Comment by goodbyegti

A really nice intro is this classic paper on numerical integration of the equation for barotropic vorticity:

http://mathsci.ucd.ie/~plynch/eniac/CFvN-1950.pdf

It's something you can solve in a few lines of python and run with minimal compute resources. A nice thing to try is to feed in some data from somewhere like below and watch what happens as you integrate it forwards in time.

http://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalys...