The development of the potential vorticity staircase in freely decaying turbulence

Recently, two papers (Scott and Dritschel, 2012; Scott and Tissier, 2012) demonstrated the role of two dimensionless parameters in controlling the tendency for forced two-dimensional turbulence on the beta plane to mix potential vorticity into a distinctive staircase-like distribution comprising piecewise uniform regions separated by strong gradients. The dimensionless parameters are obtained from the natural length scales in the problem: the Rhines scale based on total energy; a forcing intensity scale based on the energy input rate; and a forcing scale based on the spectral wavenumbers excited by the forcing. Two cross ratios give the dimensionless parameters controlling the degree of inhomogeneous mixing; the third ratio controls the nature of that mixing, whether by small-scale turbulent eddy motions or by larger Rossby-wave critical layers. In this study we consider the arguably more fundamental case of freely decaying turbulence. Based on the spectral distribution of energy in the initial conditions, two length scales are identified that can be considered analogous to the scales based on energy input in the forced case, while the Rhines scale is again based on the total energy. High resolution numerical integrations of the equations demonstrate the influence of the corresponding dimensionless parameters on the late-time development of the potential vorticity staircase and the type of mixing involved in its formation.