Plateau and Sichuan Basin, the uneven heating of the bottom layer under this terrain easily causes vertical shear of wind speed, stimulating gravity waves. The observational analysis also shows that the main source areas of the China Southwest Vortex are also active areas of gravity waves [44]. In spring and summer, the inhomogeneity of heating of the bottom of the China Southwest Vortex caused by the thermal imbalance of the underlying surface increases significantly.

Coupled with the vertical shear caused by the southwest jet, the kinetic energy of the gravity wave disturbance is large and the gravity wave disturbance frequency can easily exceed the critical frequency, maintaining the vortex or causing it to develop.

However, in autumn and winter, the inhomogeneity of heating of the bottom layer of the vortex caused by the thermal imbalance on the underlying surface is weakened, and the frequency of the gravity wave disturbance does not easily exceed the critical frequency; therefore, the vortex tends to die out in the initial stage. Moreover, the gravity wave characteristics of different types of southwest vortices are significantly different. The gravity wave energy and potential of outgoing southwest vortexes are large and vary dramatically, while those of the source southwest vortexes are small and vary slightly [44], also showing the important influence of gravity waves. In addition, it should be noted that the development here does not mean that the two-dimensional pure vortex structure can develop into a three-dimensional twisting vortex structure. In fact, the development from the primary-stage southwest vortex to the mature-stage southwest vortex requires more stringent conditions, which will be further analyzed.

The physical mechanism of the primary southwest vortex can be discussed by analyzing the stability of the radial velocity of the disturbance. Therefore, the question of whether the primary-stage southwest vortex (pure vortex) can transform into a mature-stage vortex (spiral vortex) is also an important problem, which mainly depends on the development of vertical motion. Therefore, this section will further discuss the mechanism of transition from the primary-stage southwest vortex to the mature-stage southwest vortex by analyzing the stability of the vertical disturbance. It should be noted that the development and extinction of the southwest vortex are determined according to the Rayleigh criterion, that is, the southwest vortex maintains and develops when the radial disturbance is stable, while the southwest vortex dies out when the radial disturbance is unstable. The difference here is that the stability of the vertical disturbance results in the stagnation of vertical motion, while the instability of the vertical disturbance results in the development of vertical motion.

To discuss the stability of vertical disturbances, the following equations are required. According to Equations (12) and (14), we have

$${v^{\prime }}_z=\frac{1}{r}\frac{\mathrm{d}\widehat{\psi }}{\mathrm{d}r}\exp [i(kz-\omega t)]$$

(18)

Obviously, the stability of $\varsigma (r)=\frac{1}{r}\frac{\mathrm{d}\widehat{\psi }}{\mathrm{d}r}$ must be studied to reveal the stability of the vertical disturbance velocity ${v^{\prime }}_z$. Therefore, both sides of Equation (15) are divided by $\lambda g(r)-h(r)$ and then differentiated by $r$.

$$\frac{\mathrm{d}}{\mathrm{d}r}\left\{\phi (r)\frac{\mathrm{d}\zeta }{\mathrm{d}r}\right\}+\left|\lambda \right|r\zeta =0$$

(19)

Here, $\phi (r)=\frac{\left|\lambda \right|}{\lambda g(r)-h(r)}$. The corresponding boundary conditions are (16) and (17).

According to the Sturm–Liouville eigenvalue theory, if $\phi (r)$ is constantly greater than zero during $[0,R]$, the eigenvalues ($\left|\lambda \right|$) are all positive. The system (19) has infinitely many discrete eigenvalue spectra, which suggests that $\omega =\pm \sqrt{\frac{k^2}{\left|\lambda \right|}}$; therefore, the vertical motion is stable. If $\phi (r)$ is inconsistently greater than zero during $[0,R]$, the eigenvalues ($-\left|\lambda \right|$) are all negative or negative in some regions, which suggests that $\omega =\pm \sqrt{\frac{k^2}{-\left|\lambda \right|}}$ are pure imaginary numbers, at least in some regions. According to the regular pattern Equation (18), the vertical motion is unstable. Therefore, the value of $\phi (r)$ in $[0,R]$ will be further analyzed.

