Superhelicity, helicity and potential vorticity

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Abstract
‘‘Helicity'’ density H ≡ u · ω and other pseudo-scalar fields such as P ≡ ω · Vlnρ (which is related to Ertel potential vorticity) are useful quantities in theoretical fluid dynamics and magneto-fluid dynamics. Here u denotes the Eulerian flow velocity relative to the chosen frame of reference, ω ≡ V × u is the corresponding relative vorticity and ρ the mass density of the fluid. A general expression is readily obtained for ∂H/∂t (where t denotes time) in terms of P and the ‘‘superhelicity'’ density S ≡ ω · V × ω which, in fluids of low viscosity, has its highest values in boundary layers. One need for such a relationship became evident during an attempt to interpret the findings of laboratory experiments on thermal convection in rotating fluids in containers of various geometrical shapes and topological characteristics. In electrodynamics an analogous expression can be found relating the time rate of change of ‘‘magnetic helicity'’ A · B to ‘‘magnetic superhelicity'’ B · ▿ × B (where B · ▿ × A is the magnetic field) and a scalar quantity analogous to P which involves non-Ohmic contributions to the relationship between the electric current density and the electric field.

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