Surface Temperature Depression Under a Clear Sky

Here is the puzzle this page exists to answer. The forecast says +2 °C overnight, the porch thermometer agrees in the morning, and there is frost on the windscreen and on the tops of the cabbages. Nothing was ever at freezing according to the instrument. So how did water freeze?

Because the thermometer measures the AIR, and the surface is not the air. A surface facing a clear sky loses net longwave radiation continuously — 60 to 100 W/m² is typical — and it has only one significant way to get that heat back: convection from the air touching it. Convection can only deliver heat if the surface is colder than the air, so the surface drops until the gap is big enough to balance the loss. At steady state h_c(T_air − T_s) = q_net, and the depression is simply q_net/h_c. With 70 W/m² lost and a light-breeze film coefficient of 10 W/(m²·K), that is 7 K — and a +2 °C night puts the surface at −5 °C. Frost, on a night that never froze.

Everything in that expression tells you something a grower already knows from experience. The depression is inversely proportional to h_c, so WIND KILLS FROST: raise the film coefficient from 10 to 25 with a stiff breeze and the same radiative loss produces under 3 K of depression instead of 7. That is why frost fans and helicopters are used in orchards, and why the dangerous nights are the still ones. Cloud kills it from the other end by cutting q_net toward zero. A horizontal surface with a clear view of the whole sky cools hardest, which is why the car roof frosts before the doors and why a tree, a hedge or a sheet of fleece overhead protects what is under it — not by insulating it, but by replacing the cold sky with a warm surface.

The honest caveats. This is a steady-state balance on a surface with negligible thermal mass and no heat coming up from below, which describes a leaf, a windscreen or a thin roof panel well and describes a concrete slab badly — a slab bleeds stored heat upward all night and frosts far less. Evaporation and condensation are absent from the balance, and both matter: dew forming on the surface releases latent heat and slows the cooling, which is why heavy dew and heavy frost rarely occur on the same night. And h_c near a cold surface in still air is itself weakly known, so treat a recovered h_c as an effective figure for that night rather than a number you can carry elsewhere. Use this page for the direction and the rough size of the effect, which is what actually changes a decision.