As the southwest vortex is still considered as being in its initial phase, all parameters are the same as above. According to Sturm–Liouville eigenvalue theory, there are two cases with $\mathsf{\Gamma }={\beta }^2{\omega }^2-(R_i-R_s)$:

(1)

$\left|\omega \right|<\frac{\sqrt{-(R_s-R_i)}}{\beta }$; therefore, $\mathsf{\Gamma }<0$. The only solution of Equation (15) is zero with $h(r)<0$. This means that, when the atmospheric stratification is stable and the disturbance frequency is too low, the initial southwest vortex will disperse until its disappearance.

(2)

$\left|\omega \right|>\frac{\sqrt{-(R_s-R_i)}}{\beta }$; therefore, $\mathsf{\Gamma }>0$. When the external environment flow field is divergent, the geostrophic vorticity of the external environment flow field ${\zeta }_{R_o}<0$; thus, $\left(\frac{1}{R_o}+\frac{2V}{r}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)<0$ and $\widehat{\psi }(r)$ is unstable. Therefore, the corresponding disturbance radial velocity $v_r{}^{\prime }$ is unstable and the initial southwest vortex will disappear. When $\left|\omega \right|<\sqrt{\left(\frac{1}{R_o}+\frac{2V}{r}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)}$, $\lambda g(r)-h(r)>0$, and $\phi (r)>0$, referring to the above analysis, $\varsigma (r)$ is stable. Meanwhile, when the corresponding vertical velocity of the disturbance is stable, the southwest vortex maintains a pure vortex structure until the dispersion disappears, and cannot be transformed into a twisting vortex structure with a vertical structure. Secondly, $\left|\omega \right|>\sqrt{\left(\frac{1}{R_o}+\frac{2V}{r}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)}$, $\lambda g(r)-h(r)>0$, and $\phi (r)>0$; similarly, $\varsigma (r)$ and the corresponding $v_z{}^{\prime }$ are unstable; thus, the pure vortex structure will become a twisting vortex structure during development.

The above analysis shows that the main factor for the transformation from the initial southwest vortex (pure vortex) to the mature southwest vortex (twisting vortex) is the convergence and divergence of the external environment flow field. Under the convergent peripheral environmental flow field and when the gravity wave disturbance frequency is greater than the boundary frequency $\underset{r\le R}{\max }\left\{\frac{\sqrt{-(R_s-R_i)}}{\beta },\sqrt{\left(\frac{1}{R_o}+\frac{2V}{r}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)}\right\}$, the vertical disturbance becomes unstable. Then, the initial southwest vortex will change into a twisting vortex structure and develop into a mature southwest vortex with a three-dimensional structure. Under other conditions, the initial southwest vortex either dies out by dispersion or maintains a pure vortex structure for a period of time, and it cannot develop into a mature southwest vortex. The boundary conditions are not only related to the water vapor condition, but also to external environmental fields. The gravity wave disturbance frequency must be higher during the transformation from the primary stage to the mature southwest vortex. These conclusions are consistent with the fact that gravity wave activity with greater and stronger energy variation characteristics is generated by the moving and developing vortex [44]. Therefore, stronger vertical shear from the bottom wind speed is required to generate gravity waves satisfying the structural transformation. Only more drastic bottom-heating non-uniformity can satisfy this condition with constant terrain.

5. Physical Mechanism of the Development and Extinction of the Mature Southwest Vortex

As for the southwest vortex in its mature stage, its basic flow is that of twisting vortex motion:

$$U\equiv 0,V=V(r),W=W(r),P=P(r),\mathsf{\Theta }\equiv 0.$$

(20)

As in Section 3, according to the findings of Ma Huiyang [29], the disturbance wave propagated by the vortex motion is a dispersion wave, which is a gravity wave for the southwest vortex. Considering that the horizontal scale of the southwest vortex is far greater than the vertical scale, the horizontal wave numbers of its gravity wave are less than the vertical wave numbers.

