5 Moment method for nucleation, growth, evaporation and equilibrium drift

The physical and chemical processes discussed so far (nucleation, growth, evaporation, gravitational settling and element depletion/enrichment) occur simultaneously in the atmosphere and may be strongly coupled. Therefore, our aim in this section is to derive a consistent time-dependent description of the dust component in brown dwarf or giant gas planet atmospheres. We will derive a system of partial differential equations which describes the evolution of the dust component by means of the moments of its size distribution function. This idea was originally developed by Gail ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Sedlmayr (1988) and extended by Dominik ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$(1993) to core-mantle and dirty grains. The resulting differential equations are supposed to be simple and includable into hydrodynamics or classical stellar atmosphere calculations. In contrast, we want to avoid an elaborate and time-consuming multi-component treatment of the dust component, e.g.by using discrete bins for the dust size distribution function with individual drift velocities in this paper.

Figure 5: Two surface chemical processes ($r\!=\!1$ and $r\!=\!2$) populating or depopulating an infinitesimal dust grain volume interval $[V,V\!+\!{\rm d}V]$.

The master equation for dust particles $\in\![V,V\!+\!{\rm d}V]$, $f(V)~{\rm d}V$, where $f(V)~\rm [cm^{-6}]$ is the distribution function of dust particles in volume space, is given by

$$\frac{{\rm\partial} }{{\rm\partial} t}\big(f(V)~{\rm d}V\big)... ...rc}}(V)\big]f(V)~{\rm d}V\!\Big) = \sum\limits_k R_k~{\rm d}V.$$ (55)

The r.h.s. of Eq. (55) expresses the population and depopulation of the considered volume interval $[V,V\!+\!{\rm d}V]$ with dust particles which are changing their size due to accretion or evaporation of molecules (see Fig. 5),

$$\sum\limits_k R_k~{\rm d}V = \left(R_\uparrow - R^\uparrow + R^\downarrow - R_\downarrow\right)~{\rm d}V .$$ (56)

Multiplication of Eq. (55) with V^j/3 ( $j\!=\!0,1,2,...$) and integration over V from a lower integration boundary $V\!=\!{V_{\ell}}$ to $V\!\to\!\infty$ results in

 

$$\frac{{\rm\partial} }{{\rm\partial} t}\big(\rho L_j\big) + \nabla... ...}_{\rm dr}^{\hspace{-0.6ex}^{\circ}}(V)~{\rm d}V}_{\displaystyle{\cal B}_j} \ ,$$ (57)

where the jth moment of the dust size distribution function $L_ j~{\rm [cm}^j\!/{\rm g]}$ is defined by

$$\rho L_j(\vec{x},t) = \int\limits_{\rm {V_{\ell}}}^{\infty} f(V,\vec{x},t)~V^{\rm j/3}~{\rm d}V .$$ (58)

The source term ${\cal A}_j$ expresses the effects of surface chemical reactions on the dust moments and ${\cal B}_j$ is an additional, advective term in the dust moment equations which comprises the effects caused by a size-dependent drift motion of the grains, e.g.due to gravity.

The maximum grain sizes to be expected in brown dwarf atmospheres (Fig. 3) allow us to concentrate on two major cases in the following, namely the subsonic free molecular flow and the laminar viscous flow.

   
5.1 Subsonic free molecular flow $({Kn}\!\gg\!1 \wedge \vec{v}_{\rm dr}^{\hspace{-0.6ex}^{\circ}}\!\ll\!{c}_{\rm T})$

For large Knudsen numbers, the chemical rates depicted in Fig. 5 can be expressed according to Eq. (22):

    

$$R^\uparrow~{\rm d}V \sum\limits_{\rm r} f(V)~{\rm d}V~4\pi[a(V)]^2~ n_{\rm r} v^{\rm rel}_{\rm r} \alpha_{\rm r}$$ = (59)

$$R_\uparrow~{\rm d}V \sum\limits_{\rm r} f(V\!-\!\Delta V_{\rm r})~{\rm d}V~4\pi\! \le... ...V\!-\!\Delta V_{\rm r})\right]^2\! n_{\rm r} v^{\rm rel}_{\rm r} \alpha_{\rm r}$$ = (60)

$$R^\downarrow~{\rm d}V \sum\limits_{\rm r} f(V\!+\!\Delta V_{\rm r})~{\rm d}V~4\pi [a(V)]^2~ n_{\rm r} v^{\rm rel}_{\rm r} \alpha_{\rm r} \frac{1}{S_{\rm r}}$$ = (61)

$$R_\downarrow~{\rm d}V \sum\limits_{\rm r} f(V)~{\rm d}V~ 4\pi\!\left[a(V\!-\!\Delta V_{... ...\right]^2 n_{\rm r} v^{\rm rel}_{\rm r} \alpha_{\rm r} \frac{1}{S_{\rm r}}\cdot$$ = (62)

Applying detailed balance considerations (Milne relations), Gauger ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$(1990) and Patzer ${\rm\hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.5ex}}$(1998) have shown that for simple types of surface reactions, denoted by homogeneous and heterogeneous growth, the (inverse) evaporation rates can be expressed by the related (forward) growth rates and the supersaturation ratio in the way written in Eqs. (61) and (62). For Eqs. (59) and (60), we have neglected a possible influence of a fast drift motion on the growth rates and for Eqs. (61) to (62), we have assumed thermal equilibrium (dust temperature $\equiv$ gas temperature) and chemical equilibrium among the molecules in the gas phase.

