  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.
