Astron. Astrophys. 355, 818-828 (2000)

6. Discussion

Global current-driven instabilities of magnetized jets are studied in the linear regime. For this purpose a normal mode analysis has been performed for force-free screw pinches. The stability of such equilibria is conveniently discussed in terms of the magnetic pitch, . Various magnetic pitch profiles have been probed, in particular equilibria with constant, and both radially increasing and decreasing pitch, and allow us to draw conclusions about MHD jets in general.

The stability properties can be discussed in terms of a single parameter, the pitch value on the axis, . There exist two clearly separated regimes, with a transition depending on the profile. The transition occurs at for equilibria with moderate shear. The high-pitch regime, , where the majority of fusion machines operate, has been extensively studied in the literature. It is characterized by a strong longitudinal field, and the stability properties are highly sensitive to the particular pitch profile. The constant pitch model exhibits CDI with the largest growth rate among the equilibria studied. The configuration with increasing pitch, i.e. a TOKAMAK-like pinch, shows, for small and moderate shear, a behaviour very similar to the constant pitch case, while equilibria with a decreasing pitch become stabilized at sufficiently large central pitch. Most magnetized jets are likely to possess a dominantly azimuthal field, and consequently a small pitch, . With decreasing the current becomes more and more concentrated on the axis, within a radius of the order , and the growth time and the wavelengths of the instabilities become correspondingly shorter, scaling linearly with as and , respectively. The results of Figs. 3, 5, and 7 indicate that, in this regime, both the growth rate and the wavenumber where the instability grows fastest, , are insensitive to the particular pitch profile. Therefore, the constant pitch case yields very good estimates for the growth and the wavelength of current-driven instabilities in magnetized jets, and it provides a robust and simple model for the study the linear evolution of the CDI in the regime . The growth rate becomes inversely proportional to the pitch, , while in the large pitch regime it is much more sensitive, e.g. for the constant pitch equilibrium the growth rate depends on the third power of the pitch, . For comparison we presented results for a linear force-free field. For central pitch the cylindrically symmetric linear force-free becomes unstable w.r.t. non-axisymmetric perturbations. Being a minimum energy configuration, it is not astonishing that this kind of equilibrium possesses particularly favourable stability properties, with growth rates that are nearly an order of magnitude smaller than for the other equilibria. The wavelength of the most unstable mode is larger, too, than for the other equilibria studied. The constant-pitch and the linear force-free field represent extreme situations, corresponding to the most and least unstable configurations, respectively.

For small pitch the stability properties become essentially independent of the details of the pitch profile. While the nature of the instability (resonant or not) does not affect the growth rate and the axial wavelength of the most unstable mode, an important difference consists in the radial component of the perturbed magnetic field. The latter can vanish at some resonant radius only for an increasing pitch profile. Since it is well known from the non-linear evolution of the internal kink mode in TOKAMAKs that a current sheet forms at the resonant radius, we have studied its non-linear evolution in the small pitch regime by means of a time-dependent 3D MHD code (Lery et al. 2000).

In general, both current-driven and Kelvin-Helmholtz instabilities are present in the kind of jet models considered, and in the general case it is not possible to classify them unambiguously. To this end we have studied jet models which are particularly simple, and which are characterized by a constant density and velocity. In this approximation the CDI and the KHI can be identified for supermagnetosonic jets by comparing them to the static and the current-free counterparts, respectively. A velocity shear would lead to a more complicated coupling between the KHI and the CDI. Instabilities with wavelengths which are large compared to the radial scale of the shear behave essentially as in the case of a vortex sheet. Furthermore, in configurations with small pitch the electric current, and consequently also the instability is concentrated in the vicinity of the axis. In this case a velocity shear which is likely to be largest near the jet boundary does not affect the instability very much either. Current-driven instabilities can then be studied in the rest frame of the jet plasma, and the motion only enters through appropriately chosen boundary condition.

The normal modes of static MHD plasmas can be classified according to their properties as (slow and fast) magnetoacoustic and Alfvén modes. These are "wave-like" disturbances for which a super-fast-magnetosonic shear layer essentially becomes impermeable. The situation is different for the Kelvin-Helmholtz surface modes. Here it is the (supersonic) velocity itself which by virtue of its inertia deforms the boundary of the jet. We argue that for supermagnetosonic flows CDI develop essentially as if the jet were surrounded by a rigid wall. We examined the properties of current-driven instabilities of jets with different Mach numbers using radiation boundary conditions which are most appropriate for this purpose. We find that in fact the CDI is an instability in the rest frame of the jet, and for high Mach numbers the dispersion relation and the eigenfunctions become indistinguishable from the static case, i.e. the jet boundary is not radially displaced in the linear regime. At low Mach number the dispersion relation shows a more complicated behaviour, due to the interaction with the velocity shear. In this regime it becomes difficult to disentangle the different types of instabilities. The maximum growth rate and the most unstable wavenumber, however, remain nearly unaltered, even for vanishing velocity. This was analyzed for the constant pitch field and a configuration with radially increasing pitch, i.e. for a non-resonant and a resonant mode, respectively (Table 1). It confirms that CDI in jets are instabilities which grow in the restframe of the jet, and that it is justified to study them using fixed boundary conditions, as was done in most of the numerical calculations.

The jet configurations analyzed generally possess a net electrical current. The return current either flows back on the surface of the jet, or as a more diffuse current distributed over a larger region in the ambient medium. A comparison showed that the results are not much affected by this choice. The perturbation, , in both cases essentially remains confined to the jet interior, and the ambient medium has very little influence on the CDI. Fixed BC or radiation BC in combination with an unmagnetized ambient medium at imply that the jet is confined by ambient pressure. An azimuthal field in the ambient medium, as it has been assumed in another calculation, corresponds to a magnetically confined jet. If we suppose that these results hold for the general case, this indicates that the growth of current-driven instabilities in jets is independent of the confinement mechanism.

© European Southern Observatory (ESO) 2000

Online publication: March 9, 2000
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