Development of a Cloud Convection Model for Jupiter's Atmosphere
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The microphysical processes represented by
*Q*_{cnd} in eq. (A.7) and
*Src* in eq. (A.8)
are implemented using the warm rain bulk parameterization of
Kessler (1969)
^{[2]}.
In this parameterization scheme, each
condensible species is divided into three categories ``vapor'',
``cloud'', and ``rain''; both rain and cloud are condensed phases, but
rain falls down relative to the air, whereas cloud does not
(see Fig. 2.1). The
conversion rates between each pair of categories are calculated using
bulk mixing ratios of the three categories of condensible
components. For simplicity, we assume that the condensed phase
consists of pure condensible species; solution is not
considered. Moreover, we assume that the interactions between the
three condensed materials, i.e. H_{2}O ice,
NH_{4}SH ice, and NH_{4}SH ice, are
absent.

The condensation heat *Q*_{cnd} in eq.
(A.7) is expressed as

where
*L* is latent heat or reaction heat,
*CN _{vc}* is
conversion rate from vapor to cloud due to condensation or reaction,

where
*CN _{cr}* is
conversion rate from cloud to rain due to autoconversion,

The terms in eqs. (B.5)--(B.14) associated with cloud microphysical processes are given in the following subsection.

The conversion rate from cloud to rain due to autoconversion is determined by the following equation.

The time constant of autoconversion τ_{0} is 100
sec, and the threshold cloud mixing ratio *q*^{0}
is set to zero, following
Nakajima *et al*. (2000)
^{[6]}.
These values have been selected by
roughly taking into account the effects of the larger terminal
velocity of Jupiter's precipitation
^{[15]}.

The conversion rate from cloud to rain due to collection of cloud by rain is determined by the following equation.

where *f _{j}* is ratio of
Jupiter and Earth's gravitational acceleration,
and

The conversion rate from rain to vapor due to evaporation
is given as proportional to the amount of supersaturation
*q _{vsw}* -

The tendency of rain due to precipitation is calculated by the following equation.

where the terminal velocity of rain,
*U _{r}* [m s

Condensation and evaporation between vapor and cloud are
evaluated by adjusting
*θ*, *q _{v}*, and

In the followings, we denote the variables calculated from the
integration of eqs. (A.4)--(A.8) and
eqs. (B.5)--(B.14)
without - *CN _{vc}* +

From the first low of thermodynamics, the following relationship between the quantities before and after the adjustment holds.

where
*q _{vsw}* is saturation vapor mixing ratio
and

where

From eq. (B.21), the amount of condensation *CN _{vc}* and
evaporation

because the mixing ratios of variables after adjustment
should be non-negative, and *CN _{vc}* and

Finally, *θ*,
*q _{v}*, and

The above procedure is repeated four times to ensure the precision of the saturated state.

The partial pressures of NH_{3} and H_{2}S
have to satisfy the equilibrium condition
of the NH_{4}SH production reaction,
NH_{3} + H_{2}S ↔ NH_{4}SH.
The equilibrium constant is given by

where *p*_{NH3} and
*p*_{H2S} are partial pressures of
NH_{3} and H_{2}S, respectively
(details on *K _{p}* are given in Appendix F.
Eq. (B.25) is rewritten by using the change of partial pressure

or

By solving this equation, we can obtain
*Δ p*_{NH4SH} as

Note that the minus sign of square root is selected by ensuring
*Δ q*_{NH4SH} <
*p*_{H2S} and
*Δ q*_{NH4SH} <
*p*_{H2S}.
Now we have
*CN _{vc}* and

because the mixing ratios of variables after adjustment should be
non-negative, and *CN _{vc}* and

Finally, *θ*,
*q*_{NH3},
*q*_{H2S}, and
*q*_{NH4SH} are determined by the
following equations.

where
*M*_{NH3},
*M*_{H2S}, and
*M*_{NH4SH}
are molecular weights of
NH_{3},
H_{2}S, and
NH_{4}SH, respectively.
The above procedure is repeated four times to ensure the precision of the equilibrium state.

Development of a Numerical Model for Jupiter's Atmosphere
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