Quasi-geostrophic Theory Questions and Answers

Answer: A

Quasi-geostrophic theory is the systematic expansion of the governing equations in powers of the Rossby number, retaining terms to first order. At leading order the result is geostrophic balance, which is diagnostic, and at the next order the ageostrophic residual supplies the evolution. The Richardson and Froude numbers measure quite different competitions, and while the aspect ratio does justify the hydrostatic approximation that the theory also assumes, it is not the expansion parameter of the quasi-geostrophic system itself.

Answer: B

The defining simplification is that advection is performed by the geostrophic wind alone, while the ageostrophic wind appears only in the divergence term. This division is what makes the system tractable and also what limits it, since real frontogenesis depends on ageostrophic advection. The thermal wind is a shear rather than an advecting flow, and vertical averaging belongs to other simplified systems. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.

Answer: C

The whole system reduces to the statement that quasi-geostrophic potential vorticity is advected by the geostrophic wind, with everything else recoverable by inversion. That single prognostic equation replaces the momentum, continuity and thermodynamic equations. Absolute vorticity conservation requires far more restrictive conditions, and potential temperature is conserved for adiabatic motion in the full system rather than being the QG invariant.

Answer: D

Quasi-geostrophic potential vorticity is the sum of the geostrophic relative vorticity, the Coriolis parameter and a term involving the vertical derivative of the geopotential scaled by the static stability. The third term represents vortex stretching by the differential vertical motion. Because all three are expressible through the geopotential alone, the whole quantity is a linear operator acting on a single field, which is what makes inversion so straightforward.

Answer: A

Because the QG potential vorticity is a linear elliptic operator applied to the geopotential, inverting it to recover the height, wind and temperature fields is a straightforward Poisson-type problem. Ertel's exact potential vorticity is nonlinear and its inversion requires an iterative solution. Diabatic heating destroys conservation in both cases, and the field varies with height like any other. Reading the definition literally, rather than recalling a formula, is the reliable route through questions of this shape.

Answer: B

Given boundary conditions, solving the elliptic equation that defines QG potential vorticity yields the geopotential everywhere, and from that the geostrophic wind and the temperature follow immediately. This is the invertibility principle. Because the operator is elliptic the solution depends on the potential vorticity throughout the domain, which is why a compact anomaly influences the flow at a distance. Reading the definition literally, rather than recalling a formula, is the reliable route through questions of this shape.

Answer: C

Bretherton showed that a boundary temperature distribution is dynamically equivalent to a sheet of potential vorticity at the boundary, so specifying it completes the inversion problem. This is why a surface temperature anomaly can induce a circulation just as an interior potential vorticity anomaly does. Lateral conditions are needed for a limited domain but the vertical boundaries carry the physically essential information. Most errors here come from carrying a result across from a neighbouring case where it does not hold.

Answer: D

Treating a boundary temperature gradient as a delta function of potential vorticity gradient unifies boundary and interior effects into a single framework. It is what allows the Eady model, which has no interior potential vorticity gradient at all, to satisfy the Charney-Stern criterion for instability through its two boundaries. The equivalence is a formal device with real explanatory power. Checking the units of each quantity catches a wrong formula before the arithmetic begins.

Answer: A

Eliminating omega between the QG vorticity and thermodynamic equations gives an elliptic equation for the local rate of change of geopotential, forced by vorticity advection and by the vertical derivative of thermal advection. Eliminating the time derivatives instead gives the omega equation. The pair are the two ways of combining the same two equations and together they form the working machinery of the theory. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Answer: B

Height falls are forced where cyclonic vorticity advection occurs and where the thermal advection increases with height. Note the asymmetry with the omega equation, which uses the vertical derivative of the vorticity advection and the plain thermal advection. Keeping the two arrangements straight is a standard examination point and follows from which variable was eliminated in the derivation. Framing the question as which quantity the change acts upon makes the answer follow without calculation.