Rossby Wave

A Rossby wave (or planetary wave) is a large-scale wave motion in a rotating fluid envelope—such as a planetary atmosphere or ocean—whose restoring mechanism is the variation of the Coriolis effect with latitude (the planetary vorticity gradient). First analyzed mathematically by the Swedish-American meteorologist Carl-Gustaf Rossby in 1939 within geophysical fluid dynamics, Rossby waves govern the synoptic- and planetary-scale meanders of the mid-latitude tropospheric jet streams, transport heat, momentum, and vorticity between the tropics and poles within the general atmospheric circulation, and mediate basin-scale dynamical adjustments in the global ocean.

Unlike gravity waves, which are restored by buoyancy and pressure gradients and propagate in all horizontal directions at high phase speeds, Rossby waves are quasi-geostrophic oscillations whose intrinsic phase velocity is directed westward relative to the background fluid flow.

The physical mechanism of a Rossby wave follows from the conservation of potential vorticity (PV) derived from rotating shallow-water reductions of the Navier-Stokes equations. For a homogeneous, inviscid fluid column of depth H moving horizontally on a rotating sphere of angular velocity Ω and radius a, the shallow-water potential vorticity q is conserved following horizontal parcel motion (Dq/Dt = 0):

q = (ζ + f) / H = constant

Here, ζ = ∂v/∂x − ∂u/∂y is the vertical component of relative vorticity (positive for counterclockwise cyclonic spin in the Northern Hemisphere), and f = 2Ω sin φ is the planetary vorticity (the Coriolis parameter at latitude φ).

To analyze wave propagation around a reference latitude φ₀, Rossby introduced the β-plane approximation, expanding the Coriolis parameter in a first-order Taylor series in the northward Cartesian coordinate y = a(φ − φ₀):

f(y) ≈ f₀ + βy, where f₀ = 2Ω sin φ₀ and β = ∂f / ∂y = (2Ω cos φ₀) / a

At φ₀ = 45°, β ≈ 1.62 × 10⁻¹¹ m⁻¹ s⁻¹. Consider a fluid column of constant depth H initially at rest (ζ = 0) along y = 0. If a perturbation displaces the parcel northward (y > 0), planetary vorticity f increases by +βy; to conserve total vorticity (ζ + f = constant), the parcel acquires negative (anticyclonic, clockwise) relative vorticity (ζ = −βy < 0). Conversely, a parcel displaced southward (y < 0) experiences lower f and acquires positive (cyclonic, counterclockwise) relative vorticity (ζ > 0). Across an alternating east-west chain of northward ridges (ζ < 0) and southward troughs (ζ > 0), the induced velocity field advects fluid parcels on the western flank of each ridge northward and parcels on the eastern flank southward, shifting the ridge and trough phase lines westward relative to the mean flow.

In a fluid layer with uniform eastward zonal flow U and Rossby radius of deformation L_d = √(gH)/f₀, linearizing the quasi-geostrophic potential vorticity equation for plane waves ψ′ = ψ̂ exp[i(kx + ly − ωt)] yields the Rossby wave dispersion relation:

ω = Uk − [βk / (k² + l² + L_d⁻²)]

where ω is angular frequency, k = 2π/λ_x is the zonal wavenumber, and l = 2π/λ_y is the meridional wavenumber. The zonal phase speed c_x = ω/k is:

c_x = ω / k = U − [β / (k² + l² + L_d⁻²)]

Because β > 0 and k² + l² + L_d⁻² > 0, the intrinsic zonal phase speed relative to the mean flow (c_x − U) is always negative (westward). Long planetary waves (small k) propagate westward rapidly relative to the fluid, whereas short synoptic waves (large k) propagate westward slowly and are swept eastward over the ground by the mid-latitude westerly jet (c_x > 0).

In contrast, the group velocity c_g = (∂ω/∂k, ∂ω/∂l), which governs the propagation of wave energy, has components:

c_gx = U + [β(k² − l² − L_d⁻²) / (k² + l² + L_d⁻²)²], c_gy = 2βkl / (k² + l² + L_d⁻²)²

Thus, while individual Rossby wave crests always propagate westward relative to the fluid, wave energy can propagate eastward (for short zonal scales, k² > l² + L_d⁻²) or westward (for long zonal scales, k² < l² + L_d⁻²), as well as along great-circle ray paths from the tropics into the extratropics (Hoskins & Karoly, 1981).

When the westward intrinsic phase speed balances an eastward westerly flow (U > 0), a stationary Rossby wave (c_x = 0) forms at total wavenumber K_s = √[(β / U) − L_d⁻²]. In the atmosphere, stationary planetary waves (wavenumbers 1–3) are forced by flow over continental mountain barriers—the Rocky Mountains, the Andes, and the Tibetan Plateau—and by land-sea thermal contrasts, anchoring the semi-permanent troughs over eastern North America and East Asia.

When Rossby waves amplify nonlinearly and break, they can stall for one to several weeks in atmospheric blocking patterns (omega or Rex dipole blocks) that deflect storm tracks and sustain prolonged regional drought, flooding, or heat waves. Studies continue to evaluate how high-latitude warming from Arctic amplification influences Rossby wave amplitude and persistence.

