Benjamin–Feir instability of Rossby waves on a jet

Large‐scale waves on the extratropical tropopause have been widely observed to spontaneously organize into groups or wave packets. Here, a simple paradigm for this wave packet formation is presented. Firstly, a weakly nonlinear theory of Rossby wave propagation on a potential‐vorticity front, based on small non‐dimensional wave amplitude ε, is developed. As is typical for systems allowing conservative one‐dimensional wave propagation, the evolution of the wave envelope is governed by the nonlinear Schrödinger equation. The sense of the nonlinearity is consistent with Benjamin–Feir instability, where uniform wave trains are unstable to sideband modulations, leading to the formation of wave packets. Next, numerical results from contour dynamics integrations show that the weakly nonlinear predictions for sideband growth rates are quantitatively accurate up to ε∼0.5, and that unstable sideband growth is qualitatively similar, but slower than predicted, at higher values of ε. For ε≥0.6 the formation of wave packets leads to wave‐breaking, this occuring at much lower initial wave amplitudes than for unmodulated uniform wave trains previously studied. The numerical results reveal that the length and time‐scales of the Benjamin–Feir instability are broadly consistent with observed wave packet formation in the extratropics.