Ice, liquid and vapour are the same molecules, moving slower or faster. Warm air can hold more invisible vapour: saturation vapour pressure (Tetens) e_s = 0.6108·exp(17.27·T/(T+237.3)) kPa gives 2.34 kPa at 20 °C and 4.24 kPa at 30 °C (×1.81).

Turning 1 kg of water into vapour needs L_v = 2.501 − 0.002361·T MJ/kg, about 2.45 MJ at 20 °C. One millimetre of water on one square metre is 1 kg. Evaporation in the valley uses Priestley–Taylor: ET₀ = 1.26·Δ/(Δ+γ)·R_n/L_v. For T = 20 °C and R_n = 15 MJ/m²/day that is 5.3 mm/day. Simplification: no wind term and no humidity deficit. R_n = (1 − albedo)·R_s·0.75 (the 0.75 is a Model value).

Air that rises cools. A lifted parcel cools about 9.8 K per km while its dew point falls only 1.8 K per km, so it saturates after z ≈ 125·(T − T_d) metres (Espy): 1,250 m for T = 25 °C and T_d = 15 °C. In the model each of the ≤ 60 cloud packets holds vapour and liquid; when it is pushed up a slope the air is colder, can hold less vapour (ρ_sat = 2166.8·e_s/T g/m³), and the extra becomes cloud water. Rain falls when cloud water exceeds 0.8 g/m³: P = (C − C_crit)/τ_p, τ_p = 0.25 day (Model values). Temperature falls 6.5 K per km of height (Typical). Wind speed is time-compressed (Model).

Runoff uses the SCS curve number: S = 25400/CN − 254, I_a = 0.2 S, Q = (P − I_a)² / (P + 0.8 S). For 50 mm of rain: CN 55 gives 0.33 mm, CN 75 gives 9.29 mm, CN 98 gives 44.3 mm. Forest (CN 60) lets more soak in than meadow (70) or bare rock (90). The soil is a bucket; water above field capacity drains to one aquifer, a linear reservoir dS/dt = R − kS, k = 0.05 per day, which feeds the river slowly (S = S₀·e^(−kt)). Trees also give back more water to the air (crop coefficient K_c: forest 1.05, grass 0.8).

Rain becomes snow where the air is below 0 °C; it melts by degree-day: M = 4 mm × (T − 0 °C) per day (Typical range 3–6).

dV/dt = Q_in + P·A − E·A − Q_out − Q_village. In the Terrarium nothing enters or leaves, so the sum of all drops stays the same (1 drop = 10,000 m³). In the open valley moist sea air also enters at the upwind edge and clouds can leave; both are counted so the check still adds up. Lake spill uses a weir: Q = C·b·H^1.5, C = 1.7. For b = 2 m and H = 0.5 m that is 1.20 m³/s.

Each person uses about 150 litres a day (Model; the WHO basic need is lower, about 50–100, verify before publishing). 5,000 people use 750 m³/day, about 9 % of the valley's average stream flow of 8,400 m³/day. The turbidity number (NTU) is a Model value: it rises when rain splashes bare ground and falls when trees cover the slopes.

Each row compares the game's code with a hand calculation made separately.