Therefore, the disturbance flow is assumed as follows:

$$({v^{\prime }}_r,{v^{\prime }}_{\theta },{v^{\prime }}_z,p^{\prime },{\vartheta }^{\prime })=(i{F}_r(r),F_{\theta }(r),F_z(r),F_p(r),i{F}_{\vartheta }(r))e^{i(kz-\omega t)}$$

(21)

When the canonical schema (21) is inserted into Equation (6), the amplitude equation is obtained as follows:

$$\begin{array}{l}(\omega -kW)F_r-\left(\frac{2V}{r}+\frac{1}{R_o}\right)F_{\theta }=-\frac{\mathrm{d}{F}_p}{\mathrm{d}r},\\ -(\omega -kW)F_{\theta }+\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)F_r=0,\\ -(\omega -kW)F_z+\frac{\mathrm{d}W}{\mathrm{d}r}{F}_r=-\frac{1}{{\beta }^2}(k{F}_p+F_{\vartheta }),\\ -(\omega -kW)F_{\vartheta }=(R_i-R_s)F_z,\\ \frac{1}{r}\left(F_r+r\frac{\mathrm{d}{F}_r}{\mathrm{d}r}\right)+k{F}_z=0.\end{array}$$

(22)

Equation (22) is simplified to the following equation with $F_r$:

$$\frac{\mathrm{d}}{\mathrm{d}r}\left\{\left(\gamma +\frac{R_i-R_s}{{\beta }^2\gamma }\right)\left[\frac{1}{k}\left(\frac{F_r}{r}+\frac{\mathrm{d}{F}_r}{\mathrm{d}r}\right)\right]+\frac{\mathrm{d}W}{\mathrm{d}r}{F}_r\right\}=\frac{k}{{\beta }^2}\left(\gamma -\frac{\sigma {\zeta }_{R_o}}{\gamma }\right)F_r$$

(23)

where $\gamma =\omega -kW$, $\sigma =\left(\frac{2V}{r}+\frac{1}{R_o}\right)$. Further, Equation (23) is transformed as follows:

$$D\left[\frac{1}{k}\left(\gamma +\frac{R_i-R_s}{{\beta }^2\gamma }\right)D^{\ast }{F}_r\right]+(DW)D{F}_r+(D^{2}W)F_r=\frac{k}{{\beta }^2}\left(\gamma -\frac{\sigma {\zeta }_{R_o}}{\gamma }\right)F_r$$

(24)

where $D=\frac{\mathrm{d}}{\mathrm{d}r}$, $D^{\ast }=\frac{\mathrm{d}}{\mathrm{d}r}+\frac{1}{r}$.

According to the boundary conditions (8), (9), and (21), the boundary conditions of amplitude in Equation (14) for the perturbation function can be obtained with the following conditions:

(1)

Near-field $r=0$ boundary conditions:

$$F_r(0)=0.$$

(25)

(2)

Far-field $r=R$ boundary conditions:

$$F_r(R)=0.$$

(26)

Referring to the regular expression, the values of $\omega$ and $k$ must be known to analyze the stability of the system (24).

Therefore, it is assumed that $c=\frac{\omega }{k}$, $c$ is the phase velocity; then, the phase velocity is decomposed into real and imaginary parts:

$$c=c_r+i{c}_i$$

(27)

Thus, the system (24) is stable with $c_i\le 0$. Otherwise, the system (24) is unstable with $c_i>0$.

The above items are multiplied by $r{\overline{F}}_r$ and integrated into the interval [0, R]. Hence,

$$\begin{array}{l}\left[\frac{1}{k}\left(\gamma +\frac{R_i-R_s}{{\beta }^2\gamma }\right)r{\overline{F}}_r{D}^{\ast }{F}_r\right]\left|{}_0^R\right.-{\int }_0^R\frac{1}{k}\left(\gamma +\frac{R_i-R_s}{{\beta }^2\gamma }\right){\left|D^{\ast }{F}_r\right|}^{2}rdr\\ +\frac{1}{2}r{\left|F_r\right|}^2(DW)\left|{}_0^R\right.-\frac{1}{2}{\int }_0^R{\left|F_r\right|}^{2}d(rW)+{\int }_0^R\left(DW\right){\left|F_r\right|}^{2}rdr\\ -{\int }_0^R\frac{k}{{\beta }^2}\left(\gamma -\frac{\sigma {\zeta }_{R_o}}{\gamma }\right){\left|F_r\right|}^{2}rdr=0,\end{array}$$

(28)

where ${\overline{F}}_r$ is the complex conjugate of $F_r$.

