位涡梯度、四元数和流体质点的正交标架
The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles
摘要：
位涡梯度是一个重要的量，因为位涡（表示为$q$）倾向于在海洋和大气中局部积累。作者最近的分析表明，三维不可压缩旋转欧拉方程的矢量$\bdB = \bnabla q\times \bnablaθ$与磁流体力学（MHD）中的涡量$\bom$和磁场$\bB$都是按照相同的拉伸方程发展的。因此，$\bdB$矢量的作用类似于欧拉方程中的涡量$\bom$和MHD中的$\bB$场。例如，它允许各种类比，如拉伸动力学，螺旋度，超螺旋度和交叉螺旋度。此外，使用四元数分析，$\bdB$ -向量的动力学自然允许构造附着到流体粒子的正交标架;这被指定为四元数标架。当PV梯度$\bnabla q$较大时，该框架的排列动力学与粒子在穿过流区域时所经历的三轴旋转特别相关。
英文摘要：
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 $\bdB = \bnabla q\times \bnablaθ$ 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.
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