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Astron. Astrophys. 346, 260-266 (1999)

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3. The cross-section of a mixing layer

With the equations and numerical method described in Sect. 2, we compute three models. These models correspond to cross-sections with widths [FORMULA], [FORMULA] and [FORMULA] cm of a mixing layer at the edge of a beam of velocity [FORMULA] km s-1. Both the jet and the environment are assumed to be neutral, and with singly ionized C and S. The jet temperature and density are [FORMULA] K and [FORMULA] cm-3. The corresponding variables for the stationary environment have the same values (i.e., [FORMULA] and [FORMULA]).

The temperature and density obtained from these three models are shown in panels a and b of Fig. 2. From this figure, it is clear that while for the smallest h value we obtain a centrally peaked temperature cross-section, for the larger h values we obtained a flat-topped, quasi-isothermal cross-section. This behaviour was predicted in a qualitative way by the analytic model of Noriega-Crespo et al. (1996) and Raga & Cantó (1997). As explained in the previous section, the flat region of the temperature cross-section (present for the two models with larger h values, see Fig. 2) has a temperature determined from the balance between the turbulent dissipation [FORMULA] (see Eqs. 2 and 3) and the radiative energy loss L, the first term of the energy equation (Eq. 2) only being important close to the inner and outer boundaries of the turbulent mixing layer. In this respect, mixing layer models appear to be simpler than shock wave models, where the gas generally is in non-equilibrium thermal state (with the advective term playing an important role).

[FIGURE] Fig. 2a-d. Cross-section of a turbulent mixing layer obtained from the model described in the text: a  the temperature, b  the hydrogen density, c  the ionization fraction of hydrogen assuming coronal ionization equilibrium, and d  the ionization fraction of hydrogen after solving for the non-equilibrium ionization state of the gas. Models with three values for the mixing layer width are plotted: [FORMULA] cm (solid curves), [FORMULA] cm (dashed curve) and [FORMULA] cm (dot-dashed curve).

In panel c of Fig. 2, we have plotted the ionization fraction that would be obtained from the coronal ionization equilibrium condition at the local temperature [FORMULA], i.e. from setting the turbulent diffusion term to zero in Eq. (4). A comparison of these ionization fractions with those in panel d obtained from the full solutions of Eqs. (1-4) shows that the isothermal regions of the temperature cross-sections are less ionized than predicted for coronal ionization equilibrium at the local temperature. This deviation from ionization equilibrium is due to turbulent diffusion of neutral gas from the jet and ambient medium into the layer. Although this departure from equilibrium ionization in mixing layers is less pronounced than that encountered in shock wave calculations, it is not negligible and the diffusion term cannot be neglected in Eq. (4) [even though the turbulent conductivity in Eq. (2) can be neglected with respect to the viscous heating term].

To verify the analytical argument of Eq. (7) that the temperature of radiative layers is mainly determined by the parameter [FORMULA], we have also run a broader grid of mixing layer models. All of the models have [FORMULA] K and one of the following three densities: [FORMULA], [FORMULA] or [FORMULA] cm-3, corresponding to increasing pressures of [FORMULA], [FORMULA] and [FORMULA] dyn cm-2, respectively. For each density, we ran a model sequence of increasing thickness h (and therefore of increasing [FORMULA]) with a fixed [FORMULA] km s-1. The typical range of [FORMULA] covered was [FORMULA] to [FORMULA] g cm-2. For [FORMULA] cm-3 we also ran a second sequence with a fixed [FORMULA] g cm-2 and [FORMULA] ranging from 30 to 500 km s-1.

In Fig. 3 we plot the calculated average temperature, [FORMULA], across the mixing layer as a function of the parameter [FORMULA]. Sequences of mixing layers with constant [FORMULA] but different [FORMULA] are shown as asterisks connected by a long dash line ([FORMULA] cm-3 i.e. Table 3) or as filled triangles connected by a a thick solid line ([FORMULA] cm-3). Since all three sequences overlap in such a plot, for clarity we did not include the sequence with [FORMULA] cm-3 of Table 2. The model sequence of increasing [FORMULA] but constant [FORMULA] for [FORMULA] cm-3 is shown as open diamonds connected by a dotted line. In that sequence, the temperature starts at [FORMULA] K ([FORMULA] km s-1) and reaches a maximum of [FORMULA] K at [FORMULA] km s-1 to finally come down to [FORMULA] K when [FORMULA] km s-1. From this figure, we see that, for a large range of parameters, the balance between the turbulent dissipation and the radiative cooling results in a temperature close to [FORMULA] K (which is a direct result of the very steep increase of the cooling due to collisionally excited Ly[FORMULA] and forbidden line emission around this temperature). Furthermore, the comparison of the constant [FORMULA] and constant [FORMULA] model sequences for [FORMULA] cm-3 shows that, as predicted by Eq. (6), the layer temperature (at a given pressure) is only a function of [FORMULA], provided [FORMULA] is large enough to produce a radiative layer (here, [FORMULA] km s-1).

[FIGURE] Fig. 3. Average mixing layer temperature, [FORMULA], as a function of the parameter [FORMULA] ([FORMULA]). The dotted line represent a sequence of mixing layers of increasing [FORMULA] but constant [FORMULA]. The other two sequences (with varying [FORMULA]) have a fixed [FORMULA] km s-1 but different densities (pressures): models with [FORMULA] cm-3 are shown as asterisks connected by a long dash line; models with [FORMULA] cm-3 are shown as filled triangles connected by a thick solid line. Notice the almost exact overlap of the curves with constant [FORMULA]. All models have [FORMULA] K. The short dash line represents the interpolation formula given in Eq. 8.

Above [FORMULA] g cm-2, the numerical values for any of the 3 sequences with [FORMULA] km s-1 are well reproduced with an interpolation formula of the form:


which is represented in Fig. 3 by a short dash line. This interpolation formula can be used to estimate the approximate temperature of a mixing layer as a function of the parameter [FORMULA], or conversely for determining [FORMULA] for an emitting layer of known temperature. These results suggest that the spectrum emitted by mixing layers is quite stable over a large range in [FORMULA]. This result is quantified in the following section.

The overlap in Fig. 3 of the three sequences is explained as follows: the amount of energy to be radiated in the mixing layer is [FORMULA] while the cooling rate is [FORMULA], therefore (as in plane parallel shock waves where the column of recombining gas is independent of the preshock density), the cooling history remains the same when the product [FORMULA] is kept constant (the same argument applies to the ionization history). [FORMULA] is a representation of such a product provided one compares models with the same [FORMULA], [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999