Equations of Motion in Rotating Frame Questions and Answers

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.

Answer: C

The omega equation is a derived diagnostic relation obtained by combining the quasi-geostrophic vorticity and thermodynamic equations, not a member of the primitive set. The primitive equations comprise the horizontal momentum equations, hydrostatic balance, continuity, the thermodynamic energy equation and the equation of state. Distinguishing primitive from derived relations keeps clear what is assumed and what is deduced. Most errors here come from carrying a result across from a neighbouring case where it does not hold.

Answer: D

The advective acceleration is part of the material derivative and appears in any frame whatsoever, so it is not an artefact of rotation. Coriolis, centrifugal and Euler accelerations all arise from the frame transformation, the last of these vanishing for the Earth because its rotation rate is effectively constant. Recognising which terms are geometric and which are physical is the whole point of the transformation exercise. Checking the units of each quantity catches a wrong formula before the arithmetic begins.

Answer: A

At one kilometre spacing the aspect ratio of the resolved motion approaches unity, so vertical accelerations become comparable with the residual of the vertical pressure gradient and gravity, and the hydrostatic assumption fails. Convection permitting models are therefore non-hydrostatic. Continuity is never abandoned, and the shallow atmosphere and traditional approximations remain valid since the depth is still tiny compared with the Earth's radius. Most errors here come from carrying a result across from a neighbouring case where it does not hold.

Answer: B

Divergence is a small difference between large quantities, so observational error is amplified enormously, and the kinematic method is notoriously unreliable in data-sparse regions. The usual remedy is to adjust the vertical profile so that omega vanishes at the top. The Coriolis parameter depends only on latitude, and the hydrostatic approximation is if anything better over the smooth ocean than over land.

Answer: C

The material derivative, also called the total or substantial derivative, is the sum of the local and advective parts and is the derivative to which Newton's laws apply. The local derivative is what a fixed instrument measures, a partial derivative is a mathematical operation rather than a physical rate, and the convective derivative names only the advective portion of the total rather than the whole.

Answer: D

In pressure coordinates the horizontal force per unit mass becomes minus the gradient of geopotential taken along the constant pressure surface, and density disappears altogether. That simplification is the chief reason upper air analysis is performed on pressure surfaces. Temperature enters through the thickness relation, density is precisely what has been eliminated, and potential temperature belongs to a different coordinate choice.