Square root of biharmonic viscosity

[NoraLoose]

The way the biharmonic friction operator is implemented in MOM_hor_visc is inconsistent with what Griffies & Hallberg (2000) suggest:

• MOM6: div. ( Ah grad ( div . grad U ) );
• Griffies & Hallberg (2000): div. ( sqrt(Ah) grad ( div . (sqrt(Ah) grad U ) ).

The two options are equal if Ah is spatially constant. As Griffies & Hallberg (2000) note, the first operator does not guarantee to dissipate kinetic energy (unless Ah is spatially constant), but the second operator does.

[Hallberg-NOAA]

@NoraLoose, this is a good idea to try out, but it should be considered as a new operator, rather as a modification to the existing one. I would be happy to work with you on a PR implementing these ideas.

Also note that in order to conserve linear or angular momentum in general coordinates (i.e., with varying layer thicknesses), the operator in the momentum equation needs to become something like 1/h div. (h Ah grad ( div . grad U ) ), where h is the layer thickness. So in order to get the same sort of symmetry that you are trying to exploit, the modified expression would become something like 1/h div. ( h sqrt(Ah) grad ( 1/h div . (h sqrt(Ah) grad U ) ). In addition to coming up with plausible averaging of the thicknesses to the right places, we will also need to put some effort into finding timestep-dependent limits on the magnitudes of the Ah fields so that this operator does not overdamp and drive an oscillatory divergence.

Because biharmonic viscosities do not have a strong physical analog (the way that a Laplacian viscosity does), there is not a physical argument that we can make about exactly which form is "right". The best that we can do is to look for something that has some of the physical properties we would like and doesn't do anything too bad in any limits.

[NoraLoose]

Thanks @Hallberg-NOAA for your response and ideas!

My NeverWorld2 analysis suggests that the current operator does actually a good job at dissipating EKE (and presumably also KE), so the 2 operators may not differ too much in practice. But I agree, this would be interesting to try out! I'm hoping that I will have more time in the future to work on this. Right now, I am quite busy with other CPT-related stuff. I will keep you posted - thanks!