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

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5. Numerical results

Growth 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 field

The constant pitch field, [FORMULA], is a simple analytic equilibrium with only one free parameter, [FORMULA]. It will turn out that it provides a good model for the instabilities of linear pinches in the regime, [FORMULA], applicable to most MHD jets. The constant pitch field is unstable for all values of [FORMULA]. The resonant condition Eq. (12) only holds for a particular wavenumber, [FORMULA], which corresponds to the threshold of marginal stability.

We first study the case [FORMULA] in some detail before discussing the dependence on the pitch. The growth rate for various azimuthal wavenumbers m is shown in Fig. 1 as a function of the axial wavenumber kR. The instability sets in at the wavenumber where the resonant condition (12) is satisfied. At all other unstable wavelengths the mode is non-resonant. The maximum growth rate (at [FORMULA] for the [FORMULA] mode with [FORMULA]) decreases for increasing azimuthal wavenumber. The non-axisymmetric [FORMULA] kink mode is the most rapidly growing instability, while the [FORMULA] mode is stable in this case. For the rest of the paper we will restrict ourselves to the [FORMULA] modes. The solid line in Fig. 2 displays the eigenvector, i.e. the radial displacement [FORMULA] of the fluid elements and the radial component of the perturbed magnetic field, [FORMULA], for the most unstable case.

[FIGURE] Fig. 1. Constant pitch equilibrium with [FORMULA]. Growth rate for [FORMULA] (from the left) current-driven instabilities. The instability sets in from the short wavelength side at the resonant condition [FORMULA].

[FIGURE] Fig. 2. Eigenfunction of the most unstable kink ([FORMULA]) mode for the constant pitch profile with [FORMULA]. Radial displacement [FORMULA] (in arbitrary units) as function of the normalized jet radius, for a static equilibrium ([FORMULA]) with fixed boundary condition (solid) and radiation boundary condition (dotted), and for a moving equilibrium ([FORMULA]) with radiation boundary condition (dashed). Also shown are the cases where no surface current flows on the jet boundary, for [FORMULA] (long-dashed) and [FORMULA] (dash-dotted), using fixed boundary condition at [FORMULA]. In addition, [FORMULA] is shown for the static equilibrium with fixed boundary condition. The corresponding growth rates and wavenumbers are shown in Table 1.

In Fig. 3 the maximum growth rate of the first non-axisymmetric ([FORMULA]) mode is displayed as a function of the inverse pitch, [FORMULA]. Two different regimes can be identified with a transition at [FORMULA] and [FORMULA]. 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 [FORMULA] strongly increases with decreasing pitch, [FORMULA]. For [FORMULA] 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, [FORMULA], 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 [FORMULA], and the growth rate scales with the Alfvén crossing time for the current-carrying core, [FORMULA].

[FIGURE] Fig. 3. Maximum growth rate [FORMULA] (solid) and most unstable wavenumber [FORMULA] (short-dashed) of the [FORMULA] kink for an equilibrium with constant pitch as a function of [FORMULA] with fixed BC. The hatched area covers the range of unstable wavenumbers.

The wavenumber, [FORMULA], where the kink instability sets in, is given by the resonant condition (12), [FORMULA]. For small pitch the wavenumber of the most unstable mode is [FORMULA], or equivalently, the wavelength [FORMULA] (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, [FORMULA], remains similar to that of the example with [FORMULA] (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 [FORMULA], or expressed as a wavelength, [FORMULA], for the kink. Hence, in this regime a single mode is destabilized.

5.2. The variable pitch field

The 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 [FORMULA], and therefore of magnetized jets. We will restrict ourselves to the case of the [FORMULA] 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 k, which is determined by the pitch ratio between the boundary and the axis, [FORMULA]. All modes having a wavenumber k with [FORMULA] are resonant, and for every radius there exists a wavenumber for which the corresponding magnetic surface is resonant. We consider [FORMULA] kink instabilities of equilibria with varying pitch, Eq. (5), where [FORMULA]. Fig. 4 shows the radial displacement [FORMULA] of the most unstable [FORMULA] kink instability for an equilibrium with a parabolic pitch profile ([FORMULA]) with the same central pitch value, [FORMULA], as before. Comparison with Fig. 2 shows that though the character of the mode has changed from nonresonant to resonant, the radial displacement is only little affected. The distinction plays an important role for the nonlinear evolution of the instability, however (Lery et al. 2000). As in the constant pitch case the instability sets in at [FORMULA], but now the unstable modes (long-ward) close to the marginal stability threshold become resonant (see long-dashed line in Fig. 6). This implies that the resonant modes cover the interval [FORMULA], including the most unstable mode at [FORMULA]. Calculations with an exponent [FORMULA] yield very similar results. Fig. 5 shows the maximum growth rate as a function of the central pitch value for several equilibria with quadratically increasing pitch. The configurations with strongly increasing pitch profile ([FORMULA] and [FORMULA]) have smaller growth rates for the same central pitch, i.e. a high ratio [FORMULA], or equivalently, a high shear [FORMULA] reduces the growth rate. For small pitch, the maximum growth rate approaches closely the values of the constant pitch field, independently of the equilibrium pitch profile and regardless the different nature of the mode. The same relation holds for the most unstable wavenumber as in the constant pitch case, [FORMULA]. (This is not displayed in Fig. 5, but see in Fig. 6 that even for [FORMULA] the maxima for the constant and increasing pitch with [FORMULA] are close to each other.) For small values of the central pitch the equilibrium (1) yields an electrical current which is concentrated near the axis (unless very extreme pitch variations are considered), where the pitch becomes constant to first order since [FORMULA]. Consequently, the instability properties depend essentially on [FORMULA] and are not very sensitive to the profile [FORMULA] in this case. The most unstable mode is resonant when it falls into the k-interval defined by the resonant condition, i.e. if [FORMULA], and non-resonant otherwise.

