Analogues of potential vorticity in electrically-conducting fluids

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Abstract
Exploiting the (contravariant) vectorial form for vorticity and magnetic field, and co-(variant) vectorial form for the gradient of a scalar and magnetic potential, and using the geometric Lie derivative operator, conservation of various analogues of potential vorticity are discussed for a barotropic non-dissipative electrically-conducting fluid. These analogues include the potential magnetic field, helicity, magnetic helicity, and cross helicity, together with some higher order quantities. It is noted that the volume conservation of potential vorticity continues to hold in the presence of arbitrary dissipation. However, of the analogue quantities derived for the non-dissipative system, only potential magnetic field and cross helicity have invariant integrals in the presence of dissipation. We conclude that only they are true analogues of potential vorticity. Finally, a straightforward generalisation of the method for tensorial relationships is noted.

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