A note on 1-dimensional calculations of baroclinic and barotropic instabilities

Linear instability of zonal jets are calculated within the framework of the quasi-nondivergent potential vorticity equation. The vertical structure of the flow is treated using vertical normal mode expansion. Three simple 1-dimensional cases are investigated. Firstly, the baroclinic flow on the β-plane, where instability is seen as function of the shape and strength of a jet mimicking the vertical structure of a typical zonal jet in the atmosphere. The study shows that the jet is most unstable for a low-lying jetmaximum. Secondly, a purely barotropic flow on a β-plane, with fixed boundaries in north and south and with periodic boundaries in east—west, mimicking the west wind belt is investigated. In this simple case, the linearized instability equation is solved both spectrally and using finite differences in order to illustrate the dependence on resolution. The stability of the jet is independent on the position within the channel. Finally, for each vertical normal mode, equivalent to a barotropic flow, the stability of a typical west-wind jet on the sphere is calculated. In this case, the jet is more stable the further away from the equator, thus the Coriolis force stabilizes the flow. These very simple examples are in general agreement with findings from full 3-dimensional stability analysis.