On the representation of Rossby waves on the β-plane by a piecewise uniform potential vorticity distribution

To bridge quasi-geostrophic dynamics and its discrete representation by a series of piecewise constant potential vorticity (PV), the dispersion relation for the Rossby wave in the single-layer β-plane is compared with that for the normal mode of edge waves straddling an infinite series of PV discontinuities (‘PV staircase’). It is shown that the edge waves over evenly spaced, uniform-height PV steps converge to the Rossby wave on the β-plane as Δ → 0, L → 0, Δ/L = βeff (Δ, L and βeff are the step size, step separation and the effective β, respectively), whereas they reduce to the single-step edge wave in the short-wave limit. For sufficiently small step separations, the difference in the phase velocities of the edge wave and the Rossby wave scales as O(L2). Two effects of increasing L on the zonal propagation are identified: (i) increased phase and group velocities in the short-wave limit due to an increased zonal wind at the PV steps and (ii) decreased phase and group velocities in the long-wave limit due to a decreased effective meridional tilt of the mode. The reduced tilt also severely limits the meridional group propagation. The relationship between the edge wave mode and the finite-difference approximation to the Rossby wave is also discussed.