Numerical MHD Simulations of Astrophysical JetsLaunched from Magnetised Accretion Disks

Richard Archibald

Under the supervision of: Dr. Fabien Casse and Dr. Rony Keppens

January 2004

Abstract

The results of numerical simulations with the Versatile Advection Code includinga fully magnetohydrodynamic model of the launching of astrophysical jets frommagnetised accretion disks (Casse & Keppens 2002, 2004) are presented. A briefintroduction to adaptive mesh refinement is given and examples of its use are shown.Problems with the implementation of this jet model in AMRVAC are highlightedand possible further studies are discussed.

-Instituut voor Plasmafysica Rijnhuizen

Contents

1 Introduction 1

2 Background theory 22.1 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Introduction to MHD . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Range of validity of MHD . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Magnetised accretion/ejection model . . . . . . . . . . . . . . . . . . . . 5

3 Numerical solutions 83.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.1 The Versatile Advection Code . . . . . . . . . . . . . . . . . . . . 83.1.2 Numerical simulation of accretion disks . . . . . . . . . . . . . . . 10

3.2 Adaptive mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Results of VAC simulations 124.1 Double Mach reflection solution . . . . . . . . . . . . . . . . . . . . . . . 124.2 Magnetised accretion / ejection structure . . . . . . . . . . . . . . . . . . 13

4.2.1 Reproduction of jet simulation . . . . . . . . . . . . . . . . . . . . 134.2.2 Parameter study . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Results of AMRVAC simulations 155.1 Double Mach reflection problem revisited . . . . . . . . . . . . . . . . . . 155.2 Jet launching with AMRVAC . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Conclusion and outlook 18

7 Acknowledgements 18

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1 Introduction

Accretion in astrophysics is simply the process whereby material falls onto a central mas-sive object. Due to angular momentum conservation, this cannot happen in a sphericallysymmetric fashion but instead, the in-falling material tends to form a flattened, rotatingstructure known as an accretion disk.

Accretion disks are a very common structures in the universe and it is therefore importantto understand their dynamics in astrophysics. Accretion disks are found around manydifferent types of object and cover a whole range of scales from the proto-stellar disksaround young stellar objects (YSOs), to the huge whirlpools of material being draggedinexorably into the super-massive black holes thought to power active galactic nuclei(AGN).

Another common astrophysical phenomenon which also covers vast ranges of scale is thatof jets. The name jet is assigned to those outflows which reach supersonic speeds (of theorder of 300km s−1 (e.g. Heathcote et al. 1996) in many YSOs for example) and althoughthey start off with large opening angles greater than about 60 (Ray et al. 1996), becomewell collimated to about 3 (Burrows et al. 1996). Figure 1 shows three examples ofastrophysical jets from YSOs.

Figure 1: Hubble Space Telescope images of astrophysical jets emanating from young stellar objects (YSOs). The scale ofthe white line in each frame is 1000AU[14] or about the scale of our solar system.

The detection of synchrotron radiation from many of the jets implies the existence of a

1

magnetic field and so it is plausible that the magnetic field is responsible for the launch-ing and collimation of the jet. It is a model of this kind which will be described in thisreport, largely following recent work by Fabien Casse and Rony Keppens (Casse & Kep-pens 2002, 2004).

If the gas in the inner accretion disk is hot enough to be ionised (i.e. becoming a plasma),it will be dynamically influenced by the magnetic field and conversely, can affect the evo-lution of the magnetic field. One method of treating such systems mathematically is bythe equations of magnetohydrodynamics (usually abbreviated to MHD). In essence,MHD takes the ordinary Eulerian equations of compressible gas dynamics, adds termswhich couple the magnetic field with the fluid or plasma motion and also adds an equa-tion governing the time evolution of the magnetic field. For a slightly more in-depth lookat the equations of MHD and their conditions of validity, see section 2.1.

