Contribution of vortex Rossby wave to spiral rainband formation in tropical cyclones*

RUAN Kun (阮鯤)

Key Laboratory of Virtual Geographic Environment of Ministry of Education, College of Geographic Science, Nanjing Normal University, Nanjing 210046, China

College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing 211101, China, E-mail: forward_rk@163.com

ZHA Yong (查勇)

Key Laboratory of Virtual Geographic Environment of Ministry of Education, College of Geographic Science, Nanjing Normal University, Nanjing 210046, China

HUANG Hong (黃泓)

School of Atmospheric Science, Key Laboratory of Mesoscale Severe Weather, Nanjing University, Nanjing 210093, China

College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing 211101, China

HU You-bin (胡友彬)

College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing 211101, China

Contribution of vortex Rossby wave to spiral rainband formation in tropical cyclones*

RUAN Kun (阮鯤)

Key Laboratory of Virtual Geographic Environment of Ministry of Education, College of Geographic Science, Nanjing Normal University, Nanjing 210046, China

College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing 211101, China, E-mail: forward_rk@163.com

ZHA Yong (查勇)

Key Laboratory of Virtual Geographic Environment of Ministry of Education, College of Geographic Science, Nanjing Normal University, Nanjing 210046, China

HUANG Hong (黃泓)

School of Atmospheric Science, Key Laboratory of Mesoscale Severe Weather, Nanjing University, Nanjing 210093, China

College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing 211101, China

HU You-bin (胡友彬)

College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing 211101, China

(Received May 13, 2013, Revised August 7, 2013)

The contribution of the vortex Rossby wave (VRW) to the spiral rainband in the tropical cyclones (TCs) is studied in the framework of a barotropic non-divergent TC-like vortex model. The spectral function expanding method is used to analyze the disturbance evolution of a defined basic state vortex. The results show that the numerical solution of the model is a superposition of the continuous spectrum component (non-normal modes) and the discrete spectrum component (normal modes). Only the eyewall and the rainbands in the inner core-region in a TC are related to the VRW normal modes, whereas the continuous spectrum wave components play an important role in the formation of secondary-, principal-, and distant- rainbands, especially the outer rainband, through an indirect way. The continuous spectrum can promote the development of the TC circulation for the occurrence of a mesoscale instability. The convection under a favorable moisture condition will trigger the inertial-gravitational wave to cause the formation of unstable spiral bandliked-disturbances outside of the eyewall. The complicated interaction between the basic state-vortex and the VRW disturbances will cause a positive feedback between the TC circulation and the rainband.

tropical cyclone, spiral bands, vortex Rossby wave, continuous spectrum, discrete spectrum

Introduction

The spiral cloud (rain) bands in the TCs were detected in the earliest radar observations, but their formation and thermodynamic and dynamic roles in hurricanes are still not well understood[1].

According to the wave characteristics, their formation could be explained by using the gravitational wave theory[2]and the vortex Rossby wave (VRW)[3]. When the spiral bands are divided into two classes: the inner and outer bands, they are explained with different mechanisms. The inner rainbands are characterized by the convectively coupled VRW. The movement of the outer rainbands follows the low-level vector winds associated with the azimuthally averaged low-level flow and the radially outward cross-band flow caused by the downdraft-induced cold pool in the boundary layer[4]. Huang and Zhang proposed thatonly the instability of the mixed vortex Rossby- inertial gravitational wave can interpret the generation of the eyewall, the inner and outer spiral bands at the same time[5].

The exact wave mechanisms leading to the spiral band formation remain unclear, especially, the relative contribution of the vortex Rossby and inertial gravitational wave, which is difficult to be separated within a complicated model. It might be settled with the aid of simple models. In this paper, the role of the vortex Rossby wave will be illustrated by using a baratropic non-divergent tropical cyclone-scale vortex model. This model is the most simplified model of the tropical cyclone-scale vortex and can describe the important properties of the VRW.

