The dynamic feature of Mixed Rossby-Gravity Waves on Barotropic Vortices with Eigen-frequency Analysis

Abstract

In this study, the linearized, f-plane, shallow-water equations are discretized into a matrix eigenvalue problem to examine the full spectrum of free waves on barotropic (monopolar and hollow) vortices. An eigenfrequency analysis indicates that the basic vortex is weak, the low-frequency vortex-Rossby waves (VRWs) and the two oppositely propagating high-frequency inertial-gravity waves (IGWs) are well separated, which correspond to a continuous spectrum between 0 and n½_max, in which n is the wavenumber (WN) and ½_max is the maximum of basic angular velocity, and two discrete spectral regions on both sides, respectively. However, as the vortices reach hurricane intensity, part of the high-frequency IGWs will be "red-shifted" to the continuous spectral region. On the other hand, the low-frequency VRWs on the hollow vortex will be "violet-shifted" into the discrete spectral region due to a sign change in the radial gradient of the mean absolute vorticity. Thus, the mixed vortex-Rossby-inertial-gravity waves (VRIGWs) emerge from the frequency shifts in the presence of intense rotational flows.

An eigen-functional analysis reveals three distinct radial wave structures associated with three classes of waves in the Shallow Water Vortex Perturbation Analysis and Simulation (SWVPAS, developed by Nolan et al., 2001): (i) IGWs exhibit more wavelike structures with dominant divergent flows, and their perturbation amplitude of height (h) and vorticity (ζ) fields in the core region are in phase, with unbalanced characteristics; (ii) VRWs have little radial wavelike structures, but with dominant vortical flows, and their h and ζ fields are anti-phased, implying their balanced nature. In addition, the waves have a critical radius in the core region at which the inflexion points of the radial ζ and the perturbation amplitude of radial wind (u) profiles are located and the perturbation amplitude of tangential wind (v) becomes discontinuous; (iii) mixed VRIGWs, possessing both the IGW and VRW characteristics at higher WNs, exhibit both vortical and divergent flows at a similar order of magnitude. They also have critical radii, but in the outer regions. Moreover, their h and ζ fields are out of phase in the core region, like VRWs, while changing to an in-phase relationship in the outer region, like IGWs, implying different geostrophic adjustment mechanisms between the inner and outer regions. These mixed wave characteristics are more pronounced in hollow vortices than those in monopolar vortices due to the development of inflexion instability in the former.

Eight eigenmodes for each wave class are selected and linearly superimposed in time to gain insight into different propagation characteristics of the three wave classes on the monopolar and hollow vortices. Results show that cross-isobaric flows and isobaric flows associated with the IGW and VRWs, respectively, as expected, whereas the VRIGWs exhibit more balanced flows, like the VRWs, inside the RMW, but more unbalanced characteristics outside, like the IGWs. Moreover, the IGWs propagate rapidly outward as spiral bands, in significant contrast to the slow propagation of the VRWs and VRIGWs. The latter two waves tend to be trapped within their critical radii. On the other hand, the IGWs weaken rapidly with time, whereas both the VRWs and VRIGWs show slow reductions in amplitude. This indicates the potential importance of the VRWs and VRIGWs in maintaining spiral rainbands and organizing deep convection in the eyewall.

It is shown that numerical solutions of pure classes of the composite waves on the hollow vortex are generally similar to the previous theoretical results, except for some differences in small-scale details owing to the use of Doppler-shifted frequencies and the existence of r_c. Thus, we may conclude that VRIGWs, at least at WN-2, containing both intense vortical and divergent flows, should be common in TCs, especially in intense hurricanes. In a forthcoming study, we will examine the structural evolution of these waves using real-data-simulated hurricane cases, and investigate their roles in the formation of spiral rainbands and the eyewall replacement cycle.

Acknowledgements

This work was funded by Natural Science Foundation of China (No. 41275002 and No. 41175054), the US NSF Grant ATM-0758609, Natural Science Key Foundation of China (No.41230421), and China Postdoctoral Science Foundation (No.2013M531321).