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Astron. Astrophys. 361, 303-320 (2000)

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2. Interaction between Loop I and the Local Bubble

Irrespective of the detailed knowledge of the plasma state both in the Loop I and the Local Bubble, it is fair to assume that due to ongoing supernova activity in Loop I its pressure will be higher than that in the Local Bubble. For simplicity we assume a plasma in collisional ionization equilibrium (CIE) in both bubbles. This is no restriction and is conservative in the sense that the pressure inside the Local Bubble might be smaller in case of non-equilibrium models (Breitschwerdt & Schmutzler 1994). When the two bubbles first merged, probably [FORMULA] years ago, some complicated gas dynamical processes ensued, which are sketched qualitatively in the following. The interaction started with a head-on collision of the two outer shock waves, bounding the swept-up shells of interstellar matter. The penetration of the shocks led to a mutual weakening and retardation with a tangential discontinuity occurring at the site of collision. The discontinuity separates two regions of constant pressure and flow velocity, but with different densities, temperatures and magnetic field strengths. Before an equilibrium condition could be set up, the respective shocks hit the tangential surfaces bounding each individual hot bubble, resulting in a transmitted and a reflected shock wave. The transmitted shock rapidly decays into a sound wave owing to the high speed of sound in the bubble, whereas the reflected shock interacts with the newly created tangential surface. It is well-known that such a tangential surface is unstable and will be dissolved, and the system will settle down after some time in an equilibrium configuration, characterized by a homogeneous dense interaction zone of uniform pressure and magnetic field, bounded by the tangential discontinuities of the hot bubbles on either side.

With supernova explosions in Loop I occurring on average about every [FORMULA] years (Egger 1998), the compressed shell between the two bubbles is then subject to an effective gravity [FORMULA] pointing towards the center of Loop I (see Fig. 2) in the rest frame of the expanding hot plasma.

[FIGURE] Fig. 2. Sketch of the compressed interaction zone of thickness [FORMULA] between Loop I and the Local Bubble. If the magnetic lines of force are antiparallel, magnetic reconnection will produce a neutral current sheet in which resistive dissipation occurs, thus weakening the field in the wall. The curvature of the shell boundaries is overemphasized.

Such a situation is inherently Rayleigh-Taylor unstable for all wavenumbers [FORMULA] with the largest ones growing fastest. However the presence of a magnetic field with a component parallel to [FORMULA] acts like surface tension in an ordinary fluid. In other words, the magnetic stresses tend to straighten out the corrugated lines of force.

The compressed wall can be represented by a thin and dense slab held under pressure from both sides. The plane parallel approach is justified, because the thickness [FORMULA] of the wall is much less than its radius of curvature 2, [FORMULA]. A stabilizing effect due to the curvature is negligible, as has been shown in the case of the gravitational instability of a spherical shell (Tomisaka & Ikeuchi 1983). Moreover the density perturbations satisfy [FORMULA], and the flow may therefore be considered as incompressible. In such a situation pressure and density changes can be regarded as adiabatic. In fact, the flow in an active superbubble is energy driven with occasional cooling occurring when a supernova shock wave from the interior hits the shell. The interaction zone itself is dense enough to be treated as isothermal. The sound crossing time in both the bubbles and the wall is small enough compared to the dynamical time scale for the pressure to be treated as uniform.

The hydromagnetic equations (in case of ideal MHD) to be considered are:

[EQUATION]

where [FORMULA], [FORMULA], P, [FORMULA], [FORMULA], [FORMULA], [FORMULA] and c denote the matter density, velocity, pressure, electric current density, magnetic field strength, an external force, speed of sound in the compressed interaction shell and the speed of light, respectively.

For simplicity we consider the case of [FORMULA] being parallel to the interface of the superbubbles and the shells, i.e. [FORMULA]. This is also the most likely case from physical considerations, because any frozen-in magnetic field in the swept-up interstellar medium will be wrapped around the bubble as it expands. The external force will be due to the acceleration of the shell, which will described by an "effective" inwards gravitational acceleration [FORMULA]; hence [FORMULA].

