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__Preserving Taylor's constraint in magnetohydrodynamics__

J. Niesen and P. Livermore

**Abstract**

The Earth's magnetic field is generated in the outer core,
whose behaviour is described by a combination of the Navier-Stokes
equation and the Maxwell equation. A problem when solving these
equation is that a certain number, the Ekman number, is very small
(around $10^{-15}$). Usually, people use big computers to solve the
equations, but even then the Ekman number in the simulations can only
be taken to be $10^{-6}$. The alternative is to take the Ekman number
equal to zero. This corresponds to a singular limit, in which the
equations degenerate into a partial differential algebraic system.
Taylor's constraint is an infinite number of quadratic conserved
quantities in this limit and it is thought that this constraint causes
instabilities in earlier attempt to solve the equations. I will
describe our on-going efforts to design and implement a numerical
method for solving the equations which preserves Taylor's constraint,
making it hopefully stable.