A&A 525, A100 (2011) Astronomy DOI: 10.1051/0004-6361/200913836 & c ESO 2010 Astrophysics The magnetohydrodynamic instability of current-carrying jets A. Bonanno1,2 and V. Urpin1,3 1 INAF, Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123 Catania, Italy e-mail: [email protected] 2 INFN, Sezione di Catania, via S. Sofia 72, 95123 Catania, Italy 3 A. F. Ioffe Institute of Physics and Technology and Isaac Newton Institute of Chile, Branch in St. Petersburg, 194021 St. Petersburg, Russia Received 9 December 2009 / Accepted 3 October 2010 ABSTRACT Context. Magnetohydrodynamic instabilities can be responsible for the formation of structures with various scales in astrophysical jets. Aims. We consider the stability properties of jets containing both the azimuthal and axial field of subthermal strength. A magnetic field with complex topology in jets is suggested by theoretical models and is consistent with recent observations. Methods. Stability is discussed by means of a linear analysis of the ideal magnetohydrodynamic equations. Results. We argue that in azimuthal and axial magnetic fields the jet is always unstable to non-axisymmetric perturbations. Stabilization does not occur even if the strengths of these field components are comparable. If the axial field is weaker than the azimuthal one, instability occurs for perturbations with any azimuthal wave number m, and the growth rate reaches a saturation value for low values of m. If the axial field is stronger than the toroidal one, the instability shows for perturbations with relatively high m. Key words. magnetohydrodynamics (MHD) – instabilities – stars: pre-main sequence – ISM: jets and outflows – galaxies: jets 1. Introduction of a significant toroidal component. A situation where loops of the toroidal magnetic field dominate the field distribution would The magnetic field plays a key role in the formation and propa- produce important dynamical consequences for the jet struc- gation of astrophysical jets. Polarization measurements provide ture, for instance, providing pressure confinement through mag- important information about the orientation and the strength of netic tension forces (Chan & Henriksen 1980;Eichler1993). the magnetic field, which seems to be highly organized on a large However, it is well known from plasma physics (see, e.g., scale in many jets (see, e.g., Cawthorne et al. 1993; Leppänen Kadomtzev 1966) that the toroidal magnetic field may cause var- et al. 1995; Gabuzda 1999; Gabuzda et al. 2004; Kharb et al. ious magnetohydrodynamic (MHD) instabilities even in simple 2008). To explain the observational data, simplified models of cylindrical configurations. the three-dimensional magnetic field distribution have been pro- In general, different types of instabilities can occur in as- posed (see, for example, Laing 1981, 1993; Pushkarev et al. trophysical jets. They can be caused by boundary effects, shear, 2005; Laing et al. 2006). In particular the polarization data sug- stratification and magnetic fields and are also likely to be respon- gest that the magnetic field has a significant transverse compo- sible for the observed morphological complexity of jets. For this nent in the main fration of the jet volume. In accordance with reason several analytical (see, e.g., Ferrari et al. 1980; Bodo et al. the “traditional” interpretation (see, e.g., Laing 1993), this mag- 1989, 1996; Hanasz et al. 1999; see also Birkinshaw 1997,fora netic structure can be associated with a series of shocks along review) and numerical (Hardee et al. 1992; Appl 1996; Lucek & the jet, which enhances the local transverse field. However, Very Bell 1996;Min1997; Kudoh et al. 1999) works have been de- Long Baseline Interferometry (VLBI) observations of BL Lac voted to the study of the stability properties of jets. A systematic + 1803 784 (Hirabayashi et al. 1998) revealed that the transverse study of the Kelvin-Helmholtz instability in non-relativistic jets magnetic structure can be formed by a large-scale toroidal mag- with the longitudinal magnetic field has been performed by Bodo netic field, and indeed recent observations of a radio jet in the et al. (1989, 1996) including also the effect of rotation. In par- galaxy CGCG 049-033 also reveal a predominantly toroidal ticular several unstable modes including a slow mode associated magnetic field (Bagchi et al. 2007; Laing et al. 2006). with the magnetic field and an inertial mode caused by rotation Magnetic structures with a significant toroidal field- have been found. Shear-driven instabilities can also be very im- component are predicted by several theoretical models for jet portant in jets (Urpin 2002). Numerical simulations (Aloy et al. formation (Blandford & Payne 1982; Romanova & Lovelace 1999a,b) indicate that the radial structure of jets may be more 1992; Koide et al. 1998). In particular some models of the jet complex with a transition shear layer surrounding the jet core. propagation (see, e.g., Begelman et al. 1984) propose that the The thickness of the shear layer depends on the distance