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

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 $\bdB = \bnabla q\times \bnabla\theta$ 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.