2.1. The linearized perturbation equations

We make the usual ansatz for a perturbed quantity, [FORMULA], subtract the unperturbed stationary equilibrium background state [FORMULA], and write out the components of the perturbed equations to first order. The background fluid satisfies total pressure equilibrium across the boundaries, i.e. [FORMULA]. Since the flow is dependent on z and in order to keep the treatment general, we allow for a gradient in the density and magnetic field strength.

[EQUATION]

Writing out Eq. (8) in components gives

[EQUATION]

We begin by considering linear perturbations (normal modes) varying as

[EQUATION]

and propagating along the interface, where [FORMULA] is the increment of any perturbed quantity, and

[EQUATION]

are the respective components of the wave vector and the distance from the origin of the coordinate frame (see Fig. 2). In the above definition for perturbations, instability will arise for [FORMULA] real and positive. Using Eq. (16) one obtains the following set of equations

[EQUATION]

Making use of the incompressibility of the flow we obtain

[EQUATION]

As usually, the above equations can be combined and rearranged so that all perturbation variables can be eliminated in favour of [FORMULA]. Rewriting Eqs. (20) and (21) with Eqs. (18) and (23) results in

[EQUATION]

and

[EQUATION]

Next we multiply Eq. (25) with [FORMULA] and Eq. (19) with [FORMULA] and add them up to give

[EQUATION]

Therefore the term in brackets has to vanish, which is equivalent to the z-component of the vorticity, [FORMULA], to be zero, i.e.

[EQUATION]

Thus Eq. (25) simply reads:

[EQUATION]

Combining Eq. (29) with (19), using (24) and defining [FORMULA], we obtain

[EQUATION]

Finally, we can use Eq. (30) to eliminate [FORMULA] from (26) and after some algebra arrive at

[EQUATION]

Eq. (31) will be examined now in some detail.

Disturbances with [FORMULA] will not cause any stretching and bending of field lines, because the wave front is coplanar to [FORMULA]. Eq. (31) then reduces to

[EQUATION]

which leads to the dispersion relation of the ordinary Rayleigh-Taylor instability without magnetic field. Before discussing this special case, Eq. (31) will be solved for the general case including [FORMULA].

Since [FORMULA] and [FORMULA] are constant in each region with a discontinuous jump across the boundary Eq. (31) reduces to

[EQUATION]

for regions 1, 2 and 3, respectively (see Fig. 2). Across the boundaries, the solutions have to be matched, fulfilling the conditions of continuity of the velocity z-component and total pressure, i.e.

[EQUATION]

for [FORMULA]. We note, that waves propagating along the tangential discontinuities will deform the boundaries so that the gas at different positions will experience a different gravitational acceleration, which has to be included in condition (ii).

Defining the Alfvén velocity in each region by

[EQUATION]

Eq. (33) reads

[EQUATION]

A trivial solution corresponds to Alfvén waves, satisfying the simple dispersion relation [FORMULA], and propagating along the lines of force. They do not represent unstable solutions, because [FORMULA] must be real so that for [FORMULA], the phase factor [FORMULA] remains finite; therefore [FORMULA] is imaginary, representing a purely oscillatory behaviour.

The other solution is simply given by

[EQUATION]

and the sign has to be chosen such that [FORMULA] remains finite for [FORMULA], i.e. "-" for [FORMULA] and "+" for [FORMULA]. We note that for [FORMULA] and for [FORMULA] (with [FORMULA] being the wall thickness), condition (i) is automatically satisfied.

The velocity field [FORMULA] can be generally decomposed into a rotational and into an irrotational part:

[EQUATION]

From Eq. (12) and the general form of the rotational part, [FORMULA], we deduce: [FORMULA], and [FORMULA]. Remembering that [FORMULA], we obtain

[EQUATION]

which goes to 0 for [FORMULA], respectively. Condition (i) is automatically satisfied, if we observe that e.g. [FORMULA] for [FORMULA].

