Frequency shifts of Rossby waves in the inertial subranges of β-plane turbulence

Nonlinear interactions between waves and turbulence cause systematic frequency shifts in Rossby waves. The frequency shifts in the inertial subranges of statistically steady β-plane turbulence were examined theoretically and numerically. The theoretical analysis is based on the Lagrangian closure called the Lagrangian renormalized approximation and predicts that when the β effect is small, the frequency shifts of Rossby waves are proportional to k_x/k^4/3 in the inverse energy transfer range, while they are proportional to k_x with or without a log-correction term in the enstrophy transfer range, depending on the flow conditions, where k_x is the wave number in the eastward direction. Numerical simulations using 1024² grid points of forced β-plane turbulence that exhibit the inertial subranges, show fairly good agreement with theoretical predictions.