Formation of Jets through Mixing and Forcing of Potential Vorticity: Analysis and Parameterization of Beta-Plane Turbulence

Abstract

Formation of multiple jets in forced beta-plane turbulence is studied from the perspective of nonuniform nonconservative arrangement of potential vorticity (PV). Numerical simulations are analyzed to show that mixing and forcing reinforce jets by concentrating PV gradients at the axes of prograde jets. Based on the formalism developed in the companion paper, the nonconservative driving of jets is diagnosed and parameterized through the diffusive flux of PV and the source of wave activity. It is found that the two terms nearly balance on a long time scale, and they are both strongly anticorrelated with the PV gradient, which suggests that PV controls the nonconservative processes and that these processes could be parameterized as functions of the PV gradient. The flux is modeled using the effective diffusivity formula recently obtained by Ferrari and Nikurashin. Consistent with the PV barrier concept, the nonlinear diffusivity is a decreasing function of the squared PV gradient and agrees well with the diffusivity diagnosed from the numerical simulation. The source term is assumed to be inversely proportional to the PV gradient. The parameterization gives rise to a nonlinear partial differential equation (PDE) for the mean flow. A finite-difference model of the PDE predicts formation of a piecewise linear PV (staircase) and piecewise parabolic jets from a near-uniform initial condition when anisotropy and mixing of the flow are sufficiently strong. The origin of the discontinuities is antidiffusive instability of PV gradients, and although nonlinearity allows the discrete model to integrate stably, the solution is sensitive to the initial condition and resolution. The emerging jets in the 1D model have similar characteristics to those in the numerical simulation, but the details of the transient behavior are distinct. Similar discrete models of ill-posed PDEs in which discontinuities form also appear in image processing and granular matter dynamics.