Waves propagating (at some angle in general) along the magnetic lines of force will undulate the tangential discontinuity boundary and stretch the B-field, thereby increasing the magnetic tension forces (cf. the second term on either side of condition (ii)). The undulated surface will have the general form (see Fig. 3):

[EQUATION]

Let us now have a closer look at the deformed interface between region 1 and 2. From Eq. (13) we obtain by integrating

[EQUATION]

and by exchanging differentiations with respect to x and t:

[EQUATION]

with [FORMULA]. From Eq. (23) we infer

[EQUATION]

[FIGURE] Fig. 3. Sketch of the tangential boundary between Loop I and the Local Bubble. The magnetic field is different, but constant on either side. The gravitational pull due to the weight of the fluid is a function of [FORMULA].

Now,

[EQUATION]

and thus

[EQUATION]

Condition (i) therefore requires at [FORMULA]:

[EQUATION]

and therefore

[EQUATION]

as it was already deduced for [FORMULA].

We now inspect condition (ii). As mentioned earlier we have to incorporate the weight of the fluid between [FORMULA] and [FORMULA], which physically corresponds to an extra accelerating force per unit area. Its value is simply given by [FORMULA] in region 1 and [FORMULA] in region 2. Thus condition (ii) becomes

[EQUATION]

Using the expressions (41) and (42) and from (46) the relation [FORMULA], we obtain

[EQUATION]

from which it follows that

[EQUATION]

with [FORMULA] being the growth time of the instability. From Fig. 4 it can be seen that the maximum growth rate occurs for [FORMULA] and that it increases with decreasing magnetic field strength (for definitions of [FORMULA] and [FORMULA] see below). Note that Eq. (49) also covers the cases of zero B-field in Loop I and the same magnitude of the field in both regions 1 and 2.

[FIGURE] Fig. 4. The growth rate [FORMULA] in arbitrary units as a function of wave vector k (in units of [FORMULA]) and magnetic field strength [FORMULA] in units of [FORMULA].

Considering our previous ansatz (16), unstable solutions must have real and positive [FORMULA]. Since the density in the wall is much larger than its counterpart in the Loop I bubble ([FORMULA]), this will be satisfied for any wavenumber with [FORMULA], with the critical wavenumber [FORMULA] given by

[EQUATION]

where we have used [FORMULA], with [FORMULA] being the angle between the wave vector of the perturbation and the direction of the undisturbed magnetic field lines. The term [FORMULA] introduces an upper limit in the unstable wavenumbers. Thus magnetic tension forces have a stabilizing effect, just as surface tension in an ordinary fluid.

In the classical Rayleigh-Taylor instability (see Eq. (32)) there is no limit on k as we can see by setting [FORMULA], with the largest wavenumbers growing fastest. The same is true for perturbations with [FORMULA], which do not lead to a deformation of the field lines but simply to an exchange of higher field and lower field plasma. We believe that the present topology of field and plasma does not easily promote such a process. Firstly, we have treated for simplicity the onset of instability for slab geometry whereas, on larger scales, the field lines are circular and wrapped around the bubbles or, considering the shell, may even be rooted somewhere in the undisturbed interstellar medium. Secondly, [FORMULA] and [FORMULA] are in general not parallel but oblique with respect to each other, thereby also seriously inhibiting the exchange mode.

In the following, we will therefore consider the bending mode instability and take [FORMULA] in our numerical estimates. In this case our simplifying assumptions on the field geometry will not cause any restrictions. The most unstable wavenumber, [FORMULA], is given by (cf. Appendix A)

[EQUATION]

The corresponding growth time is then

[EQUATION]

Before discussing the implications of Eq. (52) for the interaction zone between Loop I and the Local Bubble, we have a brief look at the interface between regions 2 and 3, i.e. the tangential discontinuity between the shell and the Local Bubble. It is intuitively clear, that a stratification in which a dilute and light fluid is gravitationally supported by a dense and heavy one is inherently stable. Shifting our coordinate system with [FORMULA] to the boundary separating regions with densities [FORMULA], [FORMULA] and field strengths [FORMULA], [FORMULA], respectively, we can apply the same analysis as before and find accordingly

[EQUATION]

Now [FORMULA], hence [FORMULA] is purely imaginary and all solutions will therefore be stable.

However, once the instability becomes fully nonlinear, the whole wall separating the Local Bubble and Loop I will be affected (cf. next section).

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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