Dynamic and Numerical Study of Waves in the Tibetan Plateau Vortex

Vortex motion is a very common kind of atmospheric movement; vortices commonly arise and often persist for lengthy intervals in the atmosphere, especially in circumstances influenced by the Earth’s rotation and stable density stratification (McWilliams et al., 2003). Like all the common vortices, the flow field of the Tibetan Plateau Vortex (TPV) is mainly tangential and approximately axisymmetric (McWilliams, 1989); however, it differs in that its formation is mainly due to thermal forcing and the boundary layer conditions of the Tibetan Plateau (Chen et al., 1996).

Some strongly developed atmospheric vortices can always form spiral bands. Researchers generally agree that the spiral shape reflects some inner features of vortex dynamics that are closely related to atmospheric waves, and obtaining a clear picture of the developing process of spiral bands is important for understanding vortex evolution (Tao and Li, 2008). Most studies on spiral rain bands have concentrated on tropical cyclones, and have raised two theories to explain their cause: inertial-gravity wave theory (Tepper, 1958; Kurihara,1976; Huang and Chao, 1980; Chow et al., 2002) and vortex Rossby wave theory (Macdonald, 1968; Montgomery and Kallenbach, 1997; McWilliams et al., 2003).

Researchers in China have also studied the spiral rain bands of typhoons (Yu, 2002; Zhu et al., 2002; Zhang, 2006), as well as waves in typhoon systems (Deng et al., 2004; Kang, et al., 2007) in detail. However, the spiral cloud bands of the TPV have received little attention since they were first mentioned by (Ye and Gao, 1979). (Qian et al., 1984) analyzed National Oceanic and Atmospheric Administration (NOAA) satellite data and established that some strongly developed TPVs have spiral cloud bands and that there is less cloud, or no cloud, in the center of the vortex. (Qiao, 1987) later pointed out that the cloud structure of the TPV in summer is obviously spiral in form, and similar to tropical cyclones. However, all these studies lack theoretical explanation, and many questions remain unanswered. For instance, how are the spiral cloud bands of the TPV formed, and what is the link between the TPV itself and the spiral bands? What is its dynamic mechanism and which wave characteristics does it reflect? The fact that these fundamental and important issues have not yet been properly addressed is a reflection of the deficiency at present in dynamic studies of the TPV.

The TPV is a major weather system that causes precipitation in summer over the Tibetan Plateau. It usually generates in the western part of the plateau and disappears in the east. One noteworthy feature is that a small number of TPVs can develop and move eastward out of the Tibetan Plateau when conditions are favorable, leading to heavy rain, thunderstorms and other severe weather processes in the plateau’s catchment area (Ye and Gao, 1979; Luo, 1992; Qiao and Zhang, 1994). When the vortex moves out of the Tibetan Plateau, it forms a weather situation featuring a trough in the north and vortex in the south, which is the major reason behind subsequent heavy rainfall in the northwest of China. Eastward-moving TPVs mixing with cold air on the ground often cause regional heavy rain processes over Sichuan Province in summer, and when a vortex moves farther eastward out of Sichuan Province, it can also cause heavy rain over the middle and downstream regions of the Yangtze River, as well as over the Yellow River and Huaihe River, and even over North China. For example, during the flooding of the Yangtze River in 1998, the TPV was one of the systems that played a key role in causing the heavy precipitation.

In the above context, the present study aims to establish the governing equations of the TPV in column coordinates with the balance of gradient wind, and a study of the wave dynamics of the vortex is then carried out by solving the equations. We also report upon numerical simulations carried out to verify the theoretical results. Through this research we hope to provide a better understanding of the inner structure of the TPV, as well as the basic dynamics of the evolutionary process of the vortex.

Based on the results of former research (Liu and Li, 2007) and basic features of the TPV, the vortex was assumed as vortices in the boundary layer forced by diabatic heating and friction. The governing equations for the vortex were then established in column coordinates (r,λ,z) with the balance of gradient wind. It should also be noted that the vortex system being described is incompressible and axisymmetric (∂/∂λ=0). The origin of the coordinates is at the center of the vortex, and these simultaneous equations are:

In this model, r is radius, z is height, and t is time. The dependent variables are the radial, azimuthal and vertical velocity components u, v and w, respectively. θ is potential temperature, ρ is the air density, and T is air temperature. The subscript “0” represents the state of the stationary background atmosphere. Meanwhile, f is the Coriolis parameter, g is gravitational acceleration, Q is the diabatic heating rate, c_p is the specific heat at constant pressure, and N is Brunt-Väisälä frequency.

