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Magnetic acceleration and instabilities of astrophysical jets

Moll, R.

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Rainer

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Jets

Magnetic Acceleration and Instabilities ofAstrophysical Jets

Rainer Moll

Magnetic Acceleration and Instabilities ofAstrophysical Jets

Front cover: Volume rendering of a simulated jet. The structures in the flow are caused bynon-axisymmetric instabilities.

Back cover: Magnetic field lines in a simulated jet. The panel to the right shows an enlarge-ment of the lower part, roughly up to where the flow reaches Alfven speed. Kink instabilitiesare visible in the upper part.

Magnetic Acceleration andInstabilities of Astrophysical Jets

Magnetische Versnelling en Instabiliteit van

Astrofysische Jets

Academisch Proefschrift

ter verkijging van de graad van doctoraan de Universiteit van Amsterdamop gezag van de Rector Magnificus

prof. dr. D. C. van den Boomten overstaan van een door het college voor promoties

ingestelde commissie,in het openbaar te verdedigen in de Agnietenkapel

op donderdag, 28 januari 2010, te 12:00 uur

door

Rainer Moll

geboren te Freising, Duitsland

Promotiecommissie

Promotor: Prof. dr. H. C. SpruitOverige Leden: Prof. dr. R. A. M. J. Wijers

Prof. dr. M. B. M. van der KlisDr. A. L. WattsDr. S. B. MarkoffProf. dr. A. de KoterProf. dr. A. AchterbergProf. dr. H. F. Henrichs

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

The work described in this thesis has been carried out atthe Max Planck Institute for Astrophysics in Garching,Germany.

Contents

1 Introduction 11.1 Types of Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Magnetically Driven Flows . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Magnetic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Summary of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Kink Instabilities in Jets From Rotating Magnetic Fields 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Effects of Jet Expansion . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Rationale of the Calculations . . . . . . . . . . . . . . . . . . . 132.1.3 Expected Instability Growth in Expanding Jets . . . . . . . . . 13

2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 MHD Equations and Numerical Solver . . . . . . . . . . . . . 162.3.2 Grid Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.3 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . 182.3.4 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Cases Studied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.1 Expected Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 232.5.2 Instabilities Found in the Simulation . . . . . . . . . . . . . . . 262.5.3 Impact On Dynamics and Energetics . . . . . . . . . . . . . . . 30

2.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 312.7 Appendix: Magnetic Pitch for a Conical Jet . . . . . . . . . . . . . . . 33

3 Decay of the Toroidal Field in Magnetically Driven Jets 373.1 Introduction and Rationale of the Calculations . . . . . . . . . . . . . 373.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.1 Numerical MHD Solver, Grid and Coordinates . . . . . . . . . 393.2.2 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . 403.2.3 Parameters and Units . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.1 Acceleration, Collimation and Mass Flow . . . . . . . . . . . . 45

v

CONTENTS

3.3.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3.3 Magnetic Field: Poloidal vs. Toroidal . . . . . . . . . . . . . . 483.3.4 Forces and Powers . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Comparison with Observations . . . . . . . . . . . . . . . . . . . . . . 523.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.1 Collimation and Jet Environment . . . . . . . . . . . . . . . . . 553.5.2 Disruption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.5.3 Cold Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Large Jets From Small-Scale Magnetic Fields 594.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3.1 Numerical MHD Solver, Grid and Coordinates . . . . . . . . . 624.3.2 Parameters and Units . . . . . . . . . . . . . . . . . . . . . . . 65

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.1 Transient Jets with Arcade-Shaped Initial Field . . . . . . . . . 664.4.2 Jets From Emerging Fields: Parameter Study . . . . . . . . . . 684.4.3 Jets From Emerging Fields: A Large Simulation . . . . . . . . 714.4.4 Jets From a Magn. Arcade On a Differentially Rotating Surface 75

4.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Samenvatting 815.1 Verschillende Jet-Typen . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2 Magnetisch Aangedreven Stromingen . . . . . . . . . . . . . . . . . . 84

5.2.1 Modellen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3 Magnetische Instabiliteiten . . . . . . . . . . . . . . . . . . . . . . . . . 865.4 Magnetische Rekonnektie . . . . . . . . . . . . . . . . . . . . . . . . . 875.5 Samenvatting van de Belangrijkste Resultaten . . . . . . . . . . . . . 88

References 89

A Numerical Methods 95A.1 Principles and Terminology in Computational Fluid Dynamics . . . . 95A.2 Imposing the Initial Magnetic Field . . . . . . . . . . . . . . . . . . . . 97A.3 Boundaries Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

B Coordinate Transformations 100

C Visualization 101C.1 Volume Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101C.2 Numerical Computation of the Stream Function . . . . . . . . . . . . 102C.3 Flow Visualization with Wedges . . . . . . . . . . . . . . . . . . . . . 103

vi

1 Introduction

The first record of an astrophysical jet was made by Curtis (1918), who noticed a“curious straight ray” poking out of the galaxy M87. More than three decades later,Baade & Minkowski (1954) correctly interpreted the feature as an “ejection from thenucleus” to which they referred as “jet”. Since then, many jets have been found andour understanding of the underlying physical processes has advanced considerably.

Most generally, an astrophysical jet may be defined as an elongated high-velocityoutflow of gas. One of the most distinctive features of jets is the high degree ofcollimation, which sets them apart from more isotropic outflows like stellar winds.Their properties, as well as those of the objects from which they emanate, varyconsiderably. Yet the underlying physical processes are probably very similar inall cases, involving the infall of matter (accretion) onto a central mass and theconversion of rotational energy into an outflow by means of a magnetic field. Thedifferences in the properties are then mostly a matter of scale; the most compactobjects produce the fastest and biggest jets.

Much has to happen before a jet appears on the sky. Most jets seem to be launchedby an accretion disk, which by itself is an extensive physical problem with manyyet unsolved issues. The transition region between the disk and the jet determinesthe amount of mass flowing in the jet, but is poorly understood to date. We knowhow jets can be accelerated and collimated by means of a magnetic field. However,most models do not take into account magnetic instabilities, that is to say growingperturbations which destroy the symmetry of the jet and potentially cause a decayof the magnetic field. This is one of the main topics of the present thesis. Radiationcan be produced by the jet in many ways, including synchrotron light, Comptonscattering and thermal emission. Shocks at the boundary to the external medium aswell as within the jet itself play an important role in this context.

A global numerical treatment of the (disk-)jet problem is not feasible in theforeseeable future, as most jets cover many orders of magnitude in distance fromtheir source. In a quasar jet, for example, the jet as seen at radio wavelengthshas a size on the order of 1 megaparsec, some 1010 times larger than the sourceregion (a few astronomical units, say). The expected time scales of (magnetohydro-)dynamic processes in the jet range over the same factor. Direct, time-dependent, 3-dimensional numerical simulations of such large objects are thus out of question. Yet,it is quite likely that the observed properties of jets, like their power, speed, widthand emitted radiation depend on time-dependent, 3-dimensional MHD processes.The question is thus where in the 10 decades these processes take place, and howto meaningfully isolate them. From the results presented here it is concluded that

1

1 INTRODUCTION

Figure 1.1: Some of the best-resolved jets on the sky. Left: Optical image of the protostellarjet HH 47, whose total length is about 3 pc. Credit: J. Morse/STScI and NASA. Right:X-ray image of the active galaxy Centaurus A with a 4 Mpc long, relativistic jet. Credit:NASA/CXC/CfA/R. Kraft et al.

much of the physics determining the observed properties probably happens in thefirst 3 decades from the source, but not less. A method is applied to deal with thisrange numerically with existing computational resources.

1.1 Types of Jets

Jets emanate from a variety of astronomical objects, including galactic as well as ex-tragalactic sources. The attempt to classify them according to observational featureshas produced a plethora of names. From a theoretician’s point of view, however,it is most convenient to distinguish jets by their physical origin. Observations arein general too coarse to give direct insight into the inner workings of the respec-tive sources. Still, the basic elements to be included in a physical model are wellestablished in most cases and shall be used here for a classification.

Some jets are observed at the birthplace of stars, the young stellar objects (YSOs).There, stars are produced in giant molecular (H2) clouds when gas collapses dueto self-gravity. As the central condensation of the cloud grows, it devours thesurrounding gas via an accretion disk. Jets emerge along the axis of rotation ofthe disk with typical velocities of several 100 km s−1 per second, comparable to theescape speed from the central protostar. As the jets collide with nearby clouds of gas

2

1.1 TYPES OF JETS

and dust, they create the complex nebular patches in Hα images which are knownas Herbig–Haro objects. The overall length of such a protostellar jet is on the order1 pc, the length-to-width ratio at the largest scale is typically 10 : 1 or more.

In binary systems, it can happen that the more compact component accretesmatter from its companion. An accretion disk forms around the growing componentand a jet is launched along the axis of rotation. Many such systems are luminous inthe X-ray band, for which they are called X-ray binaries (XRBs). The properties ofthe observable light emission and the jet depend largely on the composition of thebinary. Microquasars occur when the accretor is very compact, e.g. a neutron staror a black hole. They produce jets with relativistic velocities.

Relativistic jets also emanate from active galactic nuclei (AGNs), a class of lumi-nous and compact objects found in the center of galaxies. The above-mentioned jetfrom M87 is a famous example. The now-established standard picture of an AGNincludes a black hole with an accretion disk, a surrounding torus of gas and dust,clouds of gas at various distances that produce emission lines, and a well-collimatedjet along the axis of rotation of the disk, often accompanied by an opposing counter-part. The observable properties are dependent on the viewing angle. AGN jets excelin speed and extension, having typical Lorentz factors of Γ ∼ 10 and lengths of upto several 100 kpc. Some are remarkably straight and end in spots of high radioemission, others have a meandering shape and grow dimmer with distance fromthe central source.

The fastest known jets cannot be seen directly. Their existence is inferred fromobservations of γ-ray bursts (GRBs) in distant galaxies. The large observed lumi-nosities and rapid fluctuations in the light curves imply that the source of GRBswould be optically thick because of γγ → e+e− pair production, and the observedradiation would be unable to escape unless ultra-relativistic motion with Γ & 100 isinvolved. Energetic considerations and features in the afterglow (a slowly fadingemission after the burst as the jet coasts into the interstellar gas) point to a jet ratherthan a spherical outflow. If this is indeed the case, then the total energy releasein GRBs is on the order of 1051 erg, a value which is typical for supernovae. Theprevailing idea for most GRBs is that their energy is liberated when the fast-spinningcore of a massive star collapses to a black hole. The exact circumstances of howa jet with the required properties is generated in this scenario are currently notknown. The mechanism that produces the jet must be highly efficient to producethe observed energy output. Moreover, the model has to account for the rapidfluctuations in the light curves of GRBs.

Environment The nature of the environment into which a jet propagates can vary.Close to the source of the jet, the gravity of the central compact object dominatesand presumably produces a stratified atmosphere in which the density decreasesmonotonically with distance from the center. At larger distances, the jet interactswith an interstellar or intergalactic medium. In AGNs, one expects the presence

3

1 INTRODUCTION

of clouds which produce the observed line emission; these could interact with theflow (Poludnenko et al. 2002). GRB jets, which are produced in the core of a massivestar, may have to drill through a surrounding stellar envelope. Moreover, thereis evidence that protostellar jets interact with the still infalling molecular cloudthat feeds the accretion disk of the protostar (Velusamy & Langer 1998). Someprotostellar jets appear to be deflected as they hit dense clouds (Reipurth & Bally2001). The movement of the jet-producing object must also be taken into account insome cases. Many AGN jets in galaxy clusters have a bent morphology, caused bythe motion of the host galaxy through the intra-cluster medium (head-tail galaxies,Sarazin 1986). A similar situation arises when jets from XRBs are subject to the rampressure of the interstellar medium (Heinz et al. 2008).

In simulations, the ambient medium must be assumed to have a certain (i.e. mini-mal) density for numerical reasons. Interactions of the jet with the environment aretherefore always of importance. From the calculations presented in this thesis (inparticular those in Chapter 4), it seems that apart from pure hydrodynamic interac-tions, e.g. shock waves in the ambient medium, there are “wiggling” interactionsassociated with the magnetic field.

1.2 Magnetically Driven Flows

The idea of jets driven by magnetic forces goes back to Bisnovatyi-Kogan & Ruz-maikin (1976), who showed that jets could be the result of rotation acting on asuitably shaped magnetosphere of an accretion disk. To date, this is the most viableconcept to explain all the specific properties of jets like their high velocities andstrong collimation. Indirect observational support of the magnetic theory comesfrom the fact that many jets appear magnetized on the scales where this can beinvestigated. The synchrotron radio emission observed in extragalactic jets, forexample, is possible only if there is a magnetic field.

The key element for driving jets magnetically is a magnetic field that is anchoredin a rotating object, e.g. an accretion disk. In the simplest case, the anchoring fieldhas only a poloidal (in the direction of the axis of rotation or away from it, butnot around it) component and extends out to infinity. The rotation produces anadditional toroidal (azimuthal, around the rotation axis) field component. In otherwords, it twists the magnetic field. The toroidal field contains free energy whichgets transformed into a flow.

Within the framework of magnetohydrodynamics (MHD), the acceleration can beattributed to the magnetic pressure gradient associated with the toroidal field Bϕ.Under the assumption of azimuthal symmetry, the component of the Lorentz forcein the direction of the poloidal field Bp = BR eR + Bz ez is

Fp = − 18π∇pB2

ϕ −BRBp

B2ϕ

4πR(1.2.1)

4

1.2 MAGNETICALLY DRIVEN FLOWS

with ∇p = ep ·∇, where ep = Bp/Bp denotes the poloidal unit vector. r andR ≡ r sin ϑ denote the distance from the source (spherical radius) and the distancefrom the central jet axis (cylindrical radius), respectively. If the flow expands, B2

ϕdecreases and the magnetic pressure force (first term in Eq. 1.2.1) points in positive,outwards direction, whereas the magnetic tension force (second summand) points innegative, inwards direction. The two forces cancel each other in the case of ballistic(i.e. rectilinear with constant speed in all directions within the flow) expansion, inwhich case Bp = Br and Bϕ ∝ r−1 due to magnetic flux conservation. In general,one has to solve the full problem to see whether the gradient of B2

ϕ prevails over themagnetic tension force to accelerate the flow.

In a frame that corotates with the central rotator, the acceleration can alternativelybe attributed to a centrifugal force. This is an elegant view of the accelerationprocess if it can be assumed that the magnetic field is sufficiently strong to enforceapproximate corotation of the plasma, which holds typically up to the Alfvenradius in many models. The centrifugal picture is mathematically equivalent to themagnetic picture described above, but is more limited in scope. Corotation maybreak down long before the Alfven radius, for example, if the mass flow is high andthe magnetic field weak.

Further Acceleration via Dissipation The rotation-induced acceleration ceases tobe effective at a finite distance, usually when the flow reaches the Alfven or fastmagnetosonic speed, and the flow becomes ballistic. To break the balance betweenthe pressure and tension force and produce further acceleration, Bϕ must decreasemore rapidly than in a ballistic expansion. This happens if it is dissipated along theflow, so that the magnetic pressure gradient becomes steeper. Dissipation occursnaturally as a result of instabilities associated with Bϕ. An elegant view of this effectcan be made if one regards the Poynting flux in the jet as a flux of a “magneticenthalpy” rather than energy. This gives an alternative interpretation of the fact thatthe Poynting flux in MHD contains twice the value of the magnetic energy B2

ϕ/8π.As the magnetic field is dissipated, half of the magnetic enthalpy is turned into heatwhich, in turn, may be radiated away instantly (if the optical depth is low) whilethe other half does accelerative work.

1.2.1 Modeling

Stationary Models Most theoretical work on MHD outflows relies on simplifyingassumptions, each of which has shortcomings in view of the real situation. Above all,numerous authors assume that the flow is axisymmetric and stationary. Examplesare the solutions of Sakurai (1985, 1987), which describe stellar/disk winds that,albeit slowly, collimate towards the rotation axis. In the seminal work of (Blandford& Payne 1982), the jet is additionally assumed to be radially self-similar and “cold”(pressure-less). With the self-similar ansatz, the jet extends formally to infinity in

5

1 INTRODUCTION

lateral direction. In the case of relativistic jets, it is possible to neglect the Lorentzforce under certain circumstances (Fendt et al. 1995). Although this force-freeapproach does not account for the acceleration, it is argued that the obtained fieldstructure is not too different from what would be needed for that. A most convenientsimplification is to consider the poloidal field to be given and fixed at all times. Thisallows the problem to be tackled analytically to a large extent (see e.g. the review inSpruit 1996).

Simulations The stationarity assumption has limits, of course, since it does notallow a description of the jet’s head plowing through the atmosphere when it is firedoff. Even after the starting phase, many jets do not settle into a complete steady state.Rather, they contain moving structures such as “knots” of increased light emission.The time-dependent treatment of the jet problem is computationally expensive evenif axisymmetry is assumed. Therefore, some authors consider only the propagationof the jet through the ambient medium, using a nozzle with high-velocity gas asa boundary condition to inject the jet into the computational domain (e.g. Kossl &Muller 1988). To learn something about the magnetic acceleration, one has to atleast include the accretion disk as a boundary condition (e.g. Ustyugova et al. 1995,and the simulations presented in this thesis). Simulations that include a (simplified)disk (e.g. Hayashi et al. 1996) can cover only its immediate environment. Theultimate goal would be for simulations to cover both the accretion disk as well as theatmosphere which contains the jet, with all the relevant physics being included (e.g.turbulence in the disk). The main problem here is that the contrast in length andtime scales is much too high, requiring an unaffordable amount of computationalpower. For the time being, we have to content ourselves with studying parts of thewhole problem.

1.3 Magnetic Instabilities

Jets are prone to a variety of instabilities. The most important of these are causedeither by the motion relative to the ambient medium or by a strong toroidal magneticfield in the jet, see also the introduction in Sect. 2.1 and the references given there.The present work is mainly concerned with the latter class of magnetically driveninstabilities, which have also been a long-standing issue in the attempt to buildviable devices for controlled nuclear fusion.

Although better described as truncated cones in many cases, jets may be regardedas plasma columns as a first approximation. The equilibria of such columns are well-studied in the context of controlled fusion (for a sound introduction, see Freidberg1987). For the column to be in equilibrium (in the absence of external forces),conservation of momentum implies that the sum of the gas and magnetic pressureforces must be matched by the magnetic tension force of the azimuthal field Bϕ,which is directed towards the axis. Configurations with only longitudinal fields

6

1.3 MAGNETIC INSTABILITIES

unperturbed m = 0 m = 1 m = 2sausage kink filamentary

Figure 1.2: Modes of perturbations of a cylindrical surface. The instability associated with am = 1 perturbation is found to be one of the most dangerous for fusion plasmas. It is alsohighly relevant to astrophysical jets containing twisted magnetic fields.

are possible and turn out to be stable, but self-confinement requires the presence ofa non-zero Bϕ that causes instabilities. In the laboratory, an azimuthal field ariseswhen a longitudinal current is sent along the column. Presumably for this reason,the term “current-driven” has been adopted and is widely used to denote this classof instabilities.

Perturbations of a plasma column are usually described by a displacement vectorξ(ϕ, z) which is defined at any point of the perturbed surface R = const. Anarbitrary ξ can be written as a Fourier series with the modes being proportional toei(mϕ+kz) (provided periodicity in z). Fig. 1.2 shows examples. For perturbationswith m = 0, a so-called sausage instability may arise as the tension force of theazimuthal field increases by the radial contractions. In the m = 1 case, the magneticfield strength, and thus the magnetic pressure, of the perturbed Bϕ is increased atthe inside of the kinks (denser field lines) and decreased at the outside, causing

7

1 INTRODUCTION

the perturbation to grow. The susceptibility for these kink instabilities grows withincreasing

∣∣Bϕ/Bz∣∣, i.e. with decreasing pitch of the helical field lines. Also, the

linear growth time is inversely proportional to the strength of the azimuthal field.

Immediate Consequences for a Jet The consequences of MHD instabilities areusually catastrophic in the case of fusion plasmas, destroying the fragile state ofconfinement that keeps the heated gas away from the walls of the fusion reactor.In view of the large extents of astrophysical jets, this rises an immediate concernfor the “survival” of jet flows affected by such instabilities. The apprehension isthat the jet is distorted to such an extent that it dissipates into its surroundings. Ifthis were to happen at small distances, which is most likely the location where themagnetic field attains enough twist to become unstable, then the magnetic modelwould not be able to explain the large jets that are observed.

The evolution of MHD instabilities in expanding jet flows, however, is funda-mentally different from that in non-moving laboratory plasmas. The continuousexpansion of the flow works against the growth of the instabilities. Moreover, thedissipation of the toroidal magnetic field deprives the instabilities of their drivingforce, preventing an indefinite growth of the perturbations. The jet can becomesomewhat wider than it would without instabilities, but the bulk kinetic energy per-sists. It is thus possible that, due to the decay of the toroidal field, a jet of magneticorigin ends up as an ordinary hydrodynamic flow.

Not being fatal, the deformations in the jet can manifest themselves in distinctivestructures at observable scales, e.g. the wiggles seen in some protostellar jets. Itwould seem natural to identify wiggles in observed jets with the displacementsexpected from (linear) instability. The results in Chapters 2 and 3, however, showthat the wiggles have significantly larger scales as a result of nonlinear developmentof the instability. The features are frozen in the flow when the growth of theperturbations stop and are transported outwards by the then-ballistic flow.

1.4 Magnetic Reconnection

As the magnetic field structure gets perturbed by instabilities, abrupt jumps in thepolarity of the magnetic field can arise. This causes a decay of the magnetic field, inparticular of the toroidal component Bϕ. As pointed out in Sect. 1.2, the decay hasconsequences for the acceleration of the flow. The magnetic pressure gradient alongthe jet steepens and, taken as an isolated effect, boosts the acceleration of the jet. Thelateral structure of the jet, in particular the poloidal field, is also affected by the decayof Bϕ. This also has consequences for the acceleration behavior. The dynamicalevolution of a jet with non-axisymmetric instabilities is thus fundamentally differentfrom a stable, axisymmetric one.

The details of reconnection in astrophysical settings is still a somewhat openissue. Different models exist which are not easily reconciled with the observed rates

8

1.4 MAGNETIC RECONNECTION

of reconnection (Kulsrud 2001). In ideal MHD simulations such as presented inthis work, reconnection is made possible by the numerical diffusion resulting frominterpolations (see Sect. A.1 for an overview on the numerical methods). Wherefields of different direction are compressed into a grid cell by the evolution of theflow, only their average appears in the next time step. Leaving reconnection todiscretization errors thus effectively assumes it to be fast. A popular model forreconnection is that of Petschek (1964), which yields a flow towards the reconnectionpoint at some modest fraction of the Alfven speed. This is closer to rates resultingfrom numerical diffusion than the (also popular) Sweet–Parker type models (Parker1957; Sweet 1958), which yield much slower speeds. Observations of the solar coronaindicate a fast reconnection rate (exemplified by the generally modest deviationsfrom a potential field configuration). This is sometimes used as an argument ingeneral for a fast reconnection rate in astrophysical cases. We implicitly make thesame assumption in the present work. It should be noted, however, that there arewell-documented cases where this assumption is violated (e.g. the MRI simulationspresented by Fromang & Papaloizou 2007).

9

1 INTRODUCTION

1.5 Summary of Main Results

• With a specifically tailored numerical grid, magnetohydrodynamic jets havefor the first time been simulated in three dimensions with very large ranges indistance. The biggest simulations cover a length of 1000 times the source size.

• The magnetic acceleration is fairly efficient. Most of the magnetic enthalpyassociated with the twist of the magnetic field is converted to kinetic energy.

• The jets are subject to kink instabilities. The severity of the instabilities de-pends on the jet’s collimation behavior and on the rotation which producesthe twist in the magnetic field: an opening angle that decreases with distanceand rigid rotation favor their growth.

• Contrary to the apprehensions voiced in the literature, kink instabilities donot “disrupt” a jet. Rather, they cause a decay in the magnetic field and thejet continues as a ballistic flow, though they can widen the jet by a moderatefactor. The actual disruption of a jet requires a strong interaction with themedium into which it propagates.

• The dissipation of the magnetic field induced by the instabilities liberatesenergy that can be radiated away, making the jet flow visible. The distortionsin the jet flow found in the simulations may explain some of the knots andwiggles in observed jets.

