Online material

Appendix A: The Kelvin Helmholtz instability in stratified flows

A.1. Linear theory (Chandrasekhar 1961)

	Fig. A.1 Configuration of the stratified flow.
Open with DEXTER

A flow can be considered as incompressible with respect to the Kelvin-Helmholtz-instability if the Mach number of the velocity discontinuity ℳ_cd = Δv/ c_s ≲ .3 where Δv is the velocity difference at the interface and c_s the sound speed. At the centre of the binary system, the winds are subsonic and the flow is incompressible as the two winds collide head-on. We use the incompressible approximation in the following computations. However, this may not be fully applicable further away from the binary as the winds re-accelerate. That said, the interface remains marginally sonic ( in all our simulations). In this case, the evolution of the KHI is complex to determine as it depends on the binary parameters (η, β) but also on the development of the KHI closer to the binary. We have a system with a mean profile U = ± Ue_x. Above y = 0, the flow has a density ρ⁺ and ρ⁻ for y < 0 (see Fig. A.1). We neglect the Coriolis force since the local shear timescale τ_S = Δx/ ΔU ~ 10^-6   yr is much shorter than the orbital period τ_Ω ~ 10^-1   yr. In this approximation, the linearised equation of motion is: In the following, each quantity is Fourier transformed in x and t thanks to homogeneity: Q = Qexp [i(ωt − kx)]. Rewriting the equation of motions and combining them leads to (A.3)which is solved with two decaying solutions A⁺ and A⁻ being two arbitrarily chosen constants that are adjusted by jump conditions at the interface y = 0: pressure should be continuous and fluid particles should stick to the interface on both sides. The pressure condition is given by: (A.6)where σ ^±  = ω ± U. The second condition is obtained defining a displacement vector ξ(x) that follows the interface. By definition, a fluid particle located at (x,ξ(x) − ϵ) satisfies (A.7)Applying this to both side of interface (± ϵ) leads to the jump condition (A.8)Combining (A.6) and (A.8) and looking for non trivial solutions gives (A.9)where α = (ρ⁺ − ρ⁻)/ (ρ⁺ + ρ⁻). An instability arises whenever (A.10)which is always true since −1 ≤ α ≤ 1. The growth rate is . Hence, a density contrast |α| close to 1 strongly dampens the growth rate of the KHI.

In colliding wind binaries, the density and the velocity of both winds are related through the momentum-flux ratio η. Using Eq. (1) and mass conservation for both winds then and assuming the interaction occurs far enough from the binary (so that r₁ ≃ r₂), the density ratio is roughly

A.2. Nonlinear evolution

A.2.1. Two-dimensional evolution

	Fig. A.2 Snapshot of the mixing at t = 21 (in dimensionless units) for α = 0.5.
Open with DEXTER

To investigate the evolution of the KHI in the nonlinear regime, we performed numerical simulations for increasing α. The 2D setup is as follows: box size (lx = 8,ly = 4), resolution (1024 × 256), code PLUTO (Mignone et al. 2007), adiabatic equation of state P ∝ ρ^5/ 3, background pressure P = 1 in the initial state (using units scaled to the box length, density, and velocity shear). Reflective boundary conditions are enforced in y to confine the instability in the simulation box. We always have ρ⁺ > ρ⁻ i.e. the densest medium is found where y > 0.

In addition to that, we follow the mixing using a passive scalar as explained in Sect. 2.3. We performed simulations for  {α = 0,0.5,0.9,0.99} . Kelvin-Helmholtz eddies are clearly present in the density snapshot shown in Fig. A.2 for model α = 0.5. To show the diffusion of the passive scalar as a function of time, we plot the evolution of as a function of y and t in Fig. A.3. These results demonstrate that when α ≠ 0, the scalar diffusion propagates much less in the denser medium (y > 0) and that diffusion looks less efficient when |α| increases, in the sense that the region with intermediate values of the scalar s becomes smaller when α increases.

	Fig. A.3 Mixing due to the KHI. From left to right, top to bottom: α = 0,0.5,0.9,0.99.
Open with DEXTER

A.3. Three-dimensional evolution

We performed simulations for α = 0 and 0.9 in 3D to compared them to the 2D ones. They are very similar to the 2D configuration, except for the resolution, which was reduced to 500 × 100 × 100 in order to reduce computational costs. We set l_z = l_x = 4.0, where is shown in Fig. A.4. The direct comparison with the 2D cases indicates that faster diffusion into the more tenuous region still occurs in the 3D simulation.

Figure A.5 shows the mixing at different times in the 2D and 3D simulations, for α = 0.,0.9. It confirms that both in the 2D and 3D case, the KHI behaves similarly with respect to the velocity gradient.

	Fig. A.4 Mixing in the 3D simulations with α = 0 (left) and α = 0.9 (right).
Open with DEXTER

	Fig. A.5 Mixing in 2D simulations (blue solid line) and 3D simulations (red dashed line) for α = 0 (left panel) and α = 0.9 (right panel). The different curves show different timesteps separated by Δt = 5.
Open with DEXTER

Appendix B: Parameters of the simulations

© ESO, 2012