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Home/Biophysics, Fluids & Geoscience/Rossby Waves on a Beta-Plane

Rossby Waves on a Beta-Plane

Linear planetary wave ψ = A cos(kx+ly−ωt) with ω = −βk/(k²+l²+Rd⁻²): streamfunction bands and geostrophic arrows show westward phase propagation.

Rossby mode

1.4
8
5.5
12

Animation

1
0.75 x

The color field is streamfunction ψ. Arrows show geostrophic velocity (u, v) = (-ψ_y, ψ_x). The pattern moves west because β makes ω/k negative for k > 0.

Measured values

ω-0.570
phase speed cx-0.726
group speed cgx-0.262
period11.02

On a rotating planet the Coriolis parameter grows poleward. A north-south displacement changes planetary vorticity, so the restoring motion forms large-scale Rossby waves with westward phase speed.

Live graphs

About this model

Rossby waves are large-scale planetary waves produced by the meridional gradient of the Coriolis parameter, f = f0 + βy. This simulator uses a single linear beta-plane mode with streamfunction ψ = A cos(kx + ly − ωt). The dispersion relation ω = −βk/(k² + l² + Rd⁻²) is evaluated directly, so no numerical PDE stability limit is hidden in the animation. For β > 0 and eastward wavenumber k > 0, the zonal phase speed c_x = ω/k is negative: crests drift west even though the group velocity can change sign depending on the aspect ratio and deformation radius.

Who it's for: Physical oceanography, atmospheric dynamics, or geophysical fluid dynamics introductions.

Key terms

  • Rossby wave
  • Beta-plane
  • Planetary vorticity
  • Geostrophic flow
  • Dispersion relation

How it works

Linear barotropic / shallow-water Rossby wave on a beta-plane. Tune β, wavelengths, and deformation radius to see why planetary vorticity gradients give westward phase propagation.

Key equations

ψ = A cos(kx + ly − ωt), k = 2π/λx, l = 2π/λy
ω = −βk / (k² + l² + Rd⁻²), c_x = ω/k < 0 for β > 0

Frequently asked questions

Is this a full shallow-water forecast model?
No. It is one analytic linear normal mode. That keeps the key beta-effect and dispersion relation visible, but excludes forcing, dissipation, boundaries, topography, and nonlinear wave-mean-flow interactions.
Why can group velocity differ from phase velocity?
Rossby waves are dispersive: ω is not proportional to k. Crests move with c_x = ω/k, while wave-packet energy follows ∂ω/∂k, which can be eastward or westward depending on k, l, and Rd.