As the equations of MHD are highly non-linear and intertwined, we do not have a fullanalytical solution. One way to proceed therefore is by numerical integration of theequations. In the work I have undertaken here at FOM1, I have learnt and used boththe Versatile Advection Code (VAC) (see Toth 1996) and the Adaptive Mesh Refinementversion of VAC; AMRVAC (Keppens et al. 2003). Both are multi-dimensional, shock-capturing codes which solve the MHD (or ordinary HD) equations as a system of generalhyperbolic equations using a choice of several algorithms and allow for modification ofthe base equations through a modular approach.

2 Background theory

2.1 Magnetohydrodynamics

2.1.1 Introduction to MHD

The MHD model describes the motion of an electrically conducting plasma in the presenceof a magnetic field. There are actually two divisions within MHD: ideal and resistiveMHD. The main difference between the two regimes is that in ideal MHD the magneticfield cannot diffuse through the plasma as there is no mechanism for dissipation. Thismeans the magnetic field is essentially “frozen in” to the plasma, to use the words ofAlfven who pioneered the subject. In resistive MHD however, currents generated bythe magnetic field can cause energy dissipation and the magnetic field is able to diffusethrough the plasma.

In both ideal and resistive MHD, the magnetic field can exert a force on the plasma. Thisis the well known Lorentz force which in MHD is often expressed as a combination ofmagnetic pressure and magnetic tension (sometimes called “hoop” stress). Choosingunits throughout where µ0 = 1, the Lorentz force per unit volume is

J × B = (∇× B) × B ,

1FOM-Instituut voor Plasmafysica Rijnhuizen, Nieuwegein, The Netherlands.

2

which can be transformed by means of a vector identity to

J × B = (B · ∇)B −∇(

B2

2

)

.

The first term on the right can be shown to represent a magnetic tension per unit volumeof magnitude B2, parallel to the magnetic field vector. The second term representsa magnetic pressure force per unit volume of magnitude B2/2 by analogy to thermalpressure force density: −∇P . From now on in this report, since we shall be dealingwith fluid bulk properties such as density, I will simply write force when force density isgenerally implied. The above equation can be written in a component form. The radialcomponent will be useful in later discussion and is

[J × B]R = − ∂

∂R

(

B2Z + B2

φ

2

)

−B2

φ

R+ BZ

∂BR

∂Z,

where the first term represents the radial magnetic pressure and the second term the“hoop” stress. This term is always negative, regardless of the sign of the toroidal field.Therefore “hoop” stress will always act radially inwards, as a collimating force. The lastterm may be neglected as we will only consider this form in terms of jet collimation whereBR is negligible.

The above gives some idea of what kind of phenomena to expect in MHD, let us now lookat the formulation of MHD itself. The equations of MHD can be posed in many waysbut perhaps the most useful in terms of numerical solution is their conservative form.This means that any temporal variation of a quantity within a given volume, is equal tothe opposite of the flux of this quantity through the surface containing the volume. Theresistive MHD equations in a conservative form useful for this work are then:

∂ρ

∂t+ ∇ · (ρv) = 0 , (1)

∂ρv

∂t+ ∇ · (vρv − BB) + ∇

(

B2

2+ P

)

+ ρ∇ΦG = 0 , (2)

∂B

∂t= −∇ · (vB − Bv) −∇× (ηJ) , (3)

J = ∇× B . (4)

In order to close the system of equations, an energy equation is generally required butfor the moment, let us simply assume a simple polytropic relation

P = Kργ . (5)

An additional constraint on the magnetic field is given by the Maxwell equation

∇ · B = 0 .

These equations from the top down are: (1) the equation of continuity, (2) momentumconservation, (3) evolution of the magnetic field, (4) Ampere’s law2 and in general (5) an

2Note that this really is Ampere’s law and doesn’t include the displacement current term found inthe Maxwell equations. For this reason, electro-magnetic waves do not exist in MHD and in fact, theelectric field has become a secondary quantity.

