Quasi-geostrophic Theory Questions and Answers

Answer: C

Potential enstrophy is the domain integral of one half the square of the potential vorticity, and its conservation alongside energy is what constrains the direction of the cascades. Numerical schemes are often designed to conserve it discretely for exactly that reason. The other options describe kinetic energy, a measure of vertical motion and a quantity related to the energy but not the enstrophy. A single measurement chosen to respond to the one asymmetry beats several that respond to shared behaviour.

Answer: D

A scheme that fails to conserve enstrophy allows a spurious cascade of energy to small scales where it accumulates and eventually destabilises the integration. Arakawa's Jacobian was designed precisely to conserve both quantities discretely. Long climate integrations are especially sensitive to such slow spurious drifts, which is why conservation receives so much attention in model design. Reading the definition literally, rather than recalling a formula, is the reliable route through questions of this shape.

Answer: A

The Arakawa Jacobian is a particular finite difference form of the nonlinear advection term that conserves both energy and enstrophy exactly in the discrete system. It made long stable integrations of the vorticity equation possible. Semi-implicit schemes, spectral transforms and vertical differencing all address other aspects of model construction and are due to other authors. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.

Answer: B

Using a horizontally uniform reference stability keeps the omega equation linear and therefore solvable by standard elliptic methods. The cost is a poor representation of saturated ascent, where the effective stability is much lower, and of regions where the stratification varies sharply. Allowing horizontal variation is exactly what the standard formulation avoids doing.

Answer: C

The effective stability of saturated ascent is a third or less of the dry value, so using a uniform dry reference underestimates the ascent within a precipitating cloud band by a large factor. Substituting a moist stability where the air is saturated is the standard remedy. Clear regions and slow dry descent are represented perfectly adequately by the uniform assumption.

Answer: D

Friction enters the quasi-geostrophic system most naturally as a boundary condition, with the vertical velocity at the top of the Ekman layer proportional to the geostrophic relative vorticity. This produces the spin-down of vortices over about four days. Molecular viscosity is negligible, and radiative and Rayleigh damping represent quite different processes when they are included. A single measurement chosen to respond to the one asymmetry beats several that respond to shared behaviour.

Answer: A

Frictional damping of the surface temperature wave weakens the lower boundary contribution to the instability, reducing the growth rate and requiring a larger shear before growth can occur. Including it brings theoretical growth rates closer to observed values. The convergence it supplies acts on the wrong phase to assist growth, which is why the net effect is stabilising.

Answer: B

The dry Eady growth rate is somewhat larger than observed because it omits friction and the damping effect of the surface heat flux. Latent heat release does act to increase growth and so partly compensates, which is why the two effects are often discussed together. The theory does capture the regional and seasonal variation well, since it depends directly on the shear and stability. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Answer: C

Latent heat released as saturated air ascends reduces the effective stability and increases the growth rate, shifting the most unstable wavelength to shorter values as well. Moist baroclinic waves are therefore smaller and faster growing than dry ones, which matches observations of explosively deepening maritime cyclones. Evaporative cooling and cloud top radiation are real but secondary effects in this context. Most errors here come from carrying a result across from a neighbouring case where it does not hold.

Answer: D

Reduced effective stability shrinks the deformation radius, and since the most unstable wavelength scales with it the preferred scale becomes shorter. This is consistent with the compact intense structure of rapidly deepening maritime cyclones. A preferred wavelength certainly remains, and the direction of the shift follows directly from the dependence of the deformation radius on the buoyancy frequency. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.