A note on B\"odewadt-Hartmann layers

TL;DR

This paper investigates the dominance of Coriolis or Lorentz forces in axisymmetric rotating flows with magnetic fields, providing analytical and numerical solutions for boundary layer behavior across different force regimes.

Contribution

It offers new analytical and numerical solutions for the B"odewadt-Hartmann problem, analyzing force dominance and boundary layer dynamics in rotating magnetohydrodynamic flows.

Findings

Boundary layer thickness depends on the Elsasser number A.

Analytical expressions for angular velocity and decay time are derived.

Solutions cover steady flow and spin-down scenarios.

Abstract

This paper addresses the problem of axisymetric rotating flows bounded by a fixed horizontal plate and subject to a permanent, uniform, vertical magnetic field (the so-called B\"odewadt-Hartmann problem). The aim is to find out which one of the Coriolis or the Lorentz force dominates the dynamics (and hence the boundary layer thickness) when their ratio, represented by the Elsasser number $A$, varies. After a short review of existing linear solutions of the semi infinite Ekman-Hartmann problem, weakly non-linear analytical solutions as well as fully non-linear numerical solutions are given. The case of a rotating vortex in a finite depth fluid layer is then studied, first when the flow is steady under a forced rotation and second for spin-down from some initial state. The angular velocity in the first case and decay time in the second are obtained analytically as a function of $A$ using the weakly non linear results of the semi-infinite B\"{o}dewadt-Hartmann problem.