• The mechanism by which the Poynting flux of the jet is converted into kineticenergy is different in 3 dimensions from the 2-dimensional (axisymmetric)calculations that have dominated the literature, though their overall efficiencyturns out to be similar. In 3D, the flow is driven by a magnetic pressuregradient resulting from the decay of the toroidal field by instabilities, inaxisymmetry it is a “diverging nozzle” effect. The lateral structure of theaccelerated flow is different in the two cases.

• Jets can be accelerated with (twisted) large-scale poloidal fields or with small-scale magnetic loops. In the latter case, it is essential that the footpoints of theloops are rotating differentially to generate the necessary twist in the field.

10

2 Kink Instabilities in Jets FromRotating Magnetic Fields

R. Moll, H. C. Spruit, M. ObergaulingerAstronomy & Astrophysics 2008, Vol. 492, pp. 621–630

Abstract: We have performed 2.5D and 3D simulations of conical jetsdriven by the rotation of an ordered, large-scale magnetic field in a stratifiedatmosphere. The simulations cover about three orders of magnitude indistance to capture the centrifugal acceleration as well as the evolutionpast the Alfven surface. We find that the jets develop kink instabilities,the characteristics of which depend on the velocity profile imposed atthe base of the flow. The instabilities are especially pronounced with arigid rotation profile, which induces a shearless magnetic field. The jet’sexpansion appears to be limiting the growth of Alfven mode instabilities.

2.1 Introduction

Strong magnetic fields on large scales may play an essential, active role in theformation and evolution of jet-like outflows. The general idea is that a poloidalmagnetic field, embedded in a plasma and anchored e.g. in an accretion disk ora black hole, is forced into rotation at the anchor point, a toroidal field developsand the plasma is accelerated by what can be interpreted as a centrifugal force ina corotating frame (Blandford & Payne 1982; Spruit 1996; Konigl & Pudritz 2000).However, this magnetocentrifugal acceleration is only effective up until the Alfvensurface, defined as the surface where the flow velocity equals the Alfven velocity.Beyond this point, the magnetic field will be strongly wound-up. Such a fieldconfiguration is potentially unstable with respect to certain MHD instabilities.

MHD jets are susceptible to a variety of instabilities. Kelvin-Helmholtz (KH)instabilities are fed by the relative kinetic energy between the jet and the ambientmedium. They can distort the jet surface only (ordinary modes) or the wholebeam (e.g. Birkinshaw 1991), provoking shocks, mixing with ambient material andpossibly a disruption of the jet (Bodo et al. 1995, 1998). The presence of strongmagnetic fields, poloidal or toroidal, is expected to hamper the growth of KHinstabilities (Appl & Camenzind 1992; Keppens et al. 1999).

The free energy associated with the toroidal magnetic field is responsible foranother class of instabilities, which is traditionally known as current-driven (CD)

11

2 KINK INSTABILITIES IN JETS FROM ROTATING MAGNETIC FIELDS

and of notorious importance in controlled fusion devices (for an introduction, see,e.g. Freidberg 1987; Bateman 1978). The relevance for magnetized astrophysicaljets has been pointed out by Eichler (1993); Spruit et al. (1997); Begelman (1998)and others. Among CD instabilities, m = 1 kink instabilities are the most effective.An ideal kink mode is characterized by helical displacements of the cylindricalcross sections of a plasma column. It is expected to grow on a dynamical time scalewith respect to an Alfven wave crossing the unstable column. The susceptibilityis strongly dependent on the magnetic pitch, a measure for the degree of wind-up. Kink instabilities might destroy the ordered, symmetric state of a jet, leadingto its disruption or, through magnetic reconnection, the associated dissipation ofmagnetic fields and steepening of the magnetic pressure gradient, to its acceleration(Drenkhahn 2002; Giannios & Spruit 2006).

Different kinds of instability can mix and interact. For example, Baty & Keppens(2002) show how CD instabilities can stabilize KH vortices at the jet surface. Forthis work, we used conditions under which CD kink instabilities are expected todominate (low plasma-β, small magnetic pitch).

For a self-consistent study of kink instabilities, numerical simulations need to becarried out in 3D. Bell & Lucek (1996) did so using a simple model in which a toroidalmagnetic field configuration was allowed to expand into a uniform atmosphere.This generated a jet which was subject to kink instabilities. Nakamura & Meier(2004) performed 3D simulations of MHD jets in variously stratified atmospheres,finding that they can develop kink-like distortions in the trans-Alfvenic region.Laboratory experiments of MHD jets have been performed by Hsu & Bellan (2005),confirming that the magnetic pitch plays a crucial role for the formation of kinkinstabilities.

2.1.1 Effects of Jet Expansion

Jets from protostars, and especially AGN and microquasars, expand in width d byorders of magnitude after passing through their Alfven radius. In an expanding flowthere is no clean separation between time dependence due to instability and that dueto the expansion itself, making the question of stability less well defined. Analyticalstudies thus tend to focus on instabilities in a cylindrical geometry, with constantdiameter (such as in the “magnetic tower” picture of Lynden-Bell 2003). Expansionhas strong consequences on the behavior of instabilities, however, compared withjets modeled as cylinders of constant width.

First, there is the tendency for the toroidal (azimuthal, around the jet axis) com-ponent of the magnetic field to dominate in an expanding jet. From the inductionequation, the poloidal and toroidal components of the field vary as Bp ∼ 1/d2 andBϕ ∼ 1/d respectively (for constant jet velocity). Expansion thus causes a continualincrease of the ratio Bϕ/Bp. Even when dissipation were to decrease the toroidalfield at some point, the ratio increases again on further expansion. Free energyavailable in the toroidal field thus remains the dominant form of magnetic energy,

12

2.1 INTRODUCTION

and one may expect the question of stability and dissipation to remain relevant onall length scales. It also follows that the nonlinear development of instabilities in anexpanding jet is expected to be very different from the constant-diameter case.

Secondly, expansion has a strong effect even on the conditions for occurrence ofinstability. It has a stabilizing effect, since magnetic instabilities become ineffectivewhen their signal speed (the Alfven speed) drops below the lateral expansion speedof the jet. This is discussed further below.

2.1.2 Rationale of the Calculations

The aim of the calculations reported here is to study how kink instability operatesunder these conditions of expansion of the jet over several orders of magnitude inwidth.

The degree of instability to be expected in a jet driven by a rotating magnetic fieldis intimately tied to the way it is collimated. If, instead of being cylindrical, the jethas a non-vanishing opening angle ϑ, the expected incidence of instability dependson the details of the dependence of ϑ on distance r. An opening angle increasingwith distance reduces instability, while for an asymptotically vanishing openingangle instability must always set in at some distance, if it was not present alreadyfrom the start (see discussion in Sect. 2.1.3).

The setup in the simulations presented here produces jets in the intermediate caseof an (approximately) constant opening angle: a “conical” outflow. It turns out thatin this case the presence of instabilities and their amplitude depends on secondaryconditions such as the rotation profile imposed at the base of the flow, hence it is agood test case for the incidence of instabilities.

Since the observed jets travel over such large distances, even marginal forms ofinstability can become effective. An important goal of the present calculations istherefore to cover a large range in distance, about 3 orders of magnitude. This isachieved by the use of a grid adapted to the approximately conical shape of the jet.

2.1.3 Expected Instability Growth in Expanding Jets

In the following, we estimate how the sideways expansion affects the growth ofinstabilities in broadening jets. In spherical coordinates (r, ϑ, ϕ), we assume that thejet radius (distance to the jet’s central axis) is given by

R = R′( r

r′)α

with R′ = r′ sin ϑ′ (2.1.1)

where the prime stands for evaluation at a reference distance r′, for which we takea distance somewhat beyond the Alfven radius. The flow has then approximatelyreached its asymptotic speed vr ≈ const, and the magnetic field has become pre-dominantly azimuthal. In the absence of magnetic dissipation due to instabilities,

13


the field strength then varies as Bϕ ∝ R−1 (magnetic flux conservation) and thedensity as ρ ∝ R−2 (mass conservation). Since the growth rate Γg is expected toscale with the Alfven crossing rate vAϕ/R, we introduce a dimensionless instabilityrate κ of order unity:

Γg = κvAϕ

R= κ

vAϕ

R′( r

r′)−α

(2.1.2)

where vAϕ = Bϕ/√

4πρ is the azimuthal Alfven velocity. The expansion rate isestimated by

Γe =d ln R

dt=

1R

drdt

dRdr

=αvr

r. (2.1.3)

We findΓg

Γe=

κυα sin ϑ′

( rr′)1−α

(2.1.4)

with υ := vr/vAϕ ≈ const according to the ballistic approximations mentionedabove. Consequently, the instability growth rate dominates at some distance r fora collimating jet (α < 1). Decollimation (α > 1), on the other hand, thwarts thegrowth of instabilities. A conical jet (α = 1) constitutes a limiting case where alldepends on the combination of parameters κ/(υ sin ϑ′), which is of order unity. Anumerical simulation is necessary to find out whether the instability or expansionprevails.

The paper is organized as follows. In Sect. 2.2, we introduce the magnetocentrifu-gal jet model and account for the assumptions made in our simulations. A detaileddescription of the numerical setup, the coordinate system and the scale-free unitsemployed in the analysis is given in Sect. 2.3. In Sect. 2.4 we give the parametersof the simulated cases and in Sect. 2.5 we present the results. There, we start bymaking predictions on the characteristics of instabilities by examining the relevantproperties of our simulated jets. We proceed with an analysis of the instabilities thatactually appeared and complete with looking for effects on the jets’ dynamics. Wefinish with a discussion and conclusions in Sect. 2.6.

2.2 The Model

The model is construed to apply to jets produced by ordered magnetic fields an-chored in an accretion disk. This has become the default interpretation for thejets observed in AGN, microquasars and protostellar objects, though it must bekept in mind that observational evidence of the key ingredient in this model, theordered field (Bisnovatyi-Kogan & Ruzmaikin 1976; Blandford & Payne 1982), isstill somewhat indirect.

More uncertain is the shape of this field. The strength of the field anchored in thedisk is likely to scale in some way with the orbital kinetic energy (or gas pressure)in the disk, hence will decline with distance R from the rotation axis. In the absence

14

2.2 THE MODEL

of more detailed information, we consider a simple form for a field of this kind,one in which the vertical (normal to the disk) component at the surface Bz variesas Bz(R) ∝ [1 + (R/z0)2]−ν. Neglecting gas pressure and fluid motions, the fieldabove the disk would be a potential field, its shape defined uniquely by Bz. Forν = 3/2 it is the field of a monopole with the source at a depth z0 below the centerof the disk.

The initial state of the model is a gas distribution in hydrostatic equilibrium in afield of this monopolar shape. Rotation is applied at the lower boundary, in a regionR < R0 (see Sect. 2.3.3 for details). This generates an outflow with an approximatelyconstant opening angle on the order R0/z0 (a “conical” outflow). The surroundingvolume remains approximately in static equilibrium, and serves to collimate theoutflow to the desired opening angle.

The magnetic field responds to the rotation by winding up. That is, a toroidalmagnetic field Bϕ is produced and gives, together with the poloidal field Br, riseto helical field lines. The magnetic pressure gradient (minus the tension force)associated with Bϕ gives rise to a poloidal acceleration of the plasma which is ofcentrifugal nature in a corotating frame. Beyond the Alfven radius, the accelerationceases to be effective, while the field becomes predominantly azimuthal. The furtherdevelopment depends on how strongly the jet is affected by instabilities in this highlywound-up field. Possibly, they endanger the jet’s integrity and/or facilitate magneticreconnection events. Magnetic field dissipation can entail further acceleration ofthe jet (Drenkhahn 2002). As discussed above (Sect. 2.1.3), the “conical outflow”produced in our monopolar background field is of special interest as it marks theboundary between cases expected to be strongly respectively weakly unstable.

A self-consistent investigation of the problem requires a full 3D treatment, becausekink instabilities are non-axisymmetric. In addition to every 3D simulation we alsoperformed an axisymmetric (2.5D) simulation with the same boundary and initialconditions. This way, we could detect whether the jet evolves differently due to theinstabilities.

The basic parameters of the model are the magnitude of the rotation velocity, itsprofile Ω(R), the relative strength of the magnetic field as measured by plasma-β ofthe initial state, and the jet’s opening angle. The parameter values are chosen suchthat the Alfven radius of the resulting outflow is located within the computationalvolume, so that the centrifugal acceleration process is covered in the simulation,but close to the lower boundary, so that the subsequent evolution can be followedover as large a distance as numerically feasible. Increasing the imposed rotationrate moves the Alfven radius toward the lower boundary. Due to numerical lim-itations, however, vϕ could not be increased indefinitely in the simulations, anda compromise was necessary. In the results reported below the region inside theAlfven radius occupies about 10–20% of the box length.

15


2.3 Methods

We employ a spherical grid for our jet simulations. This enables us to follow jetswith opening angles over a much longer distance than is possible with a Cartesiangrid, because the jet need not be overresolved at large heights in order to properlyresolve its base. The obvious choice of letting the jet propagate along the polaraxis is numerically problematic if non-axisymmetric flows are involved, becausethe grid is singular there. We therefore let the jet flow in equatorial direction. Thecomputational volume covers a range ∆θ = ∆φ in the polar and azimuthal angles,adjusted to the opening angle of the flow.

2.3.1 MHD Equations and Numerical Solver

We numerically solved the ideal adiabatic MHD equations, including a tempera-ture-dependent temperature-control term K = K(T(t)). Explicitly, the equationsare:

∂ρ

∂t+ ∇ · (ρv) = 0, (2.3.1a)

∂v∂t

+ v ·∇v = −1ρ∇p +

14πρ

(∇× B)× B−∇Φ, (2.3.1b)

∂e∂t

+ ∇ ·[(

e + p +B2

8π

)v− 1

4πB(B · v)

]= −ρv ·∇Φ + K, (2.3.1c)

∂B∂t

= ∇× (v× B), (2.3.1d)

where

e =12

ρv2 +B2

8π+

pγ− 1

(2.3.2)

is the total energy density, γ = 5/3 is the adiabatic index, p is the gas pressure, Φ isthe gravitational potential (external, no self-gravity) and the other symbols havetheir usual meanings. A notorious problem with low-β MHD simulations in fullyconservative form, as in the code used here, is the amplification of discretizationerrors that occurs because the gas pressure is only a small contribution to the totalenergy (cf. 2.3.2), which is dominated by the magnetic energy. As in the case ofhighly supersonic flows, these errors manifest themselves in the form of “negativepressures” at occasional grid points. This problem does not occur when an equationfor the thermal energy equation is used instead of the total energy. We compute, inparallel, an alternative update of the gas pressure from the thermal energy equation,in the form

∂p∂t

+ ∇ · (pv) = −(γ− 1)p∇ · v + K. (2.3.3)

Where negative pressures appear, they are replaced by this value.

16

2.3 METHODS

Another device that turns out very useful to avoid negative pressures is thetemperature-control term K in Eq. (2.3.1c). For this we use a scheme loosely modeledafter Newtonian cooling, or an optically thin radiative loss process. After every fulltime step, we add/subtract thermal energy according to

∆p(t)p(t = 0)

= −T(t)− T(t = 0)T(t = 0)

∆tτK

, (2.3.4)

where T is the temperature and τK is a time scale chosen so as to keep the tempera-ture within about a factor 30 of the initial atmospheric value.

With the MHD Poynting vector

S = − 14π

(v× B)× B, (2.3.5)

Eq. (2.3.1c) can also be written as

∂(e + ρΦ)∂t

+ ∇ ·[(

12

ρv2 +γ

γ− 1p + ρΦ

)v + S

]= K, (2.3.6)

describing the change of total energy including gravitational potential energy. Wewill employ this form later when we look at the energy flow rates.

We used a newly developed Eulerian MHD code (Obergaulinger 2008) to solveEqs. (2.3.1a–2.3.1d). It is based on a flux-conservative finite-volume formulation ofthe MHD equations and the constraint transport scheme to maintain a divergence-free magnetic field (Evans & Hawley 1988). Using high-resolution shock capturingmethods (e.g., LeVeque 1992), it employs various optional high-order reconstructionalgorithms and approximate Riemann solvers based on the multi-stage method(Toro & Titarev 2006). The simulations presented here were performed with afifth order monotonicity-preserving reconstruction scheme (Suresh & Huynh 1997),together with the HLL Riemann solver (Harten 1983) and third order Runge-Kuttatime stepping.

2.3.2 Grid Coordinates

In the 3D simulations, our computational domain was centered around the y-axisin a “standard” spherical coordinate system (r, θ, φ) with θ being the polar anglefrom the z-axis and φ being the azimuthal angle about the z-axis in the x-y-plane.See Fig. 2.1 for an illustration. Positioning the domain in equatorial (rather thanpolar) direction yields a grid which is free of singularities and quasi-regular intransverse jet direction: Near the y-axis, the distance between two φ = const curvesis ∆x ≈ r ∆φ if we neglect terms of order (θ − π/2)2 and of order (φ− π/2)3. Thedistance between two θ = const curves is then ∆z ≈ r ∆θ. With uniform spacings∆θ and ∆φ, we thus obtain a grid whose area elements are approximately those ofan equidistant Cartesian grid in a plane normal to the y-axis.

17


Figure 2.1: Computationaldomain and coordinate nomen-clature (schematic drawing). Thejet propagates along the y-axis(θ = φ = π/2), near which thegrid is quasi-Cartesian in thenormal plane. To describe theresults of the simulations, we usethe alternate spherical coordinatesystem (r, ϑ, ϕ) shown in theupper right inset, which takes they-axis as the polar axis.

b

b

b

b

ϑ

ϕ

R

r

x

y

z

ϑ

ϕ

θ

φ

r

x y

z

For data analysis and plotting we use the “alternative” spherical coordinate sys-tem (r, ϑ, ϕ) with ϑ being the polar angle from the y-axis and ϕ being the azimuthalangle about the y-axis in the z-x-plane. It is adapted to the propagation directionof the jet and as such better suited to describe its physics. R ≡ r sin ϑ denotes thedistance to the y-axis. We refer to the r-direction with “poloidal” or “radial”, theϕ-direction with “toroidal” or “azimuthal” and the y-axis as “polar axis” or “centralaxis”.

In the 2.5D simulations, the jet propagates along the z-axis and the φ-direction istaken to be symmetric. However, to avoid confusion, we use the same nomenclatureas in the 3D simulations throughout this paper. That is, we visualize the jet aspropagating in y-direction and employ the (r, ϑ, ϕ) system for describing its physics.

For a proper resolution at all radii, it turned out to be necessary to employlogarithmic grid spacing (Park & Petrosian 1996) in r-direction. The r-left interfaceof grid cell i is situated at

ril = r0

l

(rn−1

r

r0l

)i/n

(2.3.7)

where n is the total number of cells in the domain which is bounded by r = r0l and

r = rn−1r = rn

l . The cell center of grid cell i is situated at rc = (ril + ri

r)/2.

2.3.3 Initial and Boundary Conditions

The initial magnetic field is

B(r) =gr2 er, (2.3.8)

18

2.3 METHODS

corresponding to a magnetic monopole of charge g located at the coordinate origin.The associated vector potential

A =gr

tan(

θ

2

)eφ (2.3.9)

was employed in the numerical setup to ensure solenoidality of the discretizedmagnetic field. Satisfying ∇× B = 0, the initial magnetic field is force-free.

We impose the static gravitational field of a point mass M, also located at theorigin. The stratification of gas pressure in this potential is chosen such that theplasma-β := p/pmag in the initial state is constant throughout the computationaldomain. Hence, since B ∝ r−2, p ∝ r−4. The density in the initial state is determinedfrom hydrostatic equilibrium: ρGM = −r2dp/dr ∝ r−3 where G is the gravitationalconstant. The temperature, sound speed and Alfven velocity vary as T ∝ r−1,cs ∝ r−1/2 and vA ∝ r−1/2 in this stratification.

The lower boundary of the computational volume, where the jet nozzle resides,is at a distance rb from the origin. The conditions at this surface are related to thegravitational potential by

Φ(r) = −GMr

with M =4pbrbGρb

, (2.3.10)

where the subscript b denotes the values at rb.At the sides (θ and φ) and top (upper r) of the domain, we use open boundaries

which allow for an almost force-free outflow of material: p, ρ, all components ofv and the transverse components of B are mirrored across the boundary interfaceto the opposing “ghost cells”, the normal component of B is determined by thesolenoidality condition. The open boundaries work well and cause only minimalartefacts in the form of reflections. At the bottom (lower r) of the domain, fromwhich the jet emanates, p and ρ are kept fixed at their initial values in all ghost cells.The magnetic field is determined by extrapolation from the interior of the domainusing the same scheme as for open boundaries. The velocity is prescribed to bezero except for the ghost cells below the nozzle area (R ≤ Rb at r = rb). There, anazimuthal velocity field v = vϕ eϕ is maintained, with either a Keplerian velocityprofile

vϕ =

vmaxϕ,b

√0.2Rb

R for 0.2Rb ≤ R ≤ Rb

0 elsewhere(2.3.11)

or a rigid rotation profile

vϕ =

vmax

ϕ,bR

Rbfor R ≤ Rb

0 elsewhere.(2.3.12)

19


Table 2.1: Normalization units

Quantity Symbol(s) Unitlength x,y,z,r,R,h l0

gas pressure p p0density ρ ρ0velocity v cs0 =

√γp0/ρ0

time t,τ t0 = l0/cs0energy density e p0

energy flow rate E p0l30/t0

force density F p0/l0magnetic flux density B B0 =

√8πp0

current density j j0 = B0c/(4πl0)

Note that we use the term “Keplerian” to indicate only that vϕ ∝ R−1/2. The centralmass M only serves to balance our chosen stratification and is not to be understoodas the center of an accretion disk.

2.3.4 Units

The setup described above is unambiguously determined by 6 parameters, Bb, pb, ρb,Rb, rb and vmax

ϕ,b , but they are not all independent. As units of length, pressure anddensity we use l0 ≡ 2Rb, p0 ≡ pb and ρ0 ≡ ρb. The physical quantities expressedin these units are listed in Table 2.1. Since these units are arbitrary, the number ofindependent parameters defining the problem reduces to 6− 3 = 3. These are aplasma-β value (which determines Bb), an opening angle ϑb := arcsin(l0/2rb), anda Mach number for the rotation: either vmax

ϕ,b /cs,b or vmaxϕ,b /vAb.

The sound speed, Alfven velocity and escape velocity at r = rb, expressed in thenormalization units, are cs,b = cs0,

vAb =

√2γ

BbB0

cs0 ≈ 1.1BbB0

cs0 (2.3.13)

and vesc,b =

√8γ

cs0 ≈ 2.2cs0. (2.3.14)

For the sake of clarity, we usually omit the normalization unit. For example, v = 5would denote a velocity of 5cs0, which is sonic Mach 5 at the jet nozzle.

20

2.4 CASES STUDIED

2.4 Cases Studied

In the following we present the results of two 3D simulations for two differentrotation profiles imposed at the nozzle, the Keplerian and rigid rotation profilesgiven by (Eq. 2.3.11, case K3) and (Eq. 2.3.12, case R3). These are compared with two2.5D simulations with the same initial and boundary conditions (cases K2 and R2).

In all cases, the initial state has a constant plasma-β of 1/9 (Bb = 3), the maximumrotation velocity at the boundary is vmax

ϕ,b = 0.33cs,b = 0.1vAr,b and the initial(half) opening angle is ϑb = 5.7 (rb = 5). This choice of parameters yields ajet with a magnetic pitch low enough to be unstable to kinks. At the same time,it avoids numerical problems found to arise with higher (supersonic) rotationvelocities as a boundary condition. The “grid noise” in the 3D simulations (thegrid is not axisymmetric in the rotation direction ϕ) turned out to be sufficient toexcite instabilities, so we did not need to apply a perturbation by hand. For thetemperature-control term, we used τK = 2.

In the 3D simulations, we used 384 logarithmically spaced grid cells in radialdirection and 96 uniformly spaced grid cells in each of the two angular directions.The physical extent of the simulated domain was 500 in the radial direction and33.8 in each angular direction. The ratio between the maximum and minimum ris 505/5 = 101. The 2.5D simulations were performed with the same resolution inthe radial direction and 64 grid cells in the evolved angular direction which had anextent of 16.9. The 3D simulations each ran for about one week (wall clock time)on 64 processors with MPI parallelization.

