Vertical-advection consistency criterion for log-pressure quasi-geostrophy

For scales $U,L,D$, thermal balance estimates $w\sim f_0{U}^2/(N^{2}D)$. Density-weighted continuity gives horizontal ageostrophic flow of order $Lw/\min (D,H)$. It is small relative to $U$ only ifThis condition supplements small Rossby number and small relative variation $\beta L/|f_0|$ of the Coriolis parameter.