Abstract
We report on an instability, arising in sub-surface, laterally-sheared flows in rotation. When the lateral shear of a horizontal flow in geostrophic balance is of opposite sign as the Coriolis parameter, and exceeds it in magnitude, embedded perturbations are subjected to inertial instability, albeit modified by viscosity. When the perturbation arises from the surface of the fluid, the initial response is akin to a Stokes problem, with an initial flow aligned with the initial perturbation. Perturbation then grows quasi-inertially, rotation deflecting the velocity vector, which adopts a well-defined angle with the mean flow. While the perturbation initially grows super-inertially, the growth rate then becomes sub-inertial, eventually tending back to the inertial value. The same process repeats downward as time progresses. Ekman-inertial transport aligns with the asymptotic orientation of the flow, and grows exactly inertially with time, once the initial instants have passed. Because of the strongly super-inertial initial growth rate, this instability might compete favorably against other instabilities arising in ocean fronts.
Highlights
When wind blows over the ocean surface over long periods of time, momentum diffuses down in a very different manner from Stokes’ first problem
We treat Ro as a constant, i.e., we focus on the case of linear lateral shear for v: a strong simplification in the submesoscale regime, but one that captures the essential physics of Ekman-Inertial Instability” (EII)
We complete our initial set-up by adding boundary conditions at the surface, located at z = 0, namely, a rigid lid and an initial wind stress in the y-direction only, defined as TIy = ρν vz |z=0, such that ṽ = vŷ is a steady solution of our initial system (1) and the boundary conditions above
Summary
When wind blows over the ocean surface over long periods of time, momentum diffuses down in a very different manner from Stokes’ first problem. In a first phase, which we will refer to as “viscous-inertial peeling”, tangential viscous stresses act to set the fluid in motion much faster than the expected exponential growth of InI. In this first phase, the problem is mathematically equivalent to Stoke’s first (or Rayleigh) problem, albeit for the vertical shear. In the case of a sudden wind change, it inherits its initially infinite growth rate Past this initial phase, the flow keeps accelerating in a quasi-exponential manner and draws its energy from the lateral shear of the geostrophic current, akin to InI, albeit slowed down by downward diffusion of momentum by viscosity.
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