Watch ice grow as cloud conditions change.

What happens in the parcel

As a parcel rises, pressure falls and the air usually cools. Cooling can make vapor more favorable to ice, while expansion dilutes vapor. Existing droplets and crystals exchange water with the vapor; deposition onto ice releases heat. These effects occur together, so the change in the drive toward ice cannot be inferred from cooling or expansion alone. In one experiment lifting stops after five simulated minutes, yet exchange can continue afterward.

The three quantities to watch are ice-growth drive (the vapor imbalance relative to ice), exchange capacity (how readily existing ice can exchange water), and ice exchange rate (the actual modeled mass transfer per second). A large drive need not produce fast growth when the existing ice has little capacity. At zero existing ice, capacity and ice growth remain zero.

The compact model

Drive: ψ = ln[e/e_si(T)]. Here e is vapor pressure, e_si(T) is equilibrium vapor pressure over ice, and T is absolute temperature in kelvin. Positive ψ favors ice deposition; negative ψ favors sublimation. Multiplying ψ by RT gives the chemical driving potential per mole transferred.

How the drive changes: dψ/dt = [1/T − d ln(e_si)/dT] dT/dt − (1/V) dV/dt − (Γ_l + Γ_i)/m_v. The terms are the simultaneous contributions from temperature, volume, and phase change. V is parcel volume; m_v is vapor mass. Γ_l and Γ_i are vapor-to-liquid and vapor-to-ice mass rates in kg/s. Positive rates consume vapor; negative rates return water to vapor.

Ice response: Γ_i = K_i(exp ψ − 1). K_i is exchange capacity in kg/s, computed from the fixed number and changing equivalent radii of existing crystals, with vapor and heat transport resistance. The liquid rate has an analogous saturation relation over water.

Temperature feedback: C_p dT/dt = Q̇ + V dp/dt + L_vΓ_l + L_sΓ_i, with external heat input Q̇ = 0. C_p is the parcel's total heat capacity in J/K. Water is conserved: dm_l/dt = Γ_l, dm_i/dt = Γ_i, and dm_v/dt = −Γ_l − Γ_i.

What has been checked

The browser implementation numerically reproduces the original Python parcel trajectories for steady lift, lift cessation, and pressure perturbation, with conservation and equation-identity checks. This is numerical verification, not proof that the model matches natural clouds.

For measured crystals republished by Fuchs et al. between −16 and −14°C, measured mass was a median 3.1 times our spherical prediction at ten minutes (six crystals) and 6.9 times at fifteen to thirty minutes (nine crystals). No parameters were fitted. The calculation assumes a 25 μm initial equivalent radius because the experiment did not specify starting mass. The original experiment and the AIDA chamber comparison provide context. Chamber humidity was supplied as an observed input, so that check does not independently validate our humidity prediction after lifting stops.

The crystal-growth comparison download contains the original workbook, unchanged analysis and model code, pinned requirements, expected results, and provenance. The interactive simulation, equations, and full source notes appear when JavaScript runs.

Companion: when a bubble survives

The companion page follows three illustrative moments of boiling: a small vapor pocket shrinks, a pocket at the critical radius sits at an unstable threshold, and a larger pocket grows while evaporation cools nearby water. Heating can raise the equilibrium vapor pressure; lowering surrounding pressure can reduce the opposing liquid pressure.

For an idealized spherical bubble, Δp = p_sat(T_interface) − p_ℓ − 2γ/r. The bubble grows when this difference is positive. With positive p_sat(T) − p_ℓ, the critical radius is r_c = 2γ/[p_sat(T) − p_ℓ]. The illustrations use γ = 0.06 N/m and an initial critical radius of 6 μm. The pressure-bar values are illustrative, not a measured boiling trajectory. Real nucleation often starts at surfaces or trapped gas pockets, and bubble growth involves heat transfer and fluid motion.

This threshold is separate from the cloud-parcel model, which starts with ice already present. The ideal gas law remains valid on either side of the bubble threshold. See the inertio-thermal vapor-bubble-growth analysis for the balance and thermal feedback.