I am interested in variations of the Navier–Stokes equations that describe hydrodynamical flow on the surface of a sphere. The premise is that such a formulation can be used to perhaps model ENSO and QBO.
The so-called primitive equations are the starting point, as these create constraints for the volume geometry (i.e. vertical motion much smaller than horizontal motion and fluid layer depth small compared to Earth's radius). From that, we go to Laplace's tidal equations, which are a linearization of the primitive equations.
Of course the equations are under-determined, so the only hope I had of solving them is to provide this simplifying assumption:
If you don't believe that this partial differential coupling of a latitudinal forcing to a tidal response occurs, then don't go further. But if you do, then: