[MITgcm-support] Leith hor. viscosity

[Baylor Fox-Kemper]

Hi Dmitris,
   I've been using the modified Leith and Smagorinsky on a uniform grid, 
so I have done a crude job of accounting for variable grids (similar to 
what Alistair was doing before with Leith).
   However, I was just taking a look at Steve Griffies' book the other 
day, and I think the modification necessary to make the viscosities 
work on any grid may be as easy as changing the grid length used by the 
scaling from ra^0.5 to 2dx*dy/(dx+dy).  I've got Bob Hallberg's code 
for his c-grid isopycnal model, so I can check how he did it.

> 1) Will modified C2Leith and brand-new Smagorinsky work with spherical
> polar grid and with cubed-sphere grid?
I don't know now.  I suspect that the modified Leith is no better than 
Alistair's old code.  It is possible that the Smag will work, but I am 
pretty sure I need to make the above change first.

> 2) If so, which of the six (harmonic and biharmonic) schemes would you 
> recommend and under what circumstances?

Smagorinsky has been used successfully at GFDL, I know, and they argue 
that a biharmonic Smagorinsky may be useful in marginally-resolved 
cases sometimes, too.  Griffies and Hallberg 2000 has a nice discussion 
and examples.  Theoretically, Smagorinsky is the appropriate scaling to 
use when the subgridscale turbulence is isotropic 3-d.  Smagorinsky is 
widely used in small-scale and engineering applications.

Leith is supposed to be the correct scaling when the unresolved 
turbulence is 2-d, so it is theoretically more appropriate for 
global-scale calculations where the grid scale is much larger than the 
deformation radius.  However, as far as I know Leith is relatively 
uncommon.

Biharmonic Leith scales as 2-d turbulence as well, only it is more 
scale-selective, so like biharmonic Smagorinsky it may be useful in 
marginally-resolved cases.

The modified Leith scheme I added was my own concoction which is 
probably a bit half-baked.  Nonetheless, here's the rationale.  I was 
doing calculations using Leith which had a lot of convection going on 
(I was using a particularly strong diurnal cycle in an idealized 
channel).
I was having a lot of trouble getting a smooth w velocity.  When I 
looked at the output, I saw grid-scale noise in the w-field and in u 
and v.  However, when I calculated the vertical component of vorticity, 
the noise didn't really show up.  Thus, it appeared to be a 
computational mode that was perhaps being forced by the convection.
Since Leith is proportional to grad(vorticity), these modes were 
invisible to the Leith parameterization.  I reasoned that grad(div u) 
had the same dimensions as grad(vorticity), and since Leith was 
conceived for pure 2-d flows (where div u is zero or nearly so), I 
figured it might have been an oversight.  Anyway, for large scale 
flows, the div.u is O(Rossby Number) smaller than vorticity, so the 
difference should be negligible in that case, but it certainly squashed 
the instabilties in my problem.


> 3) Have you or will you implement Alistair's anisotropic-grid scaling 
> correction?
Haven't yet, I need to check out what he wants to do exactly.  I 
suspect that the geometric average may do much of what's needed.

Cheers,
   -Baylor