SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 359, 1124-1138 (2000)

Previous Section Next Section Title Page Table of Contents

2. Tension versus magnetic pressure

2.1. Equations and notations

We 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

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The drift between ions and neutrals is a small effect in a medium with an ionization fraction around [FORMULA] like the interstellar diffuse neutral gas, and is neglected.

As usual, [FORMULA] is the density, T the temperature, P the pressure, [FORMULA] the velocity and [FORMULA] the magnetic field. [FORMULA] is the viscous stress tensor, [FORMULA] the heat flux, [FORMULA] the thermal conductivity, [FORMULA] the heat capacity, [FORMULA] the mean particle mass, [FORMULA] the Boltzmann constant, [FORMULA] the magnetic permeability ([FORMULA], [FORMULA] the adiabatic index, [FORMULA] the net loss function (cooling minus heating), [FORMULA] denotes the sound speed,

[EQUATION]

are respectively the cooling or radiation time and Field length. The viscosity and thermal diffusivity are (Lang 1974)

[EQUATION]

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

[EQUATION]

[EQUATION]

[EQUATION]

2.2. Thermal condensation in slab geometry with magnetic fields

Let 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 [FORMULA] and [FORMULA] are not zero but depend on x only. The longitudinal field [FORMULA] is uniform and constant. We consider a converging flow in a thermally bistable medium. We showed in Paper I, that in a non-magnetized gas initially at thermal equilibrium in the WNM phase, if the typical spatial scale is at least equal to [FORMULA] and if the velocity peak reaches a critical value, thermal condensation occurs. Part of the gas leaves the first equilibrium branch (WNM) and reaches the second one (CNM). To what extent does the magnetic field alter this picture?

2.2.1. Case of a transverse field

If 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 [FORMULA] ([FORMULA]), 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

[EQUATION]

hence,

[EQUATION]

A magnetic field of only [FORMULA] 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 field

Let [FORMULA] denote the angle between the magnetic and velocity fields ([FORMULA] means that [FORMULA] and [FORMULA] 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

[EQUATION]

If [FORMULA] is not vanishing, the r.h.s. can be seen as a transport term and [FORMULA] as a current of [FORMULA] 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 [FORMULA], delimited by [FORMULA] and [FORMULA]. Let [FORMULA] be its length along the z-axis. The magnetic flux through any surface is conserved

[EQUATION]

at initial time

[EQUATION]

If the longitudinal component is equal to zero, then A will always belong to the plane [FORMULA] and if [FORMULA] decreases (compression), [FORMULA] 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

[EQUATION]

Eq. (2) gives the evolution of the transverse velocity field, and in slab geometry leads to

[EQUATION]

Consequently, if [FORMULA] decreases, [FORMULA] does not necessarily increase and for sufficient values of [FORMULA] the thermal condensation keeps going on. This is demonstrated analytically in Sect. 3 and numerically in Sect. 4.

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
helpdesk.link@springer.de