- 6.1 Dimensionless analysis
- 6.2 Hierarchy of nucleation, growth and drift
- 6.3 Scaling laws
- 6.4 Control mechanisms
- 6.5 Outlook

6 Discussion

6.1 Dimensionless analysis

For the sake of analysis and discussion, we transform the dust moment equations derived in Sects. 5.1 and 5.2 into their dimensionless form by introducing reference values as , , , , , , , , , following the procedure described in Paper I. The reference values are to be chosen according to the expected order of magnitude of the respective quantities and the length and time-scales under investigation. After this substitution, all quantities are dimensionless and can be compared by number.

This allows us to identify the leading terms in the equations, e.g.in the inner and outer regions of a brown dwarf atmosphere. The remaining constants (products of the reference values) can be summarised into characteristic numbers which provide an efficient way to describe the qualitative behaviour of the dust component.

The dimensionless dust moment equations for nucleation, growth,
evaporation, and equilibrium drift write for a *subsonic free
molecular flow* (
)

and for a

The equations are valid for . The constants in squared brackets are written in terms of characteristic numbers, which are further explained in Table 2. All other quantities and terms in Eqs. (77) and (78) are of the order of unity for an appropriate choice of the reference values.

The following discussion is based on a typical structure of a brown
dwarf atmosphere with solar abundances *in the gas phase*, i.e.
neglecting the possible depletion due to dust formation (see
Table 2). As underlying ()-structure we refer to a
brown dwarf model atmosphere with
K and
which has been kindly provided by
T. Tsuji (2002)^{}. As exemplary dust species we consider solid SiO_{2}(amorphous quartz), growing by the accretion of SiO and H_{2}O
(Eq. (38)). Since the nucleation of SiO_{2} seems
dubious (the monomer is rather unstable as a free molecule and hence
not very abundant in the gas phase) we consider nucleation of TiO_{2}instead^{}.

Name |
Characteristic |
Value |
||||||

Number |
inside | outside | ||||||

Mach number | ||||||||

Froude number | 0.842 | 0.495 | ||||||

Strouhal number | ||||||||

hydrodyn. Knudsen number | ||||||||

Knudsen number (Eq. (6)) | ... | |||||||

Drift number | ... | |||||||

combined drift number ( ) | ... | |||||||

combined drift number ( ) | ... | |||||||

: 1 | : 1 | |||||||

Sedlmaÿr number | : | : | ||||||

: | : | |||||||

: | : | |||||||

Damköhler no. of nucleation | 0 | 0 | ||||||

Damköhler no. of growth ( ) | 3.95 | |||||||

Damköhler no. of growth ( ) | ||||||||

Name |
Physical Quantity |
Reference Value |
||||||

inside | outside | |||||||

temperature | [K] | 1700 | ... | 1000 | ||||

density | [g/cm^{3}] |
... | ||||||

thermal pressure | [dyn/cm^{2}] |
... | ||||||

velocity of sound | [cm/s] | ... | ||||||

velocity | [cm/s] | |||||||

length | [cm] | 10^{+4} |
10^{+6} |
10^{+4} |
10^{+6} |
|||

hydrodyn. time | [s] | |||||||

gravitational acceleration | [cm/s^{2}] |
10^{+5} |
||||||

mean particle radius | [cm] | 10^{-3} |
... | 10^{-6} |
||||

0th dust moment ( ) | [1/g] | ... | ||||||

nucleation rate | [1/s] | ... | ||||||

growth velocity ( , Eq. (67)) | [cm/s] | ... | ||||||

growth velocity ( , Eq. (76)) | [cm^{2}/s] |
... | ||||||

diffusion constant (Eq. (26)) | [cm^{2}/s] |
... | 2.43 | |||||

mean free path (Eq. (10)) | [cm] | ... | ||||||

total hydrogen number density | [1/cm^{3}] |
... | ||||||

molecular number density | [1/cm^{3}] |
... |

: According to the experience of Paper I.
: We choose the reference value of the dust-to-gas mass ratio
by considering the case when all Si is
bound in solid SiO_{2} and adapt the referece value for the total dust
particle density
according to the assumed
grain's reference size
:
(Gail
Sedlmayr 1999).
: We assume vanishing nucleation rates in this region
around 1700 K, because the supersaturation ratios
are either
too small for efficient nucleation or the nucleation species have already
been consumed by growth.
: The reference value for the nucleation rate
is chosen by considering homogeneous nucleation of TiO_{2} according to
Paper I.
: SiO is considered as key growth species.

An analysis of the characteristic numbers in front of the source terms in Eqs. (77) and (78) (see Table 2) reveals a hierarchy of nucleation growth drift:

- 1)
- In the cool outer layers, the gas is strongly
supersaturated ()
and nucleation is effective
(
). The products of the Damköhler number of
nucleation
with the Strouhal number
and the Sedlmaÿr
numbers
are as large as the products of the Damköhler numbers of growth
with the Strouhal number
(even larger for small
*j*) and much larger than the combined drift numbers, indicating that the nucleation provides an important source term in Eqs. (77) and (78). Consequently, the condensable elements will be quickly consumed by the process of nucleation before the particles can grow much further. Hence, the dust particles remain very small in this*layer of effective nucleation*.