Considering the boundary conditions (25) and (26), Equation (28) can be simplified as

$$\begin{array}{l}-{\int }_0^R\frac{1}{k}\left(\gamma +\frac{R_i-R_s}{{\beta }^2\gamma }\right){\left|D^{\ast }{F}_r\right|}^{2}rdr-\frac{1}{2}{\int }_0^R{\left|F_r\right|}^2(rDW+W)dr\\ +{\int }_0^R\left(DW\right){\left|F_r\right|}^{2}rdr-{\int }_0^R\frac{k}{{\beta }^2}\left(\gamma -\frac{\sigma {\zeta }_{R_o}}{\gamma }\right){\left|F_r\right|}^{2}rdr=0,\end{array}$$

(29)

The expression of phase velocity (27) is inserted into Equation (29); then,

$$\begin{array}{l}-{\int }_0^R\frac{1}{k^2}\left(k^2(c-W)+\frac{(R_i-R_s)(\overline{c}-W)}{{\beta }^2{\left|c-W\right|}^2}\right){\left|D^{\ast }{F}_r\right|}^{2}rdr-\frac{1}{2}{\int }_0^R{\left|F_r\right|}^2(rDW+W)dr\\ +{\int }_0^R\left(DW\right){\left|F_r\right|}^{2}rdr-{\int }_0^R\frac{1}{{\beta }^2}\left(k^2(c-W)-\frac{\sigma {\zeta }_{R_o}(\overline{c}-W)}{{\left|c-W\right|}^2}\right){\left|F_r\right|}^{2}rdr=0,\end{array}$$

(30)

The summation of the above items should be zero, that is, the real and imaginary parts should be zero. Considering that $W(r)$ is a real function, its imaginary part is

$$-c_i{\int }_0^R\left\{\frac{1}{k^2}\left(k^2-\frac{R_i-R_s}{{\beta }^2{\left|c-W\right|}^2}\right){\left|D^{\ast }{F}_r\right|}^2+\frac{1}{{\beta }^2}\left(k^2+\frac{\sigma {\zeta }_{R_o}}{{\left|c-W\right|}^2}\right){\left|F_r\right|}^2\right\}rdr=0$$

(31)

As this is the passive force, the pressure gradient force in the vortex is determined by the external environmental field. Therefore, the pressure gradient force $-\frac{\mathrm{d}P}{\mathrm{d}r}$ in the fundamental field can be regarded as the influence of the external environment field on the vortex motion. Considering the basic flow field $U\equiv 0,W=\frac{\eta v_{zB}}{R_i}$, the constraints of $V$ are balanced by the gradient wind. The vertical velocity $W(r)$ for the fundamental field is assumed to be a constant, that is, $W(r)=\frac{\eta v_{zB}}{R_i}\equiv const$. Considering that it is a cyclonic vortex, the southwest vortex rotates counterclockwise; consequently, $\mathsf{\Omega }>0$. According to the atmospheric stable stratification condition, the stability of the southwest vortex is discussed in terms of conditional instability and conditional stability for Equations (29) and (30):

(1)

Conditional instability: $R_i-R_s<0$

$R_i-R_s<0$, $k^2-\frac{R_i-R_s}{{\beta }^2{\left|c-W\right|}^2}>0$ is constantly established. If $k^2+\frac{\sigma \delta }{{\left|c-W\right|}^2}\ge 0$, the integrand function in Equation (29) is always positive and $c_i$ must always be zero. On the contrary, if $c_i$ is not zero, Equation (29) will function as follows, at least in the interval $[0,R]$:

$$k^2+\frac{\sigma \delta }{{\left|c-W\right|}^2}<0$$

(32)

The above analysis shows that, under the condition of unstable stratification, the stability condition of the mature southwest vortex is as follows:

$$k^2+\frac{\sigma \delta }{{\left|c-W\right|}^2}\ge 0$$

(33)

(2)

Conditional stability: $R_i-R_s>0$

The following formula is supposed.

$$g(r)=\frac{1}{k^2}\left(k^2-\frac{R_i-R_s}{{\beta }^2{\left|c-W\right|}^2}\right){\left|D^{\ast }{F}_r\right|}^2+\frac{1}{{\beta }^2}\left(k^2+\frac{\sigma \delta }{{\left|c-W\right|}^2}\right){\left|F_r\right|}^2$$

(34)

