Inviscid damping of asymmetries on a two-dimensional vortex

Abstract

The inviscid damping of an asymmetric perturbation on a two-dimensional circular vortex is examined theoretically, and with an electron plasma experiment. In the experiment, an elliptical perturbation is created by an external impulse. After the impulse, the ellipticity ͑quadrupole moment͒ of the vortex exhibits an early stage of exponential decay. The measured decay rate is in good agreement with theory, in which the perturbation is governed by the linearized Euler equations. Often, the exponential decay of ellipticity is slow compared to a vortex rotation period, due to the excitation of a quasimode. A quasimode is a vorticity perturbation that behaves like a single azimuthally propagating wave, which is weakly damped by a resonant interaction with corotating fluid. Analytically, the quasimode appears as a wave packet of undamped continuum modes, with a sharply peaked frequency spectrum, and it decays through interference as the modes disperse. When the exponential decay rate of ellipticity is comparable to the vortex rotation frequency, the vorticity perturbation does not resemble a quasimode; rather, it is rapidly dominated by spiral filaments. Over longer times, linear theory predicts algebraic decay of ellipticity; however, nonlinear oscillations of ellipticity emerge in the experiment before a transition to algebraic decay would occur.

A Formal Theory for Vortex Rossby Waves and Vortex Evolution

“…For ¼ 0:5 we see some discrepancy in the location of the q peak and its magnitude. The shallow-water model develops a small inner peak that we suspect is indicative of aquasi-mode component in the initial conditions (Schecter et al, 2000) that is absent in the asymptotic theory and occurs in a shallow-water initialization due to the nonasymptotic inversion of (22). However, as drops even to 0.25, the solutions match very closely, becoming visually indistinguishable at ¼ 0:125 and 0.0625.…”

“…Thus the theory is in the category of quasi-linear waves (i.e., the only retained nonlinearity is for wave/ mean-flow interaction) in a slowly varying medium (the vortex). The scale-separation and vortex-profile assumptions exclude two other linear, vortex-wave phenomena: vortex shear instability (e.g., Flierl, 1988) and large-scale, quasi-mode perturbation decay (Schecter et al, 2000;Balmforth et al, 2001). We make no attempt here to define the regime boundaries among these phenomena.…”

Critical layer and radiative instabilities in shallow-water shear flows

“…This period increases the closer the pole is to the c r -axis, i.e. the less the quasi-mode is damped; however information about Landau poles is relevant for the evolution of disturbances and processes such as cat's eye formation and mixing, even when the quasi-mode is not weakly damped (Schecter et al, 2000;Turner et al, 2008).…”

“…The corresponding eigenvalue is called a Landau pole, and is associated with a quasi-mode, which can be considered as formed from the continuous spectrum, in other words a combination of singular modes that naturally arises in the initial value problem (Briggs et al, 1970). Quasi-modes always have a damping effect on perturbations, an effect studied for vortices, and nonlinear effects can lead to the formation of structures such cat's eyes or tripoles (e.g., Rossi et al, 1997;Balmforth, 1999;Bassom & Gilbert, 1999;Schecter et al, 2000;Balmforth et al, 2001;Turner et al, 2008), also observed in experiments (Van Heijst, 1991). Growth of unstable modes whose structure includes a critical layer has been observed in experiments of a columnar vortex in a stratified fluid (Riedinger et al, 2010b).…”

Evaluation of a reduced model for investigating hurricane formation from turbulence

“…The basic pathway to genesis in each region resembles that described more fully in SD09. During the unshown transition to day 9, local positive vorticity anomalies merge and axisymmetrize, as would occur in ordinary 2D turbulence (Melander et al, 1987(Melander et al, ,1988Carnevale et al, 1991;Dritschel and Waugh, 1992;Lansky et al, 1997;Dubin, 1999, 2001;Schubert et al, 1999;Schecter et al, 2000;Möller and Montgomery, 2000;Schecter and Montgomery, 2003). As the first incipient tropical cyclone begins to intensify, an asymmetric eyewall with mesovortices forms near the radius of maximum wind.…”