2.5 Results

The jets are initially accelerated mainly by gas pressure. This holds up to thesonic surface, which lies about halfway to the Alfven surface. Then, the Lorentzforce becomes the dominant driving force. The Alfven radii are at r ≈ 30 . . . 140,depending on the simulation and the direction ϑ: near the axis (small ϑ), the poloidalfield Br is amplified and the Alfven radii are at larger distances than in the outerregions (large ϑ), where Br is attenuated. The opening angles of the jets increasesomewhat with distance, but to a first approximation the flow can still be treated asconical, see Fig. 2.2. The jet front crosses the upper boundary (r = 505) at t ≈ 260 inthe 2.5D simulations and at t ≈ 330 in the 3D simulations. The latter are subject tomore numerical dissipation of kinetic energy, because the grid there is not symmetricin azimuthal direction. This reduces the injected Poynting flux, see Fig. 2.11. Apartfrom that, 2.5D and 3D simulations give, for our purposes, comparable results.Fig. 2.3 shows how the magnetic field is wound up inside the jet in one of thesimulations. A typical density and temperature distribution is shown in Fig. 2.4.

21


Figure 2.2: Jet border, defined as the polar angle within which a given percentage of energyflows (see Eq. 2.5.5 for the definition of Etot), as a function of distance. In all cases, most ofthe energy flow is contained within the ϑ ≈ ϑb = 5.7 surface, where ϑb is the initial openingangle. A minor but increasing amount of energy flows outside this angle. The energy densityin the jet (especially the toroidal field) causes it to decollimate somewhat compared with theconical configuration in which it is embedded.

Figure 2.3: Selectedmagnetic field lines inthe 3D simulation witha Keplerian velocityprofile. The right-handplot shows the entiredomain (r = 5 . . . 505),the left-hand plot onlythe lower part up tor ≈ 200. The colorcoding represents thestrength of the az-imuthal magnetic field,scaled by the initial fieldstrength. The magneticfield lines, which werepurely radial to beginwith, wind many timesaround the central axis,rendering it susceptibleto kink instabilities.

22

2.5 RESULTS

Figure 2.4: Density and temperature in a meridional slice in simulation K3 (Keplerian rotationprofile, 3D). The density is encoded in intensity, with light colors representing regions that areoverdense with respect to the environment. The (square root of the) temperature is encodedin the hue, with blue meaning cold and red meaning hot with respect to the environment.The hoop stress associated with Bϕ squeezes the plasma towards the central axis and createsan underdense cavity around the central part of the jet. At the boundary of this cavity theenvironment exerts the stress that confines the jet. The observed opening angle for a jet likethis would be smaller than the width of the cavity.

2.5.1 Expected Instabilities

The observed instabilities can be compared with expectations from linear stabilitytheory. To do this, we extract from the axisymmetric simulations the quantitiesthat enter the stability conditions, and then compare the result with the evidenceof non-axisymmetric instability in the corresponding 3D simulation. The availablestability conditions apply only to steady or static configurations and have beenderived either in the context of controlled fusion or cylindrical jets (Appl et al. 2000;Lery et al. 2000), hence the comparison can only be indicative.

According to the Kruskal-Shafranov criterion, the longitudinal wavelength of aninstability must be at least as high as the magnetic pitch on the unstable surface,defined to be the distance covered during one revolution of a helical field line aboutthe central axis. Besides being the result of linear stability analyses in the context ofcontrolled fusion, it can be derived heuristically from geometric arguments (Johnsonet al. 1958). Therefore, it should give a convenient scale also in cases for which it

23


Figure 2.5: Magnetic pitch as a function of distance in the 2.5D simulations. The magneticpitch is also the smallest possible wavelength of an instability. The dependence of h on thepolar angle ϑ is stronger in the Keplerian case (left plot), suggesting higher stability.

Figure 2.6: The expected onset of instability depends on the ratio of the azimuthal Alfvenspeed and the expansion velocity. This ratio is shown here for the 2.5D simulations. Thehorizontal dotted lines are for two estimates of the condition under which growth is possible(see text). Below the respective line, expansion prevails and an instability cannot grow.

24

2.5 RESULTS

Figure 2.7: Alfven crossing times with ι = π in the 2.5D simulations. In both cases, τcand with it the expected instability growth time increases with radius and distance fromthe central axis. The curves for r ≈ 100 represent an underestimate, because the jet stillaccelerates at this distance.

was not originally intended, like the expanding jets studied here. Deviations can beexpected e.g. from the effect of one-ended line-tying, which in some cases has beenfound to lead to increased instability as opposed to a configuration without a freeend (Furno et al. 2006; Lapenta et al. 2006; Sun et al. 2008).

For a conical jet, the magnetic pitch is

h = 2πR∣∣∣∣ Br

Bϕ

∣∣∣∣ (2.5.1)

on the ϑ = const surface. See Section 2.7 for a derivation of this expression. In thesimulations, h decreases with r and settles to a constant value above the Alfvenradius, see Fig. 2.5 (compare also Fig. 2.3). The variation of the pitch with ϑ dependson the kind of rotation imposed at the lower boundary. In the Keplerian case, thedependence is strong, with the asymptotic pitch being approximately 10 near theaxis and 40 at the limb of the jet. In the rigid rotation case, h ≈ 25 in all directionswithin the jet.

The Alfven crossing time in a conical, unaccelerated jet, defined as the time ittakes an azimuthal Alfven wave to orbit the central axis, is given by

τc =ιR

vAϕ − ιvR(2.5.2)

25


where ι = 2π for a full revolution, vAϕ = const is the azimuthal Alfven speed andvR = vr sin ϑ is the expansion velocity. In the simulations, vAϕ ≈ const above theAlfven radius. This is as expected theoretically from conservation of mass andmagnetic flux in a conically expanding, steady axisymmetric jet. τc is finite andphysically meaningful only if the condition

vAϕ

vR> ι (2.5.3)

is satisfied. If it is not, the expansion takes place too fast for an Alfven wave tocross the jet and an Alfven mode instability cannot grow. While the critical valueof ι is arguable, we note that causal contact across the jet by Alfven waves is onlypossible if ι ≥ π. The situation in our simulations is illustrated in Figs. 2.6 and 2.7.Instabilities can grow only slowly on magnetic surfaces with large ϑ. Depending onthe ι needed for efficient growth, they may even be stalled due to the jet’s expansion.In any case, instabilities grow most rapidly if they start at small r. In regions wherethe jet is accelerating (dvr/dr > 0) or decollimating (dϑ/dr > 0 along a field line),the effective Alfven crossing time is underestimated by Eq. (2.5.2). The jet is thenstabler than condition (2.5.3) suggests.

2.5.2 Instabilities Found in the Simulation

We observe non-axisymmetric, kink-like distortions in the magnetic field and otherquantities in both 3D simulations. They emerge near the Alfven radius, propagatewith the flow and grow in amplitude along the way. It is convenient to look at thecurrent density j = c

4π ∇× B for a quantitative analysis. The radial componentjr is related to Bϕ and is as such characteristic for the distortions in the magneticfield. In the unperturbed case, it is concentrated about the central axis and alongthe outer boundary of the cavity illustrated in Fig. 2.4, with respectively oppositeorientation. In our simulations, Bϕ is directed in negative ϕ-direction and the axialcurrent, accordingly, in negative r-direction. We denote this backward current withj−r , so that jr = j+r + j−r .

In the rigid rotation case (R3), the distortions attained large amplitudes of severaldegrees. Looking at Bϕ in the r = const plane, we find that the whole jet is affectedby the kink. The number of visible radial nodes is 2–4, corresponding to wavelengthson the order of 150, i.e. several times larger than the magnetic pitch. Owing to thedistortions in the magnetic field, the axial current was perturbed as shown in Fig. 2.8on the right-hand image. j−r helically twines around the central axis in reminiscenceof “ideal” kink instabilities with an azimuthal mode number m = 1.

To analyze the unstable displacements, we determine the barycenter of the back-ward current j−r in the r = const plane, denoting its location with (ϑj, ϕj). The result,from which one can directly read off amplitudes and wavelengths, is shown inFig. 2.9. The slope of the points of constant phase ϕj in the r-t diagram corresponds

26

2.5 RESULTS

Figure 2.8: Radial component of the current density (∇× B) in the 3D simulations just beforethe jets reach the upper boundary. The rod in the middle of the jets is their central axis(y-axis). Helical distortions, characteristical for kink instabilities, can be seen in the backwardcurrent (blue) in both cases. The amplitudes and wavelengths are significantly larger in thesimulation with a rigid rotation profile (right-hand image).

27


Figure 2.9: Position of the barycenter of the backward current (blue material in Fig. 2.8) insimulation R3. The left-hand map shows the amplitude of the instabilities and the right-handone its phase. The blank region is where the jet has not been yet, its border marks the jetfront. Observers moving with the flow follow a time-position curve which is (approximately)parallel to the front, i.e. the instabilities are at rest with respect to a comoving frame.

Figure 2.10: Amplitudeof the perturbations(crosses, left-hand axis)as seen by an observer(red line, right-hand axis)located just behind thejet front (upper panel)and located 130 lengthunits behind the jetfront (lower panel) insimulation R3. Theexponential fits (solidline, left-hand axis)yield the instabilitygrowth time.

28

2.5 RESULTS

Figure 2.11: Energy flow rates through r = const in 3D (solid lines) and in the 2.5D (dottedlines) simulations with a rigid rotation profile. There is no clear evidence of additionalconversion of Poynting flux to kinetic energy. The energy flow through the lateral boundariesis virtually zero at all times. The situation is similar in the Keplerian case (K3 and K2).

with the flow velocity vr. That is, the instabilities are at rest with respect to a comov-ing frame. We estimate the growth time in such a frame by introducing an observermoving with flow and measure ϑj in doing so. We find strictly increasing, exponen-tial growth if the observer is located just behind the jet front, see upper panel inFig. 2.10. The exponential growth time τg is generally on the order of the Alfvencrossing times shown in Fig. 2.7. For observers which are farther behind the jetfront, the amplitude does not follow a simple exponential increase, see lower panelin Fig. 2.10 for an example. Rather, it saturates and even declines in some cases. Thereason for this is not clear. We cannot rule out the possibility that there is stabilizingfeedback from the upper boundary. Considering that the flow is super-Alfvenicthere, this seems unlikely though.

In the Keplerian rotation case (K3), the jet also exhibits kink-like distortions, seeleft-hand image in Fig. 2.8. However, the perturbation amplitudes are much smaller,with ϑj attaining peak values of about 1.4 directly behind the jet front and onlyabout 0.5 farther behind. Unlike in the rigid rotation case, only inner regions of thejet are affected by the kinks, the jet border is relatively unharmed. The wavelengthsare on the order of 25–50, i.e. there are more radial nodes than in the rigid rotationcase. Even for an observer traveling just behind the jet front, the amplitude is notstrictly increasing, but saturates and tapers off after an initial rise with a growth time

29


Figure 2.12: Radial velocities in the 3D (solid lines) and in the 2.5D (dotted lines) simulations.The jets do not accelerate above the Alfven radius, despite instabilities. The value of vrnear the central axis in simulation R3, though rising, does not give conclusive evidence ofacceleration, because the jet’s axis is strongly distorted by the instabilities.

of τg ≈ 50, i.e. also on the order of the relevant Alfven crossing times in Fig. 2.7.

2.5.3 Impact On Dynamics and Energetics

Instabilities release magnetic energy by transforming it into kinetic energy, but fordissipation in the sense of magnetic reconnection sufficiently small length scaleshave to develop. In ideal MHD simulations like ours, such dissipation is present inthe form of numerical diffusion due to the effect of interpolations across adjacentgrid cells. The effect cannot be modeled by Ohmic resistivity but can be quantifiedthrough secondary effects like changes in the energy fluxes.

We did not find conclusive evidence of magnetic field dissipation provoked bythe instabilities in the 3D simulations. For example, the flow of nonradial magnetic

30

2.6 DISCUSSION AND CONCLUSIONS

field ∫r=const

√B2

ϑ + B2ϕ vr dA (2.5.4)

shows asymptotically the expected ∼r behavior (as Bϕ ∼ r−1 and A ∼ r2) withminor fluctuations but no decreasing trend when compared to the 2.5D simulations.

It is helpful to look at the energy fluxes. From Eq. (2.3.6), the energy flow rate inpoloidal direction is

Etot(t, r) =∫

r=const

[(12

ρv2 +γ

γ− 1p + ρΦ

)vr + Sr

]dA. (2.5.5)

Decomposing the integral from left to right, we identify the flow rates of kineticenergy Ekin, thermal enthalpy Ethrm, gravitational potential energy Egrav and mag-netic enthalpy Emag. We plotted these in Fig. 2.11 for the simulations R3 and R2.The total energy flow Etot rises for small r due to the temperature-control term Kin Eq. (2.3.1c). The conversion of Poynting flux to kinetic energy flux in the 3Dsimulation looks qualitatively the same as in the 2.5D comparison simulation. Inparticular, there is no evidence of an additional conversion of Emag to Ekin due tomagnetic field dissipation. This agrees with the fact that there is no additionalacceleration of the jets, see Fig. 2.12.

2.6 Discussion and Conclusions

We have simulated magnetocentrifugally driven, conical jets over a range in dis-tance of 1000 times the initial jet radius, in both 3D and axisymmetric 2.5D. Thecalculations extend to a factor of about 5–10 beyond the Alfven surface. The 3D jetsdeveloped non-axisymmetric instabilities of the kink kind.

The violence of the instabilities depends on the rotation profile applied at thebase. With a rigid rotation (∝R) profile, the perturbations grow to much largeramplitudes than with a Keplerian (∝R−1/2) profile. We suspect that the reasonfor the differing behavior lies in the magnetic shear, defined as the variation ofthe magnetic pitch with distance to the axis. In the rigid rotation case, there isvirtually no shear as opposed to the Keplerian case, for which the pitch increaseswith distance from the axis, see Fig. 2.5. A shear-free configuration is expected to beunstable to non-resonant modes, whereas a configuration with increasing pitch isexpected to be unstable to modes with a resonant surface inside the jet (Appl et al.2000; Lery et al. 2000). This fits well with what we observe in the simulations, viz.that the kink is confined inside the jet in the Keplerian case. Heuristically speaking,the differing behavior could be attributed to the fact that the outer (high ϑ) layersof the jet, which are more stable (higher magnetic pitch), damp internally arisingmodes in the Keplerian case.

In both cases, the longitudinal wavelength of the instabilities is ∼5 times largerthan the value of the magnetic pitch near the axis. The relation is qualitatively

31


consistent with the findings of Appl et al. (2000) for a cylindrical jet. The growthtime of the instabilities is on the order of the Alfven crossing time. The exact relationis difficult to determine, because the crossing time as well as the location of theresonant surface can only be estimated. As the azimuthal magnetic field strengthand with it the azimuthal Alfven speed decrease past the Alfven surface, opposingparts of the jets become causally disconnected from each other. Thus, the jet expandstoo fast for Alfven mode instabilities to grow. The effect is amplified if the jet isdiverging. Recollimation, on the other hand, should boost the growth of instabilities.

As found in other studies, the conversion of magnetic enthalpy (Poynting flux)to kinetic energy is fairly efficient, on the order 70%. Dissipation of magneticfields by internal instabilities is expected to contribute additional acceleration ofthe flow (Drenkhahn 2002). The calculations do not show a clear signature of thisprocess. It seems that either the observed region is too small, and/or the numericaldissipation of magnetic fields is too weak. Also, from a macroscopic point of view,the instabilities were not violent enough to bring together fields with an antiparallelcomponent, as is necessary for magnetic reconnection to occur. Moreover, most ofthe magnetic enthalpy was already converted in the magnetocentrifugal accelerationprocess. Therefore, even if there was magnetic dissipation, the effect would notbe dramatic. Nevertheless, we found that the magnetic field gets significantlydistorted by the instabilities. This should facilitate magnetic field dissipation furtherdownstream but it may be necessary to extend the calculations to larger distances tosee the effect.

It is tempting to compare the instability-related structures in the simulationswith structures in observed jets. The 3D jet structure in Fig. 2.4, for example, isreminiscent of the semi-regular patterns seen in Hα images taken of outflows fromyoung stellar objects (YSO) like HH 34 (Reipurth et al. 2002). There are, however,other possible interpretations of the observed structure. The wiggles in YSO jetscould also be the result of a precessing or orbitally moving source (Masciadri &Raga 2002, and references therein). The symmetric nature of structures often seen injet and counterjet (e.g. HH 212 Wiseman et al. 2001) suggests a modulation of theoutflow speed or mass flux originating at the source of the outflow rather than aninstability developing further away. The irregularities caused by the instabilitiesstudied here are possibly more important for internal magnetic energy release insidethe jet than for major observable structures like the knots and wiggles in YSO jets,though they are likely to contribute to these at some level as well.

32

2.7 APPENDIX: MAGNETIC PITCH FOR A CONICAL JET

2.7 Appendix: Magnetic Pitch for a Conical Jet

h(r0)

r0

slopeBr

Bϕer

eϕ

ϑ = const

Figure 2.13: Magnetic field linewith pitch h on a conical surface.

The radial progress of the field line depicted inFig. 2.13 is determined by

drRdϕ

=∣∣∣∣ Br

Bϕ

∣∣∣∣ =: a(r) (2.7.1)

where R = r sin ϑ is the distance to the polar axisand a(r) is the unsigned slope. Assuming that Br ∝r−2 and Bϕ ∝ r−1 due to magnetic flux conservation,we can write

a(r) = a0r0

r. (2.7.2)

By integrating the resulting expression we obtainthe radial distance covered after one revolution:∫ r0+h

r0

dr =∫ 2π

0a0R0 dϕ ⇒ h = 2πa0R0.

(2.7.3)Alternatively, we could also define a local magneticpitch h by taking a = a0 = const for the slope. Theresult is

hr0

= exp (2πa0 sin ϑ)− 1. (2.7.4)

The difference between h and h turned out to be insignificant in our analysis. This isunderstandable since h → h for small a0 sin ϑ.

33


Figure 2.14: Radial magnetic field in ameridional slice (bottom, with the left-hand plot showing only the lower part ofthe computational domain) and volume-rendered density (right-hand side) in simu-lation K3. The tension force of the twistedmagnetic field component squeezes ma-terial towards the center, forming a“backbone” with increased poloidal fieldstrength and density. This may reduce theobserved width of the jet, cf. Fig. 2.4.

34

2.7 APPENDIX: MAGNETIC PITCH FOR A CONICAL JET

Figure 2.15: Parallel magnetic field in an r = const section through the jet of simulation R3 atdifferent times. The sequence shows how the jet totters away and around the central axis as aconsequence of the instabilities. The jet’s cross section pertains a roughly round shape, as ischaracteristic for an m = 1 mode.

35

3 Decay of the Toroidal Field inMagnetically Driven Jets

R. MollAstronomy & Astrophysics 2009, Vol. 507, pp. 1203–1210

Abstract: A 3D simulation of a non-relativistic, magnetically driven jetpropagating in a stratified atmosphere is presented, covering about threedecades in distance and two decades in sideways expansion. The simu-lation captures the jet acceleration through the critical surfaces and thedevelopment of (kink-)instabilities driven by the free energy in the toroidalmagnetic field component. The instabilities destroy the ordered helicalstructure of the magnetic field, dissipating the toroidal field energy on alength scale of about 2–15 times the Alfven distance. We compare the resultswith a 2.5D (axisymmetric) simulation, which does not become unstable.The acceleration of the flow is found to be quite similar in both cases, butthe mechanisms of acceleration differ. In the 2.5D case approximately 20%of the Poynting flux remains in the flow, in the 3D case this fraction islargely dissipated internally. Half of the dissipated energy is available forlight emission; the resulting radiation would produce structures resemblingthose seen in protostellar jets.

3.1 Introduction and Rationale of the Calculations

A magnetized outflow produced by a rotating magnetic object has become thedefault interpretation for objects ranging from protostellar jets to gamma-ray bursts.The outflow in this model contains a tightly wound helical magnetic field. Such anearly toroidal field represents a source of free energy that makes the flow inherentlyprone to non-axisymmetric magnetic instabilities. In this study we investigatethe longer-term development of such instabilities and their consequences for jetphenomenology.

Instabilities are not necessarily fatal for the jet. Kink instabilities of helical mag-netic field configurations typically saturate at a finite amplitude. Such instabilitiesat moderate amplitudes have been invoked to explain phenomena like the wigglyappearance of Herbig-Haro objects (Todo et al. 1993) or the orientation of VLBI jetsin AGN (Konigl & Choudhuri 1985).

37

3 DECAY OF THE TOROIDAL FIELD IN MAGNETICALLY DRIVEN JETS

The development of kink instability in a jet is not the same as in a laboratoryconfiguration. Due to the sideways expansion of the flow, the ratio of poloidal(stabilizing) to toroidal (destabilizing) field components decreases with distancealong the flow. Conditions favorable for instability are thus continually recreatedin such a flow. In Chapter 2 we presented 3D MHD simulations showing the onsetof instabilities in an expanding jet created by twisting a purely radial magneticfield. The degree of instability was found to depend on the kind of rotation thatgenerates the twist. The highest degree of instability was attained with a constantangular velocity (rigid rotation); the jet produced in this case was subject to helicaldeformations with large amplitudes, causing sideways displacements of severaldegrees. However, the magnetic structure of the jet was not disrupted within thecomputational volume, and the instability did not lead to a significant decrease ofthe Poynting flux. These results indicate the need to follow the instabilities to largerdistances from the source. We present here the results of simulations extending to adistance of 1000 times the diameter of the jet source.

In Chapter 2 we showed how the degree of instability depends on the way inwhich the jet is collimated by its environment. If collimation conditions are such thatthe opening angle of the jet increases with distance, the Alfven travel time acrossthe jet increases more rapidly than the expansion time scale. As a result, instabilitiessoon “freeze out”, and decay of the toroidal field becomes ineffective. In bettercollimating jets, such that the opening angle narrows with distance, instability isalways effective. The calculations presented here are for such a case. It is probablythe most relevant for both AGN and protostellar jets. Observations of the jets inM87 (Junor et al. 1999) and HH30 (Mundt et al. 1990), for example, show a rapidlydecreasing opening angle in the inner regions of the jet.

The dissipation of the magnetic energy in the toroidal field heats the plasma.Radiation produced by this plasma would be an alternative or complement tothe standard mechanism invoked, which relies on dissipation in internal shocks.Observations such as those of M87 and HH30 indicate that dissipation starts fairlyclose to the central object, compared with most observable length scales, but alsothat the innermost regions, perhaps comparable to the Alfven distance rA, are quiet.Inferences from VLBI observations indicate that AGN jets are not magneticallydominated on most observable scales (cf. Sikora et al. 2005, and references therein);decay of the magnetic field relatively close to the source of the jet would fit thisobservation (see also discussion in Giannios & Spruit 2006).

If dissipation takes place close enough to the source, it is possible that it can bestudied realistically by 3D simulations with current computational resources. InChapter 2, we already found indications that decay of the toroidal field may becomeimportant at distances as close as 10–30 times rA.

The release of magnetic energy by the instability may also be important foraccelerating the flow (Drenkhahn 2002). Dissipation of toroidal field causes themagnetic pressure gradient along the jet to steepen; this adds an accelerating forcethat is absent when the toroidal field is conserved (Giannios & Spruit 2006; Spruit

38

3.2 METHODS

2008).To facilitate extraction of physics from the numerical results, the 3D results in

the following are compared systematically with a 2.5D (axisymmetric) simulationcorresponding to the same initial and boundary conditions.

3.2 Methods

3.2.1 Numerical MHD Solver, Grid and Coordinates

The following is a brief summary; for details see Chapter 2 where the same numericalapproach but different initial conditions were used.

We numerically solve the ideal adiabatic MHD equations in a static external grav-itational potential Φ ∝ r−1 on a spherical grid (r, θ, φ). In the 2.5D, axisymmetricsimulation, the jet propagates along the coordinate axis θ = 0. In the 3D simulation,the jet’s axis is taken in the direction θ = φ = π/2. The jet thus propagates in equato-rial direction of the coordinate system. This avoids the coordinate singularity alongthe polar axis (θ = 0), which is numerically problematic for trans-axial flows suchas are caused by instabilities. The computational volume covers a range ∆θ = ∆φthat comprises about twice the expected opening angle of the jet. The spacing of thecomputational grid is uniform in the angular directions and logarithmic in the radialdirection. In this way, the varying numerical resolution approximately matches theincrease of the natural length scales in the expanding jet.

