PhysSandbox
Classical MechanicsWaves & SoundElectricity & MagnetismOptics & LightGravity & OrbitsLabs
🌙Astronomy & The Sky🌡️Thermodynamics🌍Biophysics, Fluids & Geoscience📐Math Visualization🔧Engineering🧪Chemistry

More from Biophysics, Fluids & Geoscience

Other simulators in this category — or see all 32.

View category →
NewUniversity / research

Geostrophic Balance & Thermal Wind

Pressure-gradient force balanced by Coriolis: isobars set geostrophic wind, while meridional temperature gradients produce vertical shear between 1000 and 300 hPa.

Launch Simulator
NewUniversity / research

Groundwater Contaminant Plume

Advection-dispersion pulse in groundwater with longitudinal/transverse spreading, retardation factor R, and a monitoring-well breakthrough curve.

Launch Simulator
NewUniversity / research

Infinite Slope Stability

Limit-equilibrium factor of safety for a shallow planar slide: slope angle, cohesion, friction, depth, and rainfall-driven pore pressure set the landslide threshold.

Launch Simulator
NewUniversity / research

Earthquake Aftershocks: Omori + Gutenberg-Richter

Modified Omori aftershock decay n(t)=K/(t+c)^p combined with Gutenberg-Richter magnitude-frequency curves, b-value, and a synthetic catalog.

Launch Simulator
NewUniversity / research

Mantle Convection Cell (Toy)

High-Prandtl-number mantle convection cartoon: Rayleigh-number vigor, thermal boundary layers, hot upwelling, and a cold subducting slab in one viscous cell.

Launch Simulator
NewSchool

Carbon Cycle (4-Box Model)

Atmosphere, ocean mixed layer, deep ocean, and land biosphere exchange linearly; add fossil emissions or GtC pulses and watch inventory split vs a toy airborne fraction.

Launch Simulator
PhysSandbox

Interactive physics, chemistry, and engineering simulators for students, teachers, and curious minds.

Physics

  • Classical Mechanics
  • Waves & Sound
  • Electricity & Magnetism

Science

  • Optics & Light
  • Gravity & Orbits
  • Astronomy & The Sky

More

  • Thermodynamics
  • Biophysics, Fluids & Geoscience
  • Math Visualization
  • Engineering
  • Chemistry

© 2026 PhysSandbox. Free interactive science simulators.

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

Rossby Waves on a Beta-Plane

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

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

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

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.