Using a Rankine vortex as the basic flow, Smith and Montgomery proposed that, in a barotropic nondivergent model, the solution is a superposition of the shear wave (continuous spectrum) component and the Rossby edge wave (discrete spectrum) component[6]. Carr and Williams[7]argued that a continuous spectrum disturbance will propagate against the vertical flow (retrogress) more rapidly at smaller radii due to the larger gradient of the symmetric vorticity closer to the center. This variable of retrogression would then tend to counteract the effect of the tilting mechanism and thus decrease the rate of symmetrization. However, the effect of the variable retrogression might be small in the tangential wind profile of a tropical cyclone vortex, decreasing relatively gradually with the radius. Based on their work, Shapiro and Montgomery pointed out that the individual normal modes on the vortex have a fixed structure, they are not affected by the tilting mechanism that ultimately reduces the amplitudes of the asymmetries associated with the continuous part of the spectrum[8].

In a tropical cyclone-like vortex, the discrete modal instability of the pure VRW can account for the generation of the eyewall and the inner spiral bands, but cannot directly account for the generation of the outer spiral bands[5]. Hence, a problem arises, whether the VRW contributes nothing to the outer spiral bands or not, that will be addressed in this paper. It will be resolved in a way of the wave spectrum analysis[9]. In this way, the continuous spectrum (non-normal modes) and the discrete spectrum (normal modes) are decomposed from the disturbed fields to obtain non-mode solutions in the atmosphere. Therefore, the role of the VRW continuous spectrum in the generation of spiral bands in the TCs can be made clear in this paper.

1. Model and methodology

1.1Non-divergent barotropic model

In this study, we use a linearized non-divergent barotropic model on the -fplane. In a cylindrical coordinate-system, the equations can be expressed as:

Utilizing Eqs.(1), (2) and (3), the following vorticity equation can be obtained

In order to obtain the eigenvalues and the eigenfunctions numerically, let where λ is the azimuth angle, n is the azimuthal wave-number, σ=σr+iσiis the frequency. Then, substitute it into Eq.(6) to get

Dividing the radius r from 0 (Vortex center) tointo N sections with an equal distance, discretize Eq.(8) and consider corresponding boundary conditions:

then, the solution of Eqs.(1)-(3) will become an eigenvalue problem in the matrix form as follows

1.2Methods to distinguish continuous spectrum from discrete spectrum

Fig.1 Radial profile of relative vorticity and velocity

After the spectral points and functions are obtained, we shall distinguish the continuous spectral points from the discrete spectral points. The order of matrix TX(=A-1B) is different with the change of the grid distance[9]. For instance, if the grid number is N, there are (N-1) computed eigenvalues and their corresponding eigenfunctions. In our study, the domainsize of all computation is 200 km in the radial direction, i.e.,=200 km , and the corresponding grid distance is Δr(=/ N). N will be changed as needed.

The number of the computed spectral points increases with the increase of resolution, but the distribution of the discrete spectral points shall not become denser, whereas the continuous spectral points become denser. In this way, the continuous spectrum can be distinguished from the discrete spectrum.

The continuous spectrum may be turned into a numerical discrete spectrum in the numerical computation, so the real continuous spectrum shall be identified firstly. Theoretically, the continuous spectrum exists when the wave equation is singular. As for the model used in this paper, when the coefficient before the differential quotient in Eq.(10) is zero, this equation will become singular, and the frequencies of the continuous spectrum shall be located in the region

Fig.2 Distribution of spectral points (σr) as the function of the radial grid number (s-1)

Taking account of both the enough exact description of spectral functions and the computation precision, N is taken as 100 in most cases of this study. Then there exist 99 spectral points and corresponding functions, and all of them are located within [0,0.0047 s-1]. The spectral points labeled with 89 and 90 are a couple of the growing (decaying) discrete spectrum modes and the remaining points are all continuous spectral points when they are sorted in the ascending order of the propagation frequency.

Fig.3 Stream function and relative vorticity of unstable disturbance when r＜100 km

2. Structure and evolution of wave spectrum

2.1Discrete spectrum

As mentioned above, the discrete spectrum only includes two spectral points, wherein the one labeled with 89 is a growing, i.e. unstable, mode. Figure 3shows the horizontal distribution, i.e., the distribution on r-λ plane, of the stream function and the relative vorticity of this unstable disturbance, wherein, the relative vorticity,is obtained from its stream function, ψ, as

Fig.4 Evolution of stream function for unstable disturbance when r＜100 km (m2/s)

where Δ is the Laplace operator in the cylindrical coordinates.

