###
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)

####

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, 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:

@article{DellarQuasi2dMHD04,

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.