As they stand, these equations can already describe the TPV almost completely, but it remains hard to study the vortex dynamics, and thus it needs to be simplified. In order to study the vortex dynamics more conveniently, the simplification will focus on 1st, 2nd, 4th formula of the simultaneous Eq. (1), which relate to the kinematics. Note that we assume p_k = constant, here h means the upper boundary of the homogeneous incompressible atmosphere and p_k is the pressure at h. First, we integrate the equation of static equilibrium ∂p/ ∂z=-ρg from z=0 to z= h and obtain p=p_k+gρ(h-z). This means pressure decreases linearly with height, and so we obtain ∂p/∂r=gρ∂h/∂r. We also integrate the 4th formula of equations (1) from z= 0 to z= h, and after all transformation we obtain the following simultaneous equations:

These equations represent the simplified model for studying the dynamics of the TPV. This model is similar to some existing models used for studying tropical cyclones (Huang and Zhang, 2008) and tropical cyclone-like vortices (Nolan and Montgomery, 2002), which have been widely used and successful because the model can better describe the vortex motion.

The method used for linearization is that of small perturbation, in which we let , and . Taking note that the basic tangential flow field satisfies the gradient wind balance, i.e., , and the tangential flow field has radial shear that is , we finally obtain the small perturbation equations as follows:

We assume that the simultaneous Eq. (3) have the wave solution and let , . We then substitute these into Eq. (3), after which we obtain the following ordinary differential equation:

Eliminating Eqs. (4) can obtain:

In this equation,

and . has been omitted from the equation, which means the third-order shear is not considered because of its small quantities.

Obviously, Eq. (5) is too complicated to be solved directly, and so it needs to be simplified. First, we analyze the dimension of Eq. (5) and reserve the maximum and the second largest items in it. Next, we multiply the equation by Br² and divide it by . Notice that in Eq. (5) there is . Finally, Eq. (5) is simplified to the following form:

Based on the basic characteristics of the TPV, we can conclude the boundary condition of Eq. (6); that is, when r =R (at the edge of the vortex), velocity equals zero; and when r =0, it is bounded. So, the solution of Eq. (6) can be written in the form of sin (x) when β_*, A, B, and r are constants, which means the following condition is satisfied:

After some simple mathematical derivation, we can obtain the wave circular frequency as:

The wave dispersion relationship [Eq. (8)] is, to a certain extent, quite complicated. However, notice that the relationship described by Eq. (8) contains the items and β_*, which clearly means the TPV may contain mixed waves and that the waves are mixed with inertial gravity (IG) and vortex Rossby (VR) waves. We thus need to explain the item β_* and the VR wave. and is the key part of the wave mechanism of VR waves. Like the β effect of planetary Rossby waves, here β_* can explain the basic vertical vorticity distribution in the radial (r) direction to be uneven; if air particles in the r direction have generated little disturbance, in order to maintain conservation of total vorticity the perturbation vorticity must change, making the air particles oscillate in the r direction.

We are naming this mixed wave the inertial gravity-vortex Rossby (IG-VR) wave, as it contains the characteristics of both IG and VR waves. It can further be categorized as a “type II” mixed wave (Lu et al., 2007), which refers to special waves that possess the characteristics of some basic waves simultaneously, and cannot be separated. It only generates in some specific background conditions. In the physical quantity field that contains type II mixed waves, we can find the coexistence of vortex and convergence-divergence motion.

The mechanism of the mixed wave can be explained by the law of conservation of potential vorticity. From Eq. (3), we can derive the following equation:

Here, ζ’_z is the vertical vorticity perturbation ζ’_x=∂v’/∂r+v’/r and D’ is the horizontal divergence perturbation D’= ∂u’/∂r+u’/r. With Eqs. (3) and (9), we can finally derive the following equation:

This is the law of conservation of potential vorticity. Under the constraints of this law, a changing environmental potential vorticity will lead to both vortex motion and convergence and divergence movement changes. The changes in vorticity cause the formation and spreading of the VR wave, and divergence will cause changes to the motion of the IG wave’s excitation and evolution.

Having studied the dynamics of the TPV, we found that it might contain a mixed wave. The key condition is that, in the vortex, there is uneven and shear basic flow vorticity and divergence. Following this, the next step was to carry out an idealized numerical experiment to validate the results of the dynamics, at least to a certain extent, and study how the vortex flow field looks and how it forms. This idealized numerical experiment was carried out with the Weather Research and Forecasting (WRF) model, which was developed by National Centers for Environmental Prediction-National Center for Atmospheric Research NCEP-NCAR and is a convenient tool for studying various problems in the atmospheric sciences. It can not only carry out simulations with real data, but also possesses a number of inbuilt ways to perform idealized tests, 3D quarter-circle shear supercell simulations, 3D baroclinic wave simulations, 3D large eddy simulations, and so on. For example, (Rotunno et al., 2009) used the WRF model to simulate a case of an idealized tropical cyclone, and studied the role of turbulence effects in the cyclone. In the present study, we took advantage of advances in computing power to use the method of 3D large eddy simulation to model the generation process of an idealized TPV.