As shown in Fig.3, both the relative vorticity () and the stream function (ψ) of the unstable disturbance have two opposite-phase extremum centers along the radius, which correspond to the RMWs and the RMV of, respectively. The maximum amplitude of the unstable disturbance appears at the RMWs and decays quickly outside of this radius. Both the stream function and the relative vorticity have a spiral bandlike structure within the region between two peak values.

The unstable discrete spectrum will grow exponentially. The evolution of its stream function or the relative vorticity can be known by using the computed growth rate (σi) and the phase velocity (cr), where cr=σr/n. It will take 2nπ/σrhours for the unstable disturbance to rotate around the center of the vortex clockwise for a positive cror anticlockwise for a negative cr. Let the azimuthal wave number n=3, its computed propagation frequency σ=2.998× 10-3s-1. As shown in Fig.4, the stream function of the unstable disturbance rotates clockwise in a cycle of about 1.75 h. The basic–state vortex rotates anticlockwise as defined, so the unstable perturbation propagates against the basic flow.

2.2Continuous spectrum

With the increase of N, the spectral points and the identifiable modes of the continuous spectrum will increase, but their radial structures change following similar laws. For simplicity, the case of =20N is considered as an example to illustrate the radial structure variation of the continuous spectrum modes. There are 19 spectral points including 17 continuous spectrum modes and 2 discrete spectrum modes labeled with 17 and 18. The continuous spectral points labeled as 1-16 and 19 are turned into the computed discrete spectral points. Their radial structures exhibit no smooth wave properties, but with critical layers at the radius where these spectral functions or their firstorder derivatives are disconnected (Fig.5). The locations of the critical layers are closer to the vortex center from the model boundary with the increase of the label index. The propagation frequencies of the discrete spectrum perturbations are in the order of those of the continuous spectrum perturbation propagating relatively faster.

A single mode of the continuous spectrum has no evident physical meaning because of discontinuity of the continuous spectrum itself and its first-order derivative, so its evolution shall be described reasonably and correctly by using the concept of wave packet[9]. When N is taken to be 100, the stream function ()rΨof each mode is discretized as an (1)N- dimensional vector. The stream function of continuous spectrum wave packet is computed by using the following formula

Fig.5 Radial structure of perturbation stream function in ascending order of propagation frequency (=20)N

Fig.6 Evolution of horizontal stream function for continuous spectrum wave packet when r＜200 km (m2/s)

3. Discussions

The theory of idealized tropical cyclone-like vortices suggests that the flow field near the center of a storm contains normal-mode disturbances of the general size and shape of the secondary rainbands in the inner-core region outside of the eyewall zone. The numerical solution shown in Section 2 corresponds well to the analytical solution proposed by Smith and Montgomery with a barotropic nondivergent model[6]. However, the analytical method can only be used to study some special basic-state vortex, such as the Rankine vortex, the numerical method used in this paper can analyze more complex and realistic basicstate vortices, which will result in more credible and valuable conclusions.

The numerical solution of the non-divergent barotropic model in this study is a superposition of the continuous spectrum component and the discrete spectrum component of the VRW. Such disturbances exhibit different structural characteristics. In order to distinguish the relative contribution of the discrete spectrum components from the continuous spectrum ones, the schematic diagram of the radar reflectivity in a North Hemisphere TC[12]is cited (Fig.7).

Fig.7 Schematic illustration of radar reflectivity in a North Hemisphere tropical cyclone with a double eyewall[12]

The development of the discrete spectrum wave packetdψ, which is computed following Eq.(13), for the two modes labeled as 89 and 90, shows the unstable growth of such modes (Fig.8). As shown in the figure, the wave packet exhibits a two-peak structure along the radial direction initially, but the energy is accumulated rapidly near the RMWs after about 2 h’s evolution. The wave packet has no evident variation outside of this radius. Such modes extract energy fromthe basic-state vortex and decrease the rate of symmetrization under the linear assumption with the nonlinear effects neglected. Compared the structure of the unstable discrete spectrum mode with the schematic illustration shown in Fig.7, it can be inferred that the developing discrete spectrum (normal mode) component most likely applies to the smaller, more transient, secondary rainbands.

