On Mesoscale Vortex Rossby Wave in Zonal Shear Flow Xinyong Shen

1. On Mesoscale Vortex Rossby Wave in Zonal Shear Flow Xinyong Shen Department of Atmospheric Sciences, Nanjing University of Information Science and Technology, Nanjing, China Yunqi Ni Chinese Academy of Meteorological Sciences, Beijing, China Tongli Shen Department of Atmospheric Sciences, Nanjing University of Information Science and Technology, Nanjing, China Deying Wang Chinese Academy of Meteorological Sciences, Beijing, China

2. Introduction • Ougra et al., 1982 :Field observations show that there often range meso-scale surface convergence systems and temperature waves with wavelengths approximately 400 km long in the direction of a cold front and thermal winds and the vorticity fields show a similar pattern, the train of disturbances ranging along a southwest jet axis in front of a 500-hPa cold vortex • Yang et al., 1994 :observational studies based on surface and upper-air conventional records as well as satellite and radar measurements reveal that a number of aligned meso-β convective cloud nuclei (or rainstorm masses) exist in the meso-α system of a Meiyu front, and move eastward so fast as to make it difficult for surface stations to forecast

3. Charney (1947) and Eady (1949) :were the first to discover the instability of quasi geostrophic synoptic-scale disturbance happening in baroclinic flows, i.e., baroclinic instability Kuo (1978) and Kuo and Seitter (1985) :addressed the structure of TTD instability in a neutral and a partly unsteady stratified atmosphere and layered it vertically through numerical differencing calculation, thereby obtaining more than one spectrum of developing extra-long waves, in addition to the long wave interception of Charney modes. Tokioka (1971) :made layering of 4-km-deep shear flow of the Meiyu front, leading to a most unsteady wavelength.

4. Zhang (1988) :By extending the Eady model into a non-geostrophic domain so as to get numerically the phase velocity, growth rate and characteristic flow pattern of disturbance under steady stratification, she obtained ageostrophic baroclinic meso modes, tens to hundreds of kilometers in scale in addition to the Eady equivalents at synoptic and sub-synoptic scales and she also showed the growth rate of her modes to be about 4 times that of the Eady modes, which, as she indicated, maybe serves as a dynamic mechanism for initiating and organizing deep convective cloud cluster. Additionally, she got the meso TTD characteristic wave structure by means of a matrix method and the targeting, discovering an asymmetric “cat’s eye” pattern of the mode on a vertical section.

5. Because of more difficulties solving theoretically an ordinary differential equation with variable coefficients, the solution to the problem of meso TTD instability depends on a numerical differencing scheme in most cases. The present work is an attempt to mathematically prove the characteristics of propagation when TTD experiences unstable development, leading to an expression of phase velocity of vortex Rossby wave, followed by exploring the physical origin of the wave’s genesis and its properties.

6. 2. Equations of Traversal Type Disturbance With no account of external forcings, i.e., surface friction, topography, external heat transport and latent heat effect, the equations of the atmospheric motion take the form

7. All physical variables are discomposed, leading to and We let the stratification and baroclinic stabilities be and meso-β weather systems are set to be arranged E – W ,all disturbance variables are assumed to be independent of y, Therefore, we find the following equation

8. and Streamfunctions are introduced because disturbance velocity is non-divergent on the plane, we get a partial differential equation containing only disturbance streamfunction as the variable, viz., In finding the characteristic wave solution to the streamfunction by means of

9. we arrive at an ordinary differential equation =0 3. Phase velocity and energy frequency-dispersion relation of vortical Rossby wave – properties of TTD instability we set basic flow to be z = 0 at surface and the flow in linear distribution at arbitrary height z follows with <<1 where is a constant, indicating that the second-order vertical background wind shear is very small. for convenience, when we discuss non-vortical fluid, i.e., (21) has the form

10. where we set which is made to satisfy we obtain the following frequency dispersion relation a) For but We have which indicates that with no second -order shear in the flow available, disturbance is an internal gravity wave propagating both east- and westward with respect to basic flow. b)For but We have phase velocity

11. (33) which actually denotes the expression for phase velocity of vortex Rossby wave. From (33) we see that the wave propagates unidirectionally, i.e., east- (westward) with respect to basic flow As a result, if, compared to the middle troposphere, basic flow has its greater horizontal velocity in the lower and higher troposphere (related to the existence of low- and high-level jet streams), then >0 is met for vortical Rossby wave propagating eastward, and even at greater velocity, i.e., c> and if, compared to the lower and higher troposphere, basic flow has its greater velocity in the middle troposphere, then is satisfied for the wave propagating westward with respect to the basic flow.

