Questions: Rossby Waves and Barotropic Instability

Question 1

An air parcel is displaced poleward from its equilibrium latitude. According to potential vorticity conservation, what does this do to the parcel's relative vorticity?

• A. The parcel increases its relative vorticity to compensate for increased planetary vorticity
• B. The parcel decreases its relative vorticity (spins up anticyclonically) to compensate for increased planetary vorticity
• C. The parcel increases its speed to conserve angular momentum
• D. The parcel cools adiabatically, generating a density anomaly that drives propagation

Answer: (B)

Potential vorticity is conserved: planetary vorticity (f) increases with latitude, so when the parcel moves poleward, relative vorticity must decrease (become anticyclonic) to keep their sum constant. This anticyclonic adjustment then pushes neighboring air equatorward, where they must compensate by spinning cyclonically — establishing the wave pattern that propagates westward. Option D describes gravity waves (buoyancy restoring), not Rossby waves, which are vorticity waves requiring no density stratification.

Question 2

In midlatitudes, a Rossby wave with long zonal wavelength can become stationary (zero ground-speed). What condition must be satisfied?

• A. The mean zonal wind U must equal β/(k² + l²), so mean-flow advection exactly cancels the westward intrinsic propagation
• B. The wave must reach critical amplitude so its phase speed matches the group velocity
• C. The Coriolis parameter must equal zero at the wave-crest latitude
• D. The meridional wavenumber l must be zero, reducing the wave to a purely zonal oscillation

Answer: (A)

From the dispersion relation ω = Uk − β/(k² + l²), a stationary wave has zero phase speed: ω/k = 0, giving U = β/(k² + l²). For fixed U and β, longer waves (smaller k) have smaller k², requiring a larger mean wind for stationarity. This is why planetary-scale (wavenumber 1–3) Rossby waves become quasi-stationary in the midlatitude jet, creating persistent large-scale blocking patterns.

Question 3

A Rossby wave in the atmosphere always propagates westward relative to the mean flow.

• T. True
• F. False

Answer: True

The intrinsic phase speed of a Rossby wave is always westward — this follows directly from the dispersion relation: the −β/(k² + l²) term always contributes westward propagation. However, a strong enough mean westerly flow U can carry the wave eastward relative to the ground, which is why Rossby waves appear to travel eastward on weather maps even though their intrinsic propagation is westward.

Question 4

Barotropic instability is triggered when the amplitude of Rossby wave perturbations exceeds a critical threshold.

• T. True
• F. False

Answer: False

Barotropic instability is a property of the mean flow, not of wave amplitude. The Rayleigh-Kuo necessary condition states that the meridional gradient of absolute vorticity must change sign somewhere in the flow — a condition on horizontal wind shear. When this threshold is met, certain wave modes lock together and extract kinetic energy from the mean shear, growing exponentially regardless of their initial amplitude.

Question 5

Rossby waves propagate by a fundamentally different restoring mechanism than gravity waves. What is the restoring mechanism for Rossby waves, and why is it fundamentally different from buoyancy?

Model answer: Rossby waves are restored by the β-effect — the latitudinal gradient of the Coriolis parameter. A poleward-displaced parcel must spin anticyclonically to conserve potential vorticity (since planetary vorticity increased), and this spin pushes neighboring parcels equatorward where they spin cyclonically, creating a chain of alternating vorticity anomalies that propagates westward. Gravity waves are restored by buoyancy: a displaced parcel becomes denser or lighter than its surroundings and gravity pulls it back. Rossby waves require no density stratification — they exist in barotropic atmospheres purely because Coriolis varies with latitude.

This distinction is critical. Rossby waves exist in barotropic (uniform density) fluids and require only the β-effect; gravity waves require stratification. The two types operate on completely different physics, have different propagation characteristics, and respond differently to the mean flow.