Calculate velocity gradient decomposition and Lagrangian averages

[rabernat]

Many of the LCS methods are based on "velocity gradient tensor"

In 2D this is

$$ \nabla \mathbf{u} = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix} $$

This can be decomposed into a couple of different components. A key one is the divergence

$$ \delta = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} $$

This is always ZERO for QG flow and for geostrophic flow in general it is approximately zero.

Another component is vorticity

$$ \zeta = -\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} $$

A final one is strain

All of these quantities can be averaged along particle paths and are the foundation for many LCS methods.

[rabernat]

I need to write some notes on this to be more specific about what we want to calculate.