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

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 $\text{bdB}=\text{bnabla}q\times \text{bnabla}\theta$ for the three-dimensional incompressible rotating Euler equations evolves according to the same stretching equation as for $\text{bom}$ the vorticity and $\text{bB}$, the magnetic field in magnetohydrodynamics (MHD). The $\text{bdB}$-vector therefore acts like the vorticity $\text{bom}$ in Euler’s equations and the $\text{bB}$-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 $\text{bdB}$-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 $\text{bnabla}q$ is large.