For presentation and discussion, we transform the results to a different coordinatesystem. This is again a spherical coordinate system, but with the polar axis alignedwith the jet. The polar and azimuthal angles in this system are denoted by ϑ and ϕ,respectively:

sin2 ϑ = 1− sin2 θ sin2 φ, (3.2.1)tan ϕ = tan θ cos φ (3.2.2)

in the 3D simulation and ϑ ≡ θ, ϕ ≡ φ in the 2.5D simulation. R := r sin ϑ is usedto denote the distance to the axis (cylindrical radius).

Dissipation of magnetic energy in the flow causes heating. In nature this wouldlead to losses by radiation; in the computations it can cause numerical problems inregions where the magnetic energy dominates. Instead of a more realistic model forsuch losses, a temperature-control term is added in the energy equation, such thatthe temperature relaxes to that of the initial state on an appropriate time scale. Inthe simulations presented here this time scale is chosen such that the temperaturestays within about a factor 100 around the initial value.

39


3.2.2 Initial and Boundary Conditions

The initial state consists of a current-free magnetic field embedded in a stratifiedatmosphere in the gravitational field of a point mass. The field configuration of thisinitial state is of a “collimating” type, the distance between neighboring field linesincreases less rapidly than the distance from the source r. In Chapter 2, we showedthat instabilities develop more strongly in such collimating configurations than in apurely radial initial field.

3.2.2.1 Initial Field Configuration

The initial field is axisymmetric around the jet axis and hence can be written as

B =1R

∇ψ× eϕ = ∇×(

1R

ψeϕ

)= ∇× A, (3.2.3)

where ψ is the stream function of the field. We construct the initial condition as alinear combination of the stream function of a monopole field, given by

ψmono ∝ 1− cos ϑ, (3.2.4)

and a field with the stream function

ψpara ∝

√(RR0

)2+(

1 +z

R0

)2−(

1 +z

R0

), (3.2.5)

(z > 0) of which the field lines have a parabolic shape (Cao & Spruit 1994). Here,z = r cos ϑ denotes the height along the jet’s central axis (which is actually the y-axisin the 3D simulation and the z-axis in the 2.5D simulation). The weighting of thetwo components is parametrized with the quantity

ζ :=Bmono

Bpara

∣∣∣∣∣r=rb,ϑ=0

, (3.2.6)

the relative strength of the constituent fields at the lower boundary r = rb. ζ → ∞corresponds to a pure monopole field and ζ = 0 corresponds to a pure parabolicfield. The radius requ at which Bmono = Bpara on the central axis follows from

ζ =r2

equ(R0 + rn)r2

n(R0 + requ). (3.2.7)

The lower the value of ζ or requ, the more collimating is the shape of the initial field.

40

3.2 METHODS

3.2.2.2 Stratification

We impose the static gravitational field of a point mass, located at the origin of thecoordinate system. The stratification is initially in hydrostatic equilibrium in thispotential. The gas pressure is chosen such that the plasma-β := p/pmag in the initialstate is approximately constant at small radii (requ), where the monopole fielddominates: p ∝ r−4, ρ ∝ r−3, T ∝ r−1, cs ∝ r−1/2 and vA ∝ r−1/2 in the initial state.At large radii, the value of the initial β is reduced as the magnetic field strengthdecreases less rapidly than r−2.

3.2.2.3 Boundaries

Boundary conditions are maintained through the use of “ghost cells” outside thecomputational domain. At the sides (θ in the 2.5D simulation, θ and φ in the 3Dsimulation) and top (upper r), we use open boundary conditions that allow for analmost force-free outflow or inflow of material, including magnetic fields.

The bottom boundary is located at a finite height rb above the origin of thegravitational potential. This distance is 1/200 of the size of the computationalvolume. The jet is generated there by a “rotating disk”1 of radius Rb around theaxis. It is implemented by maintaining a velocity field v = vϕ eϕ in the ghost zones:vϕ = vmax

ϕ,b R/Rb for R ≤ Rb and 0 elsewhere. The disk thus rotates rigidly like inthe cases R2 and R3 in Chapter 2.

All quantities at the bottom boundary except for B are fixed at their initial valuesin the ghost cells. B is extrapolated from the interior of the domain.

3.2.3 Parameters and Units

The above suggests a specification of the problem in terms of 7 parameters: the fieldconfiguration parameter ζ, a scale for the field strength, scales for the pressure anddensity, the bottom boundary location rb, the radius Rb of the rotating disk, and itsrotation rate. Because of the symmetries of the problem, it is actually determined byonly 4 parameters; the remaining 3 dependences are equivalent to scaling factors.As the 4 independent parameters we choose the following dimensionless quantities:the plasma-β of the initial state, the angle ϑb = arctan(Rb/rb) which controls theopening angle of the flow, the field configuration parameter ζ, and finally theAlfvenic Mach number MA of the rotating velocity field at the edge (R = Rb) of thelaunching disk, which controls the power of the jet. They have the values β = 1/9,ζ = 30, ϑb = 5.7, MA = 0.1.

The units used for reporting the results below are the length scale l0 ≡ 2Rb, andthe pressure and density on the axis at the bottom boundary, p0 ≡ pb, ρ0 ≡ ρb.

1Strictly speaking, because of the spherical grid, what we call “disk” here is really a spherical cap withsmall curvature.

41



Quantity Symbol(s) Unitlength x,y,z,r,R l0

gas pressure p p0density ρ ρ0

mass flow rate M ρ0l30/t0

velocity v cs0 =√

γp0/ρ0time t t0 = l0/cs0

energy E p0l30

energy flow rate E p0l30/t0

force density F p0/l0power P P0 = p0l3

0/t0magnetic flux density B B0 =

√8πp0

current density j j0 = B0c/(4πl0)

Together with the 4 model parameters β, ζ, ϑb and MA, the units for other quantitiesfollow from these as shown in Table 3.1.

The simulations cover a distance of 2000 times the initial jet radius Rb in thespherical range 5 < r < 1005. The resolution used in the 3D simulation is 768×128× 128; the corresponding domain size is 1000× 33.8 × 33.8. The resolutionused in the 2.5D simulation is 768× 96; the corresponding domain size 1000× 16.8.In both simulations, the radial width ∆r of the grid cells increases from 0.03 at thelower boundary to 6.92 at the upper boundary in logarithmic steps.

3.3 Results

The 3D simulation was done using MPI parallelization on 128 CPUs, the 2.5Dsimulation was done with OpenMP parallelization on 32 CPUs.

The jet crossed the upper boundary of the computational volume at the physicaltime t ≈ 523 on the 13th wall clock day of the 3D simulation. We stopped it after26 days, at which time t = 1055 had been reached. The 2.5D simulation ran for 24hours, reaching t = 1732.

The 3D jet is subject to non-axisymmetric instabilities, evidently of the kink(m = 1) kind. They have a disruptive effect on the magnetic field structure andcause the toroidal field to decay, see Fig. 3.1. If such a jet was observed, it wouldprobably look similar to Fig. 3.2, with bright knots and wiggles being prominentfeatures. The knots move at a substantial fraction of the flow speed, sometimesmerging or fading before leaving the computational domain. The 2.5D jet does notexhibit any form of instability.

42

3.3 RESULTS

Figure 3.1: a) Selected magnetic field linesin the 3D simulation on successively in-creasing length scales. The color codinggives the magnetic field strength relativeto its initial value. The field lines shownare the ones that are anchored in the ro-tating disk at the lower boundary. Thejet starts out with a helical magnetic field(left image) whose toroidal component be-comes increasingly stronger (middle im-age) until instabilities disrupt the orderedstructure, the toroidal field decays andthe field becomes predominantly poloidal(right image). b) Radial component of thecurrent density (∇× B) in the 3D simu-lation. The backward current, shown inblue here, is concentrated on the centralaxis until being first displaced and thendisrupted by instabilities.

43


Figure 3.2: Volume rendered image of the 3D jet that shows what it might look like in obser-vations. For the volume rendering, a simple model was used in which emissivity and opacitydepend on temperature and magnetic field strength. The jet exhibits a wiggly structure withbright knots, produced by the instabilities and the dissipation of magnetic energy.

44

3.3 RESULTS

Figure 3.3: Maximum velocity in jet direction as a function of distance. The inset shows thelocation of the critical surfaces. The flow passes first the sonic, then the Alfven and finally thefast magnetosonic surface.

Figure 3.4: Jet boundary, defined by the angle that encloses 95% of the total energy flow (seeEq. 3.3.1), as a function of distance. The unstable 3D jet is less collimated than the 2.5D jet.

3.3.1 Acceleration, Collimation and Mass Flow

Most of the acceleration takes place below r = 100, where the jet reaches &80% of itsterminal speed, see Fig. 3.3. The location of the sonic (vr = cs ≈ csvA/(c2

s + v2A)1/2,

the slow magnetosonic cusp speed), Alfven (vr = vAr) and fast magnetosonic(v2

r = c2s + v2

A) radii depends on the direction ϑ, see inset in Fig. 3.3. This is becausethe acceleration is more effective near the boundary of the jet, which is a consequenceof the rigid rotation profile used in its generation, and because the poloidal magnetic

45


field is being redistributed such that the local Alfven velocity becomes relativelyhigh near the axis. As the toroidal field energy, which determines the jet accelerationand the development of instabilities, is concentrated towards the jet boundary (asopposed to the axis), the Alfven radius there is probably the most important for thesubsequent considerations.

The central velocity tends to be higher in the 3D simulation, presumably due to amore effective transfer of momentum from the boundary to the center. This may,together with the entrainment of ambient material discussed below, also explainwhy the peak velocities are somewhat lower in the 3D simulation.

The instabilities have a noticeable effect on the collimation behavior, see Fig. 3.4.In the 2.5D simulation, the opening angle of the jet decreases by about 2 in the firsthalf of the computational volume, which is about 1 less than what is marked bythe shape of the initial magnetic field. In the second half, well beyond the Alfvensurface, the opening angle settles to a constant value. The jet in the 3D simulation isless collimated2, the location of the boundary fluctuates with time in the unstableregion. Averaged over time, we find it to be nearly conical.

The mass flow rate M(t, r) :=∫

r=const ρvr dA is somewhat higher in the 3Dsimulation and subject to strong fluctuations above r ≈ 200: the mean value overall radii at t ≈ 1060, measured in the unit for M listed in Table 3.1, is 0.068± 8%in the 2.5D and 0.082± 34% in the 3D simulation. The average over several timesteps shows a slight increase of M with r in the 3D case, whereas no trend can bededuced in the 2.5D case. This may be an indicator for an enhanced entrainment ofambient material caused by the instabilities. It may also in part explain the lowerpeak velocities in the 3D simulation.

3.3.2 Energy

A comparison of the different kinds of energy flow rates gives information aboutenergy transformations taking place in the jet. Integrating the radial energy flux

12

ρv2vr︸︷︷︸kinetic

+γ

γ− 1pvr︸︷︷︸

thermal enthalpy

+ ρΦvr︸︷︷︸grav. potential

+ Sr︸︷︷︸magnetic enthalpy

(3.3.1)

over the r = const surface, we obtain the rate of energy Etot(r) that crosses thesurface. The individual components, which we denote with Ekin, Ethrm, Egrav andEmag, are plotted in Fig. 3.5. As the values are strongly fluctuating in the unstableregion, we averaged over time in the 3D case. Emag is reduced by 80% in the 2.5Dsimulation and by 94% in the 3D simulation along the simulated distance, the gapbetween the curves widens considerably where the 3D jet exhibits instabilities.

2Note, however, that with our definition of the jet boundary, a rigid displacement away from the centeralso increases the opening angle.

46

3.3 RESULTS

Figure 3.5: Energy flow rates through the r = const surface. Magnetic enthalpy (integral overPoynting flux, blue line) is reduced in the 3D simulation as B2

ϕ decays as a consequence ofinstabilities.

We presume this to be caused by (numerical) magnetic dissipation, which turnsmagnetic energy into heat, causing an increase in Ethrm. The environmental thermalenergy is not affected much. As our calculations include an energy loss term (seeSect. 3.2.1), there is no one-to-one correspondence between the dissipated energyand the increase in Ethrm.

The final contribution of Emag to Etot is 24% in the 2.5D simulation and 9% in the3D simulation. Ekin is up to about 10% smaller in the 3D simulation and does notshow a dissipation-induced increase. Ekin arises from azimuthal and radial motion,with the relative share being highly similar in the 2.5D and 3D case: the azimuthalcontribution drops continuously to values <5% at r > 200 and the contributionfrom vϑ is insignificant throughout.Emag can be decomposed further. The radial component of the Poynting vector

has 4 terms,

Sr =1

4π

(B2

ϑvr + B2ϕvr − BϑBrvϑ − BϕBrvϕ

). (3.3.2)

Integrating these terms over the r = const surface gives the components [in orderof their appearance in Eq. (3.3.2)] E1

mag . . . E4mag which, added together, give Emag.

47


These components of the magnetic enthalpy flow rate are plotted in the lower panelof Fig. 3.5. At very small radii, the most important term is E4

mag, the work done by theazimuthal flow against the azimuthal component of magnetic stress. Its contributionto Emag decreases rapidly, however, falling below 50% at r ≈ 30 (i.e. well below theAlfven surface, near the sonic one). It is then taken over by E2

mag, which describes theflow of magnetic enthalpy stored in the azimuthal field. E1

mag has minor significancein the 3D simulation, which may be due to the perturbed toroidal field having alsoa Bϑ component (a rigid displacement of a pure azimuthal field in the r = constplane introduces a non-azimuthal component). E3

mag is insignificant in both cases.The strong decrease of Emag in the 3D simulation is caused by the decrease of E2

mag.We find a net outflow of magnetic enthalpy through the lateral boundaries of the

computational volume at the height of the jet front, with peak rates of about 0.15in the 2.5D case and 0.05 in the 3D case. The outflow is transient in the 2.5D case,vanishing quickly after the jet front leaves the computational volume. In the 3Dcase, however, it turns into an inflow of the order −0.05 which persists until the endof the simulation. The energy in the radial magnetic field increases correspondingly,mainly outside the jet at ϑ > 5.7.

3.3.3 Magnetic Field: Poloidal vs. Toroidal

The red line in Fig. 3.6 shows the magnetic flux Ξ contained within an angle ϑ < 5.7

from the axis,Ξ(r, t) :=

∫r=constϑ<5.7

Br(t) dA, (3.3.3)

divided by its initial value. For comparison, the green curve shows Ξ+, the flux ofonly those field lines that have the same direction as the initial field (the green andthe red curves coincide in the 2.5D case). Although the 2.5D jet fills only part of the5.7 cone, it causes a significant reduction of Ξ . This reflects the lateral expansionof the jet due to the pressure exerted by Bϕ. There is less reduction in the 3D case,with Ξ being near the original value at very large distances. The difference becomesevident where the 3D jet is unstable, showing that it is due to the dissipation ofthe toroidal field. There is only a small difference between the red and green curvein the 3D case, meaning that negative values of Br contribute little to the net flux.This shows that the toroidal field component dissipates without producing much“tangling” of the field lines.

Fig. 3.7 compares the mean energy in the poloidal and toroidal magnetic fieldsover the width of the jet as a function of distance, the width of the jet being definedby a suitable velocity threshold. The 3D and 2.5D jets start out similar, with the meanpoloidal and toroidal fields becoming comparable near the Alfven surface. Beyondthat distance, the mean toroidal field energy increases more strongly in the 2.5D case,roughly proportional to r−2. In the 3D case, the instability-induced destruction ofthe toroidal field causes the slope to steepen substantially at r & 200, approximately

48

3.3 RESULTS

Figure 3.6: Red: magnetic flux Ξ(r) within 5.7 from the jet axis, normalized by its initialvalue. Green: flux of B+

r > 0 (B = B+r + B−r ), see Sect. 3.3.3.

Figure 3.7: Mean magnetic energies in the poloidal and toroidal fields within the jet, normal-ized by the initial on-axis magnetic energy, and their ratio. Without instability (2.5D), thefield becomes predominantly toroidal at large distances. With instability (3D), it becomespredominantly poloidal.

Figure 3.8: Magnetic field components, integrated over the jet cross section, as a function ofdistance, on a logarithmic scale. In a spherically expanding ballistic flow these quantitieswould be constant, see text.

49


Figure 3.9: Poloidal magnetic field lines and radial velocity field, plotted as a function ofdirection ϑ (horizontal axis) and distance r (vertical axis); a conical magnetic surface wouldappear as a vertical line. The left-hand plot shows a snapshot of the 2.5D case, the right-handplot shows the 3D case with the magnetic field and velocity being averaged in azimuthaldirection as well as time. Both plots show the same field lines in the sense that they startfrom the same direction at the lower boundary. The angular separation between field lines inthe outer, high-ϑ part of the jet (here: between the 4th and 5th line from the left) increasesnoticeably in the 2.5D case, leading to acceleration by the magnetic “diverging nozzle” effectthere, see discussion.

as 〈B2ϕ〉 ∼ r−3. The mean poloidal field energy, on the other hand, stays near the

initial value in the 3D case while decreasing somewhat in the 2.5D case. Taking theratio between the energies, we find the magnetic field to be predominantly toroidalin the 2.5D case and predominantly poloidal in the 3D case at large distances.

We also experimented with means of the form∫

Xvr dA/∫

vr dA, where theintegral is performed over the whole r = const surface, and obtained similar results.Taking the mean within the static ϑ < 5.7 cone instead of using a velocity thresholdyields a smaller value for 〈B2

ϕ〉/〈B2p〉 in the 2.5D case, viz. 2± 0.2 for r > 200. This

is because an angle of 5.7 includes more of the environment of the jet (cf. Fig. 3.4).In a ballistically expanding jet (constant velocity and opening angle) the magnetic

field components vary with distance r as Br ∼ r−2, Bϑ ∼ Bϕ ∼ r−1. Integrals overthe width of the jet of Br, B2

ϕ and B2ϑ are then constants. These integrals are shown

in Fig. 3.8. The ballistic approximation works well in the 2.5D case, for Br as well as

50

3.3 RESULTS

Figure 3.10: Rate of work done in the volume delimited by the lower boundary and r by thecomponents of the Lorentz force in the direction of the flow. The power delivered by the totalLorentz force is about the same in the 3D and 2.5D cases.

for Bϕ, if the acceleration region is excluded. In the 3D case however, the integral ofBr increases by about an order of magnitude along the jet, while the integral of B2

ϕdecreases over the entire range.

Fig. 3.9 shows some poloidal magnetic field lines and the radial velocity in the2.5D and 3D jet. The latter was “axisymmetrized” by averaging over the azimuthalcoordinate ϕ. In the outer, high-ϑ, part of the 2.5D jet, where the toroidal fieldis especially strong and jet acceleration most efficient, there is an increase of theangular separation between the field lines which is absent in the 3D jet. In otherwords, the poloidal magnetic flux decreases locally faster with distance in the 2.5Dcase.

3.3.4 Forces and Powers

To identify the accelerating forces that generate the kinetic energy flow discussedin Sect. 3.3.2, we compute P =

∫F · v dV, the instantaneous power (rate of work)

delivered by a specific force F in the direction of the flow in the integrated volume.The combination of gas pressure and gravitational forces, −∇p− ρ∇Φ, accountsfor about one third of the power delivered by the sum of all forces in the wholevolume, the corresponding acceleration takes place mainly below r ≈ 30 (sonicsurface). The rest is accounted for by the Lorentz force, which we decompose as

51


follows:

14π

(∇× B)× B =1

4π

[(∇× Bp)× Bp + (∇× Bϕ)× Bϕ

+(∇× Bp)× Bϕ + (∇× Bϕ)× Bp

], (3.3.4)

with Bϕ = Bϕ eϕ and Bp = B− Bϕ. The corresponding components of P are plottedin Fig. 3.10 as a function of the upper integral limit.

The last force in Eq. (3.3.4) has only an azimuthal component. It is importantmainly below the Alfven radius (r ∼ 60), exerting a torque in the same direction inwhich rotation is applied at the lower boundary. The next-to-last force vanishes inthe axisymmetric case. In the general case, it has only non-azimuthal components.Unlike the last force, it works against the flow; the two forces largely cancel eachother in the 3D simulation. The Lorentz force associated with Bϕ [second term inEq. (3.3.4)] performs the same work in the 3D and 2.5D cases, despite the steepeningof the toroidal magnetic pressure profile caused by the dissipation. The total poweris also similar in the two cases, in agreement with the similarity of the kinetic energyflows in Fig. 3.5.

The situation is a bit different if only the radial power∫

Frvr dV is taken intoaccount. The Lorentz force associated with Bϕ does approximately 20% more workin the whole volume in the 3D case, the additional power is delivered above r & 200.The power of the net force, however, is approximately the same in both cases.

3.4 Comparison with Observations

The results have been presented in scale-free units and can thus be applied tojets of different sizes and intrinsic properties. The application to protostellar jetsseems especially worthwhile, because observations of these flows show very similarstructures as Fig. 3.2. It turns out that the results are reasonable within the limits setby observations.

Three independent values are needed to determine the normalization units (ρ0,p0, B0, . . . ) in which the results of the simulations are given. We assume 0.8 Mfor the mass M of the central object and 10−7 M yr−1 for the jet’s mass flow rateM, in accordance with the values stated in Antoniucci et al. (2008) for three Herbig–Haro objects, out of which at least two are associated with parsec-scale jets (Bally& Devine 1994; Stanke et al. 1999). As third parameter, we take the tentative valuel0 = 1 AU for the size of the source. The length covered by the simulations is thus0.005 pc, i.e. about 2–3 decades below the length of full-blown protostellar jets. Themaximum velocity of the simulated jet (cf. Fig. 3.3) is

vmax ≈ 2.5cs0 = 19(

l01 AU

)− 12(

M0.8 M

) 12 km

s. (3.4.1)

52

3.4 COMPARISON WITH OBSERVATIONS

This is a bit slower than real jets, for which velocities in excess of 100 km s−1 areobserved. The atmospheric magnetic field strength is

Bb ≡ 3B0 = 6(

l01 AU

)− 54(

M0.8 M

) 14(

M10−7 M yr−1

) 12

G, (3.4.2)

at the base and 4 mG at the upper boundary (r ≈ 0.005 pc). Measurements ofmagnetic field strengths of accretion disks are scarce. Donati et al. (2005) measureda kilogauss field near the center (0.05 AU) of a protostellar disk. Kilogauss fieldsare also found in the photosphere of several protostars (Johns-Krull et al. 2009 andreferences therein). Assuming that B ∼ R−2 from magnetic flux conservation, weobtain a value on the order Bb at the outer edge of the disk, where most of theacceleration takes place in the simulation due to the (not so realistic) assumption ofrigid rotation. Our present knowledge about magnetic field strengths inside real jetsis too sparse to make a meaningful comparison. The unit time in the simulation is

t0 = 0.6(

l01 AU

) 32(

M0.8 M

)− 12

yr, (3.4.3)

wherefore the disk has a rotation period of approximately 20 years and the jet needsabout 300 years to cross the computational domain. The atmospheric density andgas pressure at the base are

ρb ≡ ρ0 = 4 · 10−13(

l01 AU

)− 32(

M0.8 M

)− 12(

M10−7 M yr−1

)g

cm3 , (3.4.4)

pb ≡ p0 = 0.16(

l01 AU

)− 52(

M0.8 M

) 12(

M10−7 M yr−1

)dyncm2 . (3.4.5)

The corresponding temperature is Tb = 2147 K if the gas consists of ionized hydro-gen. At the upper boundary (r = 0.005 pc), we have T = 43 K and a particle densityof ∼104 cm−3 if we assume that that the gas consists of neutral molecular hydro-gen there. For comparison, the molecular clouds in which protostars form havetemperatures as low as 10 K and particle densities & 103 cm−3 (Zeilik & Gregory1998). In Herbig–Haro objects, temperatures on the order of a few thousand degreesKelvin are observed (e.g. Gredel et al. 1992). Such temperatures are also present inthe hottest regions of the simulated jet. The maximum total flow rate of energy inthe simulation is

Etot ≈ 0.37p0l3

0t0

= 3 · 10−3(

l01 AU

)−1 ( M0.8 M

)(M

10−7 M yr−1

)L, (3.4.6)

roughly 4–5 decades smaller than the accretion luminosities inferred by Antoniucciet al. (2008). If we assume that 7% of this amount is eradiated (i.e. half of thedissipated energy), we get a luminosity of 2 · 10−4 L.

