One of the paradoxes about climate change modelling is that the atmosphere behaves as a fluid and therefore is treated as a problem in fluid mechanics.
Traditionally fluid mechanics is a branch of applied mathematics and one of the most famous equations which climate models utilize is the Navier-Stokes equations.
The first equation based on the conservation of mass is.
v is the velocity.
The second equation in more detail in based on Newton's second law F=ma.
Mathematicians have never been able to prove for the three-dimensional system of equations, and given some initial conditions, that smooth solutions always exist.
A smooth solution f(x,y,z) for the equations is where the derivative of f (x,y,z) exists for each point x, y and z.
This is called the Navier–Stokes existence and smoothness problem and one of the seven great unsolved problems in mathematics constituting the
Millennium Prize.
Solve any one of the problems earns you a million dollars.
The paradox is that while mathematicians struggle with the equations, physicists have no problems in utilizing the equations in their climate change models.
A fluid is treated as a set of points which forms a vector field where each point has both a magnitude and direction such as in the case of a circulating fluid.
The mathematician will look at each point individually whereas the physicist will break down the fluid in a series of localized regions where in each region the points are averaged.
This is basically a smoothing procedure which eliminates any “stray” points.
This is an approximation and by making the regions progressively smaller a more accurate approximation is obtained.
This numerical method approach requires massive computer power and it is no coincidence the increasing accuracy of climate change models is related to improvements in computer hardware.