The symmetric instability of a simple shear flow in which the velocity is a linear function of the vertical coordinate but which varies slowly in the cross-stream direction is studied using an asymptotic analytical method. Explicit analytical solutions are found for the evolution of the envelope of the developing linear instability. Although the problem with no lateral variation yields cell-like instabilities growing in place, the lateral variation of the shear produces time dependence and cross-stream propagation of the envelope and accompanying cells. A similar solution is derived for the case of laterally uniform shear in a current whose depth slowly varies exponentially in the cross-stream direction producing similar time dependence to the otherwise stationary cell pattern.
Pedlosky, Joseph. 2014. "Symmetric instability of cross-stream varying currents." Journal of Marine Research 72, (1). https://elischolar.library.yale.edu/journal_of_marine_research/384