4. Derivation and solution of the dispersion relation
In deriving the dispersion relation for the resistive tearing instability we concentrate on two-dimensional perturbations and assume incompressibility. We apply a linear mode perturbation analysis starting from a one-dimensional equilibrium configuration (cf. preceding section) characterized by and where and are constant. The magnetic field as well as the dust velocity can be expressed in terms of the magnetic vector potential A and the velocity stream function U:
In appropriate normalized units (the velocity is normalized to the dust Alfvén velocity , the growth rate to the inverse dust Alfvén transit time , where l is the half-thickness of the equilibrium current layer; S denotes the normalized inverse diffusivity, i.e. the magnetic Reynolds number, ) the governing linearized equations (cf. Otto 1991) are the z -component of the curl of Eq. (2):
which decouple from all the other equations (the indices 0 and 1 denote equilibrium and perturbed quantities). In deriving Eq. (9) we have assumed . Eqs. (9) and (10) determine an eigenvalue problem or, to be more specific, a boundary layer problem. They have to be expanded in an appropriate scaling. The asymptotic matching of the solutions yields the dispersion relation.
where q denotes the complex growth rate and k the wave number of the mode.
Following the procedure given by Otto (1991) we obtain the dispersion relation (see Appendix for some technical details) similar to the solution given by Otto and Birk (1992) for partially ionized plasmas without any dust component
which is valid for , and in normalized units where is the width of the inner layer of the boundary problem, i.e. the region where the resistive term in Eq. (10) cannot be neglected. The growth rates of the unstable modes should be reduced with comparison to the tearing instability in totally ionized plasmas (Furth et al. 1963) due to dust-neutral friction. If the ion-neutral collision frequency is comparable to the dust-neutral one, the growth rates are expected to be even more reduced. The z -component of the magnetic field , i.e. the component parallel to the current sheet, does not influence the dispersion relation.
We have to solve numerically (by means of an iterative damped Newton method) the dispersion relation (12) for typical parameter sets relevant in the YSO context. One should note that these physical parameters can be quite variable and depend on the distance from the stellar object as well as wind and accretion properties. In order to get a feeling for the quantitative results of tearing instabilities operating in YSO magnetospheres we choose the following set of parameters which seems to be reasonable in the T-Tauri context (cf. Königl 1994; Paatz and Camenzind 1996): The neutral gas particle density as well as the dust particle density are chosen as and the ion particle density as . For the magnitude of the magnetic field we choose a relatively moderate value . Note that is the shear component of the magnetic field and not the main component of the stellar dipole field. Thus, the dust Alfvén velocity reads , if we assume, for example, that the dust grains are heavier than the ions by a factor of . Additionally, we assume the dust charge number as (the actual value is not of importance for the macroscopic dynamics under consideration).
With these parameters collisions with dust grains are more frequent than electron-ion Coulomb collisions and electron/ion-neutral collisions. The dust collision frequencies can be calculated from the appropriate Landau collision integrals (e.g. Benkadda et al. 1996). The dependence of the ion-dust collision frequency, which determines the collisional resistivity in our case, on dust charge number (the neutral gas temperature was chosen as ) and dust particle density is illustrated in Fig. 3, whereas Fig. 4 shows the functional dependence of the dust-neutral collision frequency on neutral gas particle density and temperature.
The ion-dust collisions dominate the electrical resistivity which is of the order of for the above parameters. This implies a magnetic Reynolds number of for a half-thickness of the current sheet of the order of . The Alfvénic transit time in this case reads .
The dispersion relation for the tearing mode in the considered parameter regime is solved for a variety of magnetic Reynolds numbers of , , and (Fig. 5).
A normalized growth rate of (for ) implies a growth time of the unstable tearing mode of approximately 4 hours. Higher growth rates (due to lower magnetic Reynolds numbers as a result of ion-dust collisions) lead to an even faster development of the mode. Moreover, it should be noted that for the above study a relatively small magnetic field has been chosen. A magnetic field of would instead result in a time scale of the order of which is comparable to very fast flares (e.g. Feigelson and DeCampli 1981). In principle, varying plasma parameters, e.g. due to some variation in the accretion rate, may explain temporal variety of magnetic activity phenomena from hours up to intraday variability on the grounds of tearing theory. The normalized wave numbers of the most unstable modes (cf. Fig. 5) correspond to a wave length of . This implies that the spatial extent of the emitting flux tube is of the order of a few for the parameters chosen which fits very well to VLBI radiosizes of the emitting regions around (weak) T-Tauri stars (cf. André 1996).
© European Southern Observatory (ESO) 1998
Online publication: January 27, 1998