- 2)
- In the warmer layers, the gas is almost
saturated ()
and nucleation is not effective
(
). The products of the Damköhler numbers of growth
with the Strouhal number
are large and the growth
term (the second term on the r.h.s.) is the leading source term in
the dust moment Eqs. (77) and (78).
In comparison, the influence of the drift term is small as
quantified by the combined drift numbers in
Table 2. Consequently,
*the dust growth process will substantially be completed before the dust grains start to settle gravitationally*. In these*growth-dominated layers*, a few existing particles will quickly consume all condensable elements from the gas phase and, thus, will reach much larger particle sizes. Since these particles cannot be created via nucleation here, they must have been formed elsewhere and transported into these layers by winds or drift. The growth will either be terminated by element consumption or by the loss due to gravitational settling when the particles reach their maximum size as introduced in Sect. 3.1.

- 3)
- At the cloud base, the gas is hot and saturated (). Consequently, nucleation and growth vanish and the only remaining source term on the r.h.s. of Eqs. (77) and (78) is the additional advection of the particles due to their drift motion (drift term). The combined drift numbers, however, are small indicating that the drift term has a smaller influence than the hydrodynamical advection term.

Name | Value | ||

dust material density | [g/cm^{3}] |
2.65 | |

monomer volume | [cm^{3}] |
||

lower dust grain radius | [cm] | ||

molecular radius | [cm] |

physical process | ||

time-derivative | ||

advective term | 1 | |

nucleation term | ||

growth term | ||

drift term |

: the importance of this process depends
on the considered dust moment j with the followingassociated mean dust quantity: dust particle density, dust size, dust surface area, dust mass density. |

Table 4 shows some dependencies of the combined characteristic numbers (the squared brackets in Eqs. (77), (78)), which provides scaling laws for the importance of the different processes in the different regimes:

- 1)
- The drift term scales as
(
)
or
(
)
whereas the growth term scales as
with different
for the different cases (Table 4). This
means that at a certain large mean particle size, the drift term
will start to dominate over growth, which in fact just occurs at
the maximum particle size
introduced in
Sect. 3.1.

- 2)
- Nucleation (always) and growth (
)
become increasingly important with increasing gas density
(note that typically
with ), whereas the drift term (
)
diminishes for increasing
,
i.e.the drift is
generally more important in a thin gas. However, the particles are
expected to remain smaller in such a thin gas situated in the upper
layers, which has a stronger, opposite effect on the
drift term.

- 3)
- The larger the considered length scale
,
the more stiff the dust moment equations become because of
the increasingly large nucleation and growth terms. This may
actually cause some numerical difficulties for models on macroscopic
scales. In comparison, the advective and the drift term are
independent on
,
which means that these terms gain
importance on small scales in relation to the other terms.

- 4)
- The drift term scales as , i.e.the drift is naturally more important for heavy grains in a strong gravitational field.

In Table 2, we have assumed , i.e.we have considered time-scales of the order of , appropriate for disordered (e.g.turbulent) velocity fields, where the l.h.s. terms of Eqs. (77) and (78) are of comparable importance. However, if large-scale systematic motions are stable for a long time (e.g.a circulating thunderstorm or a stable convection roll), the system may reach a quasi-stationary situation where becomes much larger and hence . In this case, the first term on the l.h.s. of Eqs. (77) and (78) vanishes (see Table 4) whereas all other terms remain unaffected.

An interesting special case occurs if additionally , i.e.when the dust-forming system reaches the static case. In this case, also the advective terms in Eqs. (77) and (78) vanish and the source terms must balance each other. In the case, this means that the gain of dust by nucleation and growth must be balanced by the loss of dust due to rain-out, which means that the gas will be depleted. In the case, just the opposite is true, i.e.the loss by evaporation must be balanced by the gain of dust particles raining in from above. Consequently, the gas in such undersaturated layers will be enriched by the condensable elements liberated by the evaporating grains.

However, both control mechanisms (in the static limit) result in an efficient transport of condensable elements from the cool upper layers into the warm inner layers, which cannot last forever. We may conclude that if the brown dwarf's atmosphere is truly static for a long time, there is no other than the trivial solution for Eqs. (77) and (78) where the gas is saturated ( ) and dust-free ( ). This situation changes, however, if the brown dwarf's atmosphere is turbulent or, in particular, when it is convective. In that case, the replenishment of the atmosphere with fresh uncondensed gas from the deep interior will counteract the downward transport of condensable elements by the formation and gravitationally settling of dust grains. Simulations of this quasi-static balance will be the subject of the forthcoming paper in this series.

A dynamical modelling of the dust component in brown dwarf atmospheres
by means of a moment method as proposed in this paper - consistently
coupled to hydrodynamics, radiative transfer and element depletion in
the scope of hydrodynamical or classical stellar atmosphere
calculations - seems straightforward as soon as two major problems can
be solved:

Such closure conditions are fastidious problems. We are optimistic, nevertheless, to find such a closure term, because the size distribution

One idea to construct such a closure condition has been developed by
Deufelhard
Wulkow (1989) and Wulkow (1992),
studying the kinetics of polyreaction systems. The size distribution
function *f*(*V*) is here approximated by a weight function
,
which describes the basic shape of *f*(*V*), and
modified by a sum of orthogonal polynomials
(*k*=0,1,2, ... ,*n*) as

is an additional parameter (dependent on ) of the weight function and

Copyright ESO 2003