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Astron. Astrophys. 353, 108-116 (2000)

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Appendix A

We consider streams of neutral and ionized gas components that flow in parallel directions on either side of the KH interface with different bulk velocities and mass densities. In the unperturbed state the interface, say [FORMULA], is horizontal. Let be [FORMULA], [FORMULA], [FORMULA], and [FORMULA] for [FORMULA] and [FORMULA], [FORMULA], [FORMULA], and [FORMULA] for [FORMULA], where [FORMULA] and [FORMULA] denote the mass density and the bulk velocity. The index n is used to indicate the neutral gas quantities. We assume that in the basic state there is a uniform, horizontal magnetic field parallel to the flow (which is the least favorable situation for the KH instability to set in) [FORMULA]. Moreover, for the analytical calculations we assume incompressibility and that the convective term in the induction equations dominates the small resistive one, for mathematical convenience. The basic set of the linearized equations is given by

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where p, B, and [FORMULA] denote the thermal pressure, the magnetic field, and the effective plasma-neutral gas elastic collision frequency, respectively. The index 1 indicates the perturbed quantities. These are assumed to be of the form [FORMULA] with the complex frequency [FORMULA] and the wave numbers of the modes [FORMULA] and [FORMULA]. With this form of perturbations we find six algebraic equations to be solved for the x- and y-components of the perturbed flows and the perturbed pressures. For the x-component of Eq. (A5), multiplied by [FORMULA] where [FORMULA] is the Doppler shifted frequency, and the y-component of Eq. (A5) multiplied by [FORMULA] we obtain

[EQUATION]

[EQUATION]

where we made use of Eq. (A7) and here and in the following we omitted the hat indicating the amplitudes of the perturbed quantities. With the help of Eqs. (A1) and (A2) we find for the z-component of Eq. (A5) multiplied by [FORMULA]

[EQUATION]

The Cartesian components of the neutral gas momentum balance equation (A6) give accordingly

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] denotes the Doppler shifted frequency [FORMULA]. We concentrate on surface waves that decay exponentially with increasing distance from the boundary layer [FORMULA] and [FORMULA] where the minus signs holds for [FORMULA] and the plus sign for [FORMULA]. We assume that the boundary layer is not a discontinuity but that the physical quantities vary continuously across the small finite width of the interface. It is reasonable to assume that the tangential velocity perturbations, the pressure perturbations as well as [FORMULA] and [FORMULA] are odd functions of y. Denoting the `jumps' by brackets we find

[EQUATION]

where the bar denotes the mean values across the interface [FORMULA], and

[EQUATION]

Eqs. (A14) and (A15) hold accordingly for the neutral gas quantities. After integrating Eqs. (A9) and (A12) over y non-trivial solutions for the homogeneous equations for [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA] require the coefficient determinate to be zero, i.e.

[EQUATION]

Eq. (A16) represent the generalized dispersion relation for the KH modes in partially ionized plasmas. For [FORMULA] is reduces to the well-known dispersion relation for fully ionized plasmas (see e.g. Woods 1987)

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999
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