Q-Vector Form of Quasi-geostrophic Omega Equation

A version of the QG-omega equation can be obtained directly from the Equations of Motion with the beta-plane approximation (p. 249). See Bluestein, pp. 350-352 for discussion and derivation.

(1)

where the Q-vector is given by

(2)

For a given static stability and isobaric level, the Q-vector is proportional to the horizontal shear of the geostrophic wind and the horizontal temperature gradients..

The divergence of the Q-vector can be shown to measure the tendency for the geostrophic wind field to locally increase or decrease thickness (alter the shape of the thickness field). It should be noted that this term includes BOTH the changes in thickness (and heights) associated with the translation of features (embedded in the differential vorticity advection term of the QG omega equation) AND the development of features (embedded in the temperature advection term).

The operation required by equation (1) is obviously not technically difficult, merely involving calculation of gradients. However, expansion of the Divergence of Q term is formidable (involving a rule of the product from hell).

Most of the "flavor" of the QG forcing can be estimated by consideration of the Divergence of Q term alone. In fact, as Bluestein points out, there is a "Trenberth" version of (1) in which the advection of planetary vorticity and the thermal deformation embedded in the far right hand term are neglected on an order of magnitude basis.

In that case, it is often useful merely to PLOT the Q-vectors at points on a chart (rather than attempting an actual calculation of their magnitudes.).

Task: Write a general rule relating Divergence of Q and the qualitative nature of the QG-induced omega:

(From blackboard discussions in early February 2005 of Q-vectors, but also from simple interpretation of the equation above):

- omega ~ Q-vector convergence

In short, quasigeostrophic forcing for upward motion occurs, at a given level, in areas of Q-vector convergence

QG Forcing Example

(Click on Graphics to See Full Size)

500 mb Heights and Absolute Vorticity

Surface Isobars and 1000-500 mb Thickness

850 mb to 500 mb thickness and 700 absolute vorticity

Q-vector Convergence (convergence>0)