To derive the Rayleigh equation for a barotropic fluid, the barotropic vorticity equation is used. This equation assumes that the absolute vorticity is conserved:  here,  is the material derivative. The absolute vorticity is the relative vorticity plus the planetary vorticity: Here,  and  are the zonal and meridional components of the flow and  is the relative vorticity induced by the perturbations on the flow ( and ).  is the mean zonal flow and  is derivative of the planetary vorticity  with respect to  .