Interactive Beta-Plane Model
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Purpose. A live two-dimensional model of the mid-latitude atmosphere at about 5–10 km altitude, where the jet stream lives. The Earth’s rotation is represented by the β-plane: the Coriolis parameter increases northward at a constant rate β, which is what makes large-scale waves (Rossby waves) drift westward relative to the wind and lets a jet meander, break and leave behind stationary highs and lows, the ingredients of atmospheric blocking. You can switch on a jet and a band of random “storms” (stirring) and disturb the flow with the mouse.
How to use. Drag across the map to push the air along the stroke. Click (tap) to drop a high-pressure vortex (anticyclone, H); hold Shift while clicking, or change “Click places”, to drop a low (cyclone, L). A strong high placed just north of the jet often stalls and splits the jet, like a block. The sliders change the jet and the stirring while the model runs; Reset starts the chosen preset again; Calm removes all eddies but keeps the jet.
Advanced: β, friction, resolution
Method notes
Equation. Barotropic vorticity equation on a β-plane, ∂ζ/∂t + J(ψ, ζ) + β ∂ψ/∂x = Fjet + Fstir + Fmouse − ζ/τf − ν∇8ζ, with ζ = ∇2ψ, u = −∂ψ/∂y, v = ∂ψ/∂x. Domain 20 000 km × 10 000 km, periodic in both directions; the y axis is labelled 0–90°N at 111 km per degree, but β is constant (default 1.62 × 10−11 m−1 s−1, its value at 45°N) and the top edge joins the bottom edge.
Numerics. Pseudo-spectral (fast Fourier transforms; exact inversion of ζ = ∇2ψ on the periodic domain), Jacobian on the grid with 2/3-rule dealiasing, third-order Adams–Bashforth time stepping (about 7–13 min steps, halved automatically when the flow is fast), friction and ∇8 hyperviscosity (damping time 1/20 day at the smallest resolved scale) applied exactly. A single Rossby wave propagates at the theoretical speed c = −β/K2 and the unforced, inviscid energy is conserved to 1 part in 107 over three days. The model runs in a Web Worker in your browser; nothing is sent to a server.
Jet. The zonal-mean vorticity is relaxed, with the jet relaxation time, towards that of U(y) = U0 exp[−(y − yj)2/w2] (repeated periodically in y). On a periodic domain the domain-average wind cannot be represented, so the profile is shown and used with its average removed (weak easterlies away from the jet). Jets narrower than about (2U0/β)1/2 violate the Rayleigh–Kuo stability criterion (U″ > β somewhere) and meander on their own.
Stirring. Fstir = A G(y) Σm−1…m+1 Re[am(t) ei km x], with G a Gaussian of half-width 800 km around the chosen latitude (the zonal-jets preset stirs the whole domain), km = 2πm / 20 000 km and am complex random amplitudes with a 2-day memory (red noise).
Mouse. Dragging adds a velocity impulse along the stroke with a Gaussian footprint (radius 400 km), which in vorticity is a dipole; clicking adds a Gaussian vortex (radius 550 km, peak wind about 25 m s−1). The panels on the right show the zonal-mean wind with the target jet (dashed), ψ minus its zonal mean along one latitude over the last 30 days (time runs downward: eastward-moving waves tilt to the lower right, westward-moving ones to the lower left), and the kinetic-energy spectrum with the stirring wavenumber and the Rhines wavenumber kβ = (β/Urms)1/2, where the turbulent energy cascade is arrested and zonal jets form.