Atmospheric Waves Questions and Answers

Q: Which wavenumbers dominate the stationary wave pattern of the northern winter?

The stationary pattern forced by the Rockies, the Tibetan Plateau and the land-sea contrasts is dominated by the longest waves, principally wavenumbers one to three. Only these are long enough to satisfy the Charney-Drazin criterion and propagate into the stratosphere. Wavenumbers five to seven describe the travelling synoptic waves, and wavenumber zero is the zonal mean rather than a wave at all. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.

Q: Which relation must a wave satisfy to be a solution of the equations?

Substituting a wave solution into the linearised equations yields an algebraic relation between frequency and wavenumber that must hold for the solution to exist. That dispersion relation encodes the whole propagation behaviour of the wave type. Continuity, hydrostatic balance and the thermal wind relation are among the equations from which the dispersion relation is derived rather than substitutes for it. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

Q: Which assumption limits the validity of a dispersion relation so obtained?

Linearisation requires the perturbation to be small enough that products of perturbation quantities may be neglected, which fails once a wave grows to large amplitude or begins to break. A moving basic state is perfectly acceptable and simply Doppler shifts the frequency. Stationarity and barotropy are special cases rather than requirements of the method.

Q: Which phenomenon lies outside linear wave theory?

Breaking occurs when the wave amplitude becomes large enough to overturn the isentropes, at which point the linear assumption fails completely and turbulence is generated. Linear theory can predict where breaking will begin but not what happens afterwards. Propagation, Doppler shifting and ducting are all well described by linear theory.

Q: Which regime does a low Froude number flow over a mountain produce?

When the kinetic energy of the approaching flow is insufficient to lift it over the barrier against the stratification, the air is deflected around the ends instead, which is the low Froude number regime. Barrier jets and gap winds result. At high Froude number the flow surmounts the ridge and generates the mountain wave field. Recognising the regime from the flow speed and stratification is a practical forecasting skill in mountainous terrain.

Q: Which local wind results from flow forced through a gap in a barrier?

Blocked flow accelerating through a gap can produce winds far stronger than the ambient flow, and such gap winds are a recognised hazard in fjords, straits and mountain passes. The foehn descends the lee slope after crossing the crest, while sea breezes and katabatic winds are driven by thermal contrasts rather than by dynamical channelling.

Q: Which resonance can amplify a mountain wave response dramatically?

When the horizontal scale of the terrain matches a trapped wave mode of the atmospheric profile, the lee wave response is resonantly amplified and can produce severe downslope windstorms. The Boulder windstorm of 1972 is the classic documented example. Buoyancy frequency matching describes a different situation and the other options have no dynamical basis.

71.

Which wavenumbers dominate the stationary wave pattern of the northern winter?

A.Wavenumbers ten and above
B.Wavenumber zero, which represents the zonally symmetric part of the flow around the latitude circle
C.Wavenumbers one to three
D.Wavenumbers five to seven, which correspond to the synoptic scale travelling waves of the storm track

Answer: C

The stationary pattern forced by the Rockies, the Tibetan Plateau and the land-sea contrasts is dominated by the longest waves, principally wavenumbers one to three. Only these are long enough to satisfy the Charney-Drazin criterion and propagate into the stratosphere. Wavenumbers five to seven describe the travelling synoptic waves, and wavenumber zero is the zonal mean rather than a wave at all. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.

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72.

Which relation must a wave satisfy to be a solution of the equations?

A.The continuity equation alone, since mass conservation is the only constraint acting upon a wave motion
B.The hydrostatic relation
C.The thermal wind relation, which links the vertical shear of the flow to the horizontal temperature gradient
D.The dispersion relation

Answer: D

Substituting a wave solution into the linearised equations yields an algebraic relation between frequency and wavenumber that must hold for the solution to exist. That dispersion relation encodes the whole propagation behaviour of the wave type. Continuity, hydrostatic balance and the thermal wind relation are among the equations from which the dispersion relation is derived rather than substitutes for it. A limit that survives making every external component ideal is an intrinsic one, and worth knowing for that reason.

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73.

Which technique is used to obtain a dispersion relation?

A.Linearising about a basic state and assuming a wave form
B.Integrating the full nonlinear equations forward in time until a steady wave solution eventually emerges
C.Averaging over many observations
D.Solving the elliptic inversion problem for the streamfunction given a specified potential vorticity field

Answer: A

Small perturbations are superposed on a specified basic state, products of perturbations are discarded, and a solution proportional to a complex exponential is substituted, giving an algebraic condition on frequency and wavenumber. This is the standard method throughout the subject. Numerical integration and observational averaging can confirm the result but do not derive it. A single measurement chosen to respond to the one asymmetry beats several that respond to shared behaviour.