being toroidal field decays more slowly than the poloidal one as a around 20% of the jet radius near the nozzle but broadening near function of the distance from the central object, therefore the the head where it is of the order of the jet radius. This shear toroidal field should eventually become dominant. Other gen- layer may have an important consequence for various properties eration mechanisms based either on the combined influence of of jets including their stability (see, e.g., Hanasz & Sol 1996, turbulence and large-scale shear (see, e.g. Urpin 2006)orondy- 1998;Aloy&Mimica2008). namo action along with magneto-centrifugal and reconnection In this paper, we consider the instability associated with the processes (de Gouveia Dal Pino 2005) also predict the occurence topology of the magnetic field in jets. Magnetic fields generated Article published by EDP Sciences A100, page 1 of 8 A&A 525, A100 (2011) by hydrodynamic motions can be topologically non-trivial and generated, for instance, by the turbulent dynamo action or by thus prone to pressure-driven or current-driven instabilities (see, hydrodynamic motions (stretching) in the process of jet propa- e.g., Longaretti 2008, for review). This problem is well stud- gation. We show that stability properties of jets with a subther- ied in the context of experiments on controlled thermonuclear mal field can be essentially different from those with a highly fusion (see, e.g., Bateman 1978) and has been discussed in re- superthermal force-free magnetic field as was considered, for lation to jets by a number of authors (see, e.g., Eichler 1993; example, by Appl et al. (2000). In particular, higher azimuthal Appl 1996; Appl et al. 2000;Leryetal.2000). A linear stabil- modes with m 1 can have the highest growth rate at variance ity analysis of supermagnetosonic jets has been performed by with the superthermal case. In our jet model, both destabilizing Appl et al. (2000) for different magnetic configurations includ- factors – the pressure gradient and electric current – are pre- ing a force-free field. It has been shown that the current-driven sented simultaneously, and the instability occurs under the com- instabilities can occur in the axial field for various profiles of bined influence of both these factors. Therefore, the instability the azimuthal field even if these components are comparable. considered has a mixed pressure- and current-driven nature. The non-linear development of the current-driven instability has The paper is organized as follows. Section 2 contains the been analysed by Lery et al. (2000), who found that this instabil- derivation of the relevant equations, and Sect. 3 shows the nu- ity can modify the magnetic structure of a jet redistributing the merical results. Conclusions are presented in Sect. 4. current density in the inner part. The current-driven instability of relativistic jets and plerions in toroidal magnetic fields has been considered by Begelman (1998), who obtained that the dominant 2. Basic equations instability in this case are the kink (m = 1) and pinch (m = 0) modes. The kink mode generally dominates, thus destroying the We consider a very simplified model assuming that the jet is concentric field structure. Apart from the current-driven insta- an infinitely long stationary cylindrical outflow and its propa- bilities, the pressure-driven ones caused by the thermal pres- gation through the ambient medium is modelled by a sequence sure gradient can also occur in jets. The modes, corresponding of quasi-equilibrium states. For the sake of simplicity, it is of- to these instabilities, are radially localized and characterized by ten assumed that the velocity of plasma within a jet, V, does not high wavenumbers. Likely, these modes are responsible for gen- depend on coordinates (see, e.g., Appl 1996; Appl et al. 2000; erating MHD turbulence in the jet core (Kersale et al. 2000). The Lery et al. 2000), and we will adopt the same assumption in our pressure-driven instabilities are well studied in laboratory pinch study. Instabilities of the magnetic configurations associated to configurations as well. Robinson (1971) considers the stability the electric current are basically absolute instabilities, i.e. they of the pinch with a large ratio of the gas and magnetic pres- grow, but do not propagate. We suppose that this is the case also sure using the energy principle and finds that stability is pos- in the rest frame of the jet, and unstable perturbations are there- sible if currents are located outside the main plasma column. fore simply advected with the flow at the jet velocity (see, e.g., The criterion of pressure-driven instabilities in cylindrical ge- Appl et al. 2000). Therefore, we treat the stability properties in ometry has been derived by Longaretti (2003). This criterion is the rest frame of the jet. related to the Alfvénic modes, which usually require less re- We explore the cylindrical coordinates (s, ϕ, z) with the unit stricting conditions for instability and can even be unstable if vectors (es, eϕ, ez).