Equations of Motion in Rotating Frame Questions and Answers

Answer: C

The material derivative is the sum of the local time derivative and the advective term, and it gives the rate of change experienced by an observer riding with the parcel. Newton's laws and the conservation laws apply to this derivative rather than to the local one. A tower measures the local derivative, a gradient is a purely spatial quantity, and flux across a surface is a related but distinct concept used in the integral forms of the equations. A single measurement chosen to respond to the one asymmetry beats several that respond to shared behaviour.

Answer: D

The advective term is the velocity multiplied by the gradient of velocity, so it is quadratic in the unknown and is the source of the nonlinearity that makes the equations intractable analytically. It is also what produces turbulence and chaos. The Coriolis term is linear in velocity, the pressure gradient depends on a different field, and the frictional term as usually parameterised is linear too, though the eddy viscosity within it may not be. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Answer: A

Nonlinear interaction transfers error from small scales to large, so a small initial uncertainty grows until the forecast is no better than climatology, which Lorenz established and which sets a practical limit of about two weeks. Numerical integration is entirely possible and energy is conserved by well constructed schemes. The predictability limit is a property of the physics rather than of the numerical method, which is why ensemble forecasting rather than better arithmetic is the response.

Answer: B

Scale analysis substitutes typical magnitudes for each variable and compares the resulting sizes of the terms, allowing small ones to be discarded. It is how geostrophic and hydrostatic balance are justified rather than assumed. Dimensional analysis addresses a different question, Fourier decomposition is a solution technique, and perturbation expansion is the formal machinery that scale analysis motivates but does not itself constitute. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Answer: C

A value of ten metres per second is representative of mid-latitude tropospheric winds and is the number used in every standard scale analysis. Jet stream cores reach several times this but are not typical of the whole flow. One metre per second is far too slow, and the two larger values belong to the jet maximum and to sound respectively. Committing the standard set of scales to memory makes the analysis reproducible without reference to a table. Reading the definition literally, rather than recalling a formula, is the reliable route through questions of this shape.

Answer: D

About 1000 kilometres is the characteristic width of a mid-latitude cyclone and gives a Rossby number near 0.1 when combined with the standard velocity and Coriolis scales. Ten kilometres describes a convective cell and ten thousand the planetary scale. The choice of length scale is what distinguishes synoptic from mesoscale and planetary dynamics, and each regime has its own set of valid approximations that follow from it. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

Answer: A

Dividing a length of \( 10^{6} \) metres by a velocity of 10 metres per second gives \( 10^{5} \) seconds, close to one day. This advective timescale is why synoptic systems evolve noticeably from one day to the next and why twice daily observations were historically adequate. The other options describe convective, planetary and numerical timescales respectively, each belonging to a different part of the problem.

Answer: B

Large scale vertical motion is of order one centimetre per second, three orders of magnitude below the horizontal wind, and it is this disparity that justifies both the hydrostatic and the traditional approximations. Convective updraughts of a metre per second or more are outside the synoptic regime. Getting this scale right is essential, because the vertical velocity is what produces cloud and precipitation despite being far too small to measure directly. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Answer: C

A depth of about 10 kilometres against a width of about 1000 gives an aspect ratio of about 1 to 100, and it is this flatness that makes the hydrostatic approximation so accurate. Deep convection with comparable depth and width is exactly the case where the approximation fails. The aspect ratio, rather than any single dimension, is the parameter that determines whether vertical accelerations may be neglected.

Answer: D

Scale analysis shows the pressure gradient and Coriolis terms are each about \( 10^{-3} \) metres per second squared while the acceleration terms are an order of magnitude smaller, so at leading order the two large terms must balance. That balance is geostrophy. Retaining the next order gives the quasi-geostrophic system in which the small imbalance drives the evolution, which is the whole basis of dynamical forecasting. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.