3

energy equation. In the equations ρ is the plasma density, v is the plasma velocity, B isthe magnetic field, P is the thermal pressure, ΦG = −GM∗/(R2 + Z2)

1

2 is the gravita-tional potential of a central object of mass M∗, R and Z are the Cartesian coordinatesin the poloidal plane (see section 2.2), η is the resistivity and J is the current density.

The equation of continuity states that mass is conserved in the system and the lack ofany source / sink terms implies that the basic form of the MHD equations do not takeinto account pair-pair production / annihilation etc. The equation of conservation ofmomentum shows that even in the absence of external forces, the plasma flow is intrin-sically linked to the magnetic field geometry as shown by the presence of the tensorsvρv and BB. Note also that there is a source term which takes gravitational forces intoaccount. The equation describing the time evolution of the magnetic filed is also coupledto the plasma motion via the tensors vB and Bv. The presence of the resistivity in thisequation will be discussed further in section 2.2.

The equations of MHD (equations 1 to 4) are highly non-linear – as mentioned already– but may still yield some analytical insight. For example, by considering small per-turbations of the plasma parameters of pressure, density etc., it can be shown that theMHD model permits three distinct types of waves. In order of velocity of propagation,they are: slow magneto-sonic, Alfven and fast magneto-sonic waves. The magneto-sonicwaves may be thought of as the equivalents of sound waves in MHD; made distinct bythe non-isotropic nature of the magnetic forces. Alfven waves are more like waves on ataught string. The strings in this case are the magnetic field lines which are restored dueto the effects of magnetic tension discussed above. The Alfven waves travel along themagnetic field lines with a velocity known as the Alfven speed which is given by

VA ≡ B0√ρ0

,

where B0 is the unperturbed magnetic field strength and ρ0 is the unperturbed localplasma density.

The equation of momentum conservation or the dynamical equation (equation 2) of MHDshows that both the magnetic pressure and thermal pressure affect the plasma flow. Auseful parameter which indicates which term will dominate the motion is the plasmabeta parameter defined as

β ≡ 2P

B2.

In this equation, P is the thermal pressure and B2/2 the magnetic pressure.

2.1.2 Range of validity of MHD

In the development of the MHD equations, it is important to realise that some assump-tions have been made. These assumptions set limits on the range of situations in whichMHD provides an accurate description. Perhaps the biggest assumption is the treatmentof plasmas as a single fluid. Therefore MHD is only valid over time-scales and dynam-ical scales such that the motion of individual electrons and ions are averaged out. The

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times and lengths of interest should also be much greater than the gyro-frequency andgyro-radii respectively of the particles. Also, it should be noted that although the plasmais composed of electrically charged particles, each particle is surrounded (due to mutualattraction) by particles of the opposite charge so that at some distance from the orig-inal charge, its electric field is negligible. The length beyond which this “shielding” isimportant in a plasma is called the Debye length and is given by

λD =

√

kTe

4πε0e2ne

.

Here k is Boltzmann’s constant, Te is the kinetic temperature of the electron ‘gas’, e isthe electronic charge and ne is the number density of electrons. Over distances muchgreater than the Debye length, the plasma is considered electrically neutral. MHD is alsoa non-relativistic theory. Despite these caveats though, MHD is still satisfied in manyastrophysical situations and its use here is justified on account of the great length scalesand periods of time in the simulations.

2.2 Magnetised accretion/ejection model

The model followed in this work is that of a magnetised accretion disk launching bipolarjets which are both accelerated and collimated by magnetic fields (Casse & Keppens 2002,2004). The model is axial-symmetric (or in other words, it is a 2.5D model) which meansthat we only consider the dynamics in a poloidal plane through the accretion disk andassume that this plane is identical to another at an arbitrary angular position. The mostnatural coordinate system is therefore a cylindrical one with R as the radial distancefrom the centre of the accretion disk, Z as the height above a line through the centre ofthe disk and φ defining the angular position around the disk from an arbitrary startingpoint (see figure 3 for clarification). The axisymmetry is in the vertical direction and isexpressed as ∂

∂φ= 0 for all physical quantities.

