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arXiv · 1006.3891

The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles

J. D. Gibbon·D. D. Holm
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Abstract

The gradient of potential vorticity (PV) is an important quantity because of the way PV (denoted as $q$) tends to accumulate locally in the oceans and atmospheres. Recent analysis by the authors has shown that the vector quantity $\bdB = \bnabla q\times \bnablaθ$ for the three-dimensional incompressible rotating Euler equations evolves according to the same stretching equation as for $\bom$ the vorticity and $\bB$, the magnetic field in magnetohydrodynamics (MHD). The $\bdB$-vector therefore acts like the vorticity $\bom$ in Euler's equations and the $\bB$-field in MHD. For example, it allows various analogies, such as stretching dynamics, helicity, superhelicity and cross helicity. In addition, using quaternionic analysis, the dynamics of the $\bdB$-vector naturally allow the construction of an orthonormal frame attached to fluid particles\,; this is designated as a quaternion frame. The alignment dynamics of this frame are particularly relevant to the three-axis rotations that particles undergo as they traverse regions of a flow when the PV gradient $\bnabla q$ is large.

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J. D. Gibbon, D. D. Holm. 2010-06-19. The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles. https://doi.org/10.1080/03091929.2010.513117

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