Stability of SQG Kolmogorov Flow

TL;DR

This work investigates the linear and nonlinear stability of Kolmogorov-type boundary-forced flows in surface quasigeostrophic (SQG) systems and their ageostrophic extension (SQG+). Using linear Floquet analysis and the nonlinear energy method, it characterizes how semi-infinite and finite-thickness layers respond to damping and hyperdiffusion, identifying a finite most-unstable length-scale for SQG and showing how increasing damping shifts instabilities to shorter scales. Ageostrophic corrections introduce long-wave instabilities in SQG+, which damping can strongly suppress, revealing a nuanced interplay between forcing, layer thickness, and drag. Overall, SQG stability properties diverge from 2D Euler in the zero-damping limit but align with Euler-like nonlinear stabilization under sufficient damping, with thickness and symmetry determining the precise instability landscape and practical oceanographic relevance.

Abstract

Stability analysis is performed on surface quasigeostrophic systems subjected to a Kolmogorov-type "shear force" on the boundaries using linear and nonlinear approaches. For a SQG system of semi-infinite depth forced on the upper boundary, the most linearly unstable mode is 2.74 the energy injection length scale. This is contrary to two-dimensional fluid systems, where the linear instability is greatest for long waves. In the presence of damping, the most linearly unstable mode shifts toward shorter length scales. The nonlinear critical Reynolds number across different damping strengths is found to be qualitatively similar to that of Euler 2D systems. For an SQG system of finite thickness being forced on both boundaries, its behaviour approaches that of a semi-infinite SQG system at the large thickness limit. In the small thickness limit, the behaviour of a symmetrically forced fluid layer approaches that of a 2D system, while an antisymmetrically forced fluid layer is not susceptible to both linear and nonlinear instabilities. With ageostrophic effects, an SQG+ system of semi-infinite depth is much more prone to instabilities than an otherwise identical SQG system in the absence of damping due to the instability of long-wave modes. However, damping significantly suppresses such instabilities. With increasing damping, the most linearly unstable mode moves toward a smaller length scale. Contrary to the zero damping case, when the damping is sufficiently large, ageostrophic effects have a small but measurable stabilising effect.

Figures (21)

• Figure 1: Illustration of the chosen system configurations. The first figure is semi-infinite, the second has finite depth.
• Figure 2: Left: real part of the growth rate at $Re_n = 5.02$. Dotted contours indicate a negative growth rate $\sigma$, solid contours indicate a positive growth rate. Right: analogous plot for a two-dimensional system at $Re_n = 1.83$. The Reynolds numbers are chosen sp that the regions of instability are equal in size. In both plots, $n=1$ and $\lambda=0$.
• Figure 3: Left: critical Reynolds number as a function of $k$ and $l = 0$ with $n=1$ and $\lambda=0$, correspondingg to the smallest critical Reynolds number. Right: critical Reynolds number as a function of the wavenumbers $k$ and $l$.
• Figure 4: Left panel: growth rate for hyperdiffusion parameter $n=4$ with Reynolds number $Re_n=3.0e-5$; $\sigma>0$ in the grey region. Right panel: contour map of critical Reynolds numbers for $n=4$ and $\lambda=0$.
• Figure 5: Growth rate diagrams for semi-infinite SQG at $n=1$ and $1.3Re_c(\lambda)$. From left to right, top to bottom, $\lambda$ is $0$, $0.05$, $0.2$, $5$. The contour spacing is $0.01$.