4 research outputs found
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 ) 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 birthda