## Thursday, May 4, Clark Bldg., Room C238, 10am

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Nonlinear Magnetoconvection in the Presence of Strong Oblique Fields

#### Keith Julien

Department of Applied Mathematics

University of Colorado, Boulder

Two-dimensional thermal convection in the presence of a strong oblique
magnetic field is studied using an asymptotic expansion in inverse powers of
the Chandrasekhar number. The linear stability problem reveals the existence
of two distinct scales in the vertical structure of the critical
eigenfunctions, a small length-scale whose vertical wavenumber k_z is
comparable with the large horizontal wavenumber k_\perp selected at
onset, and a large-scale modulation which forms an envelope on the order of
the layer depth d. The small-scale structure in the vertical results
from magnetic alignment that forces fluid motions to be (nearly) parallel
to the field lines. Using the scaling suggested by the linear theory
strongly nonlinear
steady and overstable solutions are constructed. These are characterized
by large departures of the mean temperature profile from the conduction
profile. For overstable rolls two modes of convection are uncovered.
The first ``vertical field'' mode is characterized by thin thermal boundary
layers and a Nusselt number that increases rapidly with the applied Rayleigh
number; this mode is typical of steady convection as well. The second or
``horizontal field'' mode is present in overstable convection only and has
broad thermal boundary layers and a Nusselt number that remains small and
approximately independent of the Rayleigh number. At large Rayleigh
numbers this regime is characterized by a piecewise linear temperature
profile with a small isothermal core. The ``horizontal field'' mode is
favoured for substantial inclinations of the field and sufficiently small
ohmic diffusivity. The transition between the two regimes is typically
hysteretic and for fixed inclination and diffusivity may occur with
increasing
Rayleigh number. Similar but highly asymmetric states are obtained for
depth-dependent \zeta, where \zeta is the ratio of ohmic to thermal
diffusivity. These results are obtained from a nonlinear eigenvalue problem
for the Nusselt number and mean temperature profile, and suggest a possible
explanation for the sharp boundary between the umbra and penumbra in
sunspots.

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Please direct any comments to

Gerhard Dangelmayr at gerhard@math.colostate.edu.