12. In addition, because the phase velocity of the wave bears a relation to zonal wavenumber k, its energy is frequency dispersive and its zonal group velocity takes the form c) For and Therefore, (32)-denoted TTD ismixed vortex Rossby – gravity wave. In this sense, the instability of TTD is that of the mixed vortex Rossby – gravity wave. In fact, for we get an ordinary differential equation the Weber equation of first kind

13. the frequency dispersion relation for meso TTD in the vertically second shear flow (38) is a cubic equation and has three characteristic roots, two of which denote internal inertial gravity wave, one propagating east- and the other westward with respect to basic flow, and the third of which denotes vortical Rossby wave. 4. Analysis of vorticity and divergence equations for TTD

14. Fig.1. The structures of meso-βweather systems with n = 1 disturbance propagating eastward under more weakly unsteady stratification (N2≤0).

15. we observe no disturbance of vertical velocity and potential temperature on surface (z=0) synoptic maps; that the center of the pressure field coincides with that of the vorticity field, with the high (low) pressure core corresponding to that of anticyclonic (cyclonic) vorticity; the center of surface convergence (divergence) is ahead of that of surface low (high) pressure system, i.e., ahead of (behind) the surface trough. It is true of the situation at z=H (tropopause) except for the opposite distribution of pressure fields (only in the sense of n=1 mode). That the core of divergence (convergence) does not coincide with that of the high (low) pressure system displays stronger ageostrophy of meso motions; at middle troposphere (), a warm (cold) core is over the center of a surface low (high) pressure system, with the maximum rising (sinking) occurring over the surface convergence (divergence) center, i.e., above ahead of (behind) the surface trough.

16. 5. Analysis of energy equation for TTD development the total energy for local disturbance development comes from mean effective potential energy, mean kinetic energy and the advection of the basic flow upon the total energy.

17. 6. Equation for TTD conservation and physical mechanism for generating vortex Rossby wave Yu (2002) made an excellent overview on the formation of vortical Rossby wave in typhoon spiral rainbands from the contributions of MacDonald (1968), Guinn et al. (1993), Smith and Montgomery (1995) and Montgomery et al. (1997a, b) and Wang (2002). We make attempt to explain the physical processes of the formation of vortical Rossby wave in meso TTD in order to understand possible mechanisms of the genesis and development of meso-β rainstorm masses in the Meiyu front.

19. Fig.3. Possible mechanism for the genesis and propagation of the rainstorm mass in the Meiyu front.

20. 7.Numerical experiment by means of the MM5 we undertake numerical simulation of one torrential rain event occurring in the mid-lower basins of the Changjiang from 0000 UTC, July 29 to 0000 of 30, 1998 by means of the meso-scale non-hydrostatic model MM5 developed by US PSU/NCAR Fig.4 Rainfall distribution for 0500 – 0600 UTC, July 29, 1998. Units: mm.

21. Fig.5. Divergence (10-5) pattern at 700 hPa at 0600 UTC, July 29, 1998.

22. Fig.6. as in Fig.5 but for vorticity (10-5).

23. Fig.10. A U-wind profile in 29°N, 116°E at 0600 UTC, July 29, 1998.

24. Fig.11. the same as in Fig.10 but for at 1000 UTC. Otherwise as in Fig.10.

25. Averaging U windspeeds over the study area with a vigorous rain mass results in z-dependent E–W curves of U at 0600 and 1000 UTC (Figs.10-11). Investigation shows that in the free atmosphere (≥850 hPa), by and large, U decreases linearly versus height on account of low-level jets around 850-800 hPa. It is seen from the theoretical conclusions presented in Part I that with the linearly reduced windspeed profile available the TWT disturbances (mainly inertia-gravity waves rather than VRoW) experience instability, thereby giving rise to the development and enhancement of meso-beta rainstorm masses anterior to 1400 UTC, July 29, and are kept in a quasi-stationary state.

