2. Kelvin-Helmholtz instabilities in superwinds of primeval galaxies: analytical estimates
The dynamics of partially ionized dissipation-free two-fluid plasmas in ionization equilibrium is determined by balance equations describing the conservation of mass and momentum of both plasma and neutral gas
and the induction equation that governs the temporal evolution of the magnetic field
In these equations , , p, and c denote the plasma mass density, velocity, thermal pressure, the magnetic field and the velocity of light. The lower index n identifies neutral gas quantities. The final terms in Eqs. (3) and (4) specify the rate of momentum transfer between the ionized and the neutral gas components. This is proportional to the effective plasma-neutral gas collision frequency . For the analytical calculation of the dispersion relation for KH modes we assume incompressibility.
In the subsequent numerical simulations described in Sect. 3 we will remove the restriction of incompressibility.
The KH dispersion relation can be obtained by standard linear mode analysis. Ershkovich and co-workers have extensively analyzed KH modes, with applications to partially ionized cometary ionopauses. Ershkovich and Mendis made use of a one-fluid description which holds for the strong coupling régime only (Ershkovich & Mendis 1983). A multi-fluid approach allows for an explicit treatment of collisional momentum transfer which has been investigated for both compressible (Ershkovich & Mendis 1986), and incompressible cases (Ershkovich et al. 1986a, b). Neither of the two contributions cited above takes into account the thermal pressure forces of the neutral component, or perturbations in the neutral gas bulk velocities. This is justified for the application to cometary ionopauses but not for the application to multi-phase galactic outflow winds. Also we do not restrict ourselves to uniform media with negligible thermal neutral gas pressure, as was done by Bhatia and Steiner (1974).
In this paper we take into account the full incompressible dynamics of both the neutral and the ionized gas components. Doing this we find the following dispersion relation for KH modes (for details of the corresponding analysis see Appendix):
where the superscript eq denotes equilibrium values and is the wave number parallel to the KH interface. The bar indicates mean values across the KH interface. The Doppler shifted complex growth rate is denoted by or , respectively.
For one-fluid MHD systems the stability criterion for KH modes can be stated as follows (e.g. Woods 1987)
where and denote the magnetic field on both sides of the boundary layer, and , denote the corresponding mass densities. (Note that I and II here do not refer to the neutral and ionized components.) However, in the case of interacting flows of ionized and neutral gas components this onset criterion is no longer required to be fulfilled for KH modes to be excited.
To exemplify this, Fig. 1 shows dimensionless solutions of the dispersion relation for different choices of the normalized mass densities, velocities, magnetic field strengths, and effective collision frequencies. The real part of the normalized growth rate q, which is the inverse normalized growth time, is plotted against the normalized wave number (where is the dimensionless wave length). As in the one-fluid case we find non-dispersive modes which are, however, overstable ones (at this point we are not interested in the oscillatory part of the complex growth rates).
From the results shown in Fig. 1 we can get quantitative estimates of the linear time scales of the KH modes by normalizing to physical parameters that are typical of galactic superwinds. We have chosen the following typical plasma values (indicated by the index 0); ( being the proton mass), and wind velocity . Since the velocities are formally normalized to the Alfvén speed the choice of implies G. This choice is motivated by observational data from the prototypical starburst galaxy M82 (e.g. Lehnert & Heckman 1986; Shopbell & Bland-Hawthorn 1998). One may speculate on the appropriate neutral gas density. Here, we have in mind a situation where neutral gas is still falling onto the central region of a young primeval dwarf galaxy and/or one in which the outflow wind has a significant neutral/molecular gas component. As a typical magnetic field strength we choose G (and for comparison ) which can be expected from the compressional amplification of some seed fields during the collapse dynamics (cf. Lesch & Chiba 1995). For the wavelength we choose, for example, the typical extension of observed radio filaments in M82, i.e. kpc (Reuter et al. 1992).
The typical time scales of the KH instability can now be calculated from the inverse growth rate (see Fig. 1) . This means that even for relatively large characteristic length scales of 1kpc the KH dynamics has enough time for the regeneration of the magnetic fields that are transported by the superwinds given expected lifetimes of the bursts of some yr (cf. Rieke et al. 1980; Mateo 1998). For different choices of the normalizing quantities, the inherent time scales for the linear dynamics of the KH modes can accordingly be found from Fig. 1. It should be noted that the choice of kpc results in quite a high limit for the linear growth time of the KH modes since one should expect the KH instability to operate in thinner boundary layers.
For comparison Fig. 2 shows solutions of the KH dispersion relation for the pure MHD case. It can be seen that comparable flow conditions in the MHD, and the partially ionized régime result in quite similar growth times of the unstable modes. This means that KH modes are expected to amplify the magnetic field as efficiently as in the totally ionized MHD régime. This result is important, given the considerable uncertainties, and likely variations in the physical parameters that characterize the interaction details of superwinds and infall gas in primeval galaxies.
What we can conclude is that magnetic field amplification by KH modes can occur effectively during the entire life time of the primeval dwarf galaxies or the starbursts, respectively (cf. Mateo 1998), and that it is relatively insensitive to varying degrees of ionization in the interacting gas flows. What cannot be derived from the linear analysis is the amplification factor. However it can be obtained by nonlinear simulations, which we present in following section.
© European Southern Observatory (ESO) 2000
Online publication: December 8, 1999