Station T-

Rain0 mm/h

Slope-

Front speed-

Front to stn- km

Drag: orbit · wheel/pinch: zoom · vertical scale exaggerated 10x · orange: dry warm air, white: cloudy lifted air

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Cold Front Cross-Section: Margules Wedge Slope, Gravity-Current Speed and Frontal Rain

A wedge of cold air undercuts warm moist air. Set the temperature and wind jump, the dome depth and the dewpoint; watch the lifting, the cloud base, the rain band and the weather station as the front passes.

Live measurements

Weather station 50 km ahead of the front: temperature (red), pressure change (blue dashed) and rain (bars).

Margules slope of the wedge- Same slope, exact density form- Coriolis parameter f- Front speed (gravity current)- Ascent of the warm air- Cloud base (LCL)- Condensate- Rain intensity in the band- Storm-total rain at a point- Station now-

Lifted parcel: dry adiabat up to the cloud base, then the moist adiabat.

Air masses and front

Warm air T

Cold air T

Warm dewpoint

Wind jump Δv

Latitude

Cold-air depth

Head Froude no.

Warm-air wind

Rain efficiency

Time scale

The temperature jump (warm minus cold) and the wind jump Δv across the front set the slope; the dome depth and temperature jump set the speed; the dewpoint sets where the cloud starts.

Experiment: does a steeper front rain more?

Marches a station through the front for six values of the wind jump (your other settings) and measures the peak intensity, the duration and the storm-total depth from the time series.

Δv m/s	slope	w cm/s	peak mm/h	hours	total mm

Independent numerical checks

Check	Theory	Measured

The science

A cold front is the leading edge of a wedge of dense cold air that undercuts warmer air. Two laws fix the wedge: how steep it is, and how fast it advances.

Slope (Margules 1906). For two air masses in geostrophic balance, continuity of pressure along the sloping interface gives tan α = f (ρ_w Δv) / (g (ρ_c − ρ_w)) ≈ (f T̄ / g) (Δv / ΔT), with f the Coriolis parameter, Δv the difference of the along-front wind between the two air masses and ΔT their temperature difference. Typical fronts: 1:50 to 1:200. The panel shows both the temperature form and the exact density form found by a numerical root-finder.

Speed. The cold dome advances like a gravity current: c = Fr √(g' H), g' = g ΔT / T_w, with a head Froude number about 0.7-1.1 in laboratory density currents (Benjamin 1968). This ignores rotation, surface friction and the pressure field of the parent cyclone.

Lifting and rain. Warm air meets the wedge with speed c − u_w and rises up the slope at w = (c − u_w) tan α (centimetres per second: slow, broad ascent). It cools along the dry adiabat (g/c_p = 9.8 K/km) until the lifting condensation level (Bolton 1980 formula; Espy's rule of thumb 125 m per K of dewpoint depression), then along the moist adiabat Γ_m = g (1 + L_v r_s/(R_dT)) / (c_p + L_v² r_s ε /(R_d T²)). The condensed water per kilogram is r₀ − r_s(top); the rain rate is eff · ρ w Δq.

A neat consequence: the band behind the front is (H − z_LCL)/tan α wide and takes that long to pass, so the storm-total rain eff ρ Δq (1 − u_w/c)(H − z_LCL) does not depend on the slope: a steep front rains harder but for a shorter time. The experiment verifies this by marching a station through the front.

Honest limits: this is a two-dimensional kinematic sketch, not a weather model. Only the cold-air depth (not the full frontal zone up to the tropopause) is lifted, latent heating and convective bursts at the nose are not modelled, precipitation does not drift while falling, the station pressure and wind traces (pre-frontal trough, SW to NW veer) are illustrative shapes, and the rain efficiency 0.6 is a free parameter. The 3D scene exaggerates height 10 times relative to distance (1 unit = 10 km horizontally, 1 km vertically); the town, trees and mast are enlarged icons. Reference values (slopes, Froude number) are typical numbers quoted from memory and should be spot-checked.

Models and textures: Blender 5.2, procedurally baked with Cycles; rendering Three.js r160.