The default WRF large-eddy test produces a large-eddy simulation (LES) of a free convective boundary layer (CBL), in which the environmental wind (or the initial wind profile) is set to zero. The turbulence of the free CBL is driven by the surface heat flux, which is specified in the WRF’s namelist as “tke_heat_flux” (“tke” is short for turbulent kinetic energy) and is equal to 0.24 (in MKS units). A random perturbation is initially imposed on the mean temperature field at the lowest four grid levels to start the turbulent motion. A double periodic boundary condition is used in both x and y directions. It takes at least 30 minutes of simulation time to spin up the turbulent flow field, and only after the spin-up can the turbulence inside the CBL be considered to be well established.

After understanding the default case of WRF large-eddy simulation, and having considered the importance of thermal effects on the formation of the TPV, it was clear that we only needed to make very slight changes in order to approach our problem, i.e., to simulate the formation process of the TPV and the flow field of the vortex.

The traditional theory believes the TPV is a vortex in the boundary layer of the Tibetan Plateau, and is a kind of shallow vortex. So, the top of the model domains was set to 2 km and the vertical levels to 31 layers. The scale of the TPV is about 500 km and some strongly developed ones can reach 600 to 800 km. The size of the simulated region must therefore meet the scale of the vortex. So, in this case, we used triple-nested square domains, with the coarsest having sides of 900 km, followed by 300 km, and then 100 km. The levels of resolution were thus 9 km, 3 km and 1 km, respectively. In order to consider the gradient of heat flux—that is to say, the need to reflect the heterogeneity of non-adiabatic heating—each domain had a different “tke” heat flux (0.15, 0.25 and 0.35 K m s^-1, respectively). All the grid setups are shown in Fig. 1. The initial basic flow pattern was zero, and did not consider microphysical processes. This means that, by only considering the distribution of surface heat flux as the central heating form, the plateau vortex flow could be generated. The simulation lasted for six hours.

Figure 1.  Setup of the experimental domains.

After three hours of simulation, the resulting flow field is shown in Fig. 2. We can see that, at this time, the vortex was not fully spun up and thus not completely established. Both the cyclonic flow in the lower levels (Fig. 2a) and the anticyclonic flow in the upper levels (Fig. 2b) were not clear. The convergence and divergence was also very weak in both the lower and upper levels.

Figure 2.  Stream line (simulated for 3 hours): (a) layer 9; (b) layer 19.

However, after six hours of simulation, the vortex flow field was completely established (Fig. 3). In the lower levels, it showed cyclonic (Fig. 3a) convergence (Fig. 4a) and in the upper levels it showed anticyclonic (Fig. 3b) divergence (Fig. 4b). This feature was most notable in the central area. This result reflects the belief that the TPV always develops in the early afternoon and matures before evening (Jiang and Fan, 2002; Pei et al., 2012). That is to say, when the situation is favorable, a TPV can form in about six hours.

Furthermore, we were able to confirm that the flow field is uneven and may have shear flow along the radial or tangential directions. Both the divergence (Fig. 4) and vorticity (Fig. 5) of the flow showed a positive or negative alternate distribution in the upper and lower levels. The entire feature means that, at least in this idealized test, the vortex possesses the conditions to generate mixed IG-VR waves.

Figure 3.  Stream line (simulate for 6 hours): (a) layer 9; (b) layer 19. The black box represents the central area.

To further verify the results of the dynamic study and idealized simulation, a numerical case study was examined and is reported in this section. The case was a TPV that was active on 14 August 2006, and it was chosen because it was a TPV that formed and developed strongly with an eye region and spiral cloud band.

The numerical simulation was also carried out using the WRF model. NCEP 1°×1° data were used as the initial and background fields. The domains were triple-nested (Fig. 6), with the innermost domain covering most of the TPV with a grid size of 5 km and 28 levels in the vertical direction. We thus expected the numerical solutions to capture some of the inner structure of the TPV, given sufficient resolution. We chose the WRF Single-moment 6-class scheme (Hong and Lim, 2006) for the microphysics, the Rapid Radiative Transfer Model scheme (Mlawer et al., 1997) for longwave radiation, the Dudhia scheme (Dudhia, 1989) for shortwave radiation, and the Mellor-Yamada-Janjic scheme (Janjic, 1994) for the planetary boundary layer. The simulation time was from 0800 LST 14 August 2006 to 0800 LST 15 August 2006 —a total of 24 hours, including the developing and maturation periods of the vortex.

