Fundamental Forces in the Atmosphere Questions and Answers

Answer: A

Geostrophic balance is the leading order result of expanding the momentum equation in a small Rossby number, so it holds for large-scale, slowly evolving mid-latitude flow. The Reynolds number tells us the flow is turbulent but says nothing about rotation, and the Richardson criterion concerns shear instability. Recognising geostrophy as the leading term of a systematic expansion, rather than as an assumption plucked from the air, is what makes quasi-geostrophic theory intelligible later.

Answer: B

The traditional approximation drops the terms involving the cosine of the latitude, which couple vertical motion to the horizontal equations, on the grounds that vertical velocities are smaller than horizontal ones by several orders of magnitude. It is what allows \( f \) to be written with a sine alone. Retaining latitudinal variation of \( f \) is the beta effect, a separate matter, and the centrifugal force is absorbed into effective gravity rather than neglected.

Answer: C

On an f-plane the parameter is fixed at its value for a central latitude, which is adequate for phenomena small enough that the latitudinal change is negligible. The beta-plane retains a linear variation and is needed for Rossby waves, which owe their existence entirely to that variation. Boussinesq concerns density in the buoyancy term and hydrostatic concerns the vertical balance. Choosing between f-plane and beta-plane is choosing whether the phenomenon of interest depends on the sphericity of the Earth.

Answer: D

Rossby waves depend on the meridional gradient of planetary vorticity, so on an f-plane with constant \( f \) they cannot exist at all. The restoring mechanism is that a parcel displaced polewards acquires negative relative vorticity, which turns it back. Geostrophic balance, hydrostatic balance and the Ekman spiral all survive perfectly well with constant \( f \). Recognising which results need beta and which do not is the quickest way to keep the family of approximations straight.

Answer: A

Only the pressure gradient force can accelerate air that is initially at rest, since Coriolis, centrifugal and frictional forces all require an existing velocity. Once the air is moving, Coriolis turns it and the familiar clockwise circulation around a northern hemisphere high develops. Thinking through this sequence, from a pressure field created by differential heating to a balanced circulation, is the clearest way to see why the pressure gradient force holds the primary place among the four.

Answer: B

Because pressure decreases monotonically with height under hydrostatic balance, it can serve as a vertical coordinate, and doing so removes density from the pressure gradient term and replaces it with the gradient of geopotential. That simplification is why upper air charts are drawn on constant pressure surfaces. Potential temperature is used as an isentropic coordinate for other purposes, density is not monotonic in a useful way, and geometric altitude reintroduces the density factor that pressure coordinates were adopted to remove.

Answer: C

In pressure coordinates the force per unit mass becomes minus the gradient of geopotential taken along the constant pressure surface, with density having disappeared entirely. That is a considerable simplification, since density is not directly measured, and it is why the contours on a 500 hectopascal chart can be read exactly as isobars are read on a surface chart. The height gradient measured from the geocentre would neglect the variation of gravity that the geopotential already incorporates.

Answer: D

Coriolis and centrifugal forces are consequences of writing the equations in a rotating frame and have no counterpart in an inertial description. Pressure gradient, gravitation and viscosity are genuine interactions present in any frame. The point is not that rotation is fictitious but that we choose a rotating frame because we observe the atmosphere from the rotating Earth, and the price of that convenient choice is two extra terms that must be carried through every subsequent argument.

Answer: A

A pressure change of a few hectopascals over a thousand kilometres gives an acceleration of order \( 10^{-3} \) metres per second squared, which is comparable with the Coriolis acceleration for a ten metre per second wind. That comparability is exactly why the two nearly balance. Gravity is larger by four orders of magnitude, which is why the vertical balance is so much tighter than the horizontal one. Keeping these scales in mind converts the whole subject from formulae into physical reasoning.

Answer: B

Vertical accelerations in large-scale flow are of order \( 10^{-7} \) metres per second squared against a gravitational acceleration of about 10, so hydrostatic balance holds to one part in \( 10^{8} \). Horizontally the two competing terms are of the same order, so geostrophic balance is good only to about ten per cent. Pressure in fact varies far more rapidly vertically than horizontally, and measurement accuracy has nothing to do with the physics of the balance.