The behavior of two quasi-geostrophic, β-plane formulations under two translation operations is considered. The translation operations are zonal coordinate translation, in which the coordinate frame is translated zonally with uniform velocity U0 relative to the β-plane, and zonal flow translation, in which U0 is added to the flow at all points at some initial time. The quasi-geostrophic β-plane formulations are the Type 1 set (QG 1: valid when N2H ≪ g) and the non-Doppler set (NDQ: valid when N2H≈g, where N is the buoyancy frequency, g the acceleration due to gravity and H the vertical scale of the motion). Both formulations are here defined with geometric height as vertical coordinate. QG1 motion is invariant to zonal coordinate translation, and the only effect of zonal flow translation is a Doppler shift whereby U0 is added to the flow at all points at all subsequent times. NDQ motion is invariant to zonal coordinate translation if the accompanying change in apparent vertical is allowed for, but the effect of zonal flow translation is not a simple Doppler shift. This non-Doppler property is evident in various published stability analyses, and it also appears in certain stability criteria (some of which are derived here for the first time). Zonal coordinate translation invariance is used to re-express some established stability criteria for both QG1 and NDQ flows.
The zonal coordinate translation properties of thc, β-plane formulations are substantiated by considering the corresponding zonal coordinate rotation properties of the meridional component of the hydrostatic Navier-Stokes equation on the sphere. It is shown that the usual form of this component equation exhibits the required invariance only if the accompanying change in apparent vertical is allowed for. The parameter N2H/g is interpreted for the case of certain zonal flows in thermal wind balance on the sphere. It represents the ratio of Δα (the angular range of apparent verticals seen by observers moving with the flow at heights differing by H) and α1 (the slope of the isentropes).
Examination of the behavior of the whole NDQ set under zonal coordinate translation reveals that the horizontally non-divergent flow depends strictly on the adopted coordinate frame. Further investigation of the consequences of this effect is advocated.