Rain & Puddles (3D)

Drop sizes: the Marshall-Palmer distribution

The number of raindrops per cubic metre and per millimetre of diameter is modelled as N(D) = N₀ exp(-ΛD). Marshall and Palmer (1948) found N₀ ≈ 8000 m⁻³ mm⁻¹ and Λ = 4.1 R⁻⁰·²¹ mm⁻¹ for rain rate R in mm/h. Here the slope Λ is solved numerically so that the distribution produces exactly the chosen R:

R = 6π×10⁻⁴ ∫ N(D) D³ v(D) dD  (D in mm, v in m/s, R in mm/h). With N₀ = 8000 this reproduces the published slope within a few percent (check 1). Drizzle has many small drops (large N₀), thunderstorm rain has few huge ones (small N₀): try the presets.

Other moments follow from the same integral: liquid water content W = (π/6) ρₖ ∫ N D³ dD, the radar reflectivity factor Z = ∫ N D⁶ dD (shown in dBZ; the textbook shortcut is Z = 200 R¹·⁶), and the kinetic-energy flux ∫ N v (½ m v²) dD. The distribution is cut at 0.05 and 6 mm: bigger drops break up in the air.

Terminal velocity

A falling drop speeds up until drag balances weight. For quadratic drag, m dv/dt = mg - ½ ρₐ C₍ A v² gives v(t) = vₜ tanh(g t / vₜ) and a fall distance z(t) = (vₜ²/g) ln cosh(g t / vₜ). The terminal speed vₜ(D) in the model is the Atlas-Ulbrich fit 9.65 - 10.3 exp(-0.6 D) (D in mm, D ≥ 0.5 mm) with a Stokes-like linear branch below. It agrees with the Gunn-Kinzer laboratory data within a few percent. A rigid sphere with a Reynolds-dependent drag law (Schiller-Naumann) falls up to ~14% faster than real drops because real drops flatten into oblate shapes (axis ratio about 1.03 - 0.062 D) and drag more. The test drop uses a drag coefficient calibrated so its terminal speed equals the fit, then integrates the equation numerically and compares with the closed form.

Impact: splash or ripples

The Weber number We = ρₖ v² D / σ compares the drop's kinetic energy to its surface energy (σ = 0.072 N/m). Above roughly We ≈ 300 a drop landing on a thin water film throws a crown of secondary droplets; below it, it only makes ripples. The exact threshold depends on film depth and viscosity, so 300 is a rough rule of thumb, not a sharp law.

Ripples

Ripples on shallow puddles are capillary-gravity waves: ω² = g k + (σ/ρₖ) k³. The ring front moves at the group speed cₗ = (g + 3σk²/ρₖ) / (2ω); the phase speed has a minimum of 0.23 m/s at λ = 1.7 cm. Each ring carries a packet of wavelength about 8 drop diameters (clamped to 1.2 to 6 cm), loses amplitude as 1/√r from spreading and as exp(-2νk²t) from viscosity, which is why ripples fade within seconds. The dense fine ripple texture between the big rings is a visual stand-in for the many small impacts that cannot be drawn individually.

The puddle

The puddle is a paraboloid bowl of rim radius 0.8 m and depth 10 cm: surface radius r(h) = a√(h/H), volume V(h) = πa²h²/(2H). It receives rain from a catchment area Aₑ (roof, path) and loses water by soil infiltration through its surface: dV/dt = R Aₑ - k π r(h)², integrated with RK4. Without infiltration the depth grows as h = √(2 H R Aₑ t / (π a²)); with infiltration it approaches the steady state h* = R Aₑ H / (k π a²) (check 3), and water beyond 10 cm spills over the rim. Evaporation and sloping runoff are ignored. The Puddle time-lapse speeds up the hydrology clock (and the gauge and the rain total) without changing the raindrop animation.

What is real and what is drawn

• All readouts, graphs and checks come from the equations above, not from the picture.
• The drawn drops are a thinned sample (a real shower has thousands per cubic metre). Their sizes are drawn with a visual weighting that favours bigger drops so they are visible. Wind moves all drops sideways at the wind speed, so small slow drops slant more than big fast ones.
• The impacts on the puddle are random draws weighted by impact flux N(D) v(D), but at a thinned rate; the gauge in the model collects real Monte-Carlo samples of that flux at the true rate and its reading is compared with R in the checks.
• Ripple heights are exaggerated in the shading so they can be seen.

References: Marshall & Palmer (1948), J. Meteor. 5, 165; Gunn & Kinzer (1949), J. Meteor. 6, 243; Atlas & Ulbrich (1977); Pruppacher & Beard (1970).