Relaxation and statistical equilibria in generalised two-dimensional flows

We study relaxation toward statistical equilibrium states of inviscid generalised two-dimensional flows, where the generalised vorticity $q$ is related to the streamfunction $\psi$ via $q=(-\nabla^2)^{\frac{\alpha}{2}}\psi$, with the parameter $\alpha$ controlling the strength of the nonlinear interactions. The equilibrium solutions exhibit an $\alpha \mapsto -\alpha$ symmetry, under which generalised energy $E_G$ and enstrophy $\Omega_G$ are interchanged. For initial conditions that produce condensates, we find long-lived quasi-equilibrium states far from the thermalised solutions we derive using canonical ensemble theory. Using numerical simulations we find that in the limit of vanishing nonlinearity, as $\alpha \to 0$, the time required for partial thermalisation $\tau_{th}$ scales like $1/\alpha$. So, the relaxation of the system toward equilibrium becomes increasingly slow as the system approaches the weakly nonlinear limit. This behaviour is also captured by a reduced model we derive using multiple scale asymptotics. These findings highlight the role of nonlinearity in controlling the relaxation toward equilibrium and that the inherent symmetry of the statistical equilibria determines the direction of the turbulent cascades.