Watched the hokey movie Gifted on a plane ride. Turns out that the Millennium Prize for mathematically solving the Navier-Stokes problem plays into the plot.

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.

I give a solution here, which was originally motivated by QBO.

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:

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This might be of use in winning the prize:

Singularity of Navier-Stokes Equations

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There is no singularity in the Navier-Stokes equation as stated in the Official Problem Description (OPD) of the NS Millennium Problem. In the OPD, the incompressible fluid form of the NS equation (where the divergence of the fluid velocity is constrained to zero) is what is used, and I have shown existence, smoothness, and uniqueness of solutions to the NS equation of this form, thereby solving this Millennium Problem (at least unofficially). This need not be the case, however, for other forms of the NS equation. For example, if the fluid has a finite acoustic velocity, then supersonic flow would be possible which would result in sonic shock waves.

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“zero driving-force”Will always dissipate with zero driving force?

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That is correct. The energy of fluid motion will always approach zero with finite viscosity and zero driving force. That doesn’t mean, however, that the fluid velocity approaches zero smoothly everywhere. Proving (or disproving) the smoothness of this function is the purpose of the Navier-Stokes Millennium Problem.

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Good luck with your paper!

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