## 2. Tension versus magnetic pressure## 2.1. Equations and notationsWe consider the equations of magneto-hydrodynamics for a perfect gas including thermal exchanges with the surrounding medium (equations of radiative flows). S.I. units are used throughout the paper. Conversion to cgs units is also given for easier reference. These equations are The drift between ions and neutrals is a small effect in a medium with an ionization fraction around like the interstellar diffuse neutral gas, and is neglected. As usual, is the density, are respectively the cooling or radiation time and Field length. The viscosity and thermal diffusivity are (Lang 1974) In a magnetized gas, the thermal diffusivity depends on the magnetic field and is considerably reduced perpendicularly to the field lines (Spitzer 1962, Balbus 1986, Steele & Ibáñez 1999). However, in the dynamical problem considered here, the dissipative terms are small until the quasi-isobaric regime is reached. In the present article, we focus on the dynamical aspects of the condensation process, so we neglect the dependence of thermal diffusivity on magnetic field. Typical values in the WNM are ## 2.2. Thermal condensation in slab geometry with magnetic fieldsLet us give a qualitative description of the physical process
developed in subsequent sections. Like Elmegreen (1997) and Gammie
& Ostriker (1996), we consider the problem in the slab geometry.
The transverse fields and
are not zero but depend on ## 2.2.1. Case of a transverse fieldIf a purely transverse magnetic component is added, magnetic pressure must be taken into account. From the standard interstellar thermal pressure, a local density enhancement of around 3 is necessary to reach the critical point (Paper I). This enhances the magnetic pressure by a factor 9. For (), the magnetic pressure reaches a value about three times the initial pressure in the flow. Even with a weak initial field supersonic motions must be considered in order to reach the critical point. The subsequent evolution of the thermally unstable gas is further affected by magnetic pressure and condensation is rapidly stopped. A rough estimate of the final cloud density can be made assuming mechanical equilibrium between the thermally dominated diffuse medium and the magnetically dominated dense medium. With subscript 0 denoting values for the diffuse phase and subscript 1 for the dense one hence, A magnetic field of only thus leads to a maximum density ratio of about 5, far from the value derived from observations (Kulkarni & Heiles 1987). For higher values of the magnetic field, only transient weakly constrasted structures can emerge. ## 2.2.2. Case of an oblique magnetic fieldLet denote the angle between the magnetic and velocity fields ( means that and are parallel). In such conditions a magnetic tension applies to the flow, and consequently a (heterogeneous) transverse velocity field appears and contributes to the evolution of the magnetic field. In the slab geometry Eq. (5) becomes If is not vanishing, the r.h.s. can be seen as a transport term and as a current of that contributes to the evolution of the transverse magnetic component. For sufficient values of the longitudinal magnetic field, it possibly allows for the magnetic pressure to decrease. More physically, the longitudinal velocity compresses and bends the field lines. Thus, it enhances pressure and magnetic tension. The magnetic tension, on the contrary, tends to unbend these lines and decreases the magnetic pressure. We can also formulate this process using the flux freezing concept. Let us consider a rectangle A in the plane , delimited by and . Let be its length along the z-axis. The magnetic flux through any surface is conserved at initial time If the longitudinal component is equal to zero, then A will always belong to the plane and if decreases (compression), increases in order to keep the magnetic flux constant. With a non-vanishing magnetic longitudinal component, the rectangle A is stretched in the y-direction and the expression of the magnetic flux becomes Eq. (2) gives the evolution of the transverse velocity field, and in slab geometry leads to Consequently, if decreases, does not necessarily increase and for sufficient values of the thermal condensation keeps going on. This is demonstrated analytically in Sect. 3 and numerically in Sect. 4. © European Southern Observatory (ESO) 2000 Online publication: July 13, 2000 |