Quasi-geostrophic Theory Questions and Answers

Answer: C

Advecting higher cyclonic vorticity into a region requires the geopotential to fall there, since vorticity is the Laplacian of geopotential divided by the Coriolis parameter. This is the dynamical basis of the rule that troughs deepen downstream of vorticity maxima. The relationship is the same at every level, though the magnitude of the advection varies greatly with height.

Answer: D

The two-level model predicts the streamfunction at 250 and 750 hectopascals and diagnoses the temperature from their difference, which is the minimum structure capable of representing baroclinic instability. Equal winds at both levels would remove the shear and with it the instability. The beta effect is retained and is essential to the stability criterion the model produces.

Answer: A

The two-level model gives a critical shear below which all wavelengths are stable, a result arising from the beta effect stabilising the long waves. Observed shears exceed the critical value over the winter storm tracks, which is why cyclones form there. The Eady model, which has no beta effect, finds instability at any shear, so the difference between the two models is instructive. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

Answer: B

The beta effect opposes the interior potential vorticity gradient created by the shear, and below a critical shear it cancels it so that the Charney-Stern criterion is not satisfied. Above that shear the gradient changes sign and instability follows. The interplay makes the two-level model a better guide to observed behaviour than the Eady model despite its coarse vertical resolution. Most errors here come from carrying a result across from a neighbouring case where it does not hold.

Answer: C

Short waves are stabilised because they cannot span the depth of the shear, and long waves are stabilised by the beta effect, leaving an unstable band whose centre lies at several thousand kilometres. That band matches the observed spectrum of mid-latitude cyclones well. The short wave cut-off appears in the Eady model too, but the long wave cut-off requires beta.

Answer: D

Both the Eady and two-level models give a most unstable wavelength near 4000 kilometres, which agrees well with the observed spacing of mid-latitude cyclones. That agreement is what established the theory. The deformation radius, at about 1000 kilometres, sets the scale but the most unstable wavelength is several times larger, so quoting one for the other is a common slip.

Answer: A

The maximum growth rate is proportional to the shear divided by the buoyancy frequency, giving an e-folding time near 24 hours for typical mid-latitude values. That matches the observed pace at which a cyclone develops from a barely visible wave to a mature system in two or three days. The rate depends strongly on the shear, so a theory giving a shear-independent answer would contradict observation immediately. Checking the units of each quantity catches a wrong formula before the arithmetic begins.

Answer: B

Growing baroclinic waves convert the available potential energy of the meridional temperature gradient into eddy kinetic energy by lifting warm air and lowering cold air. Barotropic instability draws on the horizontal shear of the mean flow instead. Latent heat release amplifies the process but is not the primary conversion in the dry theory, and the final option describes the decay phase of the eddy life cycle. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.

Answer: C

The tilt places rising motion in the warm air and sinking motion in the cold air, which lowers the centre of gravity of the atmosphere and releases the available potential energy. An eastward tilt reverses the conversion and describes a decaying system, while a vertically upright structure gives no net conversion at all. This is why occlusion, which brings the structure upright, ends the deepening. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

Answer: D

In a deepening system the upper trough lies upstream of the surface low, which in westerly flow means to the west, and the two become vertically aligned only as the system matures. This classical rule of thumb from synoptic analysis matches the tilt requirement derived from theory exactly, which is a satisfying convergence of empirical practice and dynamical reasoning.