$\left|D^{\ast }{F}_r\right|\approx \frac{2}{r}\left|F_r\right|$; then,

$$h(r)=\frac{4}{k^2}\left(k^2-\frac{R_i-R_s}{{\beta }^2{\left|c-W\right|}^2}\right)+\frac{1}{{\beta }^2}\left(k^2+\frac{\sigma \delta }{{\left|c-W\right|}^2}\right)r^2$$

(35)

If $h(r)$ is always unequal to zero, $c_i$ must always be zero. Conversely, if $c_i$ is not zero, the necessary condition for Equation (29) is that $h(r)$ have more than one zero-point in $[0,R]$. The above analysis shows that, under the condition of stable stratification, the stability condition of the mature southwest vortex is that $h(r)$ must never equal zero.

In Section 4, the stability of the mature southwest vortex under two different stratification conditions is analyzed by the canonical perturbation model. The influence of unstable energy and latent heat on the development of the mature southwest vortex is further discussed with the stability conditions (32) and (34), and the physical mechanisms of the development and extinction of the mature southwest vortex are given.

Only the temporal patterns of the southwest vortex are considered in this paper, as its development mainly changes with time. The wave numbers along the $z$ direction are real numbers. For the mature southwest vortex, $V\sim {10}^0 {\mathrm{ms}}^{-1}$, $R\sim {10}^5 \mathrm{m}$, and $H\sim {10}^3 \mathrm{m}$. Therefore, one can find that $R_o\sim {10}^2$, $R_e\sim {10}^7$, $P_r\sim {10}^0$, $\beta =1$, $R_s\sim {10}^{-1}$, and $c=\frac{\omega }{k}\sim {10}^0 {\mathrm{ms}}^{-1}$. When stable atmospheric stratification occurs, $R_i\sim {10}^0$. When unstable atmospheric stratification occurs, $R_i\sim -{10}^0$. The vertical motion gradually weakens during the evolution process of the mature southwest vortex. In the early stage, the vortex is conditionally unstable, that is, $R_i-R_s<0$. In the late stage, the vortex is conditionally stable, that is, $R_i-R_s>0$. Therefore, the two cases are discussed separately:

(1)

Conditional instability: $R_i-R_s<0$.

The expressions of $\sigma$, $\delta$, and the third line of Equation (5) are inserted into the inequality (32). Hence,

$${\left|\omega -k\frac{\eta v_{zB}}{R_i}\right|}^2+\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\ge 0$$

(36)

➀ When the pressure gradient force $\frac{\mathrm{d}P}{\mathrm{d}r}$ of the external environment flow field increases, $\frac{\mathrm{d}V}{\mathrm{d}r}>0$, according to the gradient wind balance. The corresponding geostrophic vorticity exceeds 0, that is, ${\zeta }_{R_o}>0$, and the external environment flow field is convergent. Therefore, inequality (36) is constantly established. The driving force for the maintenance and development of the mature southwest vortex is the pressure gradient force. This means that, when the external environmental flow field is convergent, referring to the inequality (36), the southwest vortex never dies out.

➁ Otherwise, when the pressure gradient force $\frac{\mathrm{d}P}{\mathrm{d}r}$ decreases, that is, ${\zeta }_{R_o}<0$, the external environment flow field is divergent. If the gravity wave disturbance frequency meets the condition $\left|\omega \right|<\underset{r\le R}{\max }\left\{k\frac{\eta v_{zB}}{R_i}+\sqrt{\left|\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right|}\right\}$, the inequality (36) is not constantly established. Based on the stability condition (33), the mature southwest vortex is unstable. If $\left|\omega \right|\ge \underset{r\le R}{\max }\left\{k\frac{\eta v_{zB}}{R_i}+\sqrt{\left|\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right|}\right\}$, inequality (36) is constantly established. Therefore, the mature southwest vortex is stable. The following circumstances will occur under unstable atmospheric stratification. The southwest vortex under the divergent motion field can be maintained only when the gravity wave disturbance frequency exceeds the critical frequency $\underset{r\le R}{\max }\left\{k\frac{\eta v_{zB}}{R_i}+\sqrt{\left|\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right|}\right\}$. The southwest vortex can be maintained when the gravity waves develop extremely strongly. With the attenuation of the gravity waves, the southwest vortex will disappear. Under other conditions, the southwest vortex will also die out owing to the divergent motion field.

