Inertia-gravity waves in geophysical vortices

Abstract

Pancake-like vortices are often generated by turbulence in geophysical flows. Here, we study the inertia-gravity oscillations that can exist within such geophysical vortices, due to the combined action of rotation and gravity. We consider a fluid enclosed within a triaxial ellipsoid, which is stratified in density with a constant Brunt-Väisälä frequency (using the Boussinesq approximation) and uniformly rotating along a (possibly) tilted axis with respect to gravity. The wave problem is then governed by a mixed hyperbolic-elliptic equation for the velocity. As in the rotating non-stratified case considered by Vantieghem (2014, Proc. R. Soc. A, 470, 20140093, doi:10.1098/rspa.2014.0093), we find that the spectrum is pure point in ellipsoids (i.e. only consists of eigenvalues) with smooth polynomial eigenvectors. Then, we characterise the spectrum using numerical computations (obtained with a bespoke Galerkin method) and asymptotic spectral theory. Finally, the results are discussed in light of natural applications (e.g. for Mediterranean eddies or Jupiter's vortices).

Figures (9)

• Figure 1: (a) Cross-section of the temperature anomaly (acoustic tomography) in a Mediterranean eddy offshore in the Eastern Atlantic. Adapted from figure 3 in mcwilliams2016submesoscale. (b) Picture of Jupiter's Great Red Spot (GRS) taken on 21 April 2014 with Hubble. Credits: NASA, ESA and A. Simon (Goddard Space Flight Center). Jupiter's axis of rotation is $\boldsymbol{\Omega}$.
• Figure 2: Sketch of the mathematical model for geophysical vortices (not to scale). (a) Meridional cross-section of a vortex centred at colatitude $\theta$, which is embedded within an ocean or an atmosphere (blue region). (b) Front view of a flattened ellipsoidal vortex of semi-axes $a \gg b \gg c$, which is stratified under the uniform planet's gravity $\boldsymbol{g}$ (i.e. the background density $\rho_0$ decreases with increasing $z$).
• Figure 3: Sparse upper triangular block matrices $\boldsymbol{A}_n$ (left) and $\boldsymbol{B}_n$ (right) for $n=5$ and $\boldsymbol{1}_\Omega = \boldsymbol{1}_z$.
• Figure 4: Sketch of the bounded pure point spectrum in triaxial ellipsoids (not to scale) when $N$ is constant. The spectrum is symmetric with respect to $0$ and bounded by $|\omega| \leq \omega_{+}$ (Proposition \ref{['theo:propsigma3']}). The velocity equation is either hyperbolic or elliptic in volume. The spectrum is spanned by IGMs and low-frequency (surface) modes.
• Figure 5: Spectrum $\sigma_2$ in the aligned case $\boldsymbol{1}_\Omega = \boldsymbol{1}_z$. (a) Eigenvalues $\omega_{3,5} \geq 0$ in $\sigma_{1,1}$ for a sphere with $b/a=c/a=1$ (solid purple curves) and an ellipsoid with $b/a=0.7$ and $c/a=0.1$ (dashed olive curves). Red region shows $\sigma_2$ in which the problem is hyperbolic. (b) Number of eigenvalues in $\sigma_{1,n}$ for a sphere ($b/a=c/a=1$) and a flattened ellipsoid ($b/a=0.7$ and $c/a=0.1$). Colour bar shows the value of $N/\Omega_s$.

Theorems & Definitions (8)

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