If the momentum, energy and circulation of a fluid in a periodic, quasi-geostrophic, β-plane channel are specified, then there is a minimum enstrophy implied. This minimum enstrophy flow is obtained using the calculus of variations and is found to be also a solution of the quasi-geostrophic equations. It is either a parallel flow or a finite-amplitude Rossby wave, depending on the aspect ratio of the channel and the amount of energy and momentum within it. The most geophysically relevant case is a channel whose zonal length is substantially greater than its meridional breadth. In this instance the form of the minimum enstrophy solution is decided by the ratio of the energy to the squared momentum. When this parameter is below a Critical value one has a parallel flow, while if this value is exceeded, the minimum enstrophy, solution is a Rossby wave.
Heuristic arguments based on the enstrophy cascade in two-dimensional turbulence suggest a “selective decay hypothesis”. This is that scale-selective dissipation will decrease the enstrophy more rapidly than the energy, momentum and circulation. If this is the case, then the system should approach the minimum enstrophy solution.