## 2. Interaction between Loop I and the Local BubbleIrrespective 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 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 years (Egger 1998), the compressed shell between the two bubbles is then subject to an effective gravity pointing towards the center of Loop I (see Fig. 2) in the rest frame of the expanding hot plasma.
Such a situation is inherently Rayleigh-Taylor unstable for all wavenumbers with the largest ones growing fastest. However the presence of a magnetic field with a component parallel to 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 of
the wall is much less than its radius of
curvature The hydromagnetic equations (in case of ideal MHD) to be considered are: where ,
, For simplicity we consider the case of being parallel to the interface of the superbubbles and the shells, i.e. . 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 ; hence . ## 2.1. The linearized perturbation equationsWe make the usual ansatz for a perturbed quantity,
, subtract the unperturbed stationary
equilibrium background state , and
write out the components of the perturbed equations to first order.
The background fluid satisfies total pressure equilibrium across the
boundaries, i.e. . Since the flow is
dependent on Writing out Eq. (8) in components gives We begin by considering linear perturbations (normal modes) varying as and propagating along the interface, where is the increment of any perturbed quantity, and 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 real and positive. Using Eq. (16) one obtains the following set of equations Making use of the incompressibility of the flow we obtain As usually, the above equations can be combined and rearranged so that all perturbation variables can be eliminated in favour of . Rewriting Eqs. (20) and (21) with Eqs. (18) and (23) results in Next we multiply Eq. (25) with and Eq. (19) with and add them up to give Therefore the term in brackets has to vanish, which is equivalent
to the Combining Eq. (29) with (19), using (24) and defining , we obtain Finally, we can use Eq. (30) to eliminate from (26) and after some algebra arrive at Eq. (31) will be examined now in some detail. Disturbances with will not cause any stretching and bending of field lines, because the wave front is coplanar to . Eq. (31) then reduces to 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 . Since and are constant in each region with a discontinuous jump across the boundary Eq. (31) reduces to 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 for . 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 Eq. (33) reads A trivial solution corresponds to Alfvén waves, satisfying the simple dispersion relation , and propagating along the lines of force. They do not represent unstable solutions, because must be real so that for , the phase factor remains finite; therefore is imaginary, representing a purely oscillatory behaviour. The other solution is simply given by and the sign has to be chosen such that remains finite for , i.e. "-" for and "+" for . We note that for and for (with being the wall thickness), condition (i) is automatically satisfied. The velocity field can be generally decomposed into a rotational and into an irrotational part: From Eq. (12) and the general form of the rotational part, , we deduce: , and . Remembering that , we obtain which goes to 0 for , respectively. Condition (i) is automatically satisfied, if we observe that e.g. for . 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): Let us now have a closer look at the deformed interface between region 1 and 2. From Eq. (13) we obtain by integrating and by exchanging differentiations with respect to
Now, and thus Condition (i) therefore requires at : as it was already deduced for . We now inspect condition (ii). As mentioned earlier we have to incorporate the weight of the fluid between and , which physically corresponds to an extra accelerating force per unit area. Its value is simply given by in region 1 and in region 2. Thus condition (ii) becomes Using the expressions (41) and (42) and from (46) the relation , we obtain with being the growth time of
the instability. From Fig. 4 it can be seen that the maximum
growth rate occurs for and that it
increases with decreasing magnetic field strength (for definitions of
and
see below). Note that Eq. (49)
also covers the cases of zero
Considering our previous ansatz (16), unstable solutions must have real and positive . Since the density in the wall is much larger than its counterpart in the Loop I bubble (), this will be satisfied for any wavenumber with , with the critical wavenumber given by where we have used , with being the angle between the wave vector of the perturbation and the direction of the undisturbed magnetic field lines. The term 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 In the following, we will therefore consider the bending mode instability and take in our numerical estimates. In this case our simplifying assumptions on the field geometry will not cause any restrictions. The most unstable wavenumber, , is given by (cf. Appendix A) The corresponding growth time is then 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 to the boundary separating regions with densities , and field strengths , , respectively, we can apply the same analysis as before and find accordingly Now , hence 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). © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |