The surface current of Ekman flows with time-dependent eddy viscosity

Luigi Roberti

doi:10.3934/cpaa.2022064

Article Contents

2022, Volume 21, Issue 7: 2463-2477. Doi: 10.3934/cpaa.2022064

This issue Previous Article Exact solution and instability for geophysical edge waves Next Article Geophysics and Stuart vortices on a sphere meet differential geometry

The surface current of Ekman flows with time-dependent eddy viscosity

Luigi Roberti

Faculty of Mathematics, University of Vienna, Oskar–Morgenstern–Platz 1, 1090 Vienna, Austria

Received: August 31, 2021

Revised: January 31, 2022

Published: July 2022

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Abstract

In this paper we investigate transients in the oceanic Ekman layer in the presence of time-varying winds and a constant-in-depth but time dependent eddy viscosity, where the initial state is taken to be the steady state corresponding to the initial wind and the initial eddy viscosity. For this specific situation, a formula for the evolution of the surface current can be derived explicitly. We show that, if the wind and the eddy viscosity converge toward constant values for large times, under mild assumptions on their convergence rate the solution converges toward the corresponding steady state. The time evolution of the surface current and the surface deflection angle is visualized with the aid of simple numerical plots for some specific examples.

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Mathematics Subject Classification: 35G16, 76U60, 86A05.

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Figure 1. Plots of $ \tan(\theta_0(t)) $ (left) and the (normalized) velocity components (right) for $ K(t) = c\vert\tau(t)\vert $ and $ \tau(t) = 2- \mathrm{e}^{-t^2} $

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Figure 2. Plots of $ \tan(\theta_0(t)) $ (left) and the (normalized) velocity components (right) for $ \tau(t) = 1+ \mathrm{e}^{-t^2} $

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Figure 3. Plots of $ \tan(\theta_0(t)) $ (left) and the (normalized) velocity components (right) for $ \tau(t) = 1+2t^2\left(1-\frac{3}{4}t^2\right) \mathrm{e}^{-t^2} $

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