The accretion disk is initially threaded by some kind of bipolar magnetic field (see sec-tion 3.1.2 for more details). The accretion disk has a non-zero resistivity η while in therest of the computational domain the resistivity is so low that ideal MHD is a good de-scription of the dynamics. The main reason for this duality is that without a resistivedisk, we cannot have accretion. This is easily shown as follows. Consider Ohm’s law:

E = ηJ + B × v .

If we search for a magnetic geometry which would exist in some kind of equilibrium wefind

∂B

∂t= −∇× (ηJ + B × v) = 0 .

By considering only the toroidal components, the following can be derived

η

(

∂BR

∂Z− ∂BZ

∂R

)

= vZBR − vRBZ ,

5

which if we assume that the accretion disk is thin (i.e. ∂RBZ ∂ZBR) and note that|vR| |vZ | for an accretion flow, becomes

η∂BR

∂Z= −vRBZ . (6)

From this relation it can be seen that if the resistivity is zero then – as BZ is non-zeroby definition in this model – vR would also be zero. This implies no accretion which isa trivial stationary state. The resistivity is defined in such a way that it drops off veryrapidly beyond the disk scale height but although it never actually becomes zero, it isessentially negligible outside the disk.

Another, more physical view of the action of the magnetic field on the accretion disk,is that as the plasma rotates and drags the magnetic field with it, work is done againstmagnetic pressure and magnetic tension. This means that the kinetic energy of the or-biting plasma is converted into magnetic energy and a direct consequence of this is thedecrease in orbital radius of the plasma.

The magnetic field also exerts a vertical force on the material in the accretion disk. Forthe bulk of the disk this force (along with the force of gravity) ‘pinches’ the plasma, hold-ing it in place against the thermal pressure gradient. When these forces are balanced, theaccretion disk is said to be in equilibrium and the plasma beta parameter is of the orderof one in the bulk of the disk. Towards the surface of the accretion disk however, theplasma density drops off steeply and the thermal pressure with it. This means therefore,a decrease in plasma beta which implies a change in the dynamical relation between theplasma motion and the magnetic field. In a low beta state, the magnetic field is moreresistant to advection by the plasma and tends to dominate the motion. This meansthe magnetic field is able to release some of the stored energy by toroidally acceleratingmaterial in a thin layer or corona of the disk and also exerting a now upward verticalforce. This transfer of angular momentum from the disk to the jet is an important wayof removing the rotational energy of the accreting material.

A necessary criterion to allow jet launching in this manner is that derived by Blandfordand Payne (1982). It states that for a jet to be launched from a magnetised disk in equi-librium, the field lines must make an angle of greater than 30 to the vertical measuredat the disk surface (see figure 2). A rather simplified explanation for this criterion can beseen by considering a small plasma element on the surface of the disk. The forces actingupon this element are the force of gravity, the centrifugal force and the Lorentz force. Asthe force of gravity has a downward vertical component, any extraction of material fromthe disk must first match this vertical pull. The Lorentz force can do this so long as themagnetic field is bent over such that Blandford and Payne’s criterion is met. It shouldbe noted however, this is only valid so long as β 1 in the corona.

It is possible for the magnetic field to arrive in such a configuration if the part threadingthe accretion disk is advected with the accretion flow. Consider the schematic view ofthe system shown in figure 3. This figure shows a poloidal cut of the system after it hasarrived in a ‘quasi-stationary’ state in which the magnetic field has been bent as much as

6

F

F

Accretion disk

PB

c

CF

∗M

φ

R

Z

θM

GF

Figure 2: Simplified cartoon view of the Blandford and Payne criterion [1]. FG is the gravitational force on a fluid elementat the surface of the disk, FCF is the centrifugal force and FM is the magnetic force. The Blandford and Payne criterionstates that to overcome the vertical component of the gravitational force, the poloidal magnetic field line through a fluidelement at the surface of the disk must make an angle θc, greater than 30 to the vertical at that point.

