Energetics of horizontal friction in MOM_hor_visc

[NoraLoose]

In the MOM_hor_visc module, the computation of FrictWork and MEKE%mom_src is inaccurate if the grid is curvilinear. The computation is only exact in Cartesian coordinates.

This is what is being computed:
https://github.com/NOAA-GFDL/MOM6/blob/fae570d0a0396cf0ae8960a633e998110dbdad8e/src/parameterizations/lateral/MOM_hor_visc.F90#L1562-L1576
The issue is that

(u(I,j,k)-u(I-1,j,k))*G%IdxT(i,j)

in line 1563 should be substituted by

dudx(i,j) = CS%DY_dxT(i,j)*(G%IdyCu(I,j) * u(I,j,k) - &
                                  G%IdyCu(I-1,j) * u(I-1,j,k)),

i.e., the code is missing some of the metric terms. The metric terms are also missing in the remaining lines in the code snippet above.

I think it would be most efficient to substitute
https://github.com/NOAA-GFDL/MOM6/blob/fae570d0a0396cf0ae8960a633e998110dbdad8e/src/parameterizations/lateral/MOM_hor_visc.F90#L1563-L1564
by

((str_xx(i,j)-RoScl*bhstr_xx(i,j))*sh_xx(i,j),

and in the remaining lines one could use sh_xy.

Everything I have said above also applies to FrictWork in this code snippet.