Environment · Simulator 089
On a summer day a dark road can be far hotter than a park next to it, and it stays warm long after sunset. Pick a surface, then change how reflective it is, how much shade and greenery it has, how much heat it stores, and how much waste heat the city adds. Compare its temperature through two days with a grass park.
Every square metre of ground balances the energy it receives against the energy it loses:
C dT/dt = (1 − α)(1 − shade) S + L↓ − εσT⁴ + Q_F − H − LE − G
Absorbed sunlight depends on the albedo α (the share reflected: fresh asphalt about 0.05, a white "cool roof" 0.7 or more). The surface also gains longwave radiation L↓ from the warm air and sky and emits its own, εσT⁴. Q_F is anthropogenic heat from traffic and air conditioners.
It loses heat by convection to the air, H = h (T − T_air), faster in wind; by evaporation LE, here a fraction of the net radiation set by how much vegetation and moisture there is; and by conduction G into the ground below. C is the surface layer's heat capacity: high-mass materials such as concrete warm slowly by day and release the stored heat at night.
Below the surface, a thick ground layer exchanges heat with it. Paving conducts heat down well by day and returns it at night; soil under grass and insulated roofs exchange much less. The park reference is grass (albedo 0.25, fully vegetated, no shade or waste heat) under the same weather.
Illustrative single-surface model: one layer, fixed weather, no street-canyon geometry, and evaporation simplified to a share of net radiation. Real urban heat islands also depend on building shape, wind channelling and humidity.
Two roofs absorb sunlight differently: one has albedo 0.10, the other 0.70. At noon with 900 W/m² of sunshine, how much solar power does each absorb per square metre? Why is the real temperature difference smaller than this difference in absorbed power might suggest?
Dark roof: 0.90 × 900 = 810 W/m²; white roof: 0.30 × 900 = 270 W/m², a difference of 540 W/m². As the dark roof heats up, it loses more by convection and longwave emission (which grows with T⁴) and conducts more downward, so these losses rise to balance most of the extra gain and the temperature gap settles at some tens of degrees.