the plasma beta parameter will allow. As the inner parts of the accretion disk are rotatingfastest, they bend the magnetic field the most and are therefore the most likely parts tofulfill the Blandford and Payne criterion. Another important factor in the launching ofjets in this way is the temperature structure of the disk. The inner disk will be hottestand will therefore be the most ionised region. Without sufficiently ionised plasma, themagnetic field will not couple to the plasma motion. These two points help explain whyjet launching is only observed from the innermost regions of accretion disks.

φ

Z

R

φ

B

Bp

Figure 3: Schematic view of the ‘quasi-stationary’ state reached by the model. The accretion disk is shown in cross-section(shaded) in orbit around the central object. The magnetic field has become advected towards the centre along with theaccretion flow and has reached a state of dynamical equilibrium.

Within the jet itself, the magnetic torque accelerates the material, increasing the cen-trifugal force which tends to widen the magnetic surfaces. The magnetic field is alsoexerting a vertical force which drives the jet to trans-Alfvenic speeds. The collimation isensured by magnetic tension caused by the toroidal component of the advected magneticfield which stops the centrifugal force from spreading the escaping plasma in a radialdirection.

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3 Numerical solutions

3.1 Implementation

3.1.1 The Versatile Advection Code

The Versatile Advection Code attempts to numerically integrate the full MHD equationsin 1D, 2D or 3D, solving for the conservative variables (density, magnetic field, etc.), inthe most general way possible. VAC can therefore be used for a very wide variety ofproblems.

As with any (magneto)hydrodynamic code, VAC discretises the equations to be solved, inboth space and time. The spatial discretisation is achieved in a variety of ways dependingon the algorithm selected in a parameter file. In the simulations shown in this report,only the Total Variation Diminishing Lax-Friedrich (TVDLF) scheme is employed and soI shall use this as an example. TVDLF ensures the Total Variation Diminishing property,namely that the Numerical Total Variation

TV n ≡∑

i

∣

∣∆ρni+1/2

∣

∣ ,

does not increase with time. In this notation, the superscript on variables indicatesthe discrete time step and the subscript labels the cell in the x[, y, z] direction[s]. TVDalgorithms are of second order accuracy, although they degenerate to first order accuracyat extrema. TVD is also said to be monotonicity preserving. This means that itdoes not add any artificial extrema to the initial data. A disadvantage of the scheme isthat it tends to diffuse or smear out discontinuities. The TVDLF method is simply aTotal Variation Diminishing scheme which employs a more involved variant of the basicLax-Friedrich discretisation:

ρn+1i =

1

2

(

ρni+1 + ρn

i−1

)

− ∆t

2∆xv

(

ρni+1 − ρn

i−1

)

.

TVDLF is a shock-capturing algorithm and is used in these studies of accretion diskand jet because there exists a very high density contrast (about six orders of magnitude)between the disk and the Inter-Stellar or Inter-Galactic Medium. The temporal discreti-sation is explicit and restricted by the Courant-Friedric-Levy condition on the time step,involving a multiplicative factor, the Courant parameter, which may be adjusted from aparameter file.

In MHD codes, ensuring that ∇ · B = 0 is always satisfied presents a major problem.The VAC simulations here use a Poisson solver method which is so called as it solvesthe Poisson equation

∇2φ = ∇ · B ,

for the numerically generated, non-zero divergence. The magnetic field is then correctedto

B′ = B−∇φ ,

in every time-step, such that

∇ · B′ = ∇ · B −∇2φ = 0 .

8

This correction scheme works in all MHD regimes (ideal and resistive).