(2)

Conditional stability: $R_i-R_s>0$.

On the basis of the analysis in Section 3, circumstances when $h(r)$ is not constantly zero are discussed below. The expressions of $\sigma$ and $\delta$ are brought into expression (35) and $g(r)={\left|\omega -\frac{\eta v_{zB}}{R_i}\right|}^{2}h(r)$ is supposed. Hence,

$$g(r)=\frac{4}{k^2}\left({\beta }^2{\left|\omega -k\frac{\eta v_{zB}}{R_i}\right|}^2-(R_i-R_s)\right)+\left[{\left|\omega -k\frac{\eta v_{zB}}{R_i}\right|}^2+\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right]r^2$$

(37)

$h(r)$ is a quadratic function without a one-degree term, where ${\left|\omega -\frac{\eta v_{zB}}{R_i}\right|}^2+\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)$ is treated as a parameter.

According to the stability conditions under stratification, two cases may exist:

➀ When the pressure gradient force $\frac{\mathrm{d}P}{\mathrm{d}r}$ of the external environment flow field increases, $\frac{\mathrm{d}V}{\mathrm{d}r}>0$ according to the gradient wind balance. The large-scale flow field geostrophic vorticity is greater than zero, that is, ${\zeta }_{R_o}>0$. The external environment flow field is convergent and the quadratic coefficient of $g(r)$ exceeds 0. If $g(0)>0$, $h(r)$ is never equal to zero, and the centrifugal instability conditions are satisfied.

The inequality $h(0)>0$ is solved. The following is obtained.

$$\left|\omega \right|>k\frac{\eta v_{zB}}{R_i}+\frac{\sqrt{R_i-R_s}}{\beta }$$

(38)

Equation (38) means that the maintenance and development of the southwest vortex under a large-scale convergent motion field should meet two conditions. These are ① the stratification stability conditions and ② that the gravity wave disturbance frequency exceeds the critical frequency $k\frac{\eta v_{zB}}{R_i}+\frac{\sqrt{R_i-R_s}}{\beta }$. Under other conditions, the southwest vortex will die out.

➁ When the pressure gradient force $\frac{\mathrm{d}P}{\mathrm{d}r}$ decreases, i.e., ${\zeta }_{R_o}<0$, the large-scale motion field is divergent. If the gravity wave disturbance frequency $\left|\omega \right|\ge \underset{r<R}{\max }\left\{k\frac{\eta v_{zB}}{R_i}+\frac{\sqrt{R_i-R_s}}{\beta },k\frac{\eta v_{zB}}{R_i}+\sqrt{\left|\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right|}\right\}$, $h(r)$ is not constantly zero, and the centrifugal instability conditions are satisfied. This means that the southwest vortex under the divergent motion field can be maintained only when the gravity wave disturbance frequency exceeds the critical frequency $\underset{r<R}{\max }\left\{k\frac{\eta v_{zB}}{R_i}+\frac{\sqrt{R_i-R_s}}{\beta },k\frac{\eta v_{zB}}{R_i}+\sqrt{\left|\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right|}\right\}$. The southwest vortex can be maintained when gravity waves develop extremely strongly. With the attenuation of the gravity waves, the southwest vortex will disappear. Under other conditions, the southwest vortex will also disappear owing to the divergent motion field.

