We consider herein an atmosphere in thermodynamic equilibrium.
When pressure and temperature are given,
the equilibrium composition is calculated
by minimizing the Gibbs free energy,
, under the condition
that the total number of each element is conserved.
Assuming an ideal gas and an ideal solution,
can be expressed as

where is the temperature, and are the pressure and standard pressure, and are the mole number and mole fraction, respectively, of chemical species in phase , and are the chemical potential and chemical potential at the standard pressure, respectively, and is the gas constant [see also

The chemical potential of each gas species
at the standard pressure is calculated
from the following equation:

where is the standard temperature, is the molecular enthalpy, is the molecular entropy, and is the specific heat at constant pressure. The values of , , and are adopted from

where is the saturated vapor pressure of species , which is evaluated by the Antoine equation [

The RAND method [*White et al*., 1953;
*Van Zeggeren and Storey*, 1970, *Wood and Hashimoto*, 1993]
is used to obtain equilibrium composition.
During the application of the RAND method,
we examine whether or not each condensed phase really
equilibrates with the atmospheric gas phase
under the given temperature and pressure.
This check of phase stability is required
in order to ensure non-singularity to the coefficient matrix
of the RAND method,
to avoid an optimized solution converging to local minimum,
and to accelerate the execution speed of
numerical calculation of the RAND method.
Our source code is available at
http://www.gfd-dennou.org/library/oboro/.

The vertical profiles of temperature, composition, and condensates are obtained by considering an air parcel following the pseudo moist adiabatic process. We assume that all of the condensates are removed from the air parcel, while the value of total entropy, which is the sum of those for gas and already removed condensates at each pressure level, is conserved.

The value of is estimated as described by
*Achterberg and Ingersoll* [1989].
The profiles of the temperature
and mean molecular weight
are those obtained for the pseudo moist adiabatic process.
The value of is then given
as follows:

where is the acceleration of gravity, is the mean molecular weight, and is the mean specific heat per mole. The interpretation of the moist adiabat as a rough proxy of the mean atmospheric thermal structure is based on the knowledge of the earth's troposphere (e.g.,

The given by equation () is
slightly different
from that of
*Achterberg and Ingersoll* [1989]
with respect to the definition of specific heat.
We herein adopt the mean specific heat
of an air parcel at a given pressure level,
whereas *Achterberg and Ingersoll* [1989]
adopt that of a condensible-free air parcel.
They assume that the abundances of any condensible
elements in the Jovian atmospheres are sufficiently small.
Our value of is approximately the same as
that of *Achterberg and Ingersoll* [1989] in the
parameter ranges considered in their study.

- Achterberg, R. K., and A. P. Ingersoll (1989),
A Normal-Mode Approach to Jovian Atmosphere Dynamics,
*J. Atmos. Sci.*,*46*, 2448-2462. - Atreya, S. K., and P. N. Romani (1985),
Photochemistry and clouds of Jupiter, Saturn and Uranus,
in
*Recent Advances in Planetary Meteorology*, edited by G. E. Hunt, pp. 17-68, Cambridge Univ. Press, London. - Chase, M. W. (Eds.) (1989),
*NIST-JANAF Thermochemical Tables*, 4th ed., AIP Press, New York. - Gill, A. E., (1982),
*Atmosphere-Ocean Dynamics*, Academic Press, SanDiego. - Ingersoll, A. P., H. Kanamori, and T. E. Dowling (1994),
Atmospheric gravity waves from the impact of comet
Shoemaker-Levy 9 with Jupiter,
*Geophys. Res. Lett.*,*21*, 1083-1086. - The Chemical Society of Japan (Eds.) (1993),
*Chemical Handbook (Kagaku-Binran)*, 4th ed., Maruzen, Tokyo, in Japanese. - Lewis, J. S. (1969),
The Clouds of Jupiter and the NH-HO and
NH-HS Systems,
*Icarus*,*10*, 365-378. - Nakajima, K., S. Takehiro, M. Ishiwatari, and Y.-Y. Hayashi
(2000),
Numerical modeling of Jupiter's moist convection layer,
*Geophys. Res. Lett.*,*27*, 3129-3133. - Sugiyama, K., M. Odaka, K. Kuramoto, and Y.-Y. Hayashi
(2001),
Thermodynamic calculation of the atmosphere of the Jovian
Planets,
*Proceedings of the 34 th ISAS Lunar and Planetary symposium*, 53-56. - Van Zeggeren, F., and S. H. Storey (1970),
*The Computation of Chemical Equilibria*, Cambridge Univ. Press, London. - Weidenschilling, S. J. and J. S. Lewis (1973),
Atmospheric and cloud structure
of the Jovian planet,
*Icarus*,*20*, 465-476. - White W. B., S. M. Johnson, and G. B. Dantxig (1953),
Chemical Equilibrium in Complex Mixture,
*J. Chem. Phys.*,*28*, 751-755. - Wood, J.A. and A. Hashimoto (1993),
Mineral equilibrium in fractionated nebular systems,
*Geochimica et Cosmochimica Acta*,*57*, 2377-2388

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