AbstractIt is shown that both the steady and the vacillatory states of a two‐layer, quasi‐geostrophic, forced dissipative topographic model with a high spectral resolution can be truly intransitive. There exists only an odd number of equilibria in this geophysical system, as predicted by Benjamin's general theorem for the Navier‐Stokes equation. The overall findings constitute a generalization of previous results for a barotropic model.The analyses specifically establish for the case of a single‐wave topography that, in response to a barotropic forcing, there exist: (1) a low threshold value of the topographic height required for the existence of hysteresis: (2) a higher threshold value for the occurrence of weakly vacillating states with a high frequency; (3) a detuning effect of the nonlinearity due to an increase of the equivalent resonance frequency; (4) a baroclinization effect of an asymmetric friction; and (5) a robustness of the multiple equilibria for an additional weak baroclinic forcing.Under near‐resonance conditions of the topographic waves for the case of a simple multi‐wave topography, as defined according to the linear dynamics, none of the equilibria is stable. The strong wave‐wave interaction in such a situation sustains a dynamically different pronounced vacillation with a dominant long period in the order of tens of days. The existence of multiple vacillations with distinctly different structure and period is established. Outside that range of parametric conditions, there is only one stable equilibrium state.It is demonstrated that the time‐mean wavy flow of a decidedly nonlinear time‐dependent response can be reproduced, in a high degree of resemblance, by tuning the dissipation parameters in a counterpart linearized model that makes use of the ‘known’ mean zonal flow. It would be misleading, however, to conclude, on the basis of such an agreement, that the linear dynamics really accounts for the actual time‐mean wavy flow. This example serves to reveal the potential dangers in interpreting an observed mean flow as consisting of linear planetary waves on a pre‐determined zona