Hoar Frost on a Clear Night: Radiative Cooling, Frost Point and Vapour Deposition

A fence post, a barbed wire, a tuft of grass and a dock leaf, from dusk to dawn. Thin things radiate heat to the clear sky and fall below the air temperature; once they drop below the frost point, water vapour turns straight into ice crystals. Change the sky, the wind and the humidity and watch where frost forms, and what shape it takes.

Scenario

Preset

Time warp

Night

Live numbers

Temperatures through the night

━ air · ━ frost point (dashed) · ━ wire · ━ grass blade · ━ leaf · ━ post

Frost mass on each surface (g per m²)

Experiments (measured vs prediction)

Run an experiment. They use private copies of the model and do not touch your night.

Independent numerical checks

Also run outside the browser: node src/check.js.

Educational lumped model. The crystals are drawn 3 to 5 times larger than real hoar so that you can see them, and their number follows the simulated mass; the surfaces have one temperature each (no gradients across a blade or leaf).

Controls

Drag or one finger: orbit; wheel or pinch: zoom; Wire, Grass, Leaf and Wide change the view.
Pick a preset, or set the night with the sliders; the clock runs from 17:00 to 07:00 and the sky turns pale at dawn.

The science

Why thin things freeze first. Each surface has one temperature T_s and a heat capacity per unit area C. C dT_s/dt = -ε σ (f T_s⁴ - LW_down) - h (T_s - T_air) + L J. A clear sky sends down much less long-wave radiation than the ground emits; for the sky the model uses Swinbank's formula LW_down = 9.37×10^-6 T_air² σ T_air⁴, blended with an overcast sky of emissivity 0.96. Grass (C ≈ 500 J m^-2 K^-1) cools with a time constant of under a minute, a wooden post (C ≈ 14 000) takes hours, so on a clear calm night the grass can be several degrees colder than the air at head height.

Wind. Convection h = h_s(5 + 4.5 v) pulls the surface back towards the air temperature: a breeze reduces the cooling but also renews the humid air above it, so the frost mass has a maximum at a light wind (see the experiment).

Frost point. Vapour deposits as ice when the saturation pressure over ice at the surface temperature is lower than the vapour pressure of the air: e_i(T_s) < e_air. The model uses Buck's equations for e_w and e_i (checked against Murphy and Koop 2005) and a bisection for the frost point. The mass flux is J = h_m(ρ_v,air - ρ_v,sat(T_s)) with the Chilton–Colburn analogy h_m = h / (ρ_a c_p) · Le^-2/3. Condensation releases latent heat (L_s = 2.83 MJ/kg for frost): the frost warms its own surface, which limits how cold the leaf can get.

Dew and frozen dew. While the surface is above 0 °C and below the dew point, liquid dew collects; if the surface then falls below 0 °C the dew freezes (releasing 334 kJ/kg, a short temperature plateau near 0 °C) and gives a hard, clear coating instead of feathery hoar.

Crystal shape. Whether the ice grows as plates, needles or dendrites depends on temperature and supersaturation (the Nakaya diagram). The model uses a simplified lookup: plates warmer than -3.5 °C, needles and hollow columns -3.5 to -8.5, plates -8.5 to -12, dendrites -12 to -17.5, plates colder. Real hoar on thin wires and stems is mostly needles and feathers, and the boundaries are blurred by humidity and wind.

Honest simplifications. Lumped surfaces; the heat capacities, view factors and the frost bulk density (60 + 3 × (-T_s) kg per m³) are typical values, not measurements; the air cools along a prescribed half-cosine and carries a constant vapour pressure until it saturates (fog is not modelled); no ground heat flux, no latent heat from the soil, no snow, no sublimation by sunshine after dawn. Heavy hoar frost is usually a few hundred grams per m² at most; treat the default numbers as the upper end.