The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles
| Autor: | Gibbon, J. D., Holm, D. D. |
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| Rok vydání: | 2010 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1080/03091929.2010.513117 |
| Popis: | 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 $\bdB = \bnabla q\times \bnabla\theta$ 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. Comment: Dedicated to Raymond Hide on the occasion of his 80th birthday |
| Databáze: | arXiv |
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