Just a question I dont know the answer to but maybe someone can give some idea whether it might be correct or not... would Navier Stokes work at the sort of planetary scales ? I do know that Navier Stokes is based on incompressability of fluids with movements dictated by newtons motion laws all wrapped up in those complicated equations. But it seems to me that you would have a hard time calculating currents and gyres and so forth using a pixel grid basis.

I did my GeoTerSys model using pixel grids and I think it works ok when you have rain fall from the sky and small amounts of water are washing down height maps and eroding the landscape. I think this is workable because each bit of water is mostly independent of what other bits of water is doing because its more about the terrain its flowing over. Where it breaks down more is on rivers, especially where they bend and meander, because the flow out of a part of river determines whats going on upstream because of the pressure differences and these pressure zones and then fluid speed is what causes erosion on the banks and then forms the meandering and oxbow lakes etc. All of which are not very present in my app nor any other that I have seen. If Navier Stokes had been used on the river flow then it would have produced a more accurate fluid flow, pushed up the water speed on the outsides of bends, and probably got better river bank erosion and therefore river location. Most of the rivers I was modelling were at a scale where they were only a few pixels wide and therefore did not have enough data or volume to have made using Navier Stokes modelling to give any kind of reasonable results anyway.

But when thinking planetary scale, the seas are not a few pixels wide any more and it would seem to me that what is happening at one zone in the oceans drastically affects whats going on in another quite far away. The water in a pixel cell is not at all independent any more. This would also be true of atmopspheric fluids like jet streams etc. Since ocean currents directly affect atmospheric currents and the atmospheric current directly affect rain and the other important parts of climate modelling then I would have thought that its important to consider using something like Navier Stokes to model the flow going on. I think this problem is really difficult indeed and I would assume that its the one being solved by the meteorological institutes around the world on their super computers to give us the weather reports.

So in some magical world where anything goes its all ok but if modelling such as to give earth climates then I am not sure there can be a simplified algorithm to roughly give the right answer. I think the complexity is required to be considered / computed to get anything approaching a realistic answer.