The onset of instability for zonal stratospheric flows

Abstract

We investigate some qualitative aspects of the dynamics of the Euler equation on a rotating sphere that are relevant or stratospheric flows. Zonal flow dominates the dynamics of the stratosphere and for most known planetary stratospheres the observed flow pattern is a small perturbation of an n-jet, which corresponds to choosing the Legendre polynomial of degree n as the stream function. Since the 1-jet and the 2-jet are stable, the main interest is the onset of instability for the 3-jet. We confirm long standing conjectures based on numerical simulations by proving that the 3-jet is linearly unstable if and only if the rotation rate belongs to a critical interval. Turning to the nonlinear problem, we prove that linear instability implies nonlinear instability and that, as the rotation rate goes to infinity, nearby traveling waves change gradually from a cat's eyes streamline pattern to a profile with no stagnation points.

Figures (11)

• Figure 1: The zonal wind profile of the outer planets in our solar system, measured in m/s relative to each planet's rotation speed (Credit: Open-Stax CNX). Zonal bandings are the most prominent visual features on Jupiter and Saturn, called "zones" if they have an eastward jet along their poleward boundary and a westward one on the boundary nearest the equator and "belts" if the direction of the jets along their boundary is reversed (on Jupiter, they have a strong color contrast as bright, respectively dark regions). Zonal flow also dominates the dynamics of the stratosphere of Uranus and Neptune, with a broad westward equatorial flow and an eastward flow at higher latitudes in each hemisphere. These pictures show the high altitude clouds just beneath the stratosphere (at the top of the troposphere).
• Figure 2: Zonal velocity profiles $u_n$ of the first three jets, rescaled such that $\max\limits_{s \in [-1,1]}\{u_n(s)\}=3$ for $1 \le n \le 3$, the corresponding stream functions being $\Psi_1=-3P_1$, $\Psi_2=2P_2$ and $\Psi_3=2P_3$ in terms of the Legendre polynomials.
• Figure 3: Rayleigh's criterion ensures that the $3$-jet is spectrally stable for $\omega\in(-\infty,-18]\cup[72,\infty)$, but is not helpful for $\omega\in(-18,72)$. Theorems \ref{['positive half-line critical rotation rate']}-\ref{['negative half-line critical rotation rate']} give the sharp $\omega$-range of linear stability or instability.
• Figure 4: The green interval is the essential spectrum of $\mathcal{L}_{\omega,1}$, which is the projection of $\mathcal{L}_{\omega}$ on the first Fourier mode. For $\omega$ larger than and close to ${99\over2}$, the blue bold points are the isolated eigenvalues $-i\mu_{1,\omega}$ of $\mathcal{L}_{\omega,1}$. For $\omega={99\over2}$, the red bold point is the embedded (edge) eigenvalue $12i$ of $\mathcal{L}_{{99\over2},1}$. As $\omega\to{99\over2}^+$, the isolated eigenvalue $-i\mu_{1,\omega}$ of $\mathcal{L}_{\omega,1}$ hits the embedded eigenvalue $12i$ of $\mathcal{L}_{{99\over2},1}$, where an unstable eigenvalue emerges. The embedded eigenvalue $12i$ has negative sign of the energy quadratic form.
• Figure 5: The green interval is the essential spectrum of $\mathcal{L}_{\omega,2}$, which is the projection of $\mathcal{L}_{\omega}$ on the second Fourier mode. For $\omega$ smaller than and close to $g^{-1}(-12)$, the blue bold points are the isolated eigenvalues $-2i\mu_{3,\omega}$ of $\mathcal{L}_{\omega,2}$ and the brown bold points are the isolated eigenvalues $-2i\mu_{2,\omega}$ of $\mathcal{L}_{\omega,2}$. As $\omega\to g^{-1}(-12)^-$, the two eigenvalues with opposite Krein signatures collide at the isolated eigenvalue $-2i\mu_{2,g^{-1}(-12)}$ of $\mathcal{L}_{g^{-1}(-12),2}$. After the collision, an unstable eigenvalue emerges.

Theorems & Definitions (110)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Linear to nonlinear orbital instability of general steady flows
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6: Existence of nearby traveling waves
• Theorem 1.7: Rigidity of nearby traveling waves
• Remark 2.1
• Lemma 2.2
• proof