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WRF SCM horizontal diffusion?

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I am using the WRF SCM for my experiments. The SCM uses a 2x2 grid, and I have been told that the grids are strongly coupled by the use of strong horizontal mixing (khdif in the namelist, which is the horizontal eddy diffusivity constant as far as I understood). So basically the four grids should have the same values for all the variables. This is indeed the case for the default SCM case (cu_physics = 0). However, I have found that once I switch on a cumulus scheme, the four grids diverge for some cumulus schemes, indicating they are not strongly coupled(e.g. Arakawa Schubert, Betts-Miller-Janjic),while for some scehems they are ok and remain strongly coupled (e.g. Kain Fritsch).

In my configuration, dx and dy are 100km. I have tried a few khdif values (300, 2500, 7500, … 10000). But none of them seem to work. I read up on literature and it appears that khdif might be related to grid size, but I am not sure how are they related.

Now I am left wondering if this discrepancy between the four grids is because I haven’t chosen an appropriate khdif value, or is it diagnostic of some problem in the cumulus scheme in use....

Can anyone provide any insight? If not by manipulating khdif, how else can I strongly couple the four grids?

Thanks very much!
I would first like to apologize for the long delay in response to this post. It seems that it was overlooked by our group at the time it was posted. Have you been able to solve this problem yet?