53


If we take l0 = 100 AU, the length of the computational domain is 0.5 pc, i.e.the simulation then represents the outer part of a protostellar jet. The outflowvelocity is then reduced to a more unrealistic 2 km s−1 and the energy flow is nomore consistent with the observed luminosities. This implies that the simulationsare more relevant for the inner parts of protostellar jets.

3.5 Summary and Discussion

We have simulated jets generated by twisting a parabolically shaped large-scalemagnetic field in both 3D and axisymmetric 2.5D. The shape of the initial fieldreflects itself in a fair amount of jet collimation, with an opening angle that decreaseswith distance, thus facilitating the growth of instabilities. The simulations cover theacceleration phase, where the jet passes through the critical surfaces (sonic, Alfvenand fast magnetosonic) for stationary MHD flows, as well as a substantial distancebeyond these. We thus observe the onset of kink instabilities above the Alfvensurface in the 3D simulation. The instabilities disrupt the magnetic field structure.They cause magnetic dissipation, significantly reducing the toroidal field strengthand with it the flow rate of magnetic enthalpy (surface integral of the Poynting flux)on a length scale of about 2–15 times the minimal Alfven distance in the jet.

A direct comparison of the 2.5D and 3D simulations reveals no significant dif-ference in the way the kinetic energy of the jet (integrated over its cross section)increases with distance. However, the distribution of the kinetic energy across thejet indicates differences in the acceleration process (cf. Fig. 3.9). In the axisymmetricflow, the acceleration is restricted to magnetic surfaces that diverge from each othermore rapidly than in a flow with fixed opening angles. This creates a “magneticnozzle” effect (Begelman & Li 1994) restricted to a limited range of angles withinthe flow (cf. discussion in Spruit 2008). A similar case of non-uniform expansionhas been found in relativistic flows by Tchekhovskoy et al. (2008). In the 3D case,the (time averaged) acceleration is more uniform across the jet. It thus seems thatwhile the steepening of the magnetic pressure gradient caused by the dissipation ofthe toroidal field leads to additional acceleration in the 3D case, this is made up forby a more favorably distributed poloidal magnetic flux in the 2.5D case which alsoenhances acceleration. In other words, there are different mechanism at work whichyield the same result. Whether it is coincidence that the two effects have nearly thesame strength is not clear.

The energy released by the dissipation of the toroidal magnetic field may beradiated as light. The yield depends on the details of the dissipation (reconnection)and radiation mechanisms involved. Since the magnetic energy density (B2

ϕ/8π)accounts for half of the Poynting flux (B2

ϕvr/4π), the present simulations suggestthat the available luminosity may be as much as 10% of the initial magnetic enthalpyflow rate (Fig. 3.5), provided that half of it is converted into kinetic energy andthe rest into light. As dissipation is not a smooth process, the emission will be

54

3.5 SUMMARY AND DISCUSSION

stronger in some regions. These would likely turn up as bright knots in observations.Together with the wiggles caused by the kink instabilities, the knots produce astructured appearance that is similar to what has been observed in protostellarjets (e.g. Heathcote et al. 1996; Reipurth & Bally 2001). The knots in the mockedjet image move with the flow, consistent with observational findings (Eisloffel &Mundt 1992; Hartigan et al. 2005). This provides an alternative to the “internalshock” interpretation usually invoked to explain jet knots (e.g. Hartigan & Raymond1993).

The magnetic structure of a jet undergoes a dramatic change if it becomes subjectto violent kink instabilities. The formerly ordered helical structure is largely de-stroyed and the poloidal field becomes the dominating field component. However,magnetic flux conservation still implies that the poloidal field declines faster withdistance than the toroidal one. The toroidal field may thus, beyond the regioncovered by the here-presented simulations, become again dominant. This could, inprinciple, lead to a resurgence of instabilities until the Alfven speed has droppedbelow the sideways expansion speed of the jet (times a factor of order unity). Fromthis point on the toroidal field is effectively frozen in the flow (cf. discussion inChapter 2).

3.5.1 Collimation and Jet Environment

Collimation of jets is popularly attributed to the “hoop stress” in the toroidal fieldcomponent. This is misleading: though the stress contributed by the toroidalfield can compress the configuration near the axis (as observed in simulations), iteventually has to be taken up by an external agent (for a more extended discussion,see Spruit 2008). In the simulations presented above, as well as in other work, thisagent is an external medium surrounding the jet. It is included mostly because oflimitations of the codes used, since the demands of conserving energy typicallycause numerical instabilities at low gas densities or low plasma-β.

The boundary between the jet and the external medium actually expands due topressure exerted by the toroidal field. The role of a (material) external medium inconfining the jet can be taken over by a magnetic field, if it is able to transfer stressin the jet’s toroidal field to the surface of the accretion disk. One might imaginethat a toroidal field extending around the jet might serve this role. The high Alfvenspeeds in this field, however, would make it violently unstable to non-axisymmetricinstabilities, as the early history of magnetic configurations for controlled fusion(linear pinches) testifies. This obstacle does not become evident in the axisymmetricmodels in the literature. A more realistic possibility, proposed by Shu et al. (1995),is that the disk’s poloidal field, assumed for launching the jet in the inner regions,actually extends to much larger distances in the disk. Deformation of the field cantake up the lateral stress exerted by the jet. The collimation of the jet would then bedirectly related to the properties of this poloidal field. High degrees of collimationwould be most easily achieved in disks with a large ratio of outer to inner disk

55


radius (Spruit et al. 1997). Numerical simulations at much lower plasma-β thancurrently feasible would be needed to study this form of “poloidal collimation”.

As pointed out in the introduction, instabilities may take some distance to travelto become effective. The results presented above show that a distance of the orderof 1000 times the source size of the jet is needed to capture the dissipation of thetoroidal field. I surmise this to be the reason why the effect of instabilities is notas noticeable in the works of Anderson et al. (2006) and Ouyed et al. (2003), ratherthan the lack of external confinement of these winds.

3.5.2 Disruption

The possible presence of instabilities in jets sometimes raises concern about disrup-tion. A complete dissipation of the jet into its surroundings is possible in principlethrough instabilities driven by the interaction of the jet with its surroundings, forexample by Kelvin-Helmholtz instabilities, or its termination in a “hot spot”. Theseprocesses extract their energy directly from the bulk kinetic energy of the flow. Theeffect of internal instabilities deriving from the free energy in the toroidal field ismuch less destructive, since these mainly redistribute internal energy forms withinthe jet. The comparison between the 2.5D and the 3D cases shows that the 3D jetwidens by less than a factor 2 as a result of internal instabilities (cf. Figs. 3.4 and 3.9).The non-axisymmetric nature of the instabilities thus does cause some interactionwith the environment, but its consequences remain relatively benign.

3.5.3 Cold Flows

Current 3D simulations are poorly equipped to handle flows in which the magneticor kinetic energy density, or both, dominate over the thermal energy density. Inmagnetically dominated flows driven from actual accretion disks, the temperatureof the plasma is often sufficiently low that magnetic energy density dominates overplasma pressure already at a short distance from the disk surface (Blandford &Payne 1982). As a consequence, the sonic point in the present results is furtheraway from the source than would be the case in e.g. actual protostellar disks. It ispossible that the onset of the instabilities in more strongly magnetically dominatedflows would be faster, and their consequences even stronger than in the simulationspresented here. Codes specially designed to handle such “cold” flows would beneeded to verify this.

56


Figure 3.11: Radial velocity (upper plot) and density (lower plot) in a meridional slice throughthe jet in the 3D simulation. Due to the rigid rotation at the base, the jet accelerates moreefficiently at its rim. At large distances, material and momentum gets intermixed by theinstabilities. The jet entrains material into the upper, low-density atmosphere.

57


Figure 3.12: Volume rendering of the 3D jet, with emission and absorption governed bytemperature and density, at different times. Large kinks start growing at distances of about1–3 Alfven radii.

Figure 3.13: Meridional slice through a jet simulation similar to case K3 in Chapter 2, but witha poloidal initial field (ζ = 30) instead of a purely radial one. The plot shows the temperature(hue) together with the density (brightness), compare Fig. 2.4. The jet is a bit more agitatedthan in case K3, but the internal shear in the magnetic field prevents serious instabilitieswithin the simulated distance.

58

4 Large Jets From Small-ScaleMagnetic Fields

R. MollAstronomy & Astrophysics 2009, in press

Abstract: We consider the conditions under which a rotating magneticobject can produce a magnetically powered outflow in an initially unmagne-tized medium stratified under gravity. 3D MHD simulations are presentedin which the footpoints of localized, arcade-shaped magnetic fields are putinto rotation. It is shown how the effectiveness in producing a collimatedmagnetically powered outflow depends on the rotation rate, the strengthand the geometry of the field. The flows produced by uniformly rotating,non-axisymmetric fields are found to consist mainly of buoyant plumesheated by dissipation of rotational energy. Collimated magnetically pow-ered flows are formed if the field and the rotating surface are arrangedsuch that a toroidal magnetic field is produced. This requires a differen-tial rotation of the arcades’ footpoints. Such jets are well-collimated; wefollow their propagation through the stratified atmosphere over 100 timesthe source size. The magnetic field is tightly wound and its propagationis dominated by the development of non-axisymmetric instabilities. Weobserve a Poynting flux conversion efficiency of over 75% in the longestsimulations. Applications to the collapsar model and protostellar jets arediscussed.

4.1 Introduction

Models and simulations of jets produced by rotating magnetic fields generallyassume the existence of an ordered, axially symmetric large-scale field of uniformpolarity anchored in the central engine, starting with the original models of jets fromaccretion disks by Bisnovatyi-Kogan & Ruzmaikin (1976) and Blandford (1976), orthe magnetic supernova model of LeBlanc & Wilson (1970). While such orderedfields are the most effective in producing jets, the question whether they actuallyexist in accretion disks or the core of a star is still quite open. The largest scale atwhich magnetorotational (MRI) turbulence in accretion disks shapes the magneticfield structure is set by the disk thickness, which in turn is much smaller thanjets. Collapsar cores are also small compared to the expected GRB jet, and it is

59

4 LARGE JETS FROM SMALL-SCALE MAGNETIC FIELDS

not at all clear why the magnetic fields there should be ordered and axisymmetric.Large-scale fields are not easily trapped by an accretion disk (van Ballegooijen 1989).Without a large-scale field protruding from the disk, one may still hope to launchoutflows by twisting fields inside the disk or by magnetic loops that extend intothe disk corona as e.g. in Galeev et al. (1979); Tout & Pringle (1996); Uzdensky &Goodman (2008).

The huge range of length scales involved is a major issue in jet modeling, asnumerical simulations that cover all scales are not feasible to date. Protostellar jetsmay be several parsecs long, launched by disks with sizes of ∼100 AU, i.e. about afactor 104 smaller (e.g. Shepherd et al. 2001). Assuming that the “launching scale” ismuch smaller than the disk, perhaps on the order of ∼1 AU, one obtains an evenbigger contrast in length scales. Supposing that jets in AGN are launched at afew Schwarzschild radii from the central black hole and taking Cygnus A as anexample, one obtains a ratio of 106 between the jet length and the size of the engine(Krichbaum et al. 1998; Tadhunter et al. 2003). The final jet properties are determinedbefore it becomes ballistic, probably at scales which are somewhat smaller than thatof the largest visible structures, but still considerably larger than the central engine.

There is some numerical evidence that small-scale fields can also be used to gen-erate jets. Most of these simulations are promising in terms of outflow productionbut limited to the immediate surroundings of the outflow-forming disk. Axisym-metric simulations of outflows generated with small magnetic loops were done byRomanova et al. (1998); Turner et al. (1999); Kudoh et al. (2002). In their 3D simula-tions of accretion flows, Kato et al. (2004) demonstrated that an initially poloidalmagnetic field confined within a rotating torus surrounding the accreting black holecan give rise to a transient outflow driven by accumulated toroidal fields in theform of a “magnetic tower” (Lynden-Bell 2003). De Villiers et al. (2005) showedin simulations that loops of poloidal field in an accreting torus may give rise to alarge-scale poloidal field as the field lines are stretched out in an axial outflow. Theinflation and disruption of a magnetic loop outside a disk, caused by the generationof a toroidal field through differential rotation, was observed by Fendt (2009) insimulations of outflows from star-disk magnetospheres.

The aim of the calculations presented here is to see whether small-scale fieldsin the form of loops anchored in a rotating disk can be used to produce jets ofsignificant length (compared to the size of the source). In the cases studied the flowspropagate and are confined in an external unmagnetized atmosphere (as opposedto Chapters 2 and 3, where we studied jets embedded in a large scale magneticallydominated environment). While still idealized, this addresses environments likeprotostellar jets launched into a dense cloud, or GRB jets launched by a collapsarcore. To be investigated here are the circumstances under which jetlike flows areformed that penetrate through the atmosphere instead of dissipating in it. As thisis also a question of (non-axisymmetric) stability, three-dimensional simulationsare necessary. One of the questions to be answered is whether models of jets fromsmall-scale magnetic fields are a viable alternative to those based on the twist of

60

4.2 MODELS

a)

with shear

b)

shifted, no shear

c)

diff. rotation

Figure 4.1: Considered magnetic field geometries. In case (a), the field lines are one-sidedlyanchored in a rigidly rotating disk. In case (b), the field lines’ footpoints are either bothanchored in the disk, or they are both anchored outside of it. In case (c), the field emergesfrom a limited area inside the disk which is differentially rotating.

large-scale fields.

4.2 Models

The primary ingredients in all our models are a rotating disk, implemented as aboundary condition, and magnetic field loops anchored in and sticking out of thedisk. The magnetic field geometries considered are sketched in Fig. 4.1. In case (a),some of the loops have one footpoint inside the disk while the other is anchoredoutside. The resulting shear motion of the footpoints generates a toroidal magneticfield. In case (b), the loops are arranged such that they are not sheared. Here, atoroidal field can only be produced by the inertia of the material above the disk,such as in conventional models of jets from large-scale magnetic fields. This casewould be a model for a (solidly) rotating stellar core inside a nonmagnetic envelope.Finally, we consider a case in which all loops are anchored in the disk and shear iscreated through differential rotation (c). This case would be more representative ofsmall-scale fields generated in an accretion disk.

The magnetic field loops are established either by a suitable potential field, im-posed as initial condition (setup D), or by continuous “injection” of the field frombelow the disk (setup E), see Fig. 4.3 and Sect. 4.3.1.2 for details. While the magneticfield geometry is similar in both cases, setup E (for “Emerging field”) is meant as anidealization of the field loops emerging from magnetic turbulence in an accretiondisk.

The model chosen as initial condition for the atmosphere is a hydrostatic equi-librium stratification in the gravitational potential of a point mass (the origin).Temperature T and density ρ vary with distance r from the point mass as r−1 andr−3, respectively. The temperature is thus a constant fraction of the virial tempera-ture, as in advection-dominated accretion flows. The density profile has the propertythat the amount of mass in a cone centered on the origin is constant per decadein distance r. The stratification is thus scale-free, and a prospective jet encounters

61


b

rb

Rr ϑ

θ

φ

ϕ

x

y

z

b

Figure 4.2: Sketch of the coordinate system and the computational grid. The jets propagatein equatorial direction of the grid coordinates (r, θ, φ). To describe the results, the alternatespherical coordinate system (r, ϑ, ϕ) is used, which takes the jet’s central axis as the polaraxis.

the same amount of atmosphere mass per decade traveled. With the equation ofstate used in the calculations, an ideal gas with ratio of specific heats γ = 5/3, thestratification is convectively stable.

4.3 Methods

4.3.1 Numerical MHD Solver, Grid and Coordinates

We numerically solved the ideal adiabatic MHD equations with a static externalgravitational potential Φ ∝ r−1 on a spherical grid (r, θ, φ). The jets propagate inequatorial direction along the y-axis (θ = φ = π/2), about which the computationalvolume covers a range ∆θ = ∆φ in the angular directions, see Fig. 4.2. The spacingis uniform in θ and φ and logarithmic in r. Such a grid is more economical thana Cartesian one since it expands in jet direction. Unlike a spherical grid in polardirection z, it is free of singularities which may cause numerical problems or artifacts.

As jet physics is better described in a coordinate system which takes the jet’s axisy as the polar axis, we introduce another spherical coordinate system (r, ϑ, ϕ) forthat purpose, see also Fig. 4.2. R := r sin ϑ denotes the orthogonal distance to thataxis (cylindrical radius).

The simulations were performed with a newly developed Eulerian MHD code(Obergaulinger 2008). It is based on a flux-conservative finite-volume formulation ofthe MHD equations and the constraint transport scheme to maintain a divergence-free magnetic field (Evans & Hawley 1988). Using high-resolution shock capturingmethods (e.g., LeVeque 1992), it allows a choice of various optional high-order

62

4.3 METHODS

y

zθ

r = rb

φ = π/2

Rb

Figure 4.3: Schematic of simulation setup E described in Sect. 4.3.1.2, in the variant sketchedin Fig. 4.1a. A vertical velocity field (green arrows) pushes the magnetic field (blue lines) fromthe lower boundary (r = rb) into the computational volume. An azimuthal velocity fieldwithin R ≤ Rb twists the created magnetic arcades (dotted blue lines) which are one-sidedlyanchored in the rotating region.

reconstruction algorithms and approximate Riemann solvers based on the multi-stage method (Toro & Titarev 2006). The simulations presented here were performedwith a fifth order monotonicity-preserving reconstruction scheme (Suresh & Huynh1997), together with the HLL Riemann solver (Harten 1983) and third order Runge-Kutta time stepping.

4.3.1.1 Setup with Arcade-Shaped, Exponentially Decreasing Initial Field (D)

The initial condition for the magnetic field in this case is a two-dimensional potentialfield which is independent of the x-coordinate (see Figs. 4.2,4.3 for the coordinatesystem) and falls off exponentially with height y. It is generated by the vectorpotential

A = Bbλ sin( z

λ

)exp

(rb − y

λ

)ex, (4.3.1)

where λπ is the width of the arcades and λ is the scale height of B = |B|. Bb is thefield strength at the intersection of the lower boundary r = rb with the central axis(y). To avoid numerical problems at the lateral boundaries, we limit the number ofarcs to 4, setting A = B = 0 for |z| > 2λπ and for |x| > 2λπ.

The magnetic field is embedded in a spherically stratified atmosphere with p ∝r−4 and ρ ∝ r−3, held in hydrostatic equilibrium by the static gravitational fieldΦ ∝ r−1 of a point mass at the coordinate origin. Temperature and sound speedvary as T ∝ r−1 and cs ∝ r−1/2, respectively. In the outer, unmagnetized region(|x| , |z| > 2λπ), we compensated for the absence of magnetic pressure by increasingthe gas pressure. To maintain hydrostatic equilibrium, the density is also increased

63


correspondingly.At the lower (r = rb) boundary we maintain, through “ghost cells” outside of the

computational domain, an azimuthal velocity field v = vϕ eϕ corresponding to rigidrotation: vϕ = vmax

ϕ,b R/Rb for R ≤ Rb and 0 elsewhere. All quantities except for Bare fixed at their initial values in the ghost cells; B is extrapolated from the interiorof the domain. At the sides (θ and φ) and top (upper r) of the domain, we use openboundary conditions which allow for an almost force-free outflow of material andcause no evident artifacts in the form of reflections.

λ is chosen such that in the x = 0 plane, By is positive for |z| < Rb and negativefor Rb < |z| < 2Rb, i.e. arcades which start at |z| < Rb from the r = rb surface havetheir second footpoint outside the rotating region. This is achieved by choosing thelength scale of the arcade as λ = 2Rb/π.

4.3.1.2 Setup with Magnetic Field Arcades Emerging From the Boundary (E)

In this case, the initial condition is again an equilibrium stratification with p ∝ r−4,ρ ∝ r−3 and Φ ∝ r−1. However, unlike in setup D, the atmosphere is completelyunmagnetized.

The magnetic field enters the domain through the lower boundary r = rb, theconditions of which are determined through ghost cells at r < rb. There, we imposea transverse magnetic field Bθ (≈ Bz) with constant amplitude Bb and a polaritythat alternates with z = rb cos θ in step sizes of 2Rb, the diameter of the rotatingsurface (described below). Where the polarity of Bθ changes, the solenoidality ofthe magnetic field is maintained by an appropriate Br. The field lines thus havethe shape sketched in Fig. 4.3. An approximate equilibrium is maintained throughlowering the gas pressure by the value of the magnetic pressure, as far as this ispossible (p must not be negative). For this to work, βb := 8πpb/B2

b must be greaterthan one, which limits the possible field strengths to Bb < 1 in our system of units(described below).

To model the emergence of magnetic fields into the atmosphere we impose theradial velocity field

vr = 0.9cs,b

∣∣∣∣sinzπ

2Rb

∣∣∣∣ (4.3.2)

in the ghost cells of the lower boundary. The injection velocity is too small to form ajet by itself: the maximum amounts to 41% of the escape velocity at the boundary,gas with this speed gets theoretically as far as r ≈ 1.2rb without acceleration otherthan gravity. In addition to the radial velocity field, we maintain an azimuthalvelocity field vϕ ∝ R (rigid rotation) within R ≤ Rb. The shape of the emergingmagnetic field is sketched in Fig. 4.1a. For the other boundaries, we use the sameoutflow conditions as in the model described in the preceding section.

Two variations of the above-described setup have also been studied, correspond-ing to the cases (b) and (c) described in Sect. 4.2. In the first, the positions where

64

4.3 METHODS


Quantity Symbol(s) Unitlength x,y,z,r,R l0

gas pressure p p0density ρ ρ0velocity v cs0 =

√γp0/ρ0

time t t0 = l0/cs0energy flow rate E p0l3

0/t0magnetic flux density B B0 =

√8πp0

Bθ changes its polarity, and with it the radial velocity field that injects the magneticfield, are shifted by Rb in −z direction. Thus, both footpoints of an individual fieldloop rotate with the same angular velocity. In the second case (c), the magneticfield and vertical velocity in the lower boundary are changed such that only asmall, confined arcade in 0.2 < R/Rb < 0.8 with a width of 0.2Rb emerges, andthe azimuthal velocity field has a Keplerian profile with vϕ = vmax

ϕ,b√

0.1Rb/R for0.1 ≤ R/Rb ≤ 1. The inner edge (R = 0.2Rb) of the arcade rotates twice as fast asthe outer edge (R = 0.8Rb), the difference in rotation velocity is vmax

ϕ,b /√

8.

4.3.2 Parameters and Units

The models described above contain the following 6 parameters, not all of whichare independent: rb, Rb, ρb, pb, Bb and vmax

ϕ,b . We eliminate the dependences bytaking l0 ≡ 2Rb, ρ0 ≡ ρb and p0 ≡ pb as units of length, density and pressure, andexpressing all physical quantities in terms of these, see Table 4.1. The remaining 3parameters can then be expressed as dimensionless numbers. The first of these isl0/rb, which is a measure for the curvature of the rotating surface. In all simulationspresented here, this curvature is small and probably does not influence the resultssignificantly. The parameter has technical significance, however, because it alsocontrols the opening of the lateral boundaries. The remaining two numbers arechosen to be βb := 8πp0/B2

b, which controls the strength of the magnetic field, andthe Mach number Mb := vmax

ϕ,b /cs0, which determines the speed of rotation. Notethat in the emerging field model (E), βb is a measure of the field strength but not alocal plasma-beta value, since the pressure entering its definition is not measured atthe same location as the field strength. For the sake of clarity, we usually omit theunits in the presentation of the results. The concerned quantity is then measured interms of the associated normalization unit listed in Table 4.1.

65


Figure 4.4: Selectedmagnetic field lines ina simulation with anarcade-shaped initialfield (setup D). Thelines are closing withthe lower boundary,ascending inside the jetand descending at theoutside.

4.4 Results

As explained in Sect. 4.2 above, the calculations were done with different modelsfor the rotating magnetic field configuration. It turns out that some of these casesproduce long-lived jets, others only transient flows or flows that dissipate in theatmosphere close to the source. The results presented in the following subsectionwere obtained with setup D (Sect. 4.3.1.1), those in the subsequent subsections wereobtained with setup E (Sect. 4.3.1.2).

