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The science
Travertine terraces such as those of Pamukkale or Mammoth Hot Springs grow because warm spring water is rich in dissolved CO2 and calcium and loses CO2 to the air as it runs downhill. Less CO2 means a higher pH and a lower calcite solubility, so the water becomes supersaturated and calcite (CaCO3) precipitates, fastest where the water is thin and turbulent: on the lip of each dam. The dam grows, the pool behind it deepens, and the staircase builds itself.
Equilibrium. For the reaction Ca2+ + 2 HCO3− ⇄ CaCO3 + CO2 + H2O, dilute-solution equilibrium gives the classic cube-root law between the calcium concentration and the dissolved CO2 x = KH pCO2:
ceq = a(T) x1/3, x = KH(T) pCO2, KH(T) = 33.8 exp[2400 (1/T − 1/298.15)] mol m−3 atm−1
(a(T) falls with temperature because calcite is less soluble in warm water.) Water saturated at the source pressure carries about 6 to 7 mol/m3 of calcium; water in equilibrium with the air (0.042 % CO2) could keep only about 0.7 mol/m3. The difference is the stock that the terraces can deposit.
Pool chemistry (quasi-steady). The water stays in a pool for minutes, the dams change over days, so every pool is solved as a steady well-mixed reactor with area A = w Lc and flow Q:
Q (cin − c) = R A, Q (xin − x) + R A = kg A (x − xair), R = kp(T) [ c / ceq(x) − 1 ]+
R is the calcite precipitation rate (mol m−2 s−1), kg the gas-transfer velocity, kp(T) an Arrhenius rate constant (activation energy 50 kJ/mol, a typical value). Each mole of calcite released one mole of CO2, which is why the second equation has +RA. The monotone residual is solved by bisection in c.
Lip film and dam growth. Over the lip the water forms a thin sheet of critical depth hc = (Q2/(w2 g))1/3 (weir law Q = w √g hc3/2), a few millimetres, which degasses much faster than the pool: a fraction ψ = 1 − exp(−f kg tf / hc) of the excess CO2 is lost during the fall time tf = √(2Δz/g) + 0.5 s, f being the cascade turbulence factor. The film deposits Rlip over a path of 0.6 m but the calcite lands on the narrow crest of width lc, so the dam grows at
dh/dt = (M/ρ) Rlip (0.6 m / lc), floor: dz/dt = (M/ρ) R, M = 0.1001 kg/mol, ρ = 2710 kg/m3
This focusing factor conserves mass exactly: the deposited volume equals the calcium removed from the water. The growth is capped smoothly near the maximum dam height (flow starts to bypass tall dams), and the drop Δz between a crest and the next pool is recomputed as the dams grow, which gives the terraces a self-regulating spacing.
Temperature and pH. The water cools by T = Tair + (Tin − Tair) exp(−hT A / (ρw cp Q)) in each pool (hT = 120 W m−2 K−1 for steaming water, a typical value). The pH shown is pH = pK1(T) + log10(2c / x), assuming bicarbonate = 2c and CO2 = x.
Numerical checks run in the page. (1) One pool with fixed CO2 against the closed form of the linear mass balance; (2) a very large pool, whose calcium must fall to the equilibrium with air CO2; (3) the CO2 relaxation without precipitation against (Q xin + kA xair)/(Q + kA); (4) the weir law for the film depth; (5) calcium conservation over 60 simulated days, comparing the deposited volume read from the dam and floor heights with the time-integrated calcium flux in minus out; (6) the conversion of 10−4 mol m−2 s−1 to 0.319 mm/day.
Simplifications, stated honestly. Ideal-dilute equilibrium (no activity coefficients, no ion pairs, ionic strength ignored: the real solubility of a mineral spring is up to a few times higher than ceq shown); first-order surface kinetics without the Plummer-Wigley-Parkhurst back-reaction or inhibitors such as phosphate; no evaporation or evaporative concentration; no biology (cyanobacterial mats and microbes accelerate real precipitation; the orange colour is decoration); 8 identical rectangular pools in a chain, no lateral flow or channel switching; a smooth cap instead of the flow diversion of tall dams. The constants (kp, hT, a(T), the 0.6 m film path, the 3 m cap) are typical illustrative values, not field calibrations. Real growth rates range from fractions of a mm per day to a few mm per day; here they depend on the sliders. The model teaches the mechanism and the order of magnitude, not the history of any particular site.