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%  Calculation
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\section{Calculation Method}

\subsection{Gibbs free energy and equilibrium composition}

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, 
  $G$, under the condition 
  that the total number of each element is conserved. 
Assuming an ideal gas and an ideal solution, 
  $G$ can be expressed as
%
\begin{eqnarray}
&& G(T, p) 
  = \sum_{i}\sum_{\phi} n_{i}^{\phi} \mu_{i}^{\phi}(T, p, n_{i}^{\phi}) ,
  \nonumber \\
&& \mu_{i}^{\phi}(T, p, n_{i}^{\phi}) 
  = {{\mu_{i}}^{\circ}}^{\phi}(T) + RT \ln{x_{i}^{\phi}} + \delta_{1\phi} RT \ln{p/p_{0}} ,
  \nonumber
\end{eqnarray}
%
where 
$T$ is the temperature, 
$p$ and $p_{0}$ are the pressure and standard pressure, 
$n^{\phi}_{i}$ and $x_{i}^{\phi} \equiv n_{i}^{\phi}/\sum_{i}n_{i}^{\phi}$ 
  are the mole number and mole fraction, 
  respectively, 
  of chemical species $i$ in phase $\phi$, 
$\mu_{i}^{\phi}$ and ${{\mu_{i}}^{\circ}}^{\phi}$ 
  are the chemical potential and chemical potential
  at the standard pressure, 
  respectively,   
and $R$ is the gas constant [see also {\it Sugiyama et al}., 2001]. 
Here, the gas phase is indicated by $\phi = 1$. 
  $\delta_{1\phi}$ 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 standard pressure is calculated 
  at the standard pressure is calculated 
  from the following equation: 
%
\begin{eqnarray}
{{\mu_{i}}^{\circ}}^{\phi}(T)
  &=& h_{i}^{\phi}(T_{0}) - T s_{i}^{\phi}(T_{0}, p_{0}) \nonumber \\
    &&+ \int_{T_{0}}^{T}{c_{p}}_{i}^{\phi}(T) dT 
    - T \int_{T_{0}}^{T} \frac{{c_{p}}_{i}^{\phi}(T)}{T} dT, 
    \;\;\;{\rm for} \;  \phi = 1, 
  \nonumber
\end{eqnarray}
%
where 
$T_{0}$ is the standard temperature, 
$h_{i}^{\phi}$ is the molecular enthalpy, 
$s_{i}^{\phi}$ is the molecular entropy, 
and ${c_{p}}_{i}^{\phi}$ is the specific heat 
  at constant pressure. 
The values of $h_{i}^{\phi}(T_{0})$, $s_{i}^{\phi}(T_{0}, p_{0})$, 
and ${c_{p}}_{i}^{\phi}(T)$ are adopted from 
{\it NIST-JANAF Thermochemical Tables} [1989]. 
Spline interpolation is used to give a functional 
form of ${c_{p}}_{i}^{\phi}(T)$.  
%
The chemical potential of each condensed species 
%  at standard pressure, 
  at  the standard pressure, 
  except for NH$_4$SH, is given by 
%
\begin{eqnarray}
{{\mu_{i}}^{\circ}}^{\phi}(T)
  = {{\mu_{i}}^{\circ}}^{1}(T) + RT \ln{e_{i}(T)/p_{0}}, 
    \;\;\;{\rm for} \;  \phi \neq 1, 
  \nonumber
\end{eqnarray}
%
where $e_{i}(T)$ is the saturated vapor pressure of species $i$, 
which is evaluated by the Antoine equation 
%
[{\it Chemical Handbook}, 1993]. 
%
The values of Antoine coefficients 
  are adopted from the {\it Chemical Handbook} [1993]. 
In the lower temperature range, 
  where the Antoine equation is not valid, 
  ${{\mu_{i}}^{\circ}}^{\phi}(T)$ is given 
  as a quadratic function of temperature 
  that is smoothly connected to the above expression. 
%
The chemical potential of 
  the condensed phase of NH$_4$SH 
%  at standard pressure 
  at  the standard pressure 
  is calculated from the equilibrium constant for NH$_4$SH formation reaction 
  [{\it Lewis}, 1969]. 


The RAND method [{\it White et al}., 1953; 
  {\it Van Zeggeren and Storey}, 1970, {\it 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/. 


\subsection{Moist adiabatic profile}

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. 
%
% I remove the sentence and equation because of making enough space.
%
%Total entropy $S$ is calculated from the following equation.  
%
%\begin{eqnarray}
% S(T, p)
%  & = & - \sum_{i} \sum_{\phi} 
%       \frac{\partial \mu_{i}^{\phi}(T,p,n_{i}^{\phi})}{\partial T} 
%       n_{i}^{\phi} 
%       \nonumber \\ &&
%   - \int_{p_b}^{p}
%     \sum_{i} \sum_{\phi \neq 1} 
%       \frac{\partial \mu_{i}^{\phi}}{\partial T}(T(p'),p',n_{i}^{\phi}(p'))
%       \frac{dn_{i}^{\phi}}{dp'}dp',
%       \nonumber
%\end{eqnarray}
%where $p_b$ is initial pressure of calculation, and
%$T(p)$ and $n_{i}^{\phi}(p)$ are the obtained pseudo adiabatic
%profiles. 


