Analytical study of the Ekman angle for the isothermal flow of a viscous incompressible fluid in view of the Navier boundary condition

Abstract

The Ekman angle is the angle between the tangential stress vector on the free surface and the flow velocity vector in the upper layer. Ekman showed that, for an isobaric flow in an ocean with an infinitely great depth (fluid flow in half-space), the angle is equal to 45°. The paper studies the Ekman angle occurring in an isothermal laminar flow with the Navier slip boundary condition. In order to determine the Ekman angle, an analytical solution of the Oberbeck-Boussinesq equations is constructed; the solution describes the laminar Ekman-Poiseuille isothermal flow with allowance made for two components of the Coriolis force and the Navier condition at the lower (solid) boundary. The components of the tangential stress vector are set at the upper boundary. The velocity representation in the form of linear functions of the horizontal coordinates is used. The paper studies the dependence of the Ekman angle on the pressure gradient and on the friction coefficient on the solid surface, the depth of the ocean being finite. It is shown that at a sufficiently large depth there is no friction effect, and the Ekman angle is 45°. © 2020 American Institute of Physics Inc.. All rights reserved.