What is the name of this equation? I have found this picture but I don't know the name of the equation in it. Another thing, what kind of plots are those in the picture?

I have also tried to re copy it:
$$
\frac{\overline{\partial\zeta}}{\partial t}=-\overline{\mathbf{u}'\cdot\nabla\zeta'}-\left(\overline{\mathbf{u}}\cdot\nabla\overline{\zeta}+\beta\overline{v}-f\frac{\partial \overline{w}}{\partial z}\right).
$$


 A: After a bit of searching I found the following paper: http://journals.ametsoc.org/doi/abs/10.1175/2009JPO4093.1 It has figures very similar to these, and was authored by the people who I suspect are responsible for the image you posted.
The large plot in the background is, I think, a plot of the velocity of fluid flow in the Atlantic ocean (the large white chunk on the left is north america.)
If I had to guess at what the parameters in the equation represent, I'd say $\vec{u}$ is the velocity, $\beta$ could be the "first-order effects of variations in Coriolis force with latitude in planetary dynamics" (from wikipedia), $\zeta$ could be the relative vertical vorticity (again, an educated guess from reading wikipedia), and $f$ is probably the Coriolis parameter. I'd also hazard a guess that $v$ and $w$ are the velocities in the $y$ direction (i.e. latitude) and $z$ direction (i.e. depth). Some fluid texts I've read use the prime symbol (i.e. $'$) to denote a small variation with respect to some reference.