4.4.1 Transient Jets with Arcade-Shaped Initial Field

Simulation setup D with the field geometry in Fig. 4.1a was found to produceextended, collimated outflows. However, the magnetic field at the base of the jetdecays in this case (hence the “D”) and is not replenished, for which reason these

66

4.4 RESULTS

Figure 4.5: Left: Magnetic field and temperature in a horizontal slice next to the bottomboundary in a simulation with an arcade-shaped initial field (setup D) with shear (Fig. 4.1a)after 1 (bottom left panel) and after 32 revolutions (top left panel). The texture shows the geome-try of the parallel field component and the green/yellow coloring gives the magnitude of thenormal component. Regions of increased temperature are tinted in red (low temperatures aretransparent). Right: Evolution of the radial velocity in a meridional slice. The jet is eventually“choked off” at the bottom.

jets are not permanent.

The forced rotation at the lower boundary stretches the magnetic field lines in az-imuthal (ϕ) direction around the axis of rotation (y). The gas is accelerated upwardsand the magnetic field assumes a tangled helical structure. A good momentaryacceleration was obtained with the parameters (βb = 1/4, Mb = 18). The magneticfield in this case is depicted in Fig. 4.4. The jet diameter is resolved with about 35grid cells in this simulation. The jet attains a height of about 30 times its initialdiameter, the factor of expansion being about 7. It reaches velocities that are close toMb and about a factor of 10 above the escape speed. The field lines are closing withthe lower boundary: inside the jet, the radial magnetic field has the same polarityas on the disk, whereas the net radial magnetic flux

∫Br dA, integrated over the

r = const surface, is virtually zero at all times and all radii.

67


Figure 4.6: Left-hand panel: Maximum velocity over escape speed for different parameters(βb, Mb) in otherwise similar simulations of the emerging field model (E), variant (a). Thestrength of the jet decreases with a decrease of the field strength (increasing βb, representedby different line styles) and with a decrease of the rotation velocity (Mb, represented bydifferent colors). Right-hand panel: Strength of the jets as a function of the rotation rate andfield strength parameters.

The “absolute flux”∫|Br|dA decreases linearly with time near the lower bound-

ary. Increased temperatures near the outside of the rotating surface indicate thata substantial amount of magnetic field is being dissipated there, see Fig. 4.5. Dueto the rotation, the magnetic field assumes a vortex-like structure, at the borderof which magnetic field lines of opposite polarities become entangled. This leadsto a continuous decay of the magnetic field. Such a decay of the magnetic field,through wrapping up of field lines followed by cancellation through diffusion, iscalled “convective expulsion” in overturning flows (e.g. Zel’dovich 1956; Parker1963). The magnetic field in the physical source of a successful jet evidently must beof a different nature. This calls for a modification of the boundary conditions at thebase of our simulations. Such cases are discussed in the following sections.

4.4.2 Jets From Emerging Fields: Parameter Study

Simulation setup E, in which the magnetic field loops above the rotating surface arecontinuously replenished, produced long-lasting jets that propagate considerabledistances. Before presenting big simulations with long jets in the next two sections,we discuss the influence of the parameters (βb, Mb) on the results by comparing a

68

4.4 RESULTS

Figure 4.7: Parallel magnetic field (wedges) and temperature (color) in a meridional slicethrough the jet in a simulation with shear (top panel) and in two simulations without shear(middle/bottom panel). Collimated, magnetically driven jets form only in cases where a toroidalfield develops through a shear of the magnetic field.

69


series of smaller, computationally cheaper simulations in which these parametersare varied. Although the proximity of the upper boundary possibly influencesthe results (see discussion of instabilities below), the small simulations give clearindications about which parameters yield efficient jets.

The simulations cover the relatively short distance of 20 length units (40 disk radii),with 10 < r < 30 and θ, φ = π/2± π/9, the resolution being 256× 96× 96. Theatmospheric density differs by a factor 27 between the lower and upper boundaries.The simulations were pursued until t ≈ 100. In the cases where a jet was successfullylaunched, this suffices for about 2–3 passages through the computational volume.The computational cost for this was 1–3 wallclock days on 36 processors with MPIparallelization.

To test the flow’s ability to penetrate the atmosphere in the setup with shear,Fig. 4.1a, we compare the jet velocities halfway through the simulated distanceagainst the escape velocity in Fig. 4.6. The jet velocity decreases with both decreasingrotation velocity Mb and decreasing magnetic field strength Bb (or increasing βb). Itis, however, much more sensitive to the former parameter: βb must be changed byorders of magnitude to get a significant impact on jet velocity whereas with Mb, afactor of order unity suffices. The right-hand panel in Fig. 4.6 shows which flowspass this test, with those exceeding the escape speed marked in green, the othersin red. Failed jets do not reach the upper boundary. The background atmosphere,whose equilibrium is perturbed, tends to fall down on them. The fixed conditions atthe lower boundary avert a pile-up of thermal energy: downward flowing hot gasvanishes across the boundary and the injected gas has a constant temperature. Adelayed onset of a flow due to accumulated heat is unlikely for this reason.

There are instabilities in all cases where a jet develops. The instabilities developmainly after the first passage through the computational domain, in the form ofhelical displacements and/or a change of direction of the whole jet by up to severaldegrees. The latter are likely modes with wavelengths longer than the computationaldomain, i.e. one sees only the lower part of what would be a kink if the radial extentof the domain was larger. Jets created by stronger magnetic fields (βb ≤ 4.9) tendto show this kind of incipient instability. Jets from weaker fields move slower anddevelop pronounced helical deformations already within the computational domain.The proximity of the upper boundary does not allow for more conclusive statementsabout differences in instability behavior within the limits of this parameter study. Aseries of large and expensive simulations of the kind presented in the next sectionwould be needed for that.

If the configuration is shifted such that no shear in the magnetic field occurs(Fig. 4.1b), a jet does not develop. The middle panel in Fig. 4.7 shows such a case.The magnetic field lines are unconnected to the surrounding atmosphere and thefield is not amplified by shear. This appears to be sufficient to allow them to rotatewithout producing a magnetically powered flow. However, for very large values ofMb and βb, an outflow forms at the edge of the rotating disk, see bottom panel inFig. 4.7. This outflow appears to be driven by thermal buoyancy associated with the

70

4.4 RESULTS

dissipation of rotational energy at the disk’s edge. Having the shape of a hollowcone with a large opening angle, it is clearly distinctive from the jets discussedabove and the rest of this paper.

4.4.3 Jets From Emerging Fields: A Large Simulation

The long-range behavior of jets from emerging fields (setup E) in the case withshear (Fig. 4.1a) was investigated in a large simulation that covers the radial range10 < r < 110. Along this range, the atmospheric density decreases by a factor 1300.The parameters used were (βb = 4, Mb = 3). The jet crosses the upper boundary atthe physical time t ≈ 110, which corresponds to 105 complete rotations of the disk;the simulation was stopped at t = 539. For this, 19 wallclock days on 64 processorswith MPI parallelization were needed, the resolution being 400× 160× 160. Thelateral boundaries are at θ, φ = π/2± π/6; the jet stays within these boundaries atall times.

The wound-up magnetic field rises upwards as shown in Fig. 4.8, with a colli-mated outflow forming in vertical direction. Not counting the changes in directioncaused by instabilities, the jet’s half opening angle is about 5. Inside the jet, thefield lines spiral upwards, with the radial magnetic flux having the same sign asthe radial field on the rotating surface. The total flux is, in comparison, close tozero at most radii and times. Accordingly, visualizations of the field lines give theappearance that most lines remain connected to the lower boundary with both endsuntil the jet crosses the upper boundary.

Fig. 4.10 shows the average value of various quantities across the flow, i.e. theaverage of a quantity X is calculated as

∫Xvr dA/

∫vr dA over the r = const

surface. The average velocity increases mainly below r ≈ 20, at which point itexceeds the average fast magnetosonic speed v2

fm = c2s + v2

A. The peak velocity(maximum of vr on r = const) is constant beyond r ≈ 30 and is twice as large asthe average velocity (4 instead of 2). The average density is at all radii significantlylower than the atmospheric value, the temperature (c2

s ) is increased with respectto the environment. The transversal (toroidal) magnetic field dominates over theradial component at medium and large distances. The magnetic pressure dominatesover the gas pressure by a factor of about 2–4 at all distances in the jet.

The dependence of the energy flow rates with distance gives information aboutenergy transformations taking place in the jet. The total energy flow rate Etot(t, r) isobtained from a surface integral over the components of the radial energy flux

12

ρv2vr︸︷︷︸kinetic

+γ

γ− 1pvr︸︷︷︸

thermal enthalpy

+ ρΦvr︸︷︷︸grav. potential

+ Sr︸︷︷︸magnetic enthalpy

, (4.4.1)

71


Figure 4.8: Selected magnetic fieldlines in the simulation presented inSect. 4.4.3. The color gives the mag-netic field energy. The boundaryconditions create magnetic field ar-cades above the boundary, the ar-cades are twisted in the center by theimposed rotation and a jet with heli-cally shaped field lines forms alongthe axis of rotation. The jet is subjectto occasional helical deformations.

72

4.4 RESULTS

Figure 4.9: Meridional slices through the jet presented in Sect. 4.4.3. The magnetic jet isaccelerated into a non-magnetic atmosphere, penetrating through denser material. The hot,magnetic material in the environment of the upper part of the jet are remnants of previousinstabilities.

with Sr being the radial component of the Poynting vector:

Sr =1

4π

(B2

ϑvr + B2ϕvr − BϑBrvϑ − BϕBrvϕ

). (4.4.2)

We denote the individual components of Etot by Ekin, Ethrm, Egrav and Emag in orderof their appearance in Eq. (4.4.1). The components of the magnetic enthalpy flowrate are denoted by E1...4

mag in order of their appearance in Eq. (4.4.2). The energyflow rates in the present simulation are plotted in Fig. 4.11. The plot shows howmagnetic enthalpy is converted into kinetic and potential energy. The conversion isefficient in that less than 25% of the initial Emag remains in the jet when it reaches

73


Figure 4.10: Spatial and time-averaged values of various quantities in the simulation pre-sented in Sect. 4.4.3. The dotted lines represent the initial atmospheric value, the ‖ symbolstands for “parallel to v”.

the upper boundary. Thermal energy is less important, Ethrm being reduced onlyby half as many units as Emag. The most important components of Emag are E2

mag,which is associated with the advection of the azimuthal field and, at low radii, E4

mag,the work done by the azimuthal flow against the azimuthal component of magneticstress. E3

mag is virtually zero. In summary, the qualitative behavior of the energyflow rates is similar to those in simulations of jets generated by twisting a uniformlypolarized large-scale magnetic field (Chapters 2&3).

The jet has a lot of substructure, see Fig. 4.9. Above r ≈ 20, it is affected byrecurrent instabilities that divert its course away from the y-axis by several degrees.Stirred up ambient material contributes to the unstable behavior. Occasionally, the

74

4.4 RESULTS

Figure 4.11: Energy flow rates in the jet presented in Sect. 4.4.3. Magnetic enthalpy (integralover Poynting flux, blue line) is converted into kinetic energy (green line) and potentialenergy (brown line).

jet develops a pronounced helical shape as is characteristic for kink instabilities. Thehelix makes approximately one complete turn within the computational volume,out of which it is rapidly advected. For comparison, the magnetic pitch is .10inside the jet (cf. Fig. 4.8), which is at least a factor 10 below the wavelength ofthe prominent instability. The largest deflections from the central axis amount toapproximately 10. Being continually recreated at its base, the jet survives in atime-averaged sense.

4.4.4 Jets From a Magnetic Arcade On a Differentially RotatingSurface

We also produced jets with emerging field loops (setup E) on a differentially rotatingsurface, see the sketch in Fig. 4.1c. These jets are mostly the result of thermalbuoyancy, driven by dissipative heating of the near-disk atmosphere. Nonetheless,the presence of a magnetic field is required to launch a directed flow.

The simulations cover the same physical range as the one presented in the preced-ing section. The resolution is 400× 256× 256, so that the diameter of the differen-tially rotating surface is resolved with about 25 cells. The longest simulation, whichalso produces the most efficient jet, has the parameter values (βb = 1.2, Mb = 15).

75


Figure 4.12: Meridional slices through the jets presented in Sect. 4.4.4. A jet forms only if amagnetic field is present and the rotation velocity is sufficiently large.

It ran for 14 wallclock days on 96 processors and covers almost two jet crossings,the upper boundary being reached at t ≈ 62.

In the jet of the preceding section, Poynting flux was the main source of energy toaccelerate the jet. This is clearly different here. The simulation with (βb = 1.2, Mb =15) attains a momentary quasi-stationary state, with Etot ≈ const in the energy flow,at t ≈ 80 . . . 90. In sharp contrast to Fig. 4.11, Ethrm is the dominating componentat the beginning, being a factor of ∼3 larger than Emag. The equipartition pointEkin = Emag is very close to the lower boundary, the radius where Ekin = Ethrm islarger, at r ≈ 15. Emag also diminishes (by about 80%), but it is only a small fractionof the final Ekin.

The excess temperature responsible for the high Ethrm turns up near the centralaxis next to the lower boundary and increases with the speed of rotation. Theheat is apparently produced by numerical viscosity, i.e. dissipation of kineticenergy due to the coarse resolution of the rapidly rotating inner part of the disk.

76


Unfortunately, a much higher resolution is unfeasible and a large Mb is needed foran efficient production of Bϕ. The simulations are therefore relevant for cases inwhich dissipative heating is of importance.

The appearance of the flow becomes more jetlike with increasing rotation velocity.In a simulation with (βb = 1.2, Mb = 3), corresponding to a differential rotationvelocity of about 1.1 of the outer loops’ footpoints, the jet terminates at a few diskradii from the source in a tightly wound helix, similar to the free end of a gardenhose. With Mb = 9, the flow has a turbulent, “smoke stack”-like appearance. Thejet produced with Mb = 15 is subject to non-axisymmetric perturbations, but muchmore coherent.

Despite the main energy source being thermal, the results depend greatly onwhether a magnetic field is present or not. In a simulation with (βb = 1028, Mb = 15),i.e. with virtually no magnetic field, there is no unidirectional flow. Rather, materialis ejected sideways as well as in the upward direction, see Fig. 4.12. The simulationsoon crashes, presumably due to too much heat accumulating. The same simulationwith βb = 1.2 is well collimated and plows effortlessly through the dense parts ofthe atmosphere.

4.5 Summary and Discussion

We produced collimated outflows in numerical simulations by rotating the foot-points of arcade-shaped magnetic loops. The longest of these jets cross a com-putational domain which is two orders of magnitude larger than the size of thesource.

Flow acceleration via dissipative as well as non-dissipative processes was ob-served. The latter relies on the presence of shear in the form of differential rotationof the footpoints of individual field loops, which results into the development ofa toroidal magnetic field component. The resulting jet has a helical magnetic fieldstructure with field lines running back to the source outside the jet.

The case of uniformly rotating magnetic loops has also been studied. It turnsout to be markedly less effective at producing outflows; in fact, no jetlike outflowswere observed for uniformly rotating arcades within the numerically accessibleparameter range. The shear between the rotating loop and the stationary atmospheresurrounding it also produces a toroidal field component, but this appears to bemuch less effective than direct shearing of field lines inside the rotating source.

We find that the continuation of the outflow beyond the initial transient dependson the way the magnetic field is maintained at the lower boundary. Differentialrotation acting on a non-axisymmetric field quickly dissipates the field throughconvective expulsion, with the result being that the jet is “choked off” at the base.We have compensated for this by adding an inflow of magnetic field at the base.This would represent, for example, the emergence of loops of magnetic field intothe atmosphere of an accretion disk.

77


Apart from geometric requirements, the most important parameter for efficientjets turned out to be the (differential) speed of rotation of the footpoints. In themodels calculated, the necessary speed is of the order of the escape velocity fromthe disk. This is more than what is needed in simulations with a long-scale poloidalinitial field (e.g. those presented in Chapter 2). Consequently, the critical points(sonic, Alfven , fast magnetosonic) and the equipartition point between the magneticand kinetic energy flow rates are reached at smaller distances. The conversion ofmagnetic enthalpy (and the components thereof) to kinetic energy is similar apartfrom that, being fairly efficient.

In the adiabatic calculations presented here, thermal energy from hydrodynamicand magnetic dissipation stays in the flow. This is in contrast with our earlier calcu-lations (Chapters 2&3), where we assumed optically thin environments in whichmuch of the dissipated energy is lost by radiation. The present models assume noenergy loss and are most relevant for optically thick conditions. In the stratifiedatmosphere of these models, heating contributes to the flow by thermal buoyancy.In some of the cases presented (Sect. 4.4.4), this is the dominant driving mechanism.Though a wound-up magnetic field is also present in these buoyant plume flows,they probably do not qualify as magnetically driven. The non-axisymmetric “stir-ring” by the rotating magnetic field at the base that drives them may, however, wellbe relevant for the case of a (rapidly) rotating core inside a dense stellar envelope.Even though its field configuration is not of the right kind to produce a magneti-cally powered outflow, dissipation of rotational energy by “stirring” may still havepowerful effects on the envelope.

Although the simulations presented here demonstrate possible forms of rotation-induced acceleration, they probably do not represent very realistic models of actualaccretion disks. One may wonder what happens in a more realistic scenario withmore complicated or chaotically arranged magnetic field loops emerging from thedisk. Obviously, magnetic reconnection would likely play a major role in such ascenario. Nonetheless, the surviving magnetic field is stretched out in azimuthaldirection by the rotation, and the free energy in the toroidal field is transformedinto an outflow. Temporary fluctuations are likely to occur if the supply of magneticfield loops is not continuous. The coherent length of the jet will then be determinedby the speed with which reconnection processes destroy the initial field by whichthe flow is produced. The jet may flare up anew when new loops emerge. Thiseffect could be responsible for the temporary fluctuations in gamma ray bursts.

78


Figure 4.13: Volume renderings of jets in simulations with emerging magnetic loops (modelE) and differential rotation of the footpoints (variant a). The strength of the jet decreases witha decrease of the field strength (magenta arrows at top row and right column) and with adecrease of the rotation velocity (cyan arrows at left column), compare Fig. 4.6. All jets areaffected by non-axisymmetric instabilities.

79

5 Samenvatting

In dit proefschrift gaat het om numerieke magnetohydrodynamika, toegepast op de’jets’ die in astronomische objekten worden waargenomen.

In algemene zin wordt een jet in de astrofysika gedefinieerd als een langwerpige,snelle uitstroming van gas. Een van de meest karakteristieke trekken van deze jets isde sterke mate van kollimatie (een nauwe openingshoek), waardoor ze zich onder-scheiden van meer isotrope uitstromingen zoals sterwinden. Hun eigenschappen,en die van de objekten waar ze in ontstaan, varieren aanzienlijk. Toch zijn de eraanten grondslag liggende processen in alle gevallen zeer vergelijkbaar: het gaat omde akkretie van materie op een centrale massa, en de konversie van rotatieener-gie in een uitstroming door tussenkomst van een magneetveld. De verschillen ineigenschappen zijn in eerste orde een kwestie van verschillen in lengte- en tijdschaal.

De meeste jets schijnen te worden geproduceerd door een akkretieschijf, zelfeen uitgebreid fysisch probleemgebied met nog onbeantwoorde vragen. Het over-gangsgebied van de schijf naar de jet bepaalt hoeveel massa in de jet stroomt. Ditovergangsgebied is op het ogenblik nog een bron van onzekerheid in de vergelijkingvan theorie met waarnemingen.

We weten daarentegen wel goed hoe jets versneld en gekollimeerd kunnen wor-den door een magneetveld. De meeste bestaande modellen houden echter geenrekening met 3-dimensionale instabiliteiten die in dit magneetveld kunnen ont-staan, de jet storen, en wie weet het hele model ten val brengen. Dit is een van dehoofdthemas van dit proefschrift. Straling kan door de jet worden geproduceerddoor verschillende processen, daaronder synchrotronstraling, Comptonstrooiing enthermische straling. Schokgolven aan de rand met het medium dat de jet omringt enin het inwendige van de jet spelen hierbij een belangrijke rol. Deze wisselwerkingmet de omgeving is het hoofdonderwerp van de berekeningen in hoofdstuk 4.

Een globale numerieke behandeling van een realistische (schijf +) jet is tot inafzienbare toekomst niet uitvoerbaar, aangezien jets zich gewoonlijk uitstrekkenover vele ordes van grootte in afstand van de bron. In het geval van een quasar-jetbijvoorbeeld, heeft de jet zoals te zien bij radiogolflengten een afmeting in de ordevan 1 megaparsec, ca. 1010 keer zo groot als het gebied waar de jet ontstaat (eenpaar astronomische eenheden, zeg). De te verwachten tijdschalen van (magneto-)hydrodynamische processen in de jet strekken zich uit over een vergelijkbare faktor.Direkte, tijdsafhankelijke 3-dimensionale numerieke simulaties van zo grote objek-ten is uitgesloten. De waargenomen eigenschappen van jets, zoals hun vermogen,snelheid, breedte en de uitgezonden straling worden echter in hoge mate doortijdsafhankelijke, 3-dimensionale MHD-processen bepaald. De vraag is waar in de

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10 decaden in afstand deze verschillende processen plaatsvinden, en of ze zinvol inbeperktere numerieke simulaties geısoleerd kunnen worden. Uit de resultaten diehier voorgesteld worden is de konklusie dat veel van de fysika die de waargenomeneigenschappen bepaalt, plaatsvindt in de eerste drie dekaden vanaf het brongebied.Ook die drie dekaden zijn met de huidige computationele middelen een uitdagingvoor 3D MHD simulaties. Berekeningen over zo’n grote faktor in de lengteschaalwaren mogelijk door middel van een, aan de jet-geometrie aangepaste, methode diede auteur voor het onderzoek van dit proefschrift heeft ontwikkeld (hoofdstuk 2).

5.1 Verschillende Jet-Typen

Jets worden waargenomen in verschillende astronomische objekten, daarondergalaktische zowel als extragalaktische bronnen. Het streven ze te klassificerennaar waargenomen eigenschappen heeft geleid tot een oerwoud van namen. Uittheoretisch gezichtspunt ligt het echter voor de hand jets te onderscheiden naar hunfysische oorsprong. Waarnemingen zijn in het algemeen te grof om direkt inzicht televeren over de gebeurtenissen in het inwendige van de bron. De basale processendie in een fysisch model in rekening gebracht moeten worden zijn echter in velegevallen bekend, zodat een benadering middels numerieke simulaties voor de handligt.

Sommige jets worden waargenomen in de geboorteplaatsen van jonge stellaireobjekten (YSOs). Hier worden sterren gevormd in molekulaire (H2) reuzenwolkenals het gas onder zijn eigen gravitatiekracht samentrekt. Naarmate de centralekondensatie in de wolk groeit slokt hij het omringende gas op via een akkretieschijf.Jets ontstaan langs de rotatieas van de schijf met typische snelheden van een paarhonderd km/s, vergelijkbaar met de ontsnappingssnelheid van het oppervlak vande centrale protoster. Waar de jets met gaswolken in de buurt botsen veroorzakenze de komplexe nevelgebieden die uit Hα-opnamen bekend zijn als Herbig–Haroobjekten. De lengte van zo’n protostellaire jet is van de orde van 1 parsec, deverhouding van lengte tot breedte is typisch 10 : 1 of meer.

In dubbelstersystemen kan het voorkomen dat de kompaktere van de twee sterrenmassa van zijn begeleider akkreteert. Een schijf vormt zich om de groeiende steren een jet vormt zich langs de rotatieas. Als de kompaktere ster een zwart gat ofneutronenster is, is zo’n systeem gewoonlijk helder in Rontgenstraling, en wordtdaarom Rontgendubbelster genoemd (XRB). De eigenschappen van het waargeno-men licht van de jet hangen voornamelijk af van de samenstelling van de dubbelster.Wanneer de akkreterende ster zeer kompakt is (d.w.z. een neutronenster of zwartgat) en een jet produceert heet het systeem microquasar. De snelheid van zulke jetsis relativistisch.

Relativistische jets ontstaan ook in aktieve galaktische kernen (AGN), een klassevan lichtkrachtige en kompakte objekten in de centra van melkwegstelsels. De jetvan M87 is een beroemd voorbeeld. Het nu algemeen geaccepteerde beeld van

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een AGN behelst een zwart gat met een akkretieschijf, daaromheen een torus vanstof en gas, gaswolken op verschillende afstanden die emissielijnen produceren,en twee nauwe jets in tegengestelde richtingen langs de rotatieas van de schijf.De waargenomen eigenschappen hangen af van de hoek waaronder men de jetwaarneemt. AGN-jets maken bijzonder lange en snelle jets, met LorentzfaktorenΓ ∼ 10.

