Equations of Motion in Rotating Frame Questions and Answers

Answer: A

Initialisation adjusts the analysed fields so that they satisfy an appropriate balance condition, preventing the excitation of large amplitude gravity waves at the start of the integration. Modern systems achieve this within the assimilation itself through incremental or digital filter techniques. Assimilation produces the analysis, output filtering is cosmetic, and ensembles address uncertainty rather than imbalance.

Answer: B

The Coriolis term is generated purely by rewriting the inertial acceleration in terms of quantities measured in the rotating frame, so it has no existence in the inertial description. The reason therefore explains the assertion exactly. This does not make the effect unreal for an observer on the Earth, any more than the reality of a curved path depends on which frame it is described from, and the deflection of a shell is measurable either way.

Answer: C

Phillips demonstrated that the two approximations form a consistent pair whose joint application preserves the conservation laws, whereas either alone introduces spurious sources. The reason is therefore precisely why the assertion holds. This is a good example of approximations that are individually plausible but must nonetheless be tied together, and it is why textbooks introduce them in the same breath rather than separately.

Answer: D

The reason is correct and is exactly why the assertion fails: a purely diagnostic relation containing no time derivative cannot generate an evolution. Prediction requires the ageostrophic component, which is what quasi-geostrophic theory retains at the next order of the expansion. This pattern, in which a true reason establishes the falsity of the assertion, appears repeatedly in this subject and rewards careful reading.

Answer: A

The material derivative is defined as the local derivative plus the advective term, so the first statement holds and the second does not. The local derivative is the rate of change at a fixed point and contains nothing else. Keeping the two apart is essential, because Newton's laws apply to the material derivative while observations from a fixed station measure the local one, and the difference between them is often the larger part of the total.

Answer: B

Both are correct: omega is the material derivative of pressure, which decreases upward, while the geometric vertical velocity is measured in the direction of increasing height. The two are related approximately by omega equal to minus density times gravity times the vertical velocity. The opposing signs are a frequent source of error when reading model output, so it is worth fixing the convention firmly at the outset.

Answer: C

The first two statements are correct: Coriolis and centrifugal accelerations appear, and the latter is combined with gravitation into effective gravity. The third fails because the Coriolis force is perpendicular to the velocity and therefore does no work. That property is what allows the force to organise circulations without supplying any of the energy that drives them. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

Answer: D

The first two statements follow from the standard scales of ten metres per second, one thousand kilometres and \( f \) about \( 10^{-4} \). The advective timescale is length divided by velocity, giving \( 10^{5} \) seconds or roughly one day rather than one hour, so the third fails. Carrying the standard scale set in memory makes all three of these judgements immediate rather than requiring a fresh calculation each time.

Answer: A

Only the last pairing is wrong: Boussinesq neglects density variation except in the buoyancy term and has nothing to say about rotation. The other three are correctly matched. Setting the approximations out by what each discards, rather than by what each keeps, is the clearest way to hold the family in mind and to judge which are appropriate to a given problem.

Answer: B

The timescale obtained by dividing the length scale by the velocity scale is about one day, not one hour, so the last pairing fails. The other three are the standard values used in every textbook scale analysis. Because every subsequent approximation rests on these numbers, learning them as a set is more efficient than deriving them repeatedly from first principles.