Quasi-Geostrophic (Q-G) Omega Equation

            The Quasi-Geostrophic Approximation assumes, among other things, geostrophic and hydrostatic balance. Noting that advection is overshadowed by the geostrophic contribution, only allowing limited departures from the geostrophic balance, is reason it is not simply geostrophic and, instead, quasi-geostrophic. However, the advection of vorticity and thermal gradients usually disturb the geostrophic and hydrostatic balance, which is where the quasi-geostrophic equation could come in handy. To put it briefly, the Q-G equation, on a hypothetical vertical motion field, restores the geostrophic and hydrostatic balance accurately and instantaneously. In other words, the vertical motion could be considered a response to the disrupting factor of geostrophic advection on a system. Although, more importantly, this should be thought of as a hypothetical scenario due to the fact that there is no physical manifestation thus cannot be measured.

The Quasi-Geostrophic Omega Equation represents a method for diagnosing midlatitude, synoptic-scale vertical motions at a specific time. Neglecting diabatic processes, it implies that vertical motion can be calculated from a series of geopotential height analyses at different pressure levels—it is a diagnostic measure of vertical motion based on geopotential height.

For 3-D laplacian of omega, ω (vertical motion) it is important to remember that the sign of the term is proportional to the negative of ω. It is, also, common to assume the dominance of vertical motion is sinusoidal: approximately zero at both the surface and tropopause, and attaining a max/min value in the mid-troposphere hence, qualitatively, like a minus sign.

The vertical differential of geostrophic absolute vorticity advection term is proportional to the rate of increase of geostrophic absolute vorticity advection with increasing height. Overall, vorticity advection increasing with height forces synoptic-scale upward motion. However, vorticity advection at some pressure (mb) alone does not force the vertical motion, it is the change of vorticity advection with height that does.

The 3-D laplacian of thickness (thermal) advection relates to the laplacian of (horizontal) temperature advection to vertical motion, ω—which are greatest when the gradients of temperature advection are large. The dot product, within the brackets, is proportional to the negative of geostrophic advection of thickness.

Nonetheless, warm air advection also plays a role in the Q-G equation because it will increase the thickness of the layer, resulting in higher heights aloft compared to below. Which implies the formulation of anticyclonic vorticity aloft and cyclonic below. In the absence of vorticity advection there is divergence aloft and convergence below. Which, thanks to the vorticity equation, we know that in order to decrease vorticity there has to be negative vorticity advection or divergence.