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74.

Which assumption limits the validity of a dispersion relation so obtained?

A.The atmosphere must be barotropic so that the wave has no vertical structure within the layer concerned
B.The wave amplitude must remain small
C.The basic state must be at rest so that no advection of the perturbation can occur during the analysis
D.The wave must be stationary

Answer: B

Linearisation requires the perturbation to be small enough that products of perturbation quantities may be neglected, which fails once a wave grows to large amplitude or begins to break. A moving basic state is perfectly acceptable and simply Doppler shifts the frequency. Stationarity and barotropy are special cases rather than requirements of the method.

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75.

Which phenomenon lies outside linear wave theory?

A.Doppler shifting
B.The trapping of a wave within a duct formed by a stable layer beneath a region of increasing wind speed
C.Wave breaking
D.The propagation of a small amplitude gravity wave upward through a uniformly stratified layer of air

Answer: C

Breaking occurs when the wave amplitude becomes large enough to overturn the isentropes, at which point the linear assumption fails completely and turbulence is generated. Linear theory can predict where breaking will begin but not what happens afterwards. Propagation, Doppler shifting and ducting are all well described by linear theory.

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76.

Which parameter measures how close a gravity wave is to breaking?

A.The Rossby number, which compares the inertial acceleration against the acceleration due to rotation
B.The Froude number
C.The ratio of the horizontal wavelength to the vertical wavelength at the level being considered here
D.The ratio of wave amplitude to the saturation amplitude

Answer: D

Saturation is reached when the wave induced vertical displacement is large enough to make the total potential temperature gradient vanish, and the ratio of actual to saturation amplitude measures how close that is. Parameterisation schemes limit the wave flux to keep the ratio at or below one. The Rossby and Froude numbers measure quite different competitions, though the Froude number does govern whether flow surmounts a barrier at all. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

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77.

Which regime does a low Froude number flow over a mountain produce?

A.Flow around rather than over the barrier
B.Flow directly over the crest with a strong lee wave train developing downstream of the range concerned
C.Uniform flow
D.Flow that separates from the surface upstream of the barrier and reattaches well beyond the crest of it

Answer: A

When the kinetic energy of the approaching flow is insufficient to lift it over the barrier against the stratification, the air is deflected around the ends instead, which is the low Froude number regime. Barrier jets and gap winds result. At high Froude number the flow surmounts the ridge and generates the mountain wave field. Recognising the regime from the flow speed and stratification is a practical forecasting skill in mountainous terrain.

Discuss in Forum

78.

Which local wind results from flow forced through a gap in a barrier?

A.A katabatic wind, driven by the drainage of radiatively cooled air down a slope during the night hours
B.A gap wind
C.A foehn wind, which is instead the warm dry flow that descends the lee slope after crossing the crest
D.A sea breeze

Answer: B

Blocked flow accelerating through a gap can produce winds far stronger than the ambient flow, and such gap winds are a recognised hazard in fjords, straits and mountain passes. The foehn descends the lee slope after crossing the crest, while sea breezes and katabatic winds are driven by thermal contrasts rather than by dynamical channelling.

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79.

Which resonance can amplify a mountain wave response dramatically?

A.Alignment of the ridge with the Coriolis force
B.Equality between the wind speed at the crest and the speed of sound within the air above the barrier
C.Matching of the forcing scale to a trapped wave mode
D.Coincidence of the wave frequency with the buoyancy frequency of the layer through which it is passing

Answer: C

When the horizontal scale of the terrain matches a trapped wave mode of the atmospheric profile, the lee wave response is resonantly amplified and can produce severe downslope windstorms. The Boulder windstorm of 1972 is the classic documented example. Buoyancy frequency matching describes a different situation and the other options have no dynamical basis.

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80.

Which severe local phenomenon results from that amplification?

A.A tornado, which forms where the rotating updraught of a supercell storm is stretched towards the surface
B.A microburst
C.A squall line, which is an organised line of convective cells advancing ahead of an approaching cold front
D.A downslope windstorm

Answer: D

Downslope windstorms with gusts exceeding 50 metres per second occur on the lee slopes of major ranges when the wave response is amplified and breaks, and they cause substantial damage in places such as Boulder and the Alpine valleys. Tornadoes, microbursts and squall lines are all convective phenomena with quite different dynamics.

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