Effect of scalar nonlinearity on zonal flow generation by Rossby waves

Effects of scalar nonlinearity on the generation of zonal flow by Rossby waves in shallow rotating fluid are considered. Zonal flows are generated via the action of Reynolds stress due to vector nonlinearity together with the effects of scalar nonlinearity. It is shown that the scalar nonlinearity reduces the amplitude threshold of the zonal flow instability. In addition, it increases the range of wave vectors of unstable modes subjected to the instability. The growth rate of the instability as a function of the spectrum of primary waves is calculated. The spectrum is assumed to be arbitrary with emphasizing the case of two monochromatic waves.