## 5. Numerical resultsGrowth rates, eigenfunctions, dispersion relations, etc. are presented for the various linear pinches in the rest frame of the jet using boundary conditions which correspond to a conducting wall, Eq. (11). In the last subsection we a posteriori justify this choice by a comparison with calculations in the rest frame of the ambient medium using radiation boundary conditions. ## 5.1. The constant pitch fieldThe constant pitch field, , is a simple analytic equilibrium with only one free parameter, . It will turn out that it provides a good model for the instabilities of linear pinches in the regime, , applicable to most MHD jets. The constant pitch field is unstable for all values of . The resonant condition Eq. (12) only holds for a particular wavenumber, , which corresponds to the threshold of marginal stability. We first study the case in some
detail before discussing the dependence on the pitch. The growth rate
for various azimuthal wavenumbers
In Fig. 3 the maximum growth rate of the first non-axisymmetric () mode is displayed as a function of the inverse pitch, . Two different regimes can be identified with a transition at and . The large pitch is represented on the left of the diagram, where the magnetic field is dominantly longitudinal. This is the regime where most fusion devices such as the TOKAMAK operate. Growth rates in this limit have been calculated by Goedbloed & Hagebeuk (1972). In this regime the growth rate strongly increases with decreasing pitch, . For a diffuse electrical current is distributed over the entire jet diameter, and the growth of the instability becomes sensitive to the detailed pitch profile. For a dominantly azimuthal field, , as it is likely to be the case for many jets, the growth rate increases only linearly with decreasing pitch. For small values of the pitch the electric current flows only within a radius of the order , and the growth rate scales with the Alfvén crossing time for the current-carrying core, .
The wavenumber, , where the kink instability sets in, is given by the resonant condition (12), . For small pitch the wavenumber of the most unstable mode is , or equivalently, the wavelength (dashed line in Fig. 3), which is of the order of the circumference of the current-carrying core. The instability affects mainly the innermost region of the plasma column. The shape of the dispersion relation, , remains similar to that of the example with (Figs. 1 and 6), where the growth rates and wavenumbers change according to Fig. 3. The interval of unstable wavenumbers (indicated by the hatched area) becomes larger for decreasing pitch, destabilizing modes over a wide range of wavelengths. For large pitch equilibria, on the other hand, this interval becomes very narrow, and , or expressed as a wavelength, , for the kink. Hence, in this regime a single mode is destabilized. ## 5.2. The variable pitch fieldThe constant pitch field may be considered as a special case, since the unstable modes do not contain resonant surfaces within the plasma. We therefore extend our studies to equilibria with radially varying pitch, considering both monotonically increasing and decreasing pitch profiles. The results of this subsection may give an indication whether the behaviour found from the constant pitch fields is representative of the regime , and therefore of magnetized jets. We will restrict ourselves to the case of the kink mode, since it is the most unstable. Unlike the constant pitch case the resonance condition (12) for
increasing pitch is satisfied for an interval in
The decreasing pitch profiles studied are of the form of Eq. (5), with . The interval for resonant modes is given by . In the same line of reasoning as above, the resonant modes are now located to the right, i.e. at wavenumbers larger than . Consequently, instead of the sharp cutoff and stabilization at , as it is the case for the constant pitch and the monotonically increasing equilibrium, the range of instability is extended towards shorter wavelengths (Fig. 6, short-dashed line). The unstable modes with are resonant, their growth rate, however, is small. Otherwise the dispersion relation is very similar to the previous cases. Fig. 7 shows the maximum growth rate for equilibria with monotonically decreasing pitch as a function of the central value . As it is well known from fusion research, for large pitch the stability behaviour is very sensitive to the profile chosen, and a stabilization is observed for sufficiently large pitch on the axis, corresponding to condition (13). In the small pitch regime, however, the results are essentially indistinguishable from the constant pitch case.
## 5.3. The linear force-free fieldFor comparison we consider now the linear force-free field. While the constant-pitch field provided an upper limit for the growth rate, we expect the linear force-free field to represent the lower limit, being a configuration of minimum energy. The CDI for such an equilibrium has been studied for a particular value of (Appl 1996a), which has been motivated mainly by the requirement of having a net zero current. The growth rate for the most unstable () mode, which was resonant in this case, was comparatively low with at . We now abandon this restriction and calculate growth rates for a range in , corresponding to a central pitch value of . It should be noted, however, that unlike the previous equilibria, the BFM is not characterized by a pitch profile with fixed radial dependence, where simply the central value is modified. The profile possesses a varying number of zeroes and poles, depending on the value of , and consequently on . The dash-dotted line in Fig. 7 shows that indeed the linear force-free field possesses growth rates nearly an order of magnitude smaller, for the same central pitch value. For numerical reasons we only give results for . ## 5.4. The internal character of the CDIGrowth rates and wavenumbers presented so far have been calculated using fixed boundary conditions. In this subsection we verify numerically whether these boundary conditions are appropriate for current-driven instabilities in jets. Studying the eigenvalue problem Eqs. (7) and (8) with the (correct) radiation boundary conditions we demonstrate that for superalfvénic jet velocities, the current-driven instabilities behave as absolute instabilities in the rest frame of a jet surrounded by a rigid conducting wall. For the particular example of the linear force-free field it has been shown previously (Appl 1996a) that were practically identical for the jet and the static configuration bounded by a conducting wall, in particular for supermagnetosonic velocities. For a resonant mode the instability affects mainly the region interior to the surface. This makes it plausible that the boundary has only little influence on the mode. One may suspect, however, that non-resonant instabilities are much more affected by the choice of boundary conditions. We therefore chose representatives of resonant and non-resonant modes. As examples we take an equilibrium with increasing pitch, with , , and , and a constant pitch configuration with and compare the properties of the mode with the largest growth rate for various Mach numbers and boundary conditions (Table 1). The former is a resonant mode, while in the constant pitch case the mode is non-resonant. We applied rigid wall boundary conditions at and radiation boundary conditions both at and . For the latter case calculations were performed with , which all produce the same results. We therefore only present those corresponding to . Table 1 shows that their phase velocity in a reference frame where the ambient medium is at rest, , is equal to the jet velocity, as indicated by the Mach number, . CDI are therefore absolute instabilities in the rest frame of the jet. We further note that the most unstable wavenumber, , as well as the maximum growth rate, , are only little affected by the jet velocity, and for high Mach number they become indistinguishable from a jet with rigid boundary. The same is true for the eigenfunctions (Figs. 2 and 4).
A comparison over a much wider range in Mach numbers and wavenumbers, shown in Fig. 8 for the constant pitch case, confirms that in fact the most unstable mode is unaffected by the choice of boundary conditions, even at relatively low Mach number. With radiation BC the unstable wavenumbers extend to larger wavelengths, and for Mach numbers, the CDI is modified by the velocity gradient (not displayed). We take the results of this subsection as justification that current-driven instabilities in jets are correctly modeled by boundary conditions describing a perfectly conducting rigid wall.
Both the conducting wall BC and the radiation BC used so far correspond to a return current that flows on the surface of the jet. The jet is therefore confined by external pressure. In order to study whether the results are sensitive to the surface current sheet, we compare them to a constant pitch equilibrium where the magnetic field extends to , while the jet, defined by the plasma motion, is restricted to . The results remain unaffected (see Table 1 and Fig. 2), and come even closer to the case with fixed BC at . For the eigenfunction possesses a kink at the jet boundary at , and remains very small at (not visible in Fig. 2). © European Southern Observatory (ESO) 2000 Online publication: March 9, 2000 |