Vertically, the Charney-Drazin theorem (1961) shows that Rossby waves propagate upward from the troposphere into the stratosphere only when stratospheric winds are westerly (U > 0) and below a cutoff speed U_c that decreases with wavenumber. In winter, wavenumbers 1 and 2 penetrate into the polar stratosphere; when they break and deposit westward momentum flux, they can decelerate and reverse the polar night jet, triggering a Sudden Stratospheric Warming (SSW) that raises polar stratospheric temperatures by 30–50 K.

In physical oceanography, oceans support fast barotropic Rossby waves (crossing basins in days to weeks) and slow baroclinic Rossby waves, which displace the internal thermocline by tens of meters with sea surface height anomalies of 5–15 cm. Because the internal Rossby radius L_d is small (20–150 km), long baroclinic Rossby waves satisfy k² + l² ≪ L_d⁻², with westward phase speed:

c_x ≈ −βL_d² = −(βc_e²) / f₀²

where c_e ≈ 2–3 m/s is the first-mode internal gravity wave speed. Because f₀² ∝ sin²φ₀, baroclinic Rossby waves travel at ~20–25 cm/s near 5°–10° latitude (crossing the tropical Pacific in 6–12 months) but slow to 1–3 cm/s at 35°–45° latitude (taking 5–15 years). Imaged globally by satellite radar altimetry (Chelton & Schlax, 1996), oceanic Rossby waves concentrate wave energy and momentum flux along western basin margins to drive the westward intensification of the Gulf Stream and Kuroshio, while equatorial Rossby waves couple with Kelvin waves to pace El Niño–Southern Oscillation (ENSO) cycles.

Why does the phase of a Rossby wave always propagate westward relative to the background flow?

In both hemispheres, the Coriolis parameter f increases toward the North Pole (β = ∂f/∂y > 0). Consequently, any fluid parcel displaced northward acquires anticyclonic (clockwise in the Northern Hemisphere) relative vorticity, while a parcel displaced southward acquires cyclonic (counterclockwise) vorticity. The velocity field induced between an alternating east-west array of ridges and troughs always pushes fluid northward on the western flank of each ridge and southward on the eastern flank, shifting the ridge and trough phase lines westward relative to the fluid.

How can Rossby wave energy travel eastward while individual wave crests move westward?

Because Rossby waves are dispersive, their phase velocity (ω/k) and group velocity (∂ω/∂k) have different mathematical dependencies on wavenumber k. For short synoptic wavelengths (where k² > l² + L_d⁻²), the intrinsic zonal group velocity is eastward even as individual troughs and ridges drift westward relative to the mean flow. In the atmosphere, this allows a localized disturbance—such as cyclogenesis over the North Pacific—to radiate downstream wave energy eastward across North America and into the North Atlantic over several days.

Why do stratospheric sudden warmings happen primarily in the Northern Hemisphere winter?

By the Charney-Drazin theorem, only ultra-long planetary Rossby waves (wavenumbers 1 and 2) can propagate upward into the westerly winter stratospheric polar vortex. The Northern Hemisphere possesses large mid-latitude mountain chains (the Rockies and the Tibetan Plateau) and pronounced continent-ocean thermal contrasts that force strong wavenumber 1 and 2 stationary waves. The Southern Hemisphere mid-latitudes are dominated by the continuous Southern Ocean with weaker topographic wave forcing, leaving the Antarctic stratospheric polar vortex stronger and rarely disrupted.

Why did satellite altimeters reveal mid-latitude oceanic Rossby waves moving faster than linear theory predicted?

When Chelton and Schlax (1996) analyzed TOPEX/Poseidon sea-surface height observations, they discovered that mid-latitude baroclinic Rossby waves propagated westward at 1.5 to 2 times the speed predicted by the standard flat-bottom, resting-ocean linear formula c_x = −βL_d². Subsequent theoretical work demonstrated that background baroclinic vertical shear in the mean ocean gyre circulation, seafloor topography, and nonlinear mesoscale eddy interactions modify the effective potential vorticity gradient and accelerate westward propagation.

1. Rossby, C.-G., & Collaborators (1939). Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. Journal of Marine Research, 2(1), 38–55. doi:10.1357/002224039806649023
2. Charney, J. G., & Drazin, P. G. (1961). Propagation of planetary-scale disturbances from the lower into the upper atmosphere. Journal of Geophysical Research, 66(1), 83–109. doi:10.1029/JZ066i001p00083
3. Hoskins, B. J., & Karoly, D. J. (1981). The steady linear response of a spherical atmosphere to thermal and orographic forcing. Journal of the Atmospheric Sciences, 38(6), 1179–1196. doi:10.1175/1520-0469(1981)038<1179:TSLROA>2.0.CO;2038<1179:TSLROA>2.0.CO;2)
4. Chelton, D. B., & Schlax, M. G. (1996). Global observations of oceanic Rossby waves. Science, 272(5259), 234–238. doi:10.1126/science.272.5259.234