On the other hand, by a similar manipulation or the same two equations, we obtain the quasi-geostrophic height tendency equation for $\chi := \partial \Phi /\partial t $
Both equations have are related to "vertical motion" ($\omega, \chi$), are they telling more or less the same? I marvel at the amazing similarities and the subtle differences of the two terms on the right. What are the applications of both equations in particular - when to use the first, when the second?