Evolution of Mean-Flow Fofonoff Gyres in Barotropic Quasigeostrophic Turbulence

Abstract

Numerical experiments are performed over a wide range of parameters to show that mean flows in the form of Fofonoff gyres, characterized by a linear relationship between streamfunction and potential vorticity, are universally produced in the statistically steady state of inviscid unforced barotropic quasigeostrophic turbulence, provided that the initial state is sufficiently well resolved. Further, as the resolution is increased, the mean-flow energy approaches the total energy, and the mean-flow potential enstrophy reaches a minimum value, which is lower than the value with no flow. This is in agreement with the predictions of the theory of equilibrium statistical mechanics. The timescale for the appearance of these flows is on the order of 5?10τ?, where τ? is a mean eddy turnover time. When viscosity is turned on, the mean-flow Fofonoff gyres become internally homogenized and eventually disappear entirely as the flow decays to zero. This evolution of the gyres can be universally scaled with a timescale τ? = δ2/?, where δ is the Rhines scale and ? is the viscosity coefficient. There is an initial period of very rapid adjustment on a timescale of ?0.005τ? at the enstrophy accumulated at very high wavenumbers is dissipated, followed by an intermediate period with a timescale of ?0.04τ? during which the gyres are homogenized, and finally a period of gyre decay on a timescale of ?0.3τ?. In general, there is a competition between the statistical tendency to organize the mean flow into Fofonoff gyres and the tendency for homogenization, with the tendency to form Fofonoff gyres being always overwhelmed given a sufficiently long time. Thus, the issue of whether statistical mean flows, such as Fofonoff gyres, emerge and play a role depends on the relative magnitude of the two timescales, τ? and τ?.