Before using the Versatile Advection Code to model accretion / ejection scenarios itis important to learn what the code is capable of by picking some well known (mag-neto)hydrodynamical problems and attempting to solve them. The examples I choosewere collected in a paper by Toth and Odstrcil (1996). Although I did several of theseproblems, only one is shown in the results section: the problem of a Mach 10, planarshock-front reflecting from a wedge at 60 leading to the formation of a self-similar struc-ture known as a lambda point. This problem was first proposed by Woodward andColella (1984) and has been singled out as it illustrates most of the more difficult prob-lems encountered in learning to use the code such as realising special initial conditionsand split boundary conditions.

The problem is solved in a physical domain [0,4]×[0,1] (dimensionless coordinates). Theunits are made dimensionless such that the adiabatic sound speed is unity. The adia-batic index γ = 1.4. The shock is represented by a density level ρ = 8 and an inter-nal energy density e = 563.5. The momentum of the shocked material is (ρvx, ρvy) =(8 × 8.25sin60,−8 × 8.25cos60). Everywhere else the density ρ = 1.2, the internalenergy density e = 2.5 and the momentum ρvx = ρvy = 0. The wedge is represented bya reflective boundary located at x ∈ [1/6, 4] and y = 0. The remaining section of thebottom boundary is set to the post-shock conditions along with the left boundary andthe portion of the top boundary which is to the left of the moving shock-front as definedby xs(t) = 10t/sin60 + 1/6 + 1/tan60. The region of the top boundary x > xs(t) is setto the pre-shock values, as is the right boundary.

The initial conditions in VAC are controlled by a combination of a initial parameter fileand, in the case of non-uniform initial values, a file containing user modifiable subroutineswritten in Fortran. This source file (vacusr.t.problem name) also contains a subroutinecalled by VAC to deal with special boundary conditions. In this problem, I was able touse the fact that VAC also contains several simple boundary conditions (which may becalled through the main VAC parameter file), to take care of the right hand boundarywhich is continuous. The other boundaries were marked as ‘special’ in the parameter fileand this is the cue to call the appropriate subroutine in the problem dependent vacusr

file. The subroutine uses two ghost cells around the computational domain to enforceboundary conditions. Two cells are used as the numerical algorithm I used for this andother simulations is spatially second order (TVDLF scheme, see above). For the leftboundary, the ghost cells are simply set to the values of the shock. This is also true forthat part of the bottom boundary x < 1/6 and the top boundary for x < xs(t). Theremaining part of the top boundary was set to the pre-shock values while the x > 1/6part of the bottom boundary was made reflective. This means making the ghost cellvalues mirror those in the bottom two cells of the computational domain exactly. Everyphysical value is therefore copied with the exception of the y-component of momentumwhich undergoes a sign reversal on account of the mirroring of a vector quantity. Thesimulation was run with a resolution of 480×120 with a final output at time-step T = 0.2(dimensionless unit).

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3.1.2 Numerical simulation of accretion disks

With some experience in using VAC, I next began my first experiments with the modelpresented by Casse and Keppens (2002). In this implementation, the computationaldomain represents a poloidal section through the accretion disk but, because of the sym-metry in the model, only one quadrant of the disk is calculated. In other words, thecomputational domain spans the region R = [0, 40] and Z = [0, 80]. An IDL subroutineis used to recreate the full system. Once again the shock-capturing, second order accu-rate TVDLF scheme (Toth & Odstrcil 1996) is used with the Poisson solver mentionedin section 3.1.1 enforcing a solenoidal magnetic field.

Boundary conditions are enforced with two ghost cells on either side of the computationaldomain. The boundaries are: continuous at the top and right, except for a small regionof the lower right boundary which is used to simulate the continual advection of newmaterial into the inner disk, by holding the poloidal components of momentum constantin these ghost cells. The left boundary and the bottom boundary (equatorial plane) areboth symmetrical and so the primitive variables are either symmetric or antisymmetricacross in these ghost cells. Table 1 shows the imposed conditions on each vectorial vari-able.

