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
is the temperature,
and are the pressure and standard pressure,
are the mole number and mole fraction,
of chemical species in phase ,
are the chemical potential and chemical potential
at the standard pressure,
and is the gas constant [see also Sugiyama et al., 2001].
Here, the gas phase is indicated by .
is the Kronecker delta.
We assume that only the chemical potential
of the gas phase depends on pressure.
The chemical potential of each gas species
at the standard pressure is calculated
from the following equation:
is the standard temperature,
is the molecular enthalpy,
is the molecular entropy,
is the specific heat
at constant pressure.
The values of
are adopted from
NIST-JANAF Thermochemical Tables .
Spline interpolation is used to give a functional
The chemical potential of each condensed species
at the standard pressure,
except for NHSH, is given by
where is the saturated vapor pressure of species ,
which is evaluated by the Antoine equation
[Chemical Handbook, 1993].
The values of Antoine coefficients
are adopted from the Chemical Handbook .
In the lower temperature range,
where the Antoine equation is not valid,
as a quadratic function of temperature
that is smoothly connected to the above expression.
The chemical potential of
the condensed phase of NHSH
at the standard pressure
is calculated from the equilibrium constant for NHSH formation reaction
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
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 .
The profiles of the temperature
and mean molecular weight
are those obtained for the pseudo moist adiabatic process.
The value of is then given
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., Gill ).
Although dynamics may alter the detailed profiles of
and , as is exemplified
by Nakajima et al. ,
we believe that this formula is still applicable
to arguments concerning the order of magnitude.
The given by equation () is
from that of
Achterberg and Ingersoll 
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 
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  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
Icarus, 10, 365-378.
- Nakajima, K., S. Takehiro, M. Ishiwatari, and Y.-Y. Hayashi
Numerical modeling of Jupiter's moist convection layer,
Geophys. Res. Lett., 27, 3129-3133.
- Sugiyama, K., M. Odaka, K. Kuramoto, and Y.-Y. Hayashi
Thermodynamic calculation of the atmosphere of the Jovian
Proceedings of the 34 th ISAS Lunar and Planetary
- 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,
$B$3$NJ8=q$O(BLaTeX2HTML $BK]Lu%W%m%0%i%`(B Version 2002-2-1 (1.70)
Copyright © 1993, 1994, 1995, 1996,
Computer Based Learning Unit, University of Leeds,
Copyright © 1997, 1998, 1999,
Mathematics Department, Macquarie University, Sydney.
2002-2-1 (1.70) JA patch-1.8 $BHG(B)
Copyright © 1998, 1999,
Copyright © 2001, 2002,
latex2html -no_footnode -local_icons -short_extn -bottom_navigation -no_contents_in_navigation -no_auto_link -up_url /arch/deepconv/index.htm -up_title deepconv -split 3 -show_section_numbers oboro.tex.
$BK]Lu$O(B SUGIYAMA Ko-ichiro $B$K$h$C$F(B $BJ?@.(B18$BG/(B2$B7n(B2$BF|(B $B$K