Fig.8 Time evolution of the discrete spectrum wave packet as the function of radius

The primary eyewall and some of the rainbands in the inner core of a TC are related to the VRW normal modes. The unbalanced property of the wave outside the stagnation radius of the VRW is one of the important causes for the formation of the unstable outer spiral bands in the TCs. Accordingly, the outer spiral band in the actual TCs can be identified to possess properties of the inertial-gravitational wave[5].

Fig.9 Time evolution of the continuous spectrum wave packet as the function of radius

In the general circulation, the west winds can be nourished by continuous spectrum disturbances[13]. Similarly, it can be inferred that the continuous spectrum of the VRW may enhance the basic state vortex, which will favor the development of mesoscale instabilities. Such assumption is verified by the time evolution of the continuous spectrum wave packetcψ as shown in Fig.9. It can be seen that the amplitudes of the continuous spectrum wave packet are reduced ultimately with time on the whole, which shows an energy transfer from the asymmetries associated with the continuous spectrum to the symmetric basic-state vortex. Such energy transfer mainly occurs in the regions corresponding to the eyeall, principal-, and distantrainband zone shown in Fig.7. Such energy transfer is dominant, especially in the distant rainband region.

The spiralband-liked region exhibited by the continuous spectrum wave packet (as shown in Fig.6) often corresponds to the potential instability region in the actual TCs. In such a zone, the convection will occur under a favorable moisture condition. As a result, the inertial-gravitational wave will be triggered[10]. In fact, the inertial-gravitational wave triggered by the direct effect of the latent heat release will cause the formation of the unstable spiral bandlike disturbances outside of the eyewall, which was verified in the numerical simulations[14].

4. Conclusions

The contribution of the VRW to the spiral rainband in the TCs is studied in the framework of a barotropic non-divergent vortex model. In contrast to previous similar studies, a more complex profile of the basic flow, which is similar to the radial wind profile of some real TCs, is defined using empirical formulas. Then the spectral function expanding method is used to analyze the disturbances for this basic state vortex.

The results show that the numerical solution is a superposition of the shear wave (continuous spectrum) component and the Rossby edge wave (discrete spectrum) component. All stable spectral points belong to the continuous spectrum, whereas the discrete spectrum only includes the growing and decaying spectral points. The numerical results are consistent with the analytical solution for a simpler basic flow.

The eyewall and some of the rainbands in the inner core of a TC are mainly related to the VRW normal modes. The unstable perturbation propagates against the cyclonic basic-state vortex and extracts energy from the symmetric basic-state vortex. In such a zone, the continuous spectrum wave packet decays evidently as shown in Fig.9, so the energy is transferred from the disturbances associated with these modes into the basic-state vortex. Moller and Montgomery proposed that, for the disturbance amplitudes of 40% of the basic-state PV at the radius of the maximum wind, a discrete normal mode propagating cyclonically around the vortex is excited as a by-product of the process by which the energy is transferred from the asymmetries into the basic state (axisymmetrization)[15]. Such a process is also observed in our simple model. The continuous spectrum part of the VRW can help developing the discrete normal mode of the VRW and facilitate the formation of unstable inner spiral bands.

The continuous spectrum components also playan important role in the formation of spiral bands outside of the inner core of a TC, especially the distant rainband, through an indirect way. They can promote the development of the TC circulation favorable for the occurrence of mesoscale instability. Under a desired moisture condition, the real potential instability will be triggered to favor the development of convection. Then the unstable spiral bands will form as a result of the propagation of the inertial gravity wave. At the same time, in such a rainband zone, the energies transfer from the asymmetries associated with the continuous spectrum to the symmetric basic-state vortex, which will result in an asymmetry-induced intensification of the basic flow.

It can be concluded that the complicated interaction between the basic state-vortex and the VRW disturbances, including the normal modes and the continuous spectrum components, will result in a positive feedback between the TC circulation and the spiral rainband. However, the detailed process and the dynamic mechanism remain a challenge for further studies.

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10.1016/S1001-6058(14)60081-0

* Project supported by the National Nature Science Foundation of China (Grant No. 40905021) the Chinese Postdoctoral Science Foundation (Grant No. 2011M500894).

Biography: RUAN Kun (1978-), Male, Ph. D. Candidate,

Lecturer

HUANG Hong,

E-mail: hhong7782@163.com