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

P. J. Dellar (2004) Quasi-two-dimensional liquid metal magnetohydrodynamics and the anticipated vorticity method J. Fluid Mech.  515 197-232 DOI: 10.1017/S0022112004000217 (BibTeX entry)

Preprints available in PDF format (Quasi2dMHD.pdf 988K)


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, and Moreau (2000) [J. Fluid Mech. 424 75-100 henceforth 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 spatiotemporal derivatives, in addition to the algebraic drag term found previously. The current 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 spatiotemporal 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 spatiotemporal derivatives in PSM's original equation lead to a system of differential-algebraic equations, instead of ordinary differential equations, after discretising 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. Solutions of these equations thus appear much more organised 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-organising behaviour and appearance of global modes seen in laboratory experiments with thin liquid metal layers and magnetic fields may be attributed to self-organising properties of the unmodified two-dimensional Navier--Stokes equations.

A. Pothérat, J. Sommeria and R. Moreau (2000) An effective two-dimensional model for MHD flows with transverse magnetic field J. Fluid Mech. 424 75-100

The anticipated vorticity method first appeared in:

C. Basdevant and R. Sadourny (1983) Parametrization of virtual scale in numerical-simulation of two-dimensional turbulent flows J. Mech. Theor. Appl. (special issue) pp 243-269

while its modification for a proposed lattice Boltzmann implementation is best described in:

R. Benzi, S. Succi and M. Vergassola (1992) The lattice Boltzmann equation: theory and applications  Phys. Rept. 222 145-197

BibTeX citation information:

author = "P. J. Dellar",
title = "Quasi-two-dimensional liquid metal magnetohydrodynamics and the anticipated vorticity method",
year = "2004",
journal = "J. Fluid Mech.",
volume = "515",
pages = "197--232",
DOI = "doi:10.1017/S0022112004000217"

Many journal articles now list a Digital Object Identifier (DOI). This is intended to provide a uniform citing and linking mechanism across journals and publishers,  see www.doi.org for details. Any paper with a listed DOI may be linked to using a URL of the form http://dx.doi.org/DOI. The DOI resolver will translate this URL into a valid URL for the paper.