What does this equation mean?

A model of the barotropic vorticity equation used in that 1950 work can be written, in simplified form, as $\frac{\partial \zeta}{\partial t} + J(\psi, \zeta) = 0, \qquad \zeta = \nabla^2 \psi$. where $\psi$ is the stream function describing the horizontal flow, $\zeta$ is the relative vorticity, and J($\psi$,$\zeta$) is the Jacobian term expressing advection of vorticity by the flow itself. This is stated here not to reproduce the numerics of the original run but because the equation exposes exactly what made the problem hard to do by hand and tractable by machine: it is a single nonlinear evolution equation, and marching it forward requires repeating the same arithmetic over many small time steps across many grid points — precisely the kind of repetitive, exact, high-volume calculation electronic computation was built for and hand computation was not.