  The partition of enstrophy between zonal (ordered) and wavy (turbulent)
components of vorticity has been studied for the beta-plane model of
two-dimensional barotropic flow. An analytic estimate of the minimum value for
the zonal component has been derived. The energy, angular momentum,
circulation, as well as the total enstrophy are invoked as constraints for the
minimization of the zonal enstrophy. The corresponding variational principle
has an unusual mathematical structure (primarily because the target functional
is not a coercive form), by which the constraints work out in an interesting
way. A discrete set of zonal enstrophy levels is generated by the energy
constraint; each level is specified by an eigenvalue that represents the
lamination period of zonal flow. However, the value itself of the zonal
enstrophy level is a function of only angular momentum and circulation, being
independent of the energy (and total enstrophy). Instead, the energy works in
selecting the "level" (eigenvalue) of the relaxed state. The relaxation occurs
by emitting small-scale wavy enstrophy, and continues as far as the nonlinear
effect, scaled by the energy, can create wavy enstrophy. Comparison with
numerical simulations shows that the theory gives a proper estimate of the
zonal enstrophy in the relaxed state.
