1 Introduction

Astrophysical jets emerge from a variety of astrophysical sources, ranging from protostellar objects, to micro-quasars, active galactic nuclei (AGN) and probably Gamma Ray Bursts (GRBs). Currently, the observations of jets from radio-loud AGN may be counted among the most interesting ones as they constitute test laboratories for studying the spatial structure in relativistic jets. Today there is convincing evidence that the central engine in these AGN is a rotating supermassive black hole surrounded by a geometrically thin accretion disk, which gives rise to the formation of a pair of relativistic jets. The observation of superluminal motions and theoretical opacity arguments indicate that the bulk plasma in these jets moves at relativistic speeds along the jet axis. Models based on the assumption of a one-dimensional velocity structure have thus allowed useful insights into the emission properties of AGN jets. However, as real jets are generally expected to show a significant velocity shear perpendicular to their axes, such models may be adequate only for a first approximation. In particular, several independent arguments suggest that AGN jet flows might be characterized by an additional rotational

velocity component:
(1) The analysis of the jet energetics in extragalactic radio sources have revealed a remarkably universal correlation between a disk luminosity indicator and the bulk kinetic power in the jet (Rawlings & Saunders 1991; Celotti & Fabian 1993) and supported a close link between jet and disk. The successful application of models within the framework of a jet-disk symbiosis (Falcke & Biermann 1995; Falcke et al. 1995) indicates that for radio-loud objects the total jet power may approach 1/3 of the disk luminosity so that a considerable amount of accretion energy, and hence rotational energy of the disk (cf. virial theorem), is channelled into the jet leading to an efficient removal of angular momentum from the disk.
(2) Observational findings, including helical motion of knots or periodic variabilities, seem to provide additional evidence for intrinsic rotation in AGN jets (cf. Biretta 1993 for M87; Camenzind & Krockenberger 1992 for 3C273; Schramm et al. 1993 for 3C345).
(3) From a more theoretical point of view, intrinsic jet rotation is generally expected in magnetohydrodynamical (MHD) models for the formation and collimation of astrophysical jets (e.g. Begelman 1994; Sauty et al. 2002). In such models, intrinsic rotation with speeds up to a considerable fraction of the velocity of light is a natural consequence of the assumption that the flow is centrifugally accelerated from the accretion disk. It should be noted however, that the rotation profile in the jet does not necessarily have to be disk-like, i.e. the set of available jet rotation profiles could be much wider and might include, for example, rigid, flat and Keplerian profiles (e.g. Hanasz et al. 2000). In particular, rigid rotation inside a well-defined light cylinder might be related to foot points of the magnetic field lines concentrated near the innermost stable orbit (e.g. Camenzind 1996; Fendt 1997a), while more generally differential rotation would be intuitively expected if there is an intrinsic connection between jet motion and the rotating disk (cf. also Fendt 1997b; Lery & Frank 2000).

In this paper, we are interested in the influence of such rotation and shear profiles on the acceleration of particles in AGN jets. So far, several authors have contributed to our understanding of particle acceleration by shear. A kinetic analysis was used in the pioneering approach of Berezhko (1981, 1982) and Berezhko & Krymskii (1981). Their results showed that the particle distribution function in the steady state might follow a power law if the mean interval between two scattering events increases with momentum according to a power law. Later on, particle acceleration in the diffusion approximation at a gradual shear transition in the case of non-relativistic flow velocities was studied independently by Earl et al. (1988). They re-derived Parker's equation (i.e. the transport equation including the well-known effects of convection, diffusion and adiabatic energy changes), but also augmented it with new terms describing the viscous momentum transfer and the effects of inertial drifts. Jokipii & Morfill (1990) used a microscopic treatment to analyse the non-relativistic particle transport in a moving, scattering fluid which undergoes a step-function velocity change in the direction normal to the flow. They showed that particles may gain energy at a rate proportional to the square of the velocity change. Matching conditions in conjunction with Monte Carlo simulations for shear discontinuities were derived by Jokipii et al. (1989). The Monte Carlo analysis was extended by Ostrowski (1990, 1998), who also studied the acceleration at a sharp tangential velocity discontinuity, including relativistic flow speeds. He found that only relativistic flows can provide conditions for efficient acceleration, resulting in a very flat particle energy spectra which depends only weakly on the scattering conditions. The relevance of such a scenario for the acceleration of particles at the transition layer between AGN jets and their ambient medium was stressed in recent contributions by Ostrowski (2000) and Stawarz & Ostrowski (2002).

The work on gradual shear acceleration by Earl et al. (1988) was successfully extended to the relativistic regime by Webb (1989, 1992). Assuming the scattering to be strong enough to keep the distribution function almost isotropic in the comoving frame (so that the diffusion approximation applies), he derived the general relativistic diffusive particle transport equation for both rotating and shearing flows. Subsequently, Green's formula for the relativistic diffusive particle transport equation was developed by Webb et al. (1994). Applying their results to the cosmic ray transport in the galaxy, they found that the acceleration of cosmic rays beyond the knee by means of galactic rotation might be possible, but not to a sufficient extent. In its form presented however, it rather remains an essentially theoretical approach which could not be easily related to observations.

In a previous contribution (Rieger & Mannheim 2001b) we developed and applied a new model, that utilizes the relativistic transport theory advanced by Webb and that permits the analysis of the acceleration of energetic particles in rotating and shearing AGN jets. By studying velocity profiles known to be typical for such jets, we obtained results indicating that the resultant particle energization is in general a consequence of both centrifugal and shear effects. The formation of power law particle spectra under a wide range of conditions reveals the significant potential of shear and centrifugal acceleration as a natural explanation for the origin of the extended, continuous emission recently observed from AGN (e.g. Meisenheimer et al. 1997; Jester et al. 2001). In the present paper we aim to provide a more detailed investigation of this model, including the derivation of the relevant particle transport equation (cf. Appendix B).

The paper is organised as follows: after a short review of the underlying theoretical background in part 2 (see also Appendix A), we present several applications to relativistic jet flows with rigid, Keplerian and flat intrinsic rotation profiles in part 3, leaving the detailed derivation of the particle distribution function to Appendix C. Part 4 provides a discussion of our results. The observational relevance of the model presented is pointed out in part 5 with reference to recent observations. The paper finally closes with a short conclusion.