Absorption and reflection of inertial waves by a geostrophic vortex

TL;DR

The paper analyzes how inertial waves interact with a rapidly rotating axisymmetric geostrophic vortex, deriving a cylindrical-wave equation on top of differential rotation and identifying a cylindrical critical layer where the wave Doppler-shifted frequency vanishes. It shows that monochromatic waves do not exchange momentum with the mean flow unless absorbed at the critical layer, and that absorption depends on the sign of angular momentum relative to the vortex, with opposite-sign waves being reflected at turning surfaces. By mapping to a rectilinear shear model, the authors obtain Airy-type behavior near the layer, derive wave-train evolution with wave-action conservation, and quantify transmission and reflection in both inviscid and viscous regimes. Viscosity regularizes the singularity, yields a finite transmitted amplitude, and introduces dissipation that governs the long-time decay of wave trains, with implications for momentum transfer and the maintenance of large-scale vortices in rotating flows.

Abstract

We study interaction of inertial waves with geostrophic flow in a rapidly rotating fluid system. In accordance with experimental conditions in [1, 2], we consider inertial waves, which were excited by a source being near side boundary of the flow and enter the region where geostrophic vortex flow is present. The wave equation is derived and analyzed in the paper that describes the propagation of convergent and divergent cylindrical waves on the background of mean vortex flow, which is considered as an axisymmetric differential rotation. We show that a monochromatic wave does not exert any torque on the vortex flow in the inviscid limit until it is absorbed inside the critical layer. Among convergent waves those only are absorbed which carries angular momentum of the same sign as one's of the rotation in the vortex. Convergent waves with the opposite sign of angular momentum are just reflected from the vortex. The absorption of a wave is possible only if the vortex flow is characterized by fast enough angular velocity there. The behavior of the wave near the critical layer can be described by the well-known model where the mean flow is rectilinear shear flow. We show that the conventional wave train approximation for the short-wave limit is not applicable in the vicinity of the layer and revise it, deriving the proper equation and reformulating the conservation law of wave action. For the vicinity of critical layer, a model which accounts for the viscous dissipation is derived; viscous effects are studied for absorption both of monochromatic wave and wave train.

Figures (3)

• Figure 1: Schematics for inertial wave propagation: absorption with some reflection at the critical layer $r=r_\ast$ and total reflection at the reflection surface $r=r_t$. For the rectilinear model from Section \ref{['sec:rect_flow']} we pass to the inverted $Oy$ axis, keeping subscript notations for special points.
• Figure 2: Plot of numerical solution of (\ref{['visc:v_pm-eq']}) for wave $v_{\sigma}$ with $\sigma=1$ passing the viscous inner layer. The solution is obtained by integration of (\ref{['visc:v-sol_real']}) with $\beta=4$. Red and blue lines correspond to real and imaginary parts of the solution respectively. Dashed lines in both regions $\left|\eta/\eta_{\mathrm v}\gg1\right|$ denote the scaling $|v_+|\propto \sqrt{|\eta|}$ for the absolute value, as per the asymptotic (\ref{['eq:02']}). The vertical bias between dashed asymptotics corresponds to the absolute value of the transmission coefficient, its numerical value is 2% less than its theoretical prediction (\ref{['eq:transmission']}).
• Figure 3: Wave train evolution in the vicinity of critical layer in the inviscid limit obtained by numerical integration of (\ref{['visc:train-sol']}). Taking parameters of width in (\ref{['quadratic']}) as $l_0/\overline \eta_0\approx 0.4$ and $\beta=25$, $\mathrm{Re }\:v_+$ function is plotted at times $|\Sigma k_x\overline \eta_0| t = 0, \,15, \, 75$. The train center's position is approaching critical layer and its wavelength is shortening (by $2.2$ and $7$ times respectively, vertical dashed lines represent the center positions, the location of $v$-axis is $\eta=0$) in agreement with (\ref{['eta-bar']},\ref{['quadratic']}). The amplitude is decreasing in agreement with $t^{-3/2}$ power-law from Eq. (\ref{['fin-booker']}) for times in between blue and red curves (decreased by $3.3$ and $19$ times respectively). Meanwhile, the train's width remains almost unchanged, which follows from (\ref{['quadratic']}).