Equations of Motion in Rotating Frame Questions and Answers

Answer: C

The omega equation diagnoses vertical motion from the vorticity advection and the temperature advection in the flow, and it is far less sensitive to observational error than the kinematic method. It is derived by combining the quasi-geostrophic vorticity and thermodynamic equations to eliminate the time derivatives. The thermal wind and hydrostatic relations are diagnostic relations among other fields, and the vorticity equation is one of the ingredients rather than the result.

Answer: D

Ascent is forced where cyclonic vorticity advection increases with height and where warm advection occurs, which is why the region ahead of an upper trough is a region of cloud and rain. Friction and diabatic heating appear as additional terms in extended versions but not in the classical adiabatic form. Learning to read the two classical terms off a chart is one of the most useful skills the quasi-geostrophic system provides. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.

Answer: A

Boussinesq treats density as constant everywhere except where it multiplies gravity, which is legitimate when the density variation across the depth of the motion is small. It is excellent for the ocean and for shallow atmospheric layers but poor for motion spanning the whole troposphere, where the anelastic approximation is used instead. The final option states the condition under which Boussinesq is valid rather than the approximation itself. Most errors here come from carrying a result across from a neighbouring case where it does not hold.

Answer: B

The anelastic system retains the exponential decrease of the reference density with height while still filtering sound waves, which is what deep convection spanning many scale heights requires. Boussinesq would misrepresent the expansion of a rising plume, and hydrostatic balance fails outright in a convective updraught. Geostrophic balance is irrelevant at convective scales, where the Rossby number is very large.

Answer: C

Sound waves are meteorologically irrelevant yet travel at over three hundred metres per second, so retaining them forces a numerical time step far shorter than the physics requires. Boussinesq, anelastic and hydrostatic systems each remove them by a different route. Rossby waves, gravity waves and inertial oscillations are all meteorologically significant and are deliberately retained, which is the whole art of choosing an approximate equation set. Checking the units of each quantity catches a wrong formula before the arithmetic begins.

Answer: D

The traditional approximation discards the Coriolis terms proportional to the cosine of the latitude, which are precisely those coupling vertical velocity into the zonal equation and zonal velocity into the vertical equation. Its justification is the small aspect ratio of large-scale motion. The radial replacement is the shallow atmosphere approximation, which must be applied alongside it for consistency.

Answer: A

Phillips showed that the shallow atmosphere and traditional approximations must be adopted as a pair if the resulting system is to conserve angular momentum and energy correctly. Applying one alone produces a system with spurious sources. Potential temperature conservation follows from adiabatic thermodynamics rather than from these geometric approximations, and mass conservation is guaranteed by the continuity equation regardless. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

Answer: B

Operational models integrate the primitive equations, being the full set with hydrostatic, shallow atmosphere and traditional approximations applied, though the most recent convection permitting models are non-hydrostatic. Quasi-geostrophic and barotropic systems were used in the earliest numerical forecasts and remain valuable for understanding. Fully unapproximated equations are used only in research at very small scales. A single measurement chosen to respond to the one asymmetry beats several that respond to shared behaviour.

Answer: C

Charney, Fjortoft and von Neumann integrated the barotropic vorticity equation on the ENIAC in 1950, producing a twenty four hour forecast in about twenty four hours of computation. The single equation was chosen precisely because it filters gravity waves, which had wrecked Richardson's earlier attempt with the primitive equations. Multi-level primitive equation models came a decade later as computing power grew.

Answer: D

Richardson's initial pressure field contained an imbalance that excited large amplitude gravity waves, producing a predicted surface pressure change of 145 hectopascals in six hours. The equations and the arithmetic were essentially correct. The episode led directly to the development of initialisation techniques and to the use of filtered equation sets, and it is the standard illustration of why balance in the initial conditions matters so much. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.