Geophys. Astrophys. Fluid Dynamics **100** (2006) 121-137.
doi:10.1080/03091920600565595
## Mean flow instabilities of two-dimensional convection in strong magnetic
fields

Alastair M. Rucklidge(1),
M.R.E. Proctor(2) and J. Prat(3).

(1) Department of Applied Mathematics,

University of Leeds, Leeds, LS2 9JT, UK

(2) Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Cambridge, CB3 0WA, UK

(3) Departament de Matematica Aplicada IV,

Universitat Politecnica de Catalunya (EPSEVG), 08800 Vilanova, Spain

**Abstract.**
The interaction of magnetic fields with convection is of great importance in
astrophysics. Two well-known aspects of the interaction are the tendency of
convection cells to become narrow in the perpendicular direction when the
imposed field is strong, and the occurrence of streaming instabilities
involving horizontal shears. Previous studies have found that the latter
instability mechanism operates only when the cells are narrow, and so we
investigate the occurrence of the streaming instability for large imposed
fields, when the cells are naturally narrow near onset. The basic cellular
solution can be treated in the asymptotic limit as a nonlinear eigenvalue
problem. In the limit of large imposed field the instability occurs for
asymptotically small Prandtl number. The determination of the stability
boundary turns out to be surprisingly complicated. At leading order, the linear
stability problem is the linearisation of the same nonlinear eigenvalue
problem, and as a result, it is necessary to go to higher order to obtain a
stability criterion. We establish that the flow can only be unstable to a
horizontal mean flow if the Prandtl number is smaller than order B_0^{-4/3},
where B_0 is the imposed magnetic field, and that the mean flow is concentrated
in a horizontal jet of width B_0^{-1/6} in the middle of the layer. The
result applies to stress-free or no-slip boundary conditions at the top and
bottom of the layer.

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