Some points along these lines will be made here. In a sample of thirteen forecasts done with the Eta Model in both its eta and its sigma mode, much greater negative geopotential height errors were obtained in sigma, reminiscent of the strikingly ubiquitous cold bias of atmospheric models. For some of these results, and references to others, see Mesinger et al.
Indications are thus compelling that a real disadvantage of the eta is at hand.
Reexamination of the eta formulation is thus warranted. This differed from the usual practice of defining the coordinate independently of and prior to discretization. Notice, though, that, using the same definition, the eta coordinate can also be defined prior to discretization. At that point, shape of the topography is arbitrary and at the ground surface air flow through eta coordinate surfaces is permitted. Preliminary forecast results from a step-mountain eta coordinate regional model.