3. Stability analysis
A global normal mode stability analysis is performed. The radial displacement, , of the fluid elements from their equilibrium positions are of the form, , and similarly for the remaining perturbed quantities. The linearized equations can be cast into a system of two first order ODE for and the total perturbed pressure ,
with functions of the equilibrium quantities and the Fourier parameters (Appl & Camenzind 1992).
Jets propagate through a compressible medium of high conductivity. Since one is interested only in instabilities due to the jet itself, radially outgoing decaying waves for large radii define the boundary condition (BC),
The are the Hankel functions of the first kind and , with the sound speed. The ambient medium has been assumed to be unmagnetized. The perturbed jet is assumed to remain in equilibrium with its surroundings. Regularity on the axis provides the other boundary condition,
The normal mode equations (7) and (8) describe all types of instability that may develop in a cylindrical magnetized outflow, and in the general case it is not possible to clearly separate different types of instabilities. In a setup as simple as described in the previous section, we can attribute the KHI to the vortex sheet at the jet surface. The most dangerous instability is the so-called surface mode, which displaces the entire flow. The body modes give rise to oblique shocks within the jet in the nonlinear regime. The KHI are particularly sensitive to the fast magnetosonic Mach number, and are moderately affected by the magnetic structure. Their growth for the long wavelength instabilities differs at most by a factor of two in the supermagnetosonic regime (Appl & Camenzind 1992). Experience from CTR shows that the behaviour of the CDI strongly depends on the choice of the magnetic profile, and having chosen a constant velocity throughout the jet, we assume that the CDI develops essentially independently from the KHI. CDI of static equilibria are absolute instabilities, i.e. they grow but do not propagate. We further suppose that this is the case also in the rest frame of the jet, and the CDI are therefore simply advected with the flow at the jet velocity. Consequently, no signal can communicate the interior of a supermagnetosonic jet with its surroundings, and we expect that the instabilities behave as if the jet were bounded by a rigid conducting wall, as long as only CDI are considered. This will be verified numerically in Sect. 5.4. For most calculations of CDI we therefore employ conducting wall boundary conditions,
Previous calculations for the BFM with a particular value of indicate that this is indeed appropriate for CDI provided that the jet velocity is supermagnetosonic (Appl 1996a). We will demonstrate a posteriori in more detail that this choice of boundary condition is justified also for the configurations under consideration. We will also demonstrate that it does not matter whether the return current flows on the jet surface or as a more diffuse current in the ambient medium. Fusion plasmas, on the other hand, are surrounded by a vacuum, with high resistivity and no inertia. These properties are mainly responsible for the destructive behaviour of external modes in CTR. This is one reason that results from fusion research cannot be applied without modification to the case of astrophysical jets. For internal modes that affect mainly an inner plasma region, the boundary conditions are of minor importance, and these instabilities can be supposed to behave in a similar way in jets and fusion plasmas.
A temporal approach is taken: the axial wavenumber k is chosen as real and the differential equations (7,8) formulate an eigenvalue problem for the complex frequency . The imaginary part, , is the growth rate of a mode specified by . Without loss of generality only the case is considered in the following. The temporal approach is appropriate for magnetically confined laboratory plasmas and allows an immediate comparison with those results. When calculating eigenvalues, , and eigenvectors, , in the comoving frame using fixed boundary conditions (11), the unstable modes have purely imaginary eigenvalues, . In Sect. 5.4 the calculations are performed in a reference frame which is at rest w.r.t. the ambient medium, and radiation boundary conditions (9) are applied. The phase velocity of the instability is then given by .
© European Southern Observatory (ESO) 2000
Online publication: March 9, 2000