2. Formulation of the Problem and Numerical Technique
|
|
Figure 1
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[11] In order to model the process of magnetic field generation in a
stellar convection zone, we consider a spherical shell of
thickness
d = ro - ri (where
ro and
ri
are the outer and inner radii of the shell), full of
electrically conducting fluid and rotating with a constant angular
velocity
W about a fixed axis
ez,
as shown in Figure 1.
[12] We follow the standard formulation used in earlier work by
Tilgner and Busse [1997],
Busse et al. [1998],
Grote et al. [1999, 2000],
Busse [2002],
and
Simitev and Busse [2002, 2005],
but we assume a more general form of the static
temperature distribution,
 | (1) |
where the radial coordinate
r is measured in units of
d,
c is the thermal diffusivity,
cp is the specific heat at
constant pressure,
q is the mass density of uniformly
distributed heat sources,
h
ri/ro is
the inner-to-outer radius ratio of the shell, and
T0 is a
constant. The quantity
DT is related to the difference
between the constant temperatures of the inner and outer spherical
boundaries,
Ti and
To, as
 | (2) |
and reduces to
Ti - To in the case of
q=0 (also
dealt with in some simulations). The shell is self-gravitating,
and the gravitational acceleration averaged over a spherical
surface
r = const can be written as
g = - (gd) r, where
r is
the position vector with respect to the center of the sphere; as
specified above, its length
r is measured in units of
d. In
addition to
d, the time
d2 / n, the temperature
n2 / gad4 (where
a is the volumetric coefficient
of thermal expansion), and the magnetic induction
n ( mr )1/2 /d are used as scales for the dimensionless description
of the problem; here,
n denotes the kinematic viscosity of the
fluid,
r is its density, and
m is its magnetic
permeability (we set
m=1 ).
[13] We use the Boussinesq approximation in that we assume
r to be
constant except in the gravity term, where, in addition to the
standard linear dependence
r(T) (according to which
r-1(dr/dT) = -a = const ),
we introduce a small quadratic term in most cases. Once a cellular
pattern has developed, the presence of this term and of the
volumetric heat sources should not radically modify the properties
of the dynamo; however, both these factors favor the development
of polygonal convection cells
[Busse, 2004]
similar to the
cells observed on the Sun, rather than meridionally stretched,
banana-like convection rolls. Without these essential
modifications, polygonal cells could only be obtained at much
smaller rotational velocities; in this case, the process would
develop very slowly, and the computations would be extremely time
consuming.
[14] Thus the equations of motion for the velocity vector
u,
the heat equation for the deviation
Q from the static
temperature distribution, and the equation of induction for the
magnetic field
B are
 | (3a) |
 | (3b) |
 | (3c) |
 | (3d) |
 | (3e) |
where
p is an effective pressure.
[15] Six nondimensional physical parameters of the problem appear in
our formulation. The Rayleigh numbers measure the energy input
into the system,
 | (4) |
and are associated with the internally distributed heat sources
q and the externally specified temperature difference
Ti - To (see equation (2)), respectively. The Coriolis number
t, the Prandtl number
P, and the magnetic Prandtl number
Pm describe ratios between various timescales in the
system,
 | (5) |
( nm is the magnetic viscosity, or magnetic
diffusivity). Finally,
is the small constant that
specifies the magnitude of the quadratic term in the temperature
dependence of density (see equation (3b)).
[16] Since the velocity field
u and the magnetic induction
B are solenoidal vector fields, the general
representation in terms of poloidal and toroidal components can be
used,
 | (6a) |
 | (6b) |
By taking the (curl)2 and the curl
of the Navier-Stokes equation (3b) in the rotating system by
r, we obtain two equations for
v and
w,
 | (7a) |
 | (7b) |
where
j denotes the azimuthal angle ("longitude") in the
spherical system of coordinates
r, q, j, and the
operators
L2 and
Q are defined by
The heat equation (3b) can be rewritten in the form
 | (8) |
Equations for
h and
g can be obtained multiplying the equation
of magnetic induction (3e) and its curl by
r,
 | (9a) |
 | (9b) |
[17] We assume stress-free boundaries with fixed temperatures,
 | (10) |
For the magnetic field, we use electrically insulating boundaries
such that the poloidal function
h must be matched to the
function
h( e) that describes the potential fields outside
the fluid shell
 | (11) |
[18] The numerical integration of equations (7)-(11) proceeds with a
pseudospectral method developed by
Tilgner and Busse [1997]
and
Tilgner [1999],
which is based on an expansion of all
dependent variables in spherical harmonics for the
q and
j dependences; in particular, for the magnetic scalars,
 | (12a) |
 | (12b) |
(with truncating the series at an appropriate maximum
l ), where
Plm denotes the associated Legendre functions. For the
r dependences, truncated expansions in Chebyshev polynomials are
used. The equations are time stepped by treating all nonlinear
terms explicitly with a second-order Adams-Bashforth scheme
whereas all linear terms are included in an implicit
Crank-Nicolson step.
[19] For the computations to be reported here, a minimum of 33
collocation
points in the radial direction and spherical harmonics
up to the order 96 have been used. In addition to the geometric
parameter
h and the above mentioned physical parameters, we
specified a computational parameter, namely, the fundamental
(lowest nonzero) azimuthal number
m0. Thus only the following
azimuthal harmonics were really considered:
In other words, we imposed an
m0 -fold symmetry in the
j direction. If
m0 1,
this reduces the computation time.
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