A note on helicity

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Abstract
Theoretical work on the generation of magnetic fields by fluid motions, by the so-called “hydromagnetic dynamo” process, has called attention to the importance of the role of helicity H (defined as the scalar product of the velocity and vorticity vectors), implying that the discovery of general relationships between the helicity and other properties of the fluid should facilitate the future study of specific dynamo models. It is shown in the present note that H in a rapidly-rotating Roussinesq fluid satisfies where t denotes time, u is the Eulerian flow velocity relative to a frame which rotates with angular velocity ω, [pbar] + p′ is the density of the fluid and [pbar] the mean density, and g is the acceleration due to gravitational and centrifugal effects. The first term is proportional to the speed of vertical motion and is not in general zero when the kinetic energy of the fluid is produced by buoyancy forces; the second is proportional to the adjective term in the equation of heat transfer, which vanishes in those regions where the flow can be regarded as steady and isentropic but not elsewhere; and the third is associated with processional effects due to time variations in the basic rotation vector ω.

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