The atmospheric Ekman spiral for piecewise-uniform eddy viscosity

TL;DR

This paper analyzes the atmospheric Ekman spiral with piecewise-uniform eddy viscosity, providing solutions, proofs of existence and uniqueness, and exploring how the boundary layer angle varies with different viscosity profiles.

Contribution

It introduces a method for solving Ekman flows with arbitrary step-function eddy viscosity and proves the existence and uniqueness of these solutions for multiple viscosity jumps.

Findings

Solutions exist and are unique for one and two viscosity jumps.

The boundary layer angle can vary from 0° to 90°, not just 45°.

The method extends to multiple viscosity jumps through induction.

Abstract

We investigate the boundary-value problem of atmospheric Ekman flows with piecewise-uniform eddy viscosity. In addition we present a method for finding more general solutions by considering eddy viscosity as an arbitrary step-function. We discuss the existence and uniqueness of the solutions obtained through this method, providing detailed proofs for cases with one and two "jumps" in eddy viscosity. For scenarios with more "jumps," we establish results inductively. Furthermore, we examine the angle between the bottom surface of the Ekman layer and geostrophic winds by extremizing variables such as the eddy viscosity and its point of change. These calculations reveal how the angle can differ from $45^\circ$, demonstrating that the extreme values of $0^\circ$ and $90^\circ$ are achievable, indicating the potential range of the deflection angle.