Mon. Not. R. Astron. Soc. **387** (2008) 1445-1462.
doi:10.1111/j.1365-2966.2008.13359.x
## Numerical simulations of rotating axisymmetric sunspots

G.J.J. Botha(1),
F.H. Busse(2),
N.E. Hurlburt(3) and
A.M.Rucklidge(1)

(1) Department of Applied Mathematics,

University of Leeds, Leeds LS2 9JT, UK

(2) Institute of Geophysics and Planetary Physics,

UCLA, Los Angeles, CA 90024, USA

(3) Lockheed Martin Solar and Astrophysics Laboratory,

Organization L9-41 Building 252, Palo Alto, CA 94304, USA

**Abstract.**
A numerical model of axisymmetric convection in the presence of a vertical magnetic flux
bundle and rotation about the axis is presented. The model contains a compressible plasma
described by the non-linear MHD equations, with density and temperature gradients simulating
the upper layer of the Sun's convection zone. The solutions exhibit a central magnetic flux
tube in a cylindrical numerical domain, with convection cells forming collar flows around the
tube. When the numerical domain is rotated with a constant angular velocity, the plasma forms
a Rankine vortex, with the plasma rotating as a rigid body where the magnetic field is strong,
as in the flux tube, while experiencing sheared azimuthal flow in the surrounding convection
cells, forming a free vortex. As a result, the azimuthal velocity component has its maximum
value close to the outer edge of the flux tube. The azimuthal flow inside the magnetic flux tube
and the vortex flow is prograde relative to the rotating cylindrical reference frame. A retrograde
flow appears at the outer wall. The most significant convection cell outside the flux tube is the
location for the maximum value of the azimuthal magnetic field component. The azimuthal
flow and magnetic structure are not generated spontaneously, but decay exponentially in the
absence of any imposed rotation of the cylindrical domain.

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