Does anyone know some of the contemporary mathematical challenges associated with the fluid dynamics behind climate modelling? Of course the Navier-Stokes equations are relevant, but even with these I am unsure of what approaches mathematicians who are particularly interested in atmospheric or oceanic fluid dynamics are taking when it comes to studying them (as opposed to those with interests in complex fluids for engineering purposes). I am looking for mathematicians who research these things, but since I don't know what exactly they would be researching it is difficult to find them.
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In geophysical fluid dynamics, one cares in particular about planetary rotation and gravity in combination with a fluid model (such as the Navier-Stokes equations) in a spherical geometry. So for mathematicians in atmospheric and oceanic fluid dynamics, the wellposedness of the Navier-Stokes and/or Euler equations is a central topic. However, since this problem is very difficult in three dimensions, approximations are often made.
An important achievement was the paper by Cao and Titi (2007) in the Annals of Mathematics where they proved that the primitive equations (which follow as a hydrostatic approximation of the 3D Navier-Stokes equations with gravity and rotation) are globally well-posed in a cylindrical domain given regular enough initial data and appropriate boundary conditions.
A good starting point for finding related work is by seeing who has cited this paper. This may give you a good overview of what mathematicians in geophysical fluid dynamics are working on.
Erwin
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