5 Local dynamic

Interstellar chemistry is known to be sensitive to density since some destruction processes (photo-ionisation and/or destruction by cosmic rays) proceed as the density, whereas chemical reactions proceed as the square of the density. Le Bourlot et al. (1995) have shown that under some fairly ordinary physical conditions, two stable chemical phases may exist. So, depending on initial conditions, some parts of the cloud may evolve towards one phase as others evolve towards the other phase. Interfaces between those phases lead to reaction-diffusion fronts where unusual chemical abundances may prevail for long times in a manner similar to reaction-diffusion fronts in a thermally bistable fluid studied by Shaviv & Regev (1994).

Thus, a minimal local dynamic should at least exhibit bistability. This can be achieved with a 3-variable model which is the minimal non-passive scalar model possible. By turning on or off turbulent mixing, we can test the effects of that mixing on mean abundances along the line of sight and on time and length scales for each variable within the cloud.

Figure 10: Dispersion of a passive scalar (after 1024 iterations of ) by our synthetic turbulent field and by a Gaussian field (mean and standard deviation of the velocity field are the same for both fields). Point source released at t=0 at x=256.

Figure 11: Temporal evolution of the mean and standard deviation of the passive scalar density distribution for our two fields. As expected for diffusion by a Gaussian field, the mean and variance are proportional to time.

This is an extension to intrinsically scale-dependent models of the work of Xie et al. (1995) and Chièze & Des Forêts (1989). However, full-size interstellar chemical schemes are still beyond our reach.

As a test model, we chose the following set of chemical reactions (inspired from Gray & Scott 1990):

$$\left\{ \begin{array}{cccc} R & \rightarrow & A & (k_{1})\\ ... ..._{3})\\ B & \rightarrow & P & (k_{4})\cdot \end{array}\right.$$ (12)

Such a model can be seen as an excerpt from a larger chemical network with $R=R'\rho$, being a production term proportional to the density, and where the product(s) P returns to the rest of the gas.

If we suppose that k₁ has the following temperature dependence $k_{1}=k_{10}\exp (-\frac{E_{a}}{k_{b}T})$ and that reaction (4) is exothermic, then thermal balance is governed by: $\Delta U=k_{4}\Delta t\Delta Hn_{B}-k_{5}k_{b}\Delta t(n_{A}+n_{B})T$ (with $U=\rho c_{\rm p,\rho }T=n_{\rm R}c_{\rm p,R}T$). Here the cooling term mimics radiative cooling by both A and B after collisional excitation.

We can reduce the problem to a simple dynamical system with three differential equations and four parameters  $(r,\varepsilon ,k,\gamma )$:

$$\left\{ \begin{array}{ccc} \frac{{\rm d}\alpha }{{\rm d}\tau ... ...ta )\frac{\displaystyle u}{\displaystyle r} \end{array}\right.$$ (13)

with

$$\left\{ \begin{array}{rcl} \alpha & ~=~ &\sqrt{\frac{k_{3}}{... ...5}k_{b}k_{10}}{c_{\rm p,R}~ k^{2}_{4}}\cdot \end{array}\right.$$ (14)

This simple dynamical system leads to many different situations: bistability or limit cycle with Hopf bifurcation (see Fig. 12) for different parameters. In the following, we use k=0.001, r=1, and as our reference bistable model. The chemical time scale is 1/k₄, so that the ratio of turbulent to chemical time scales is . In the following, we usually use ( being the crossing time at the smallest scale we resolve).

Figure 12: An example of bistability and limit cycle with Hopf bifurcation. The parameters values are $k=0.001,\: \gamma =1,\: \varepsilon =0.01$. We plotted the values of $\alpha$ and $\beta$, at the equilibrium, for different values of the rparameter (x axis).