Vorticity and Circulation Questions and Answers

Answer: C

The reason is correct but the conserved quantity is the absolute vorticity, not the relative vorticity, because a parcel changing latitude exchanges relative for planetary vorticity. That exchange is precisely the beta effect and the mechanism of Rossby waves. The assertion would be true only on an f-plane where the planetary contribution is fixed, which is a special case rather than the general result.

Answer: D

Positive vorticity means anticlockwise rotation by convention, and cyclonic means anticlockwise in the north but clockwise in the south, so the sign of cyclonic vorticity reverses between hemispheres. This is why the word cyclonic is preferred to any statement about sign when a result is meant to apply globally, and why southern hemisphere charts of vorticity are often plotted with the sign reversed to preserve visual familiarity.

Answer: A

Both statements are correct, and Stokes' theorem connects them: the circulation around a curve equals the integral of the vorticity over any surface bounded by it. That connection is what allows a theorem about the circulation around a material loop to be translated into a statement about the vorticity of the enclosed fluid, which is how Kelvin's and Bjerknes' results are applied in practice.

Answer: B

The first two statements are correct. The solenoidal term is in fact very small when pressure is used as the vertical coordinate, because pressure surfaces are then coordinate surfaces and the term nearly vanishes. That is one of the practical advantages of pressure coordinates for synoptic work, and it is why the synoptic vorticity budget is usually written with only advection, stretching and friction retained. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted. Most errors here come from carrying a result across from a neighbouring case where it does not hold. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.

Answer: C

Conservation and invertibility are the two properties that make potential vorticity so powerful, but latent heat release is a diabatic process and therefore destroys the conservation. Indeed the low level potential vorticity anomaly generated by condensational heating is an important contributor to rapid cyclogenesis, so the third statement is not merely wrong but wrong in an interesting and useful way.

Answer: D

The last pairing fails: the solenoidal term arises from intersecting pressure and density surfaces and has nothing to do with surface friction, which appears as a separate dissipation term. The other three are correctly described. Knowing which term does what is what allows the equation to be simplified appropriately for a given scale of motion rather than being carried around in full.

Answer: A

A hurricane core reaches about \( 10^{-2} \) per second, three orders of magnitude above the synoptic value rather than one order below it, so the last pairing is wrong. The other three are correct. Carrying this ladder of magnitudes in mind makes it immediately obvious which terms of the vorticity equation matter for a given phenomenon.

Answer: B

Buoyancy appears in the vertical momentum equation rather than in the vorticity equation, where its effect enters indirectly through the vertical velocity in the stretching and tilting terms. Advection, stretching and tilting are genuine terms, along with the solenoidal and frictional contributions. Keeping the inventories of the different equations separate prevents a common confusion when they are first written out together. Most errors here come from carrying a result across from a neighbouring case where it does not hold.

Answer: C

Relative vorticity changes as a parcel changes latitude, because the planetary contribution changes and only the sum is conserved. Absolute vorticity, circulation and potential vorticity are all conserved under these conditions. The exchange between relative and planetary vorticity is exactly the beta effect and is the mechanism by which Rossby waves propagate through the flow.

Answer: D

Compression of the column as it climbs the windward slope reduces the absolute vorticity, giving anticyclonic curvature, and the subsequent stretching in the lee restores and overshoots it, giving the cyclonic lee trough. The column then oscillates downstream as a standing Rossby wave train. This sequence explains the semi-permanent lee troughs east of the Rockies and the Andes and the cyclogenesis regions associated with them. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.