Cyclone and Coriolis Effect: Gradient Wind, Pressure Gradient and Friction

What is modelled

A map-view patch of ocean (700 km radius) with a circular pressure system. Hundreds of air parcels move under the pressure-gradient force, the Coriolis force and surface friction; they are drawn as arrows coloured by speed. A weather buoy at an adjustable distance reads pressure and wind. You can pick a low (cyclone) or high (anticyclone), the hemisphere, the strength of the Coriolis effect, and see what happens when it is switched off. The eyewall cloud and rain bands are a visual kit that follows the radius of maximum wind and the sense of rotation; they do not feed back on the physics.

Equations

• f-plane momentum equations for a parcel: du/dt = −(1/ρ) ∂p/∂x + f v − κ u, dv/dt = −(1/ρ) ∂p/∂y − f u − κ v, with f = 2Ω sin φ (Ω = 7.292×10⁻⁵ s⁻¹) and κ the friction rate. Deflection is to the right of the motion for f > 0 and to the left for f < 0.
• Pressure profile (Holland-type): p(r) = penv − s Δp (1 − exp(−(Rm/r)B)), s = +1 for a low and −1 for a high.
• Gradient wind (steady circular flow): v2/r + f v = (1/ρ) dp/dr, so v = −f r/2 + √((f r/2)2 + (r/ρ) dp/dr). At small radius the centrifugal term dominates (cyclostrophic balance, Rossby number v/(f r) much greater than 1); far away it approaches geostrophic balance v = (1/ρ f) dp/dr.
• Without friction two quantities are exactly conserved: absolute angular momentum M = r vθ + f r2/2 and energy E = v2/2 + p/ρ. A free parcel with no pressure force moves in an inertial circle of radius v/f and period 2π/f.
• Friction removes angular momentum, so the wind falls below the gradient wind and parcels cross the isobars inward toward a low, spiralling in; in a high they spiral outward.

Checks run in the page

• Inertial circle: radius, period and speed from the integrator.
• Gradient-wind balance residual, and a parcel launched in that wind keeps its radius.
• Angular momentum conservation for a free parcel in the vortex.
• Energy conservation (the Coriolis force does no work).
• Northern/southern hemisphere mirror symmetry.

Honest simplifications

• The vortex is a prescribed, steady, axisymmetric pressure field on a flat f-plane at a fixed latitude; a real tropical cyclone has a warm core, moisture, heat release and a moving centre that create its pressure field, which is not computed here.
• Air density is constant (1.15 kg/m3) and the flow is two-dimensional; there is no vertical motion, no boundary-layer structure and no wind shear.
• Friction is a simple linear (Rayleigh) drag with one rate, not a wind-speed-dependent surface stress.
• The Coriolis strength slider is a thought experiment (a planet spinning faster or slower), and "near the equator" shows why cyclones do not form there: f is too small to balance the pressure force.
• Map objects (buoy, island, clouds) are exaggerated in size for visibility. Time runs at several hours per second.

What to try

• Switch off the Coriolis force: the air falls straight into the low with no rotation.
• Compare the northern and southern presets: opposite sense of rotation, mirror-image paths.
• Raise the friction and watch the inflow angle grow and the measured wind drop below the gradient wind.
• Move the buoy outward and read how pressure and wind change with distance.