De snelste jets zijn niet direkt zichtbaar. Ze worden aangenomen als verklaringvan gamma-ray bursts (GRBs) in ver verwijderde melkwegstelsels. De grote waar-genomen lichtkracht en snelle variatie in de lichtkromme betekenen dat de bronvan een GRB optisch dik wordt tengevolge van γγ → e+ee− paarproduktie, ende waargenomen straling daardoor niet in staat is te ontsnappen, tenzij ultrare-lativistische snelheden (Γ & 100) een rol spelen. Energetische overwegingen endetails van het ‘nagloeien’ (een na de burst langzaam afvallende emissie van stralingdie ontstaat als de jet in het interstellaire gas ploegt) wijzen erop dat de strominggebundeld is: een jet in plaats van een meer sferische expansie. Als dit juist is, isde totale energie die in een GBR vrijkomt van de orde 1051 erg, vergelijkbaar meteen supernovaexplosie. Het algemeen aangenomen model voor de meeste GRBsis dat hun energie geproduceerd wordt wanneer de snel roterende kern van eenzware ster in elkaar stort tot een zwart gat. De preciese omstandigheden die in ditscenario tot de vorming van een jet met de gewenste eigenschappen leiden zijn ophet ogenblik nog niet geheel duidelijk. Om de waargenomen energie van de jette leveren moet het mechanisme zeer efficient zijn. Bovendien moet het model desnelle variaties in de lichtkrommen van GRBs kunnen verklaren.

De berekeningen voor dit proefschrift zijn niet-relativistisch. Hoewel ze daaromniet direkt van toepassing zijn op AGN- en GRB-jets, is te verwachten dat deversnellings- en instabiliteitsprocessen die hier beschreven worden ook in deze jetseen rol spelen.

De Omgeving van de Jet De omgeving waar de jet in beweegt kan van ver-schillende oorsprong zijn. Dicht bij de bron domineert de zwaartekracht van hetcentrale objekt; gas dat in deze potentiaal aanwezig is vormt een atmosfeer waarinde dichtheid monotoon met de afstand afneemt. Op grotere afstanden komt dejet in kontakt met een interstellair of intergalaktisch medium. In AGN vermoedtmen wisselwerking met ’clouds’, die de waargenomen lijn-emissie veroorzaken (e.g.Poludnenko et al. 2002). GRB jets die in de kern van een zware ster geproduceerdworden, moeten zich een gat door de omringende ster boren. Bij protostellaire jetszijn er aanwijzingen dat ze wisselwerken met de nog invallende materie van demolekulaire wolk die de akkretieschijf van de protoster voedt (Velusamy en Langer1998).

In numerieke simulaties komt de wisselwerking van de jet met de omgevingvanzelf aan de orde, aangezien om numerieke redenen een omringend medium meteen zekere (zij het minimale) dichtheid aangenomen moet worden. In dit proefschrift

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(vooral in hoofdstuk 4) blijkt dat naast de interakties die ook bij hydrodynamische(niet-magnetische) jets bekend zijn (zoals schokgolven in het omringend medium)er ook bijzondere ‘kronkelende’ interakties zijn die direkt met het magneetveldsamenhangen.

5.2 Magnetisch Aangedreven Stromingen

Het idee van een door magnetische krachten aangedreven jet gaat terug tot Bisnova-tyi-Kogan en Ruzmaikin (1976), en Blandford en Payne (1982), die lieten zien datjets kunnen ontstaan door de rotatie van een akkretieschijf waar een magneetveldvan passende vorm in verankerd is. Tot nu toe is dit het nuttigste koncept geblekenter verklaring van de meest karakteristieke eigenschappen van jets, zoals de hogesnelheden en hoge mate van kollimatie. Indirekte steun uit waarnemingen wordtgeleverd door het feit dat veel jets op de (grotere) afstanden waar dit gemeten kanworden, gemagnetiseerd blijken; de synchrotronstraling van extragalaktische jetsbijvoorbeeld is alleen mogelijk als er een magneetveld is.

Het essentiele element van een magnetisch aangedreven jet is een in het roterendeobjekt verankerd magneetveld, bijvoorbeeld een akkretieschijf. In het eenvoudigstegeval heeft het ankerveld alleen poloidale komponenten: in de richting van derotatieas en ervan weg, maar niet eromheen, en strekt het zich uit tot oneindig.De rotatie produceert daaruit een azimuthale veldkomponent, in andere woorden,’wikkelt het veld op’. Dit toroidale1 veld bevat vrije energie die in een uitstromingomgezet wordt.

De versnelling kan begrepen worden als een gevolg van de magnetische drukgra-dient geassocieerd met het toroidale veld Bϕ. Deze kracht wordt ten dele gekom-penseerd door de magnetische ’krommingskracht’ die in de tegengestelde richtingwijst. De twee krachten heffen elkaar precies op als de stroming ballistisch is, d.w.z.in alle richtingen rechtlijnig met konstante snelheid.

In een stelsel dat met de centrale rotator meedraait kan de versnelling als alterna-tieve interpretatie ook aan een centrifugale kracht toegeschreven worden. Dit is eenelegante beschrijving als aangenomen mag worden dat het magneetveld voldoendesterk is dat het plasma in de stroming met de bron meeroteert (korotatie). Dit ismeestal het geval ongeveer tot aan de Alfven-afstand. Dit centrifugale model ismathematisch ekwivalent2 met het magnetische beeld hierboven, maar beperktervan toepassing. Korotatie is bijvoorbeeld niet van toepassing als de massaflux hoog,of de magnetische veldsterkte van de bron laag is.

Versnelling door Dissipatie De versnelling door rotatie houdt op effektief te zijnwanneer de stromingssnelheid de Alfven- of de magnetosonische snelheid bereikt,

1Hier gebruikt als synoniem met azimuthaal.2De huidige literatuur van numerieke simulaties vertoont overbodige verwarring op dit punt.

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5.2 MAGNETISCH AANGEDREVEN STROMINGEN

en de stroming daarna ballistisch wordt (d.w.z. als bij de brandweer). Er is daneen evenwicht tussen de magnetische druk- en krommingskrachten die verdereversnelling verhindert. Om dit te verbreken moet Bϕ sneller afvallen dan in ballisti-sche expansie. Dit is het geval als het veld (in een meebewegend systeem) vervalt,zodat de magnetische drukgradient steiler wordt. Zo’n verval gebeurt als gevolgvan 3-dimensionale (kink) instabiliteiten. In een elegant beeld van dit effect kan dePoynting flux vergeleken worden met een flux van ’magnetische enthalpie’ in plaatsvan magnetische (interne) energie. Dit geeft een alternatieve interpretatie van hetfeit dat de Poynting flux twee keer de waarde van de magnetische energie B2

ϕ/8πbevat. Als de magnetische energie van het Bϕ-veld dissipeert, wordt dus de helft inwarmte omgezet (die dan bijvoorbeeld uitgestraald kan worden), de verblijvendehelft draagt bij tot versnelling van de stroming.

In hoofdstuk 3 worden deze effekten aan de hand van de simulaties nader onder-zocht, en de versnellingsmechanismen in 3-dimensionale en axiaalsymmetrischeberekeningen met elkaar vergeleken.

5.2.1 Modellen

Stationaire Modellen Het meeste theoretische werk over MHD-stroming maaktgebruik van vereenvoudigende aannamen, vooral die van stationariteit en axialesymmetrie, die tot een toegepast-wiskundig aantrekkelijke theorie leiden. Voor-beelden zijn de oplossingen van Sakurai (1985, 1987). In het bekende werk vanBlandford en Payne (1983) wordt bovendien aangenomen dat de stroming ’selfsi-milar’ en koud (gasdruk verwaarloosbaar) is. In het geval van relativistische jetskan in sommige omstandigheden een ’krachtvrije’ benadering gebruikt worden(e.g. Fendt et al. 1995). Hoewel krachtvrije modellen niet de versnelling verklaren,wordt geargumenteerd dat de hiermee berekende veldkonfiguratie wel de juistevorm heeft om tot versnelling te leiden. Een klassieke vereenvoudiging in hetniet-relativistische geval (Weber en Davis, Mestel, voor een introduktie zie Spruit1996) is aan te nemen dat het poloidale veld niet verandert door het toroidale velddat door de rotatie ontstaat.

Simulaties De aanname van stationariteit heeft natuurlijk zijn beperkingen; hetis daarin o.a. niet mogelijk te beschrijven hoe de kop van de jet door de omgevingbeweegt (in veel waargenomen jets mooi te zien). Een tijdsafhankelijke berekeningvergt ook in axiale symmetrie al veel rekentijd. Sommige auteurs beperken zichdaarom tot de beweging van de jet door de omgeving, vanaf het ogenblik dathet versnellingsproces al is afgelopen (e.g. Kossl en Muller 1988). Om iets overde magnetische versnelling te leren, is het nodig dat het model op zijn minst alsrandvoorwaarde een akkretieschijf bevat (e.g. Ustyugova et al.1995). Dit wordtgedaan in de berekeningen van dit proefschrift. Met simulaties waarin ook deakkretieschijf zelf onderdeel van het rekenrooster is (e.g. Hayashi et al.1996) kan

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alleen de direkte omgeving boven de schijf, de basis van de jet, berekend worden.Het grootste probleem hierbij is dat het kontrast in lengte- en tijdschalen te groot is.Voorlopig moeten we ons nog tevreden stellen met onderzoek van delen van hethele probleem.

5.3 Magnetische Instabiliteiten

Jets zijn onderhevig aan diverse instabiliteiten. De belangrijkste daarvan wordenofwel door zijn beweging langs het omringende medium veroorzaakt, ofwel doorhet sterke toroidale veld in de jet zelf. Zie hiervoor de introduktie in 2.1 en dereferenties daarin. Het werk van dit proefschrift houdt zich in eerste instantie bezigmet de laatste klasse van instabiliteiten. Dit zijn dezelfde die een centrale rol spelenbij het ontwerp van fusiereaktoren.

Hoewel in vele gevallen beter te beschrijven als afgeknotte kegels worden jetsin berekeningen vaak als eerste benadering in cylindrische geometrie beschreven.Het evenwicht van zulke kolommen is een goed onderzocht gebied in de kontextvan fusiereaktoren (voor een gedegen introduktie zie Freidberg 1987). Evenwicht inafwezigheid van externe krachten betekent dat de som van gasdruk en magnetischedruk, die naar buiten werken, gekompenseerd wordt door de (naar de as gerich-te) krommingskracht geassocieerd met de azimuthale veldkomponent Bϕ. Dezekomponent maakt de kolom echter ook instabiel. In het laboratorium wordt dezekomponent gerealiseerd door middel van een elektrische stroom langs de kolom;de instabiliteit ervan wordt in dit verband ook ’current driven’ genoemd. In deastrofysische MHD zijn er geen stroomleverende batterijen; het Bϕ veld ontstaatin plaats daarvan door een makroskopische beweging: het ’opwikkelen’ van eenveld door rotatie van het brongebied. We vermijden daarom liever de naam ’currentdriven’, en gebruiken de (ook algemeen gangbare, en geometrisch inzichtelijkere)naam ’kink instabilities’. Voor een plaatje van dit soort instabiliteiten zie Fig. 1.2.

Gevolgen voor de Jet De konsekwenties van MHD instabiliteiten zijn gewoonlijkfataal voor fusieplasmas, daar ze met hun hoge groeisnelheid de breekbare toestandvan insluiting die het plasma van de wanden weg houdt zeer snel verstoren. Dit zoueen reden voor bezorgdheid kunnen zijn over de levensvatbaarheid van magnetischejets, die het over grote afstanden zonder steun van stabiliserende wanden moetendoen. De zorg is dat door instabiliteit de jet uitelkaar zou vallen en daarmee inzijn omgeving ’op zou lossen’. Als dit plaats zou vinden op korte afstand van debron, waar het magneetveld al genoeg opgewikkeld is om inderdaad instabiel teworden, zou het magnetische model niet in staat zijn de waargenomen lange jets teverklaren.

De ontwikkeling van MHD instabiliteiten in expanderende jets is echter funda-menteel verschillend van de statische laboratorium situatie. Dit wordt beschrevenin hoofdstuk 2. Enerzijds beperkt de zijdelingse expansie van de jet de groei van

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instabiliteiten. Anderzijds neemt de azimuthale magneetveldsterkte die de insta-biliteit aandrijft in het verloop van de instabiliteit af. De jet ondergaat een zekerezijdelingse verbreding als gevolg van de instabiliteit, maar de kinetische energiedie de jet gekregen heeft blijft bestaan. Door het verval van het azimuthale veld ishet mogelijk dat de jet, hoewel magnetisch van oorsprong, zich uiteindelijk als eengewone hydrodynamische jet gaat gedragen.

De instabiliteiten kunnen zich uiten in kenmerkende strukturen op waarneembareschalen, bijvoorbeeld de ’wiebels’ in sommige protostellaire jets (hoewel daar ookandere verklaringen voor zijn voorgesteld). Het ligt voor de hand deze strukturendan te identificeren met de verplaatsingen die je van de (lineaire) instabiliteit zouverwachten. De resultaten van hoofdstuk 2 en 3 laten echter zien dat de wiebelsveel langer worden dan de snelst groeiende instabiliteiten; dit is een gevolg van deniet-lineaire ontwikkeling van de instabiliteit.

5.4 Magnetische Rekonnektie

Gedurende de niet-lineaire groei van de instabiliteiten kunnen gebieden met ver-schillende richting van het magneetveld dicht tegen elkaar komen te liggen, en kan’rekonnektie’ tussen die gebieden tot verval van het azimuthale veld leiden. Zoalshierboven vermeld, leidt dit verval tot een toename van de magnetische drukgra-dient langs de jet. Dit leidt ook, ietwat anti-intuitief, tot een versnelling van de jet.Tegelijk wordt de jet door het effekt van de instabiliteiten, gemiddeld in de tijd, wathomogener over zijn doorsnee (zo je wilt ’turbulente diffusie’). Daardoor wordt hetmechanisme dat in axiale symmetrie de jet versnelt minder belangrijk. De evolutievan een 3-D jet verschilt dus in essentiele punten van die in een axiaalsymmetrische.

Het funktioneren van rekonnektie in astrofysische kontext is nog een onderwerpvan debat in de literatuur. Diverse gedetailleerde modellen zijn niet makkelijk metwaargenomen rekonnektie-snelheden in overeenstemming te brengen (Kulsrud2001). In de MHD berekeningen in dit proefschrift gebeurt rekonnektie implicietdoor de numerieke diffusie die inherent is aan het diskretisatieschema. Waar veldenvan verschillende richting in verloop van de evolutie in een roostercel bijelkaargebracht worden, verschijnt in de volgende tijdstap alleen hun gemiddelde. Rekon-nektie aan het numerieke schema overlaten neemt dus aan dat het een snel procesis. Een bekend model voor rekonnektie is dat van Petschek; het leidt tot een in-stroming van veldlijnen naar het rekonnektiegebied, met een goede fraktie van deAlfvensnelheid. Dit ligt dichter bij de snelheid die in simulaties door numeriekediffusie ontstaat dan het (eveneens populaire) Sweet-Parker model, dat veel lagererekonnektiesnelheden levert. Waarnemingen van de zonnekorona wijzen op eentamelijk hoge rekonnektiesnelheid (aangezien de veldlijnen in de korona meestal opdie van de minimum-energietoestand, een potentiaalveld, lijken). Dit wordt somsals argument voor een hoge rekonnektiesnelheid in algemene astrofysische situatiesgenomen. In de hier gepresenteerde berekeningen is impliciet ook deze aanname

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gemaakt. Opgemerkt moet echter dat er goed gedokumenteerde gevallen zijn waardeze aanname niet opgaat (zie bijvoorbeeld de MRI simulaties van Fromang enPapaloizou, 2007).

5.5 Samenvatting van de Belangrijkste Resultaten

• Met behulp van een speciaal aangepast numeriek rooster konden jets in driedimensies gesimuleerd worden over zeer grote afstanden. Met de grootstesimulaties in dit proefschrift werden voor het eerst afstanden van 1000 maalde diameter van de bron gerealiseerd.

• Versnelling van jets door een roterend magneetveld is tamelijk efficient. Hetgrootste deel van de magnetische enthalpie die met het azimuthale veldsamenhangt wordt in kinetische energie omgevormd.

• Magnetische jets zijn onderhevig aan magnetische instabiliteiten. Hoe sterk heteffekt van deze instabiliteiten is hangt nauw samen met de aard van kollimatievan de jet, en met de konfiguratie van het roterende magneetveld aan de basisvan de jet. De instabiliteiten zijn sterker als de graad van kollimatie met deafstand toeneemt (de openingshoek afneemt).

• In tegenstelling tot in de literatuur geuitte bezorgdheid betekenen kink-insta-biliteiten niet het einde van een jet. Ook als het aandrijvende magneetveldvolledig vervalt, kan de jet ballistisch voort blijven bewegen, zij het zijdelingswat verbreedt. Om een jet werkelijk te stoppen is een sterke wisselwerkingmet het omringende medium nodig.

• De dissipatie van het magneetveld door de 3-dimensionale instabiliteitenlevert interne energie die uitgestraald kan worden, zoals in waarnemingenvan jets ook te zien is. De vervormingen van de jet die de simulaties laten zienkunnen ook sommige van de ’knopen’ en ’wiebels’ verklaren die in (met nameprotostellaire) jets waargenomen worden.

• Het mechanisme dat de Poynting flux van de jet in kinetische energie omzetverschilt in drie dimensies van de 2-dimensionale (axisymmetrische) bereke-ningen die de literatuur beheerst hebben. De netto efficientie ervan is echtervergelijkbaar. In drie dimensies wordt de stroming aangedreven door eenmagnetische drukgradient die door het verval van het magneetveld ontstaat.In axiale symmetrie is het een ’divergerende nozzle’ effekt.

• Jets kunnen zowel door een bron met een grootschalig geordend veld versneldworden als door een bron bestaande uit kleinschalige magnetische lussen.In het laatste geval werkt het mechanisme het beste als de bron differentieelroteert en de lussen daardoor ’opgewikkeld’ worden.

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Bibliography

Anderson, J. M., Li, Z.-Y., Krasnopolsky, R., & Blandford, R. D. 2006, ApJ, 653, L33

Antoniucci, S., Nisini, B., Giannini, T., & Lorenzetti, D. 2008, A&A, 479, 503

Appl, S. & Camenzind, M. 1992, A&A, 256, 354

Appl, S., Lery, T., & Baty, H. 2000, A&A, 355, 818

Baade, W. & Minkowski, R. 1954, ApJ, 119, 215

Bally, J. & Devine, D. 1994, ApJ, 428, L65

Bateman, G. 1978, MHD Instabilities (MIT Press)

Baty, H. & Keppens, R. 2002, ApJ, 580, 800

Begelman, M. C. 1998, ApJ, 493, 291

Begelman, M. C. & Li, Z.-Y. 1994, ApJ, 426, 269

Bell, A. R. & Lucek, S. G. 1996, MNRAS, 283, 1083

Birkinshaw, M. 1991, in Beams and Jets in Astrophysics, ed. P. Hughes (CambridgeUniversity Press), 278–341

Bisnovatyi-Kogan, G. S. & Ruzmaikin, A. A. 1976, Ap&SS, 42, 401

Blandford, R. D. 1976, MNRAS, 176, 465

Blandford, R. D. & Payne, D. G. 1982, MNRAS, 199, 883

Bodo, G., Massaglia, S., Rossi, P., et al. 1995, A&A, 303, 281

Bodo, G., Rossi, P., Massaglia, S., et al. 1998, A&A, 333, 1117

Cao, X. & Spruit, H. C. 1994, A&A, 287, 80

Curtis, H. D. 1918, Publications of Lick Observatory, 13, 55

De Villiers, J.-P., Hawley, J. F., Krolik, J. H., & Hirose, S. 2005, ApJ, 620, 878

Donati, J.-F., Paletou, F., Bouvier, J., & Ferreira, J. 2005, Nature, 438, 466

89

BIBLIOGRAPHY

Drenkhahn, G. 2002, A&A, 387, 714

Eichler, D. 1993, ApJ, 419, 111

Eisloffel, J. & Mundt, R. 1992, A&A, 263, 292

Evans, C. R. & Hawley, J. F. 1988, ApJ, 332, 659

Fendt, C. 2009, ApJ, 692, 346

Fendt, C., Camenzind, M., & Appl, S. 1995, A&A, 300, 791

Freidberg, J. P. 1987, Ideal Magnetohydrodynamics (Plenum Press, New York)

Fromang, S. & Papaloizou, J. 2007, A&A, 476, 1113

Furno, I., Intrator, T. P., Ryutov, D. D., et al. 2006, Physical Review Letters, 97, 015002

Galeev, A. A., Rosner, R., & Vaiana, G. S. 1979, ApJ, 229, 318

Giannios, D. & Spruit, H. C. 2006, A&A, 450, 887

Gredel, R., Reipurth, B., & Heathcote, S. 1992, A&A, 266, 439

Harten, A. 1983, J. Comput. Phys., 49, 357

Hartigan, P., Heathcote, S., Morse, J. A., Reipurth, B., & Bally, J. 2005, AJ, 130, 2197

Hartigan, P. & Raymond, J. 1993, ApJ, 409, 705

Hayashi, M. R., Shibata, K., & Matsumoto, R. 1996, ApJ, 468, L37

Heathcote, S., Morse, J., Hartigan, P., et al. 1996, AJ, 112, 1141

Heinz, S., Grimm, H. J., Sunyaev, R. A., & Fender, R. P. 2008, ApJ, 686, 1145

Hsu, S. C. & Bellan, P. M. 2005, Physics of Plasmas, 12, 2103

Johns-Krull, C. M., Greene, T. P., Doppmann, G. W., & Covey, K. R. 2009, ApJ, 700,1440

Johnson, J. L., Oberman, C. R., Kulsrud, R. M., & Frieman, E. A. 1958, Physics ofFluids, 1, 281

Junor, W., Biretta, J. A., & Livio, M. 1999, Nature, 401, 891

Kato, Y., Mineshige, S., & Shibata, K. 2004, ApJ, 605, 307

Keppens, R., Toth, G., Westermann, R. H. J., & Goedbloed, J. P. 1999, Journal ofPlasma Physics, 61, 1

90

BIBLIOGRAPHY

Kirby, R. M., Marmanis, H., & Laidlaw, D. H. 1999, in IEEE Visualization, 333–340

Konigl, A. & Choudhuri, A. R. 1985, ApJ, 289, 173

Konigl, A. & Pudritz, R. E. 2000, Protostars and Planets IV, 759

Kossl, D. & Muller, E. 1988, A&A, 206, 204

Krichbaum, T. P., Alef, W., Witzel, A., et al. 1998, A&A, 329, 873

Kudoh, T., Matsumoto, R., & Shibata, K. 2002, PASJ, 54, 267

Kulsrud, R. M. 2001, Earth, Planets, and Space, 53, 417

Laidlaw, D. H., Kirby, R. M., Jackson, C. D., et al. 2005, IEEE Transactions onVisualization and Computer Graphics, 11, 59

Lapenta, G., Furno, I., Intrator, T., & Delzanno, G. L. 2006, Journal of GeophysicalResearch (Space Physics), 111, 12

LeBlanc, J. M. & Wilson, J. R. 1970, ApJ, 161, 541

Lery, T., Baty, H., & Appl, S. 2000, A&A, 355, 1201

LeVeque, R. J. 1992, Numerical Methods for Conservation Laws, 2nd edn., ETHZurich: Lectures in mathematics (Birkhauser)

Lynden-Bell, D. 2003, MNRAS, 341, 1360

Masciadri, E. & Raga, A. C. 2002, ApJ, 568, 733

Mundt, R., Buehrke, T., Solf, J., Ray, T. P., & Raga, A. C. 1990, A&A, 232, 37

Nakamura, M. & Meier, D. L. 2004, ApJ, 617, 123

Nolting, W. 2002, Grundkurs Theoretische Physik 1: Klassische Mechanik, 6th edn.(Springer)

Obergaulinger, M. 2008, PhD thesis, Max-Planck-Institut fur Astrophysik, Garchingbei Munchen

