## Modeling of rapidly rotating thermal convection using vorticity
and vector potential

**Zdenek Mistr** ** & **
**Ctirad Matyska** ** & **
**David A. Yuen**
We have used a numerical scheme based on higher-order
finite differences to investigate
effects of adiabatic heating and viscous
dissipation on 3-D rapidly rotating thermal convection in a
cartesian box with an aspect-ratio of 2x2x1.
Although we omitted coupling with the magnetic field,
which can play a key role in the dynamics of the Earth's core,
the understanding of non-linear rotating convection including
realistic thermodynamic effects is a necessary prerequisite for
understanding full complexity of the Earth's core dynamics.
The system of coupled
partial differential equations has been solved in terms of
the principal variables vorticity $\bh \omega$, vector potential $\bh A$
and temperature $T$.
The use of the vector potential $\bh A$ allows the velocity field
to be calculated with one spatial differentiation in contrast
to the spheroidal and toroidal function approach.
The temporal evolution is governed by
a coupled time-dependent system consisting
of $\bh \omega$ and $T$.
The equations are discretized in all directions by using
an eighth-order, variable spaced scheme.
Rayleigh number $Ra$ of 10$^6$, Taylor number $Ta$ of
10$^8$ and a Prandtl number $Pr$ of 1 have been employed.
The dissipation number of the outer core was taken to be 0.2.
A stretched grid
has been employed in the vertical direction
for resolving the thin shear boundary layers at the top and bottom.
This vertical resolution corresponds to
around 240 regularly spaced points with an eighth-order accuracy.
For the regime appropriate for the Earth's outer
core, the dimensionless surface temperature $T_0$ takes on a large
value, around 4. This large value in the adiabatic heating/cooling
term is found to cause stabilization of both the temperature and
velocity fields.

*Keyword(s):*High Rayleigh number, 3-D convection, higher-order finite
differences, rotating fluid, finite Prandtl number.

### Whole paper

The paper is available in PDF (392 kB).

*Stud. geophys. geod.*,**46** (2002),59-81.