I think there is a problem here with an implicit assumption about the numbering convention for whether k=N is the surface or the bottom. In ocean_sigma_coordinate is does not matter if a user numbers from the surface sigma(1) = 0 to bottom sigma(N) = -1, or as in the ROMS model for example sigma(1) = -1 at the bottom and sigma(N) = 0 at the surface (k increasing in the same direction as z). As written, the definition above requires k=1 be sigma=0 and z=eta. The reverse order is not supported.
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