Heat and Gas Diffusion in Comet Nuclei (pdf file 5.5 MB) - ISSI

6.4. Simultaneous Solution for Transfer of Heat and Mass 109
and the total production rate is obtained by summing the contributions of
such wedges over one spin period. This is, essentially, a 2.5-dimensional
calculation. The next step is to take into account both diurnal and latitudinal
solar flux variations (Huebner and Boice, 1992; Gutiérrez et al.,
2000; Julian et al., 2000; Cohen et al., 2003, De Sanctis et al., 2005), considering,
however, only radial heat conduction, that is, neglecting lateral
conduction. This quasi 3-D approach is amply justified by the extremely
low heat conductivity of cometary material; the characteristic heat diffusion
time between equator and pole (as between surface and centre) is of
the order of the life-time of a comet (see Table 5.1 above). The different
models are shown schematically in Fig. 6.4; however, calculations use much
finer meshes than the ones shown.
6.4 Simultaneous Solution for Transfer of Heat
and Mass
We note that the evolution equations are coupled through the source terms
and the gas fluxes, which are functions of both temperature and pressure,
and hence must be solved simultaneously. A flowchart of an evolution code
that uses an implicit numerical scheme is shown in Fig. 6.5; the energy and
mass balance equations are solved in an alternating sequence, until they
converge. The question marks represent the question whether convergence
has been achieved in the corresponding process. The question mark following
n or m represents the question whether the number of iterations has
exceeded the allowed limit (since there is no point in continuing an iterative
process that does not converge).
The simultaneous solution of the heat and mass conservation equations is
extremely time consuming, keeping in mind that the equations are strongly
nonlinear. Simplifying approximations may be used under special conditions.
If the effective permeability of the medium is sufficiently high, the left
side of the mass conservation equation for the gas phases (Eq. 4.9) becomes
negligible. Neglecting it is tantamount to a quasi-steady state approximation,
where gas densities and production rates change only as far as the
temperature distribution changes. Thus Eqs. (4.9 – 4.11) are replaced by
∇ · J n = q n
∂ρ s,n
∂t
= −q n (6.23)
In this way we have to solve only one time-dependent equation, supplemented
by structure (space-dependent) equations. This constitutes a large
computational advantage, particularly in a long-term evolution calculation,