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The quasi-geostrophic $\omega$-Equation

enter image description here is a diagnostic equation for $\omega$ (=dp/dt = vertical motion) in terms of geopotential. It is derived in a few steps by manipulating vorticity and temperature equation.

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 $ enter image description here

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?

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