Ouyed, R., Clarke, D. A., & Pudritz, R. E. 2003, ApJ, 582, 292

Park, B. T. & Petrosian, V. 1996, ApJS, 103, 255

Parker, E. N. 1957, Physical Review, 107, 830

Parker, E. N. 1963, ApJ, 138, 552

Pen, U.-L., Arras, P., & Wong, S. 2003, ApJS, 149, 447

91

BIBLIOGRAPHY

Petschek, H. E. 1964, NASA Special Publication, 50, 425

Poludnenko, A. Y., Frank, A., & Blackman, E. G. 2002, in ASP Conference Series,Vol. 255, Mass Outflow in Active Galactic Nuclei: New Perspectives, ed. D. M.Crenshaw, S. B. Kraemer, & I. M. George, 285

Priest, E. 1982, Solar Magnetohydrodynamics, Vol. 21 (D. Reidel, Dordrecht)

Reipurth, B. & Bally, J. 2001, ARA&A, 39, 403

Reipurth, B., Heathcote, S., Morse, J., Hartigan, P., & Bally, J. 2002, AJ, 123, 362

Romanova, M. M., Ustyugova, G. V., Koldoba, A. V., Chechetkin, V. M., & Lovelace,R. V. E. 1998, ApJ, 500, 703

Sakurai, T. 1985, A&A, 152, 121

Sakurai, T. 1987, PASJ, 39, 821

Sarazin, C. L. 1986, Reviews of Modern Physics, 58, 1

Shepherd, D. S., Claussen, M. J., & Kurtz, S. E. 2001, Science, 292, 1513

Shu, F. H., Najita, J., Ostriker, E. C., & Shang, H. 1995, ApJ, 455, L155+

Sikora, M., Begelman, M. C., Madejski, G. M., & Lasota, J.-P. 2005, ApJ, 625, 72

Spruit, H. C. 1996, arXiv:astro-ph/9602022v1

Spruit, H. C. 2008, arXiv:0804.3096 [astro-ph]

Spruit, H. C., Foglizzo, T., & Stehle, R. 1997, MNRAS, 288, 333

Stanke, T., McCaughrean, M. J., & Zinnecker, H. 1999, A&A, 350, L43

Suresh, A. & Huynh, H. 1997, J. Comput. Phys., 136, 83

Sun, X., Intrator, T. P., Dorf, L., Furno, I., & Lapenta, G. 2008, Physical ReviewLetters, 100, 205004

Sweet, P. A. 1958, in IAU Symposium, Vol. 6, Electromagnetic Phenomena in Cosmi-cal Physics, ed. B. Lehnert, 123

Tadhunter, C., Marconi, A., Axon, D., et al. 2003, MNRAS, 342, 861

Tchekhovskoy, A., McKinney, J. C., & Narayan, R. 2008, MNRAS, 388, 551

Todo, Y., Uchida, Y., Sato, T., & Rosner, R. 1993, ApJ, 403, 164

Toro, E. F. & Titarev, V. A. 2006, J. Comput. Phys., 216, 403

92

BIBLIOGRAPHY

Tout, C. A. & Pringle, J. E. 1996, MNRAS, 281, 219

Trac, H. & Pen, U.-L. 2003, PASP, 115, 303

Turner, N. J., Bodenheimer, P., & Rozyczka, M. 1999, ApJ, 524, 129

Ustyugova, G. V., Koldoba, A. V., Romanova, M. M., Chechetkin, V. M., & Lovelace,R. V. E. 1995, ApJ, 439, L39

Uzdensky, D. A. & Goodman, J. 2008, ApJ, 682, 608

van Ballegooijen, A. A. 1989, in Accretion Disks and Magnetic Fields in Astrophysics,ed. G. Belvedere, 99–106

Velusamy, T. & Langer, W. D. 1998, Nature, 392, 685

Wiseman, J., Wootten, A., Zinnecker, H., & McCaughrean, M. 2001, ApJ, 550, L87

Zeilik, M. & Gregory, S. 1998, Introductory Astronomy and Astrophysics, 4th edn.(Brooks/Cole)

Zel’dovich, Y. B. 1956, JETP, 31, 154

93

Appendix A

Numerical Methods

A.1 Principles and Terminology in ComputationalFluid Dynamics

Below, I give a rough overview on how the (M)HD equations are solved numericallyon a grid without going into much detail. For a clear introduction to the subjectmatter of grid-based numerical HD, the reader is referred to Trac & Pen (2003). Thegeneralization to MHD is explained in the same manner in Pen et al. (2003). Detailsabout the computer code used in the present work can be found in Obergaulinger(2008).

The first step is to discretize space by dividing it into a finite number of small,disjoint volumes called grid cells or zones. It suffices to consider only one dimensionwhose coordinate shall be x, noting that with more dimensions, the algorithmoutlined below is successively applied on the different dimensions using the sameinitial data. The cell which goes from xi− 1

2to xi+ 1

2shall be denoted with Zi.

Except for the induction equation (2.3.1d), which is treated separately, the MHDequations (2.3.1) can be written in the conservation form

∂tu + ∇ · F = S (A.1)

where the conserved variable u is one of (ρ,ρv,e), the flux F is accordingly a vector ora rank 2 tensor and S is an optional source term, e.g. gravity. For each cell Zi, theset of conserved variables represents its physical state, either as volume averageor as the value at the cell center xi. This state may change due to fluxes acrossthe cell’s boundary, i.e. the cell interfaces at xi± 1

2, or the source term. The fluxes

are functions of the conserved variables. Hence, we can compute the fluxes at thecell interfaces if we reconstruct the conserved variables there by making use of anappropriate interpolating function φi(x). The grid cells used in this process arecalled reconstruction stencil. Several methods exist.

The piecewise-constant method (PCM) uses φi(x) = ui, that is, it simply uses thevalue at the cell center as an approximation to the value at the cell interface. Ob-viously, this method is not very accurate, but it is the numerically most stable one.The piecewise-linear method (PLM) uses linear interpolation between the value ui of

95

APPENDIX A NUMERICAL METHODS

the considered cell and the values ui±1 of the neighboring cells. This yields twofunctions, φ−i at the left interface xi− 1

2and φ+

i at the right interface xi+ 12, which may

be different in general. It suggests itself to use φ−i for calculating the flux through theleft interface and φ+

i for calculating the flux through the right interface. However,as this is problematic at discontinuities according to experience, one is forced to findsuitable combinations of the two. This task is dealed with by total variation diminish-ing (TVD) slope limiters. Higher-order reconstruction schemes employ polynomialsof 2nd order or more for the interpolation. Having found φi from such methods,one can take either φi(xi± 1

2) directly (point-value formulation) or build the volume

average of φi over the staggered cells Zi± 12

(volumetric formulation).

After reconstructing the conserved variables to the cell interfaces, we can computethe fluxes. In general, the flux at the interface xi+ 1

2resulting from the reconstruction

centered at Zi will not be the same as the one resulting from the reconstructioncentered at Zi+1, and one speaks of left and right flow states. Whenever a well-definedupwind direction can be identified, the obvious choice is to take the upwind flux,i.e. the one from the left cell if the velocity is directed to the right. However, thisis not the case in general and one is left with a Riemann problem. For example, anAlfven wave might propagate to the right while a sound wave travels to the left.Depending on the system (e.g. HD or MHD), an exact solution to this problem isfairly complicated, numerically expensive or even unknown. Therefore, approximateRiemann solvers are used to determine the “best” flux from the left and right states.If necessary, the accuracy of approximate solvers can be improved e.g. by the multistage (MUSTA) method, which uses “virtual” time steps to find better solutions.

Having determined the interface fluxes, the solution can be advanced in timee.g. using a Runge–Kutta method. The timestep must small enough to satisfy theCourant–Friedrichs–Lewy (CFL) condition, which states that ∆t must be smaller thanthe time needed for the fastest wave to cross the cell, at any cell in the whole grid.

As mentioned above, the induction equation requires special treatment. In con-trast to Eq. (A.1), it has the form ∂tu + ∇× F = 0 where F is c times the electricfield. It is important that the magnetic remains solenoidal at all times. This can beachieved via the technique of constrained transport, which relies on the magnetic fieldbeing defined at the center of the cell faces: Bx at (xi± 1

2, yj, zk), By at (xi, yj± 1

2, zk)

and Bz at (xi, yj, zk± 12) for the cell Zi,j,k. The electric field is reconstructed at the

(center of the) cell’s edges, the various components being distributed such that eachis parallel to its edge1. ∂tB can then calculated. How the initial magnetic field canbe installed in a staggered grid in a way that guarantees ∇ · B = 0 is described inthe next section.

1Note that there are several possiblities. One may first reconstruct v to the cell face, compute the electricfield E and then reconstruct E at the cell edges, or one may first reconstruct v and B at the cell edgesand compute E there.

96

A.2 IMPOSING THE INITIAL MAGNETIC FIELD

A.2 Imposing the Initial Magnetic Field

Setting up the initial magnetic field for a simulation is not trivial. Every grid cellmust be solenoidal in the magnetic field as best as possible, ideally up to machineprecision. A practical way to do this is by means of the corresponding vectorpotential (Evans & Hawley 1988). First of all, A is prescribed at the center of thegrid cells’ edges. The components are distributed such that they point along therespective edge: Ax is defined at (i, j− 1

2 , k − 12 ), Ay is defined at (i − 1

2 , j, k − 12 )

and Az is defined at (i− 12 , j− 1

2 , k) for all i, j and k. The magnetic field components,which are defined at the center of the cell faces with normal orientation, can nowbe computed from the vector potential components of the surrounding edges. InCartesian coordinates, we have simply

Bi− 12 ,j,k

x = ”(∇× A)x = ∂y Az − ∂z Ay”

=1

∆y

(Ai− 1

2 ,j+ 12 ,k

z − Ai− 12 ,j− 1

2 ,kz

)− 1

∆z

(Ai− 1

2 ,j,k+ 12

y − Ai− 12 ,j,k− 1

2y

)(A.1)

and so on, with ∆ denoting the line elements (edge lengths). In general, one has toresort to the definition of the curl as circulation per unit area. For example, the xcomponent of B follows from

Si− 12 ,j,kBi− 1

2 ,j,kx = ”

∮∂S

dl · A”

= li− 12 ,j,k+ 1

2y Ai− 1

2 ,j,k+ 12

y − li− 12 ,j,k− 1

2y Ai− 1

2 ,j,k− 12

y

+ li− 12 ,j− 1

2 ,kz Ai− 1

2 ,j− 12 ,k

z − li− 12 ,j+ 1

2 ,kz Ai− 1

2 ,j+ 12 ,k

z (A.2)

with S and l denoting the face area and edge length, respectively. Following thisprocedure, the set magnetic field is automatically solenoidal. Special care has to betaken at coordinate singularities.

A.3 Boundaries Conditions

Experience has shown that the success of simulations stands or falls with the im-plementation of the boundary conditions, hence some remarks are in order. Thestandard boundary conditions are realized via ghost cells as follows. As an example,the x-left boundary shall be considered, with i = 1

2 being the index of the boundaryinterface.

Open (“Outflow”) The objective here is to allow material and waves to cross theboundary smoothly, without spurious forces at the boundary interface. Thisis done by setting U0,j,k = U1,j,k, U−1,j,k = U2,j,k and so forth for the hy-drodynamic quantities U = (ρ, p, v) and for the gravitational potential Φ.

97

APPENDIX A NUMERICAL METHODS

Figure A.1: Results of the test described in Sect. A.3 with By = 4 (β = 1/16, vA,y ≈ 4.4)and vx = 0.3 at the y-left boundary, of which the first ghost cell is included in the plot. Inthe left-hand case all quantities were held at their initial values in the boundary ghost cells,whereas in the right-hand case open conditions were used for B. The horizontal dotted linesshow the imposed vx and the expected Bx (multiplied with 10). The vertical dotted line is aty = vA,yt.

The transverse magnetic field components, By and Bz in the given case, are

treated likewise: B0,j−1/2,ky = B1,j−1/2,k

y , B0,j,k−1/2z = B1,j,k−1/2

z etc. The normalmagnetic field component is computed from the constraint that ∇ · B shouldvanish also in the ghost cells.

Reflective These are realized just as open boundaries, except that the signs of thenormal velocity and the transverse magnetic field are reversed.

Periodic The values from the cells next to the opposing boundary are copied to theghost cells: U0,j,k = UNx ,j,k, U−1,j,k = UNx−1,j,k etc.

It has proven itself beneficial to fix as few quantities as possible to avoid spuriousoscillations in the solution. This can be demonstrated in 2D Cartesian simulationsthat are in some sense analogous to the conditions at the lower boundary of thejet simulations. The setup consists of a homogeneous magnetic field and a theretoorthogonal velocity field at the boundary. The initial conditions are: B = By ey,ρ = ρ0 = const, p = p0 = const and vx = vy = 0 inside the computational domain.The boundary conditions are periodic in x and open at the y-right. At the y-left, vx isfixed at a nonzero value and the other quantities are either held at their initial valuesor copied from the computational domain like in open boundaries. The differingresults from the two approaches can be seen in Fig. A.1 (as usual, in normalizedunits). The expected value for the orthogonal field is Bx = −vxBy/vA,y as theperturbation propagates with Alfven velocity. Obviously, the results are much better

98

A.3 BOUNDARIES CONDITIONS

Figure A.2: Switch-on shock with maximal deflection in the shock frame (left) and in thelaboratory frame (right). In the shock frame, the shock rests at y = 5. The vertical line showsthe theoretical position of the shock front in the laboratory frame.

if open conditions are used for B. It also helps if vx is prescribed in the cells directlyabove the boundary. In all the tests with subsonic velocities, ρ, p and vy do not showa significant departure from their initial values. By remained constant in all tests.

It may still be necessary to fix some values, either in the ghost cells or next tothe boundary in the domain, for the sake of numerical stability. The problemsseem to correlate with the “harshness” of the conditions in many cases. The jetsimulations presented above get along without such fixes as long as the imposedrotation velocity is moderate. Boundary effects can also be minimized by puttingthe boundaries as far away as possible from the region of interest, for instance withgrid cells that grow exponentially in size. Unfortunately, this cannot be applied tothe jet inlet boundary and it is of limited use in spherical grids.

In an attempt to mimic the jet front, one can also prescribe the post-shock statein the ghost cells. If the propagation velocity of a disturbance exceeds the fastmagnetosonic speed, a fast shock may form. Fast shocks have the special propertyof refracting the magnetic field away from the shock normal. When a fast shockpropagates in the direction of the magnetic field, it has the effect of kinking themagnetic field lines. That is, in a setup similar to the test simulation described above,it creates a normal field component Bx. Such a shock is called switch-on shock andexists only when the Alfven speed exceeds the sound velocity in the unshockedplasma or, equivalently, if the fast magnetosonic velocity is the Alfven speed. Thenumerical solver handles fast shocks well in Cartesian 2D simulations, see Fig. A.2,with jump conditions from Priest (1982).

99

Appendix B

Coordinate Transformations

In this work, several coordinate systems are used alternately: a Cartesian system(x, y, z) as the basis, a traditional spherical coordinate system (r, θ, φ) in which the z-axis is the polar axis, an alternative spherical coordinate system (r, ϑ, ϕ) in which they-axis is the polar axis and the cylindrical coordinate systems (Rz, φ, z) and (Ry, ϕ, y)where R denotes the perpendicular distance to the respective axis. The (r, θ, φ)system is connected to the (r, ϑ, ϕ) system by the substitution (x, y, z) → (z, x, y).The conversion formulae are

x = r sin θ cos φ = Rz cos φ = r sin ϑ sin ϕ = Ry sin ϕ (B.1a)

y = r sin θ sin φ = Rz sin φ = r cos ϑ (B.1b)z = r cos θ = r sin ϑ cos ϕ = Ry cos ϕ (B.1c)

To convert vector fields from the (r, θ, φ) to the (r, ϑ, ϕ) system, one needs to knowthe versors eϑ and eϕ (er is unchanged) in the (r, θ, φ) system, i.e. expressed as alinear combination of er, eθ and eφ. The spherical versors er, eθ and eφ, expressed inCartesian coordinates (i.e. as a linear combination of ex, ey and ez), can be readilycalculated from Eqs. (B.1) or looked up in textbooks such as Nolting (2002). Fromthem, one can determine the Cartesian versors expressed in spherical coordinatesby adding up the projections along the three coordinate directions:

ex = (ex · er)er + (ex · eθ)eθ + (ex · eφ)eφ

= sin θ cos φ er + cos θ cos φ eθ − sin φ eφ (B.2a)

ey = sin θ sin φ er + cos θ sin φ eθ + cos φ eφ (B.2b)

ez = cos θ er − sin θ eθ (B.2c)

Having these, one can write the azimuthal versor

eϕ = cos ϕ ex − sin ϕ ez (B.3)

as a combination of eθ and eφ (the er component vanishes). eϑ can be found in asimilar manner or, more easily, via the relation eϑ = eϕ × er. Finally,

eRy = sin ϕex + cos ϕez. (B.4)

100

Appendix C

Visualization

C.1 Volume Rendering

Volume rendering is one of the most practical ways to visualize a three-dimensionalscalar field. The basic idea is to assign light emission and absorption properties,depending on the field(s) to be visualized, to every voxel in the volume. Thecontributions of all cells along a particular line-of-sight are then added up to give a2D image. Volume rendering is a standard feature of most visualization softwareand the implementation is technically complex in general (arbitrary viewing angle,perspective view). However, a volume renderer for lines-of-sight parallel to one ofthe coordinate axes is easy to implement and the results are in some cases betterthan with standard software, see e.g. Fig. 3.2. The starting point for creating avolume renderer is the equation of radiative transfer,

dIds

= −κ I + ε, (C.1)

whereby the light intensity along the line-of-sight is reduced by absorption, parame-trized by the field κ, and augmented by emission, parametrized by the field ε. Ifthe total volume is divided by discrete cells in which κ and ε are constant, then thesolution is

I(si+ 12) = I(si− 1

2)e−κi∆si +

εiκi

(1− e−κi∆si

), (C.2)

where ∆si is the intersection length of the line-of-sight with the i-th cell. A Taylorexpansion for small ∆si yields

I(si+ 12) = I(si− 1

2) · (1− κi∆si) + εi∆si. (C.3)

For practical purposes, it is important [and essential if Eq. (C.3) is used] that κ∆sis not larger than one in any cell, i.e. only the cell with the largest value shouldbe optically dense. Therefore, the field which controls the absorption should benormalized to the range [0, 1]. Moreover, exponentiating the field controls the levelof detail shown in the rendered image. It is possible and often useful to associatedifferent fields with absorption and emission.

101

APPENDIX C VISUALIZATION

C.2 Numerical Computation of the Stream Function

Field lines (streamlines) in a 2D vector field can be found by determining the levelsets of a scalar field Ψ, the stream function, whose derivatives give the vector field.This works for all solenoidal fields, such as the magnetic field or an incompressiblefluid flow. If the 2D field comprises of the projection of a 3D field onto a 2D plane,then the field must be symmetric in plane’s normal direction. An application to thismethod is the plot in Fig. 3.9.

In Cartesian coordinates (x, y, z), the field u = u(x, y)ex + v(x, y)ey + w(x, y)ez isconnected to the stream function Ψ = Ψ(x, y)ez by u = ∇×Ψ + wez. The level setsdΨ = 0 of such a stream function coincide with the field lines of the parallel field inthe z = const plane:

0 != dψ =∂ψ

∂xdx +

∂ψ

∂ydy = −vdx + udy ⇐⇒ dx

dy=

uv

. (C.1)

Numerically, we seek a solution to u = ∂yΨ, v = −∂xΨ. It is straightforward toshow that the following two functions are independent solutions, provided that∇ · u = 0:

Ψ1 =∫ y

y0

dy′ u(x, y′)−∫ x

x0

dx′ v(x′, y0), (C.2a)

Ψ2 = −∫ x

x0

dx′ v(x′, y) +∫ y

y0

dy′ u(x0, y′) (C.2b)

where x0 and y0 are constant coordinates, conveniently chosen to be at the borderof the grid. The integrals can be found with a cumulative version of the compositerectangle or of the Simpson rule. Eventually, the two solutions may be combined toa single one: Ψ = (Ψ1 + Ψ2)/2.

In spherical coordinates (r, θ, φ), the field u = u(r, θ)er + v(r, θ)eθ + w(r, θ)eφ isconnected to the stream function Ψ(r, θ) by ∇Ψ× eφ/R where R = r sin θ. Analo-gous to the Cartesian case, the level sets dΨ = 0 coincide with the field lines of theparallel field in the φ = const plane. We need to solve u = −∂θΨ/(rR), v = ∂rΨ/R.The following two functions are independent solutions, provided that ∇ · u = 0:

Ψ1 = −∫ θ

θ0

dθ′ r2 sin θ′ u(r, θ′) +∫ r

r0

dr′ sin θ0 v(r′, θ0), (C.3a)

Ψ2 =∫ r

r0

dr′ r′ sin θ v(r′, θ)−∫ θ

θ0

dθ′ r20 sin θ′ u(r0, θ′). (C.3b)

102

C.3 FLOW VISUALIZATION WITH WEDGES

C.3 Flow Visualization with Wedges

b

OC

P

Q

S

L

R

M

N

ϕ

h

¯h

h = h + ¯h

c = OP

b = LR

a = MN

The visualization method presented inthis section proved itself to be a viabletool for visualizing vector fields in 2Dslices. The main advantage of it overmore simple methods like drawing ar-rows at equidistant positions is that itgives a good impression of the fieldlines’ course without the need to calcu-late them explicitly. The method is ap-plied in Kirby et al. (1999) and in Laid-law et al. (2005). However, the proce-dure is only outlined there. The follow-ing is a possible implementation. It wasused to draw Fig. 4.7.

Consider the vector field u(x, y)ex +v(x, y)ey. The local (at the centroid S) field strength and direction shall be encodedin the size and orientation of a filled isosceles triangle (wedge) like the one drawn inthe adjacent figure. Taking A = α

√u2 + v2 for the area, where α is an appropriate

scaling parameter, we get the triangle’s base c from A = βc2/2, where the parameterβ := h/c determines the sharpness of the wedge. The height of the triangle ish = βc = h + ¯h = h/3 + 2h/3 and the tilt angle is ϕ = arctan(v/u). From this, wecan calculate the points Q, C, P and O:

xQ = xS + ¯h cos ϕ, xC = xS − h cos ϕ, xP,O = xC ±c2

sin ϕ,

yQ = yS + ¯h sin ϕ, yC = yS − h sin ϕ, yP,O = yC ∓c2

cos ϕ.

The isosceles trapezoid spanned by the points M, N, L and R marks the region wherethere must be no overlap with another wedge. It is determined by the parameter η,from which b = c + 2ηc and a = c ¯h/h + 2ηc. The positions of the points are

xM,N = xS ∓a2

sin ϕ, xR,L = xC ±b2

sin ϕ,

yM,N = yS ±a2

cos ϕ, yR,L = yC ∓b2

cos ϕ.

The procedure is now as follows:

1. Randomly select a point (x, y).

2. Calculate the coordinates (O, P, M, N, . . .) of a possible new wedge.

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APPENDIX C VISUALIZATION

3. Determine whether the trapezoid (L, R, N, M) of the new wedge would over-lap with existing ones. If it doesn’t, draw it and go back to step 1. If it does,discard it and continue with step 4.

4. Determine whether a preassigned maximum number of consecutive fails hasbeen reached. If not, go back to step 1. If yes, stop.

Viable (starting) values for the parameters are α = 100/(median field value) pixels,β = 4, η = 0.7 and 250 for the maximum number of consecutive fails.

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Danksagung

Allen voran mochte ich Henk Spruit fur seine kompetente, engagierte und geduldigeBetreuung ganz herzlich danken. Mit seiner von physikalischer Einsicht gepragtenund zugleich pragmatischen Herangehensweise hat er es immer wieder geschafft,mir den Wald zu zeigen wenn ich nur noch Baume sah.

Ein ganz großer Dank gilt auch Martin Obergaulinger. Naturlich dafur, dass ermir seinen MHD Code zur Verfugung gestellt hat. Vor allem aber auch fur seinenexzellenten “Support” und den nimmermuden Einsatz bei der Beseitigung vonFehlern und beim Einbau von Verbesserungen.

Meinem Burokollegen Jens Jasche mochte ich fur die nette Arbeitsatmospharedanken. Durch unsere zahlreichen Diskussionen konnte ich meinen Horizont auchuber die Doktorarbeit hinaus erweitern.

Schließlich danke ich meinen Eltern dafur, dass sie mich trotz anfanglicher Skepsisauf dem langen Weg des Studiums bestandig unterstutzt haben.

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