Numerical experiments are performed to directly test the emergence of the Fofonoff solution in an inviscid closed barotropic domain, and to explore its significance to the weakly dissipative system. The Fofonoff solution, characterized by a linear relationship between absolute vorticity and streamfunction, is generally realized as the time mean state of inviscid simulations over a fairly broad parameter range of varying (β-plane) Rossby number and resolution, in different geometrical domains, and with and without topography. The relevance of the Fofonoff solution to the viscous, decaying system is examined by numerical experiments with two different forms of viscosity, namely, biharmonic and harmonic, as well as with various boundary conditions. It is found that the boundary condition is generally more important than the order of the viscosity in determining the time mean fields. All of the frictional forms and boundary conditions prevented the complete realization of the Fofonoff state to a greater or lesser extent. Of the various boundary conditions used, the super-slip condition is most conducive to realizing a Fofonoff state. In this case, at high enough resolution the timescale of energy variability is much longer than a dynamical timescale, and the Fofonoff flow may be considered a ‘minimum enstrophy’ state. At high Reynolds number and high Rossby number an almost linear q — ψ relationship can be achieved. For lower Rossby numbers, absolute vorticity tends to become homogenized, preventing the Fofonoff solution from arising. In the case of a free slip condition, it is still harder to reach a quasi-equilibrium. The time mean fields, after spin-up, generally show a two-gyre structure with homogenization in the absolute vorticity fields. In the no slip case, neither a quasi-equilibrium nor any well formed time mean field can be reached. As a slight generalization of the flow on β-plane, the inviscid topographic experiments also ultimately yield a linear relationship between absolute vorticity and streamfunction.
Wang, Jian, and Geoffrey K. Vallis. 1994. "Emergence of Fofonoff states in inviscid and viscous ocean circulation models." Journal of Marine Research 52, (1). https://elischolar.library.yale.edu/journal_of_marine_research/2089