A Model of a Mesoscale Lens in Large-Scale Shear. Part I: Linear Calculations

Abstract

A simple model is used to study the behavior of a lens-shaped eddy in a background flow with uniform horizontal and vertical shear. The primary motivation for the work is to understand the influence of large-scale shear on Mediterranean salt lenses in the Canary Basin. The model eddy is represented by an isolated three-dimensional patch characterized by anomalous potential vorticity, and its evolution is governed by stratified, f-plane quasigeostrophic dynamics. Guided by the available observations, the potential vorticity field is chosen to be horizontally piecewise constant and is a linear function of depth within the lens. This produces a flow field characterized by a depth-dependent solid body rotation within the core of the eddy, with speeds decreasing monotonically outside the core, in good agreement with observations. A family of linearized solutions is discussed, representing a lens-shaped eddy with a large trapped fluid region, which is deformed due to interactions with external shear. The lens may propagate through the surrounding waters in the presence of external vertical shear, providing a possible explanation for the observed translation of Mediterranean sail lenses. These results generalize those of Hogg and Stommel to encompass three-dimension stratified dynamics, with a realistic, nonsingular representation of the potential vorticity field. The analysis predicts the form of the steady boundary deformation, the precession rate of boundary perturbations in the absence of external flow, and the propagation speed of the eddy as a function of external shear and core baroclinicity. It is found that there is a maximum differential rotation rate within the core beyond which no small amplitude solutions exist. General integral expressions are derived relating the propagation speed to the eddy potential vorticity and the external shear.