Quasi-two-dimensional liquid-metal magnetohydrodynamics and the anticipated vorticity method

Abstract

The flow of liquid metal in a magnetic field may become almost two-dimensional because the magnetic field inhibits velocity variations along the field lines. Two-dimensionality must break down near rigid boundaries to satisfy no-slip boundary conditions, leading to a quasi-two-dimensional flow comprising a two-dimensional core between Hartmann boundary layers. Flow in the Hartmann layers is dominated by viscosity and the Lorentz force. Pothérat, Sommeria & Moreau (J. Fluid Mech. vol. 424, 2000, p. 75, referred to herein as PSM) recently proposed a two-dimensional equation for the vertically averaged horizontal velocity to describe such flows. Their treatment extends previous work to account for inertial corrections (such as Ekman pumping) to the flow in the Hartmann layers. The inertial corrections lead to extra nonlinear terms in the vertically averaged equations, including terms with mixed spatio-temporal derivatives, in addition to the algebraic drag term found previously. The present paper shows that many of these terms coincide with a previously postulated model of two-dimensional turbulence, the anticipated vorticity method, and a subsequent modification restoring linear and angular momentum conservation that might be described as an anticipated velocity method. A fully explicit version of PSM's equation is derived, with the same formal accuracy but no spatio-temporal derivatives. This explicit equation is shown to dissipate energy, although enstrophy may increase. Numerical experiments are used to compare the effect of the various different equations (without linear drag or forcing) on both laminar and turbulent initial conditions. The mixed spatio-temporal derivatives in PSM's original equation lead to a system of differential-algebraic equations, instead of ordinary differential equations, after discretizing the spatial variables. Such systems may still be solved readily using existing software. The original and explicit versions of PSM's equation give very similar results for parameter regimes representative of laboratory experiments, and give qualitatively similar results to the anticipated velocity method. The anisotropic diffusion of vorticity along streamlines that is present in all equations studied except the Navier–Stokes equations has comparatively little effect. The additional terms in PSM's equation, and also the anticipated velocity method, that arise from Ekman pumping are much more significant. These terms lead to an outward transport of vorticity from coherent vortices, so solutions of equations with these extra terms appear much more organized and have less fine-scale structure than solutions of the Navier–Stokes equations, or even the anticipated vorticity method, with the same initial conditions. This has implications for the extent to which the self-organizing behaviour and appearance of global modes seen in laboratory experiments with thin liquid-metal layers and magnetic fields may be attributed to self-organizing properties of the unmodified two-dimensional Navier–Stokes equations.