\subsection{Static stability}

The value of $N^{2}$ is estimated as described by
  {\it 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 $N^{2}$ is then given 
  as follows: 
%
\begin{eqnarray}
N^{2}
 =
  \frac{g}{T}
  \left(
   \frac{d T}{d z}
   + \frac{M g}{c_{p}}
  \right)
  - 
  \frac{g}{M} \frac{dM}{dz},
\label{def_stab}
\end{eqnarray}
%
where $g$ is the acceleration of gravity, 
$M$ is the mean molecular weight, 
and $c_{p}$ 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 studies of 
%  is based on the experience of 
  is based on the knowledge of 
  the earth's troposphere (e.g., {\it Gill} [1982]). 
Although dynamics may alter the detailed profiles of 
  $T$ and $M$, as is exemplified 
  by {\it Nakajima et al.} [2000], 
  we believe that this formula is still applicable 
  to arguments concerning the order of magnitude. 
%

The $N^{2}$ given by equation (\ref{def_stab}) is 
  slightly different 
%  from that reported in
  from that of 
  {\it 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 {\it 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 $N^{2}$ is approximately the same as 
  that of {\it Achterberg and Ingersoll} [1989] in the 
  parameter ranges considered in their study. 




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%  REFERENCE LIST AND TEXT CITATIONS
%
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\section{References}

\begin{description}

 \item
	    Achterberg, R. K., and A. P. Ingersoll (1989),  
	    A Normal-Mode Approach to Jovian Atmosphere Dynamics, 
	    {\it J. Atmos. Sci.}, {\it 46}, 2448--2462. 

 \item
	   Atreya, S. K., and P. N. Romani (1985), 
	   Photochemistry and clouds of Jupiter, Saturn and Uranus, 
	   in {\it Recent Advances in Planetary Meteorology}, 
	   edited by G. E. Hunt, pp. 17--68, 
	   Cambridge Univ. Press, London. 

 \item
	    Chase, M. W. (Eds.) (1989), 
	    {\it NIST-JANAF Thermochemical Tables}, 4th ed., 
	    AIP Press, New York. 

 \item
           Gill, A. E., (1982), 
           {\it Atmosphere-Ocean Dynamics}, Academic Press, SanDiego. 


 \item
	   Ingersoll, A. P., H. Kanamori, and T. E. Dowling (1994), 
	   Atmospheric gravity waves from the impact of comet
	   Shoemaker-Levy 9 with Jupiter, 
	   {\it Geophys. Res. Lett.}, {\it 21}, 1083--1086.

 \item
	    The Chemical Society of Japan (Eds.) (1993), 
	   {\it Chemical Handbook (Kagaku-Binran)}, 4th ed., 
	   Maruzen, Tokyo, in Japanese. 


 \item
	   Lewis, J. S. (1969), 
	   The Clouds of Jupiter and the NH$_3$-H$_2$O and
	   NH$_3$-H$_2$S Systems, 
	   {\it Icarus}, {\it 10}, 365--378. 

\item
	   Nakajima, K., S. Takehiro, M. Ishiwatari, and Y.-Y. Hayashi
	   (2000), 
	   Numerical modeling of Jupiter's moist convection layer, 
	   {\it Geophys. Res. Lett.}, {\it 27}, 3129--3133. 

 \item 
	    Sugiyama, K., M. Odaka, K. Kuramoto, and Y.-Y. Hayashi 
	   (2001), 
	    Thermodynamic calculation of the atmosphere of the Jovian
	    Planets, 
	   {\it Proceedings of the 34 th ISAS Lunar and Planetary
	   symposium}, 53--56. 
	    
 \item
	    Van Zeggeren, F., and S. H. Storey (1970), 
	   {\it The Computation of Chemical Equilibria}, 
	    Cambridge Univ. Press, London. 

 \item
	    Weidenschilling, S. J. and J. S. Lewis (1973), 
	    Atmospheric and cloud structure 
	   of the Jovian planet, 
	   {\it Icarus}, {\it 20}, 465-476. 
	    
 \item
	    White W. B., S. M. Johnson, and G. B. Dantxig (1953), 
	    Chemical Equilibrium in Complex Mixture, 
	    {\it J. Chem. Phys.}, {\it 28}, 751--755. 

 \item
	   Wood, J.A. and A. Hashimoto (1993), 
	   Mineral equilibrium in fractionated nebular systems, 
	   {\it Geochimica et Cosmochimica Acta}, {\it 57}, 
	   2377--2388
\end{description}

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