[MITgcm-support] application of Gauss (or Stokes) theorem to vorticity eon on C-grid

[Paola Cessi]

Dear colleagues,

I am trying to calculate the circulation balance over a semi-enclosed area, by taking the vorticity balance and integrating over said area. One of the terms in the vorticity balance is div (fu, fv), which, when integrated over the area, should give the integral over the boundary of (fu,fv) \cdot n, where n is the normal to the boundary. Over the closed portion of the boundary, i.e. on the solid walls, (fu,fv) \cdot n should be zero. However, because the vorticity points are not on the velocity boundary faces, these terms do not vanish. The residual left from this contribution dominates the balance, so it is not a small error.

I have tried different versions of the vorticity definitions, i.e. the volume flux form as in  (equation 13) of Adcroft et al https://journals.ametsoc.org/view/journals/mwre/132/12/mwr2823.1.xml <https://journals.ametsoc.org/view/journals/mwre/132/12/mwr2823.1.xml>, or the simple derivatives and neither of them work. 

Does anyone have any words of wisdom or references that could be helpful?

Many thanks,
Paola Cessi

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Paola Cessi                                                         Tel: +1 858 534 0622 
Scripps Institution of Oceanography                  Fax: +1 858 534 8045
9500 Gilman Drive #0213                             e-mail: pcessi at ucsd.edu 
La Jolla, CA 92093-0213          Web: http://pordlabs.ucsd.edu/pcessi
USA               

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