26. Fig.14. The U-wind profile (29°N, 119°E) at 1000 UTC, July 22, 1998.

27. Fig.15. as in Fig.14 but at 1500 UTC.

28. Height-dependent U-wind curves in the E – W direction are presented for 1000 and 1500 UTC, July 22, 1998, as shown in Figs.14 and 15, respectively. Their analysis indicates that in the free atmosphere (850-200 hPa) U changes in a way that satisfies the condition of the second derivative of it with respect to height, with greater velocity around 850 and 200 hPa (in relation to the low- and high-level jets, respectively) and low easterly and westerly speeds in the neighborhood of 400 hPa. As shown by the theoretical findings given in Part I, with the effect of Uzz ≠ 0, TWT disturbance (dominantly VRoW) will display instability, thereby driving the meso- rain mass to travel eastward, as portrayed in Figs.16-19 where the isopluvials ≥5 mm are plotted, indicating that the meso-β-scale rain mass shifts at the rates of more than 50 km an hour between 0900 to 1200 UTC, July 22.

29. 8. Concluding remarks To investigate the physical mechanism of the genesis/development, and the structure, of a meso-β rainstorm mass (convective nucleus) in the Meiyu front and explain the physics of TTD vortical Rossby wave (VRoW), the approximate Boussinesq equations for 2D meso TTD in basic flow are used to make theoretical study of the physical representation of VRoW propagating in a zonal shear flow, thus revealing the fields of disturbance physical variables and energy sources. The main conclusions are as follows: 1) Theoretical analysis shows that vertical wind shear in basic flow causes the unsteady development of meso TTD and the consideration of second-order shear in the flow would lead to the expression of phase velocity of VRoW indicating that the wave is unidirectional in propagation with respect to the basic flow.

30. 2) The physical origin of the vortex Rossby wave lies in the second shear of the basic flow plays a role similar to the effect of β component of midlatitude planetary Rossby wave. The wave’s phase velocity is functionally related to zonal wavenumber k, with the energy being frequency dispersive and group velocity available in the x direction. 3) For the second-order shear of wind , the TTD instability is the instability of mixed vortex Rossby –gravity wave. With wind subject to linear shear rather than a second shear , the instability of TTD is that of internal inertial gravity wave. 4) With constant flow and (weaker unsteady stratification) available, we see on the surface synoptic chart that the center of the pressure coincides with that of the vorticity field, the core of

31. the high (low) field corresponds to that of the anticyclonic (cyclonic) vorticity field; the center of surface convergence (divergence) is ahead of that of the surface low (high), i.e., in front (at the back) of the trough; in the mid troposphere, a warm (cold) core is over the surface low (high) pressure core and the maximum rising (sinking) motion occurs over the center of surface convergence (divergence), i.e., above ahead of (behind) the surface trough. 5) The total energy for disturbance development on a local basis originates from mean effective potential energy, mean kinetic energy and advection of basic wind upon disturbance total energy. Under no account taken of basic flow the total energy is conserved, and the disturbance kinetic energy comes only from disturbance effective potential energy, with no energy provided by the background field for disturbance development.

32. 6) The environmental field has its mean vorticity changing with z, viz., , leading to the fact that during the vertical down- and upward movement, to keep constant, the air parcel is bound to exhibit vibrations in vertical, resulting in the formation of a VRoW that propagates in a zonal direction. When displaying unsteady development (TTD instability), the wave takes energy from the shear flow, resulting in amplitude that gets progressively bigger to such an extent as to produce a weather system of meso-β rainstorm mass in the Meiyu front.

33. Thank you, all of scientists!