[FIGURE] Fig. 4. Eigenfunction of the most unstable kink ([FORMULA]) mode for increasing pitch with [FORMULA], [FORMULA], [FORMULA]. Radial displacement [FORMULA] (in arbitrary units) as function of normalized jet radius [FORMULA], for static equilibrium ([FORMULA]) with fixed boundary condition (solid) and radiation boundary condition (dotted) and for moving equilibrium ([FORMULA]) with radiation boundary condition (dashed). In addition, [FORMULA] is shown for the static equilibrium with fixed boundary condition. See Table 1 for growth rates and wavenumbers.

[FIGURE] Fig. 5. Maximum growth rate [FORMULA] of the [FORMULA] kink as a function of the central pitch [FORMULA] for a parabolically increasing ([FORMULA]) pitch profile with [FORMULA] (short-dashed), [FORMULA] (long-dashed), [FORMULA] (dash-dotted), and the constant pitch (solid) for comparison.

[FIGURE] Fig. 6. Growth rate of the [FORMULA] mode as a function of the wavenumber for an equilibrium with (a) constant pitch (solid), (b) an increasing pitch ([FORMULA], [FORMULA]; long dashed), and (c) a decreasing pitch ([FORMULA], [FORMULA]; short dashed) profile. The central pitch is [FORMULA] in all three cases. Resonant modes are those with (a) [FORMULA], (b) [FORMULA], and (c) [FORMULA].

The decreasing pitch profiles studied are of the form of Eq. (5), with [FORMULA]. The interval for resonant modes is given by [FORMULA]. In the same line of reasoning as above, the resonant modes are now located to the right, i.e. at wavenumbers larger than [FORMULA]. Consequently, instead of the sharp cutoff and stabilization at [FORMULA], 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 [FORMULA] 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 [FORMULA]. 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.

[FIGURE] Fig. 7. Maximum growth rate [FORMULA] of the [FORMULA] kink as a function of the central pitch [FORMULA] for a parabolically decreasing ([FORMULA]) pitch profile for [FORMULA] (short-dashed), [FORMULA] (long-dashed), and the [FORMULA] mode for a linear force-free field (dash-dotted). The [FORMULA] constant pitch (solid) case is displayed for comparison.

5.3. The linear force-free field

For 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 [FORMULA] (Appl 1996a), which has been motivated mainly by the requirement of having a net zero current. The growth rate for the most unstable ([FORMULA]) mode, which was resonant in this case, was comparatively low with [FORMULA] at [FORMULA]. We now abandon this restriction and calculate growth rates for a range in [FORMULA], corresponding to a central pitch value of [FORMULA]. 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 [FORMULA], and consequently on [FORMULA]. 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 [FORMULA].

5.4. The internal character of the CDI

Growth 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 [FORMULA] 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 [FORMULA] 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 [FORMULA], [FORMULA], and [FORMULA], and a constant pitch configuration with [FORMULA] 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 [FORMULA] and radiation boundary conditions both at [FORMULA] and [FORMULA]. For the latter case calculations were performed with [FORMULA], which all produce the same results. We therefore only present those corresponding to [FORMULA]. Table 1 shows that their phase velocity in a reference frame where the ambient medium is at rest, [FORMULA], is equal to the jet velocity, as indicated by the Mach number, [FORMULA]. CDI are therefore absolute instabilities in the rest frame of the jet. We further note that the most unstable wavenumber, [FORMULA], as well as the maximum growth rate, [FORMULA], 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).


Table 1. Wavenumber [FORMULA], growth rate [FORMULA], and phase velocity [FORMULA] of the instability with largest growth rate for equilibria EQ with central pitch [FORMULA] and (a) increasing profile ([FORMULA] and [FORMULA]) and (b) for constant pitch [FORMULA]. Radiation (rad) and fixed (fix) boundary conditions at radius r have been applied for various fast Alfvén Mach numbers [FORMULA]. In the last two cases the magnetic field extends to [FORMULA], and the jet surface does not carry a current sheet. The jet velocity is zero for [FORMULA] in all cases. The last column refers to the figure displaying the eigenfunction.

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, [FORMULA] 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.

[FIGURE] Fig. 8. Growth rate for the [FORMULA] mode for the constant pitch profile with [FORMULA] for various Mach numbers (indicated by the labels) and radiation boundary condition (dotted) and for a static equilibrium ([FORMULA]) with fixed boundary condition (solid).

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 [FORMULA], while the jet, defined by the plasma motion, is restricted to [FORMULA]. The results remain unaffected (see Table 1 and Fig. 2), and come even closer to the case with fixed BC at [FORMULA]. For [FORMULA] the eigenfunction possesses a kink at the jet boundary at [FORMULA], and [FORMULA] remains very small at [FORMULA] (not visible in Fig. 2).

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Online publication: March 9, 2000