The fundamental equations describing weather and climate are the Navier-Stokes partial differential equations of fluid motion.
http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations
These are solved in computer models over a finite grid by approximate numerical methods. The equations are non-linear. Small changes in initial conditions cause the solution to diverge. Changes in boundary conditions change the probability distributions – the probability of the occurrence of one state or another – they shift either metrically or topologically. The numerical values change or the shape of the distribution changes.
On the left hand side the solutions diverge exponentially to the limits of intrinsic variability. On the right hand side the metrical and topological changes in frequency distribution are shown.
Fluid motion – both atmosphere and oceans – is the essence of weather and climate. There are 2 great systems driving both weather and climate. The first is at the polar vortices pushing turbulent eddies in atmosphere and ocean into lower latitudes. The second involves the upwelling of cold water in the eastern Pacific and the resultant feedback in wind, currents and cloud.
Perhaps as interesting in the longer term is the dynamics governing thermohaline circulation and the potential for runaway ice and snow feedbacks. We have a few very theoretical ideas for what has driven the glaciations of the past 2.58 million years – but are worryingly clueless as to what might cause the next and when it will be.