Fundamental Forces in the Atmosphere Questions and Answers

Answer: A

The advective term, the velocity dotted with the gradient of velocity, expresses the fact that a parcel carries its momentum with it, so the wind at a point can change simply because different air has arrived. Together with the local derivative it forms the material derivative, which is the acceleration actually experienced by the parcel. The nonlinearity of this term is the source of most of the mathematical difficulty in fluid dynamics, and it is what makes the equations impossible to solve in closed form for realistic flows.

Answer: B

The material derivative combines the local rate of change with the advective rate of change, giving what an observer riding with the parcel would measure. Newton's second law applies to this derivative, not to the local one, which is why the equations of motion are written in terms of it. The distinction between the Eulerian view at a fixed point and the Lagrangian view following the parcel underlies every subsequent manipulation in this subject, and confusing the two produces errors that are hard to trace afterwards.

Answer: C

Continuity states that the rate of change of density following a parcel is balanced by the divergence of the velocity field, so mass is neither created nor destroyed. It is what links horizontal divergence to vertical motion and hence what makes convergence at the surface imply ascent. Momentum conservation is the equation of motion, energy conservation is the thermodynamic equation, and vorticity conservation is a derived result rather than a primitive statement. The set of primitive equations comprises all of these together.

Answer: D

If air converges horizontally into a column and mass is conserved, the column must expand vertically, so convergence at low levels implies ascent above it. This is the single most used consequence of continuity in synoptic meteorology, and it is why surface lows produce cloud while surface highs produce clear skies. Hydrostatic and geostrophic balance and the thermal wind relation are all diagnostic relations among the fields rather than statements about how mass moves between them.

Answer: A

The primitive equations are the horizontal momentum equations, the hydrostatic relation, mass continuity, the thermodynamic energy equation and the equation of state, and they form the basis of every operational forecast model. They are the full equations with the hydrostatic and shallow atmosphere approximations applied. The vorticity and divergence forms are equivalent rewritings, the quasi-geostrophic system is a further filtered approximation, and the unapproximated Navier Stokes equations retain terms the primitive set has already discarded.

Answer: B

The shallow atmosphere approximation replaces the distance from the centre of the Earth by the Earth's radius wherever it appears, on the grounds that the atmosphere is only about one thousandth as deep as the planet is wide. It goes hand in hand with the traditional approximation. Boussinesq, incompressible and anelastic approximations concern the treatment of density rather than of geometry, and each filters a different class of motion out of the system for its own reasons.

Answer: C

Imposing hydrostatic balance removes the vertical acceleration term and with it the vertically propagating acoustic modes, which are meteorologically irrelevant but would otherwise impose a crippling limit on the time step of a numerical model. Rossby and Kelvin waves survive untouched, and gravity waves survive in modified hydrostatic form. Filtering unwanted fast modes while retaining the meteorologically important slow ones is the central design principle of every approximation in this subject.

Answer: D

Numerical stability requires that a wave should not cross more than one grid box per time step, so retaining sound waves at three hundred metres per second would force a time step some thirty times shorter than the meteorologically relevant motion requires. The fast modes carry negligible energy, which is precisely why removing them costs nothing. They are not unstable, and the observational argument concerns initialisation rather than the choice of equation set.

Answer: A

A parcel displaced vertically in a stably stratified atmosphere oscillates about its equilibrium level at the buoyancy frequency, whose square is proportional to the vertical gradient of potential temperature. A negative square indicates instability and convection. The inertial frequency, which involves \( f \) rather than stratification, describes horizontal oscillations instead, and wind turning with height is the Ekman spiral. Both frequencies appear together in the dispersion relation for internal gravity waves.

Answer: B

A parcel moving with no pressure gradient force describes a circular inertial circle at the frequency \( f \), completing one revolution in a pendulum day of about seventeen hours at forty five degrees latitude. The buoyancy frequency governs vertical oscillations instead. The Earth's own rotation rate is smaller than \( f \) by a factor of twice the sine of the latitude, and the orbital period of a streamline depends on the flow rather than on the rotation of the frame.