Figure 4.  Divergence in the central area (simulated for 6 hours): (a) layer 9; (b) layer 19 (units: 10^-5 s^-1).

Figure 5.  Vorticity in the central area (simulated for 6 hours): (a) layer 9; (b) layer 19 (units: 10^-5 s^-1).

Overall, the simulation results seemed to accurately represent the TPV. Figure 7 depicts the 300-hPa flow field at 1900 LST, in which we can clearly see the vortex showing anticyclonic flow. Figure 8 shows the time-averaged vorticity profile along 31°N from 1700 to 2000 LST (cross section through the center of the TPV). This represents the radial variation of vorticity, and we can see that from the vortex eye region (31°N, 86°E) outward, the average vorticity increased slightly, with the maximum value appearing in the region (86.3°-86.4°E). Thereafter, in the outer region, the vorticity gradually decreased. This radial gradient of average vorticity is the root cause of the VR waves, i.e., the basic flow vorticity needs to be uneven and shear.

In order to further study the waves in the vortex, we cut a concentric circle with the vortex of 50 km radius to analyze the wave conditions in the tangential direction. The best evidence for the mixed wave was that both the vorticity and divergence showed fluctuation in the circle. Figure 9 presents the vorticity and divergence profile in the azimuthal direction at different times, and we can clearly see (Fig. 9a) there was vorticity fluctuation; and, furthermore, in Fig. 9b it can be seen that the divergence and convergence were changing alternately. In addition, we can infer from Fig. 9 that the moving tangential direction of waves was clockwise, and the speed was about 8 km h^-5.

Figure 6.  Setup of the model domains for the TPV on 14 August 2006.

Figure 7.  The 300-hPa flow field at 1900 LST 14 August 2006.

Figure 8.  Time-averaged vorticity profile for the period 1700-2000 LST 14 August 2006 along 31°N (units: 10^-4 s^-5).

The status in the radial direction was also analyzed and, in summary, the features of vorticity fluctuations in this direction first spread inward, and then spread out after the mature stage of the vortex. Figure 10 shows the spreading process in the radial direction, demonstrating that the time between 1900 and 2000 LST was the period of change in terms of when the vorticity fluctuations spread inward and outward. The feature of divergence was not so clear in this period.

Following a dynamic study, idealized simulation and case study of a TPV, we can draw the following preliminary conclusions:

• TPVs possess both vortex motion and convergence and divergence movement. Changes in the vorticity cause the formation and spreading of VR waves, and divergence will cause changes to the motion of IG wave excitation and evolution. Thus, the vortex may contain mixed IG-VR waves.

• The idealized experiment not only confirmed the divergence and vorticity of the vortex is uneven and may have shear flow along the radial or tangential directions, but also showed the flow field of the vortex is cyclonic and convergent in the lower levels and anticyclonic and divergent in the upper levels.

• Having simulated a vortex case that took place on 15 August 2006, we further confirmed the existence of the mixed wave and, moreover, we found that the mixed wave spreads both in the tangential and radial directions. The features of the wave spreading in the radial direction involve an initial inward spread, and then a spreading out after the mature stage of the vortex.

Finally, it is important to note the significance of studying waves in the TPV, as has been done here, because it enables us to more precisely predict its influence on regional weather. Furthermore, the results of the present study serve to remind us that some strongly developed TPVs that do not move east also need to be studied in the future, because the wave-spreading process can also affect weather in downstream areas. And this study attempts to investigate this type of TPV other than eastward-moving TPVs that are most studied. However, the present conclusions have their limitations and need to be further tested.

Figure 9.  (a) 500-hPa vorticity profile of the azimuth (x-axis is azimuth; y-axis is vorticity). Angles: 0°, 90°, 180° and 270° are for east, north, west and south, respectively. The lines with small circles (black), solid circles (green), small squares (yellow) and solid squares (red) represent the vorticity profiles at 1700, 1800, 1900 and 2000 LST 14 August 2006, respectively (units: 10^-4 s^-1). (b) 500-hPa divergence profile of the azimuth.

Figure 10.  Perturbation vorticity zonal-time profile.

This research was supported by the National Key Basic Research and Development Project of China (Grant No. 2012CB417202), the National Nature Science Fund of China (Grant No. 41175045), the Special Fund for Meteorological Research in the Public Interest (Grant Nos. GYHY201006014, GYHY201206042 and GYHY201106003), and the Sichuan Meteorological Bureau Fund for Young Scholars (Grant No. 2011-YOUTH-02).