The main factors for the development and disappearance of the southwest vortex are the convergence and divergence of the external environmental flow field in the mature stage, but this depends on different periods. In the former period, the atmosphere creates conditional instability with vigorous convection development owing to the significant heating effect of latent heat in condensation. The southwest vortex can maintain and develop as long as the external environmental flow field converges. The gravity wave disturbance frequency must exceed the critical frequency $\underset{r\le R}{\max }\left\{k\frac{\eta v_{zB}}{R_i}+\sqrt{\left|\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right|}\right\}$ for the southwest vortex to exist, and it will ultimately disappear under the divergence conditions of the external environmental flow field. In the later stages, the atmosphere is conditionally stable with weakening convection due to latent heat release due to precipitation. The gravity wave disturbance frequency must exceed the critical frequency $\underset{r<R}{\max }\left\{k\frac{\eta v_{zB}}{R_i}+\frac{\sqrt{R_i-R_s}}{\beta },k\frac{\eta v_{zB}}{R_i}+\sqrt{\left|\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right|}\right\}$ for the mature southwest vortex to be maintained under convergence conditions of the external environmental flow field. The southwest vortex cannot be maintained and eventually dissipates with extremely strong gravity waves under the divergence conditions of the external environmental flow field. These conclusions are consistent with the observations in [44]. In conclusion, the convergent environmental flow field promotes the development of the mature southwest vortex. However, the divergent environmental flow field hinders the development and does not induce the rapid dissipation of the mature southwest vortex. Influenced by the Qinghai–Tibet Plateau and Hengduan Mountains, the southwest airflow and gravity wave activities prevailing at the edge of the western Sichuan Plateau and Sichuan Basin are often prone to producing disturbed convergence flow fields, which create positive vorticity for the development of the southwest vortex. The impact of the warm and humid southwest airflow and the convergence effect is more obvious when the western Pacific subtropical high extends westward to the Sichuan Basin. Further analysis shows that the abovementioned critical frequency is related to condensation heating and latent heat in condensation with large-scale ascending motion. Under the conditions of stable stratification and an unstable divergence environment flow field, the heating effect becomes a factor inhibiting the development of the southwest vortex, because heating increases the critical frequency of gravity waves.

6. Conclusions and Discussion

Using the method of vortex motion stability, the physical mechanisms of the development and extinction of the southwest vortex are further studied for the primary and mature stages of the southwest vortex, combined with two condensation heating methods—large-scale upward motion and cumulus convection—based on existing research [39]. The research conclusions are consistent with observations of the evolution of the southwest vortex and the activity of gravity waves in the western Sichuan Plateau and Sichuan Basin. The main conclusions are as follows:

(1)

For the primary stage of the southwest vortex, the convergence of the external environment flow field can overcome the inertial centrifugal force to develop the primary-stage southwest vortex when the gravity wave disturbance frequency exceeds the critical frequency $\frac{\sqrt{-(R_s-R_i)}}{\beta }$. When this is not the case, the primary southwest vortex tends to disappear. When the gravity wave disturbance frequency exceeds the critical frequency $\underset{0<r\le R}{\max }\left(\frac{\sqrt{-(R_s-R_i)}}{\beta },\sqrt{\left(\frac{1}{R_o}+\frac{2V}{r}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)}\right)$, the primary southwest vortex takes on a twisted vortex structure and develops into a mature southwest vortex with a three-dimensional structure under the convergence of the external environmental flow field. Under other conditions, the primary southwest vortex disappears either through dispersion or after maintaining a pure vortex structure for a period of time, as this structure cannot develop into a mature vortex.

(2)

For the mature southwest vortex, in the first period, the atmosphere creates conditional instability with vigorous convection development as a result of the significant heating effect of latent heat in condensation. The southwest vortex can maintain and develop as long as the external environmental flow field converges. The gravity wave disturbance frequency must exceed the critical frequency $\underset{r\le R}{\max }\left\{k\frac{\eta v_{zB}}{R_i}+\sqrt{\left|\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right|}\right\}$ for the southwest vortex to exist; it will ultimately disappear under divergence conditions of the external environmental flow field. In the later stages, the atmosphere creates conditional stability through weakening convection owing to latent heat release from precipitation. The gravity wave disturbance frequency must exceed the critical frequency $\underset{r<R}{\max }\left\{k\frac{\eta v_{zB}}{R_i}+\frac{\sqrt{R_i-R_s}}{\beta },k\frac{\eta v_{zB}}{R_i}+\sqrt{\left|\left(\frac{2V}{r}+\frac{1}{R_o}\right)\left(\frac{\mathrm{d}V}{\mathrm{d}r}+\frac{V}{r}+\frac{1}{R_o}\right)\right|}\right\}$ for the mature southwest vortex to be maintained under convergence conditions of the external environmental flow field. Under divergence conditions of the external environmental flow field, the southwest vortex cannot be maintained and eventually dissipates with extremely strong gravity waves.

Some aspects of this research should be improved, for instance, there are some differences between the approximant treatments and the reality of the spiral and asymmetrical structures of the southwest vortex. Using asymmetric and nonlinear methods, the physical mechanisms of the development and extinction of the southwest vortex can be studied systematically to strengthen the theoretical basis for the evolution of the southwest vortex and improve weather forecasting.