Solving Eq. (19) with the frictional force according to Eq. (13, subsonic case) the equilibrium drift velocity is

$$\vec{v}_{\rm dr}^{\hspace{-0.6ex}^{\circ}}= - \frac{\sqrt{\pi} g \rho_{\rm d}~a}{2 \rho~c_{\rm T}}~\vec{e}_{\rm r}.$$ (63)

By means of the Eqs. (59) to (63), the integrals on the r.h.s. of Eq. (57) can be evaluated as shown in more detail concerning the term ${\cal A}^{\rm lKn}$ in (Gail ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Sedlmayr 1988). After some algebraic manipulations, including $a\!=\!(3V/4\pi)^{1/3}$, the approximation $\Delta V_{\rm r}\ll V$ and partial integration, the results are

$${\cal A}^{\rm lKn}_j {V_{\ell}}^{~j/3} J({V_{\ell}}) ~+~ \frac{j}{3}~\chi_{\rm lKn}^{\rm net} ~\rho L_{j-1}$$ = (64)

$$*[1.2ex] {\cal B}^{\rm lKn}_j -~\xi_{\rm lKn}~\nabla\! \left(\frac{L_{j+1}}{c_{\rm T}}~\vec{e}_{\rm r}\right) \ ,$$ = (65)

where $J({V_{\ell}})\!=\!f({V_{\ell}})\left.\frac{{\rm d}V}{{\rm d}t}\right\vert_{V={V_{\ell}}}$ is the current of dust particles in volume space at the lower integration boundary. In case of net growth, $J({V_{\ell}})~\rm [cm^{-3}s^{-1}]$ can be identified with the stationary nucleation rate $J_\star$ (Gail ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Sedlmayr 1988). The characteristic growth speed $\chi_{\rm lKn}^{\rm net}~\rm [cm/s]$ (an increase of radius per time) and the characteristic gravitational force density $\xi_{\rm lKn}~\rm [{\rm d}yn/cm^3]$ are given by

  

$$\chi_{\rm lKn}^{\rm net} \sqrt[3]{36\pi}~\sum\limits_{\rm r} \Delta V_{\rm r}~n_{\rm r} v^{\rm rel}_{\rm r} \alpha_{\rm r} \left(1-\frac{1}{S_{\rm r}}\right)$$ = (66)

$$\xi_{\rm lKn} \frac{\sqrt{\pi}}{2} \bigg(\frac{3}{4\pi}\bigg)^{\!\!1/3}\!g~\rho_{\rm d}\ .$$ = (67)

   
5.2 Laminar viscous flow $({Kn}\!\ll\!1 \wedge {Re}_{\rm d}\!<\!1000)$

The derivation of the moment equations for the case of small Knudsen numbers is analogous to the previous subsection. We express the surface chemical rates according to Eq. (32) by

$$R^\uparrow~{\rm d}V \sum\limits_{\rm r} f(V)~{\rm d}V~4\pi a(V)~ n_{\rm r} D_{\rm r}$$ = (68)

$$R_\uparrow~{\rm d}V \sum\limits_{\rm r} f(V\!-\!\Delta V_{\rm r})~{\rm d}V~4\pi a(V\!-\!\Delta V_{\rm r})~ n_{\rm r} D_{\rm r}$$ = (69)

$$R^\downarrow~{\rm d}V \sum\limits_{\rm r} f(V\!+\!\Delta V_{\rm r})~{\rm d}V~4\pi a(V)~ n_{\rm r} D_{\rm r} \frac{1}{S_{\rm r}}$$ = (70)

$$R_\downarrow~{\rm d}V \sum\limits_{\rm r} f(V)~{\rm d}V~4\pi a(V\!-\!\Delta V_{\rm r})~ n_{\rm r} D_{\rm r} \frac{1}{S_{\rm r}} ,$$ = (71)

again neglecting the influence of drift on growth and assuming thermal and chemical equilibrium. The equilibrium drift velocity results from Eq. (19) with the frictional force according to Eq. (16, small $Re_{\rm d}$ case)

$$\vec{v}_{\rm dr}^{\hspace{-0.6ex}^{\circ}}= -\frac{2~g~\rho_{\rm d}~a^2}{9~\mu_{\rm kin}} ~\vec{e}_{\rm r}.$$ (72)

By repeating the procedure of the last subsection, the r.h.s. terms of the dust moment equations are

$${\cal A}^{\rm sKn}_j {V_{\ell}}^{~j/3} J({V_{\ell}}) ~+~ \frac{j}{3}~\chi_{\rm sKn}^{\rm net} ~ \rho L_{j-2}$$ = (73)

$$*[1.2ex] {\cal B}^{\rm sKn}_j -~\xi_{\rm sKn}~\nabla\! \left(\frac{\rho L_{j+2}}{\mu_{\rm kin}}~\vec{e}_{\rm r}\right) \ .$$ = (74)

The characteristic growth speed $\chi_{\rm sKn}^{\rm net}$ has now the unit [cm² s^-1] (an increase of surface per time) whereas the characteristic gravitational force density $\xi_{\rm sKn}$ remains the same except for a different geometry factor.

  

$$\chi_{\rm sKn}^{\rm net} \sqrt[3]{48\pi^2}~\sum\limits_{\rm r} \Delta V_{\rm r}~n_{\rm r}~D_{\rm r} \left(1-\frac{1}{S_{\rm r}}\right)$$ = (75)

$$\xi_{\rm sKn} \frac{2}{9}\bigg(\frac{3}{4\pi}\bigg)^{\!\!2/3}\!g~\rho_{\rm d}.$$ = (76)