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Condensation and rainout

VULCAN treats condensation and evaporation through a particle growth rate, and lets the resulting particles fall under gravity (Section 2.6 of Tsai et al. 2021 1).

Growth rate

For the schematic condensation/evaporation reaction \(A_\mathrm{(gas)} \leftrightarrow A_\mathrm{(particle)}\), the growth rate in the continuum regime (particles larger than the mean free path, Knudsen number \(\mathrm{Kn} < 1\)) follows the mass-balance expression (Seinfeld & Pandis 2016 2):

\[\frac{\mathrm{d}n_A}{\mathrm{d}t} = -\frac{D_A\,m_A}{\rho_\text{p}\,r_\text{p}^{2}}\left(n_A - n_A^\mathrm{sat}\right) n_A \tag{20}\]

where \(D_A\) and \(m_A\) are the molecular diffusion coefficient and mass of gas \(A\), \(\rho_\text{p}\) and \(r_\text{p}\) the particle density and radius, and \(n_A^\mathrm{sat}\) the saturation number density. A negative value (when \(n_A > n_A^\mathrm{sat}\)) is condensation; a positive value (when \(n_A < n_A^\mathrm{sat}\)) is evaporation.

Difference from earlier models

Hu et al. (2012) 3 and Rimmer & Helling (2016) 4 use the kinetic-regime growth rate. VULCAN uses the continuum form because, for most applications, condensation occurs in the lower atmosphere with micron-size or larger particles. The code does contain a commented-out kinetic-regime expression (rate_c) for reference.

Particle settling

Once gas condenses to particles, they fall at the terminal velocity from Stokes' law:

\[v_\text{s} = \frac{2\,r_\text{p}^{2}\,\rho_\text{p}\,g}{9\,\mu} \tag{21}\]

with \(\mu\) the atmospheric dynamic viscosity. The slip-correction factor is taken as unity (large-particle limit).

Saturation vapor pressures

build_atm.Atm.sp_sat precomputes the saturation vapor pressure (dyne cm\(^{-2}\)) for each condensable species:

Species Parameterization
H\(_2\)O Ackerman & Marley (2001), separate ice (\(T<0\) °C) and liquid branches
NH\(_3\) Weast (1971)
H\(_2\)SO\(_4\) Ayers form
S\(_2\), S\(_8\) Zahnle et al. (2016), refit from Lyons (2008) 5; piecewise at 413 K
S\(_4\) Lyons (2008)
C NIST fit
H\(_2\)S Giauque & Blue (1936), separate ice/liquid branches

The dynamic viscosity used in Eq. (15) is taken per background gas (atm_base) from the Cloutman (2000) combustion-coefficient table in build_atm.Atm.f_mu_dz.

Implementation in VULCAN

Condensation is driven from op.Integration and applied within the transport operator:

  • Integration.conden updates the condensation/evaporation rate coefficients each step using the continuum form of Eq. (14), one branch per species (H2O → H2O_l_s, NH3 → NH3_l, H2SO4 → H2SO4_l, S2/S4/S8 → *_l_s, C → C_s).
  • Settling enters the transport operator and Jacobian through the diffdf_settling / lhs_jac_settling variants when use_settling = True; the settling velocity is stored in atm.vs (computed in f_mu_dz).
  • An alternative relaxation scheme (h2o_conden_evap_relax, nh3_conden_evap_relax) is used when use_relax is set, removing super-saturated vapor with an implicit-Euler step.
  • Because condensation operates on a short timescale, the code can fix the condensing species and condensates after dynamic equilibrium is reached (fix_species, fix_species_from_coldtrap_lev) — a quasi-steady-state approach that decouples fast and slow processes. The cold-trap level is found and species are held fixed above it.

Non-gaseous (condensate) species are excluded when forming mixing ratios from number densities (atm.gas_indx), so hydrostatic balance is maintained over the gas phase only.

Relevant parameters

Parameter Meaning
use_condense enable condensation chemistry
use_settling enable gravitational settling
condense_sp species allowed to condense
non_gas_sp condensate (non-gaseous) species
r_p, rho_p per-condensate particle radius (cm) and density (g cm\(^{-3}\))
humidity relative-humidity multiplier applied to the H\(_2\)O saturation
fix_species, start_conden_time, stop_conden_time quasi-steady-state fixing controls

  1. Tsai, S.-M., Malik, M., Kitzmann, D., et al. (2021). A comparative study of atmospheric chemistry with VULCAN. The Astrophysical Journal, 923(2), 264. https://doi.org/10.3847/1538-4357/ac29bc ↩

  2. Seinfeld, J. H., & Pandis, S. N. (2016). Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, 3rd ed. Wiley. ↩

  3. Hu, R., Seager, S., & Bains, W. (2012). Photochemistry in terrestrial exoplanet atmospheres. I. The Astrophysical Journal, 761(2), 166. https://doi.org/10.1088/0004-637X/761/2/166 ↩

  4. Rimmer, P. B., & Helling, C. (2016). A chemical kinetics network for lightning and life in planetary atmospheres. The Astrophysical Journal Supplement Series, 224(1), 9. https://doi.org/10.3847/0067-0049/224/1/9 ↩

  5. Lyons, J. R. (2008). An estimate of the equilibrium speciation of sulfur vapor over solid sulfur and implications for planetary atmospheres. Journal of Sulfur Chemistry, 29(3–4), 269–279. https://doi.org/10.1080/17415990802208743 ↩