Potential Vorticity

Last semester, we derived the simplified vorticity equation from the Principle of Conservation of Absolute Circulation. Recall that vorticity is circulation per unit area, or

(1)

If there are no solenoids, and no friction, assume that absolute circulation is conserved. Thus, (1) becomes

(2)

or

(3)

Equation (3) can be used to derive the Simplified Vorticity Equation. Then by assumption of non-divergence, the equation reduces to Conservation of Absolute Vorticity, which can then be used to understand Rossby Waves.

We now use Equation (3) to develop another important conservation principle. Conservation of Mass states that the volume of an air column is constant, or

Volume = Area of Air Column X Depth = Constant (4a)

Or

Area of Air Column = (Constant)/Depth (4b)

Put Equation (4b) into (3) to obtain a relationship between the absolute vorticity and the depth of the air column by embedding both constants on the right side of the equation.

(5)

Equation (5) states that the ratio of the absolute vorticity to the depth of the air column (in a barotropic system) is constant. This ratio is known as Potential Vorticity.

Since this derivation was predicated on Conservation of Absolute Circulation, it is important to note that Equation (5) will help you to understand characteristics of only the features in the large scale flow that are barotropic or equivalent barotropic (such as Rossby Waves), and not baroclinic waves.

With that in mind, consider a zonal jet stream in which there is no northward variation in u (no horizontal speed shear) approaching a mountain range.

As the depth of the air column decreases, Equation (5) states that it's absolute vorticity must also decrease. Since we are at the core of the jet stream in which there is no horizontal shear, this decrease in absolute vorticity must show up as either anticyclonic curvature relative vorticity and/or a decrease in latitude (f). In either case, a southward turn will develop (a ridge). Downwind of the mountain range the opposite occurs, leading to troughs down wind of major mountain ranges.

In order for the topography to have this effect, the mountain range must have a width and depth of synoptic-scale dimensions and be oriented at right angles to the flow. That is to say, 1000 km or so in diameter and a good fraction of the troposphere in depth. The complex of the highlands of western North America and eastern Asia fulfill these criteria. Studies have shown that the level above which the underlying topography has no effect on flow patterns (the so-called "nodal" surface) is around the 200 mb level, or the top of the troposphere.

Note that, in this model, the air stream approaches its original latitude downwind of the mountains at a 45 degree angle. This means that it will overshoot its original latitude and produce a train of Rossby Waves, from our discussion of Tuesday 25 February.

Ertel"s Potential Vorticity

Conversion of (5) into isentropic coordinates yields

(6)

the factor ∂q /∂p  (finite differenced as ∆q /∆p) is related to the depth of the air column as shown in the figure below

The factor -∆q /∆p is also directly related to the static stability parameter. In the diagram above vertical shrinking and horizontal divergence will lead to in increase in an increase in -∆q /∆p. This is consistent with what we learned last semester: that the more stable the atmosphere, the more closely spaced the isentropes in vertical cross-section.