Table 1:

Location ρ VR ΩR VZ BR Bφ BZ

R-axis symm symm symm asymm asymm asymm symmZ-axis symm asymm asymm symm asymm asymm symm

A problem is encountered however, due to the Newtonian gravitational potential whichhas a singularity at the origin. To avoid this and the steep gradient of the potential nearthis point, a sink region is defined at R = [0, 1] and Z = [0, 1]. This is implementedby splitting the treatment of the left and bottom boundary conditions with a user writ-ten special boundary subroutine. In this region, the values of the ghost cells are copieddirectly from the first row just above the sink and further – since the idea of a “sink” isthat no material should flow out again – the poloidal velocity is controlled so that neitherVR nor VZ have positive values.

The initial conditions for the simulation place a linear accretion disk around the equatorialplane, with scale height H(R) = εR. The scaling parameter ε is 0.1 in all simulations inthis report. The magnetic field in this early simulation is entirely vertical (BR = Bφ = 0)with intensity dropping off in the radial direction. In later simulations however, the initialmagnetic field has a more open topology but remains poloidal. Figure 4 shows a view ofthe initial system. See Casse and Keppens (2002) for more details.

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Figure 4: The initial setup of the magnetised accretion disk simulation showing: mass density profile in the accretion disk,magnetic field topology and magnetic field strength gradient. The mass density scale is coded with white corresponding toρ = 1 and dark blue to ρ = 10−5 .

3.2 Adaptive mesh refinement

Adaptive mesh refinement (AMR) is an h-refinement method for use in solution-adaptivecomputations (Keppens et al. 2003). A simple overview is that after a set number oftime steps have elapsed, the solutions for all variables are compared to the results foundby computing the same time length but with (say) twice as many spatial points. If thedifference in these solutions is greater than some predefined tolerance, then the higherresolution is used until the next evaluation point. This refinement process can be “nested”such that the resolution increases in steps towards the most computationally difficult re-gions of the computational domain. For an example of such nesting, see section 5.1.

Adaptive mesh refinement has relatively recently been added to the Versatile AdvectionCode project and is available as AMRVAC. There are great advantages in using AMRVACfor the jet launching simulations already mentioned; mainly the increase of computationalefficiency as the higher resolutions are only used where they are needed. In a simulationwith a lot of regions where the density and other variables is effectively uniform, a rela-tively low resolution can be used over much of the computational domain, with a muchhigher resolution used in areas where there are shocks or boundaries between the jet andthe surrounding medium, for example. The improved computational efficiency wouldenable both an increase in the time-scale that can be observed as well as the structureof the jet at greater distances from the accretion disk. This could help answer questionsabout the stability of the jet and its long term collimation.

As with any piece of new technology however, there must be some challenges with theimplementation of AMRVAC to the study of jet launching from accretion disks. Someof these challenges, such as the control of the numerical value of ∇ · B for example, are

11

addressed in the paper on AMRVAC by Keppens et al. (2003).

4 Results of VAC simulations

4.1 Double Mach reflection solution

Figure 5 shows the final snapshot from a VAC simulation of the Woodward and Colella(1984) double shock reflection problem. The snapshot is for a dynamical time of T = 0.2and shows the density profile of the fluid with white representing ρ = 22.4 and blackrepresenting the pre-shocked material with ρ = 1.4. The shock is visible leaving the plotat the right hand side. It started off as a straight line but note how the reflection fromthe wedge has increased the density even further and driven part of the shock faster. Thestructure is often called a lambda point – due to the resemblance of the triple pointedregion to the Greek letter – and it grows in a self-similar fashion.

Figure 5: Density profile from a VAC solution to the problem of a Mach 10 shock reflecting from a wedge [13]. Thestructure formed is known as a lambda point due to the resemblance to the Greek letter. Resolution for the domain shownis 360 × 120. The axes show dimensionless coordinates.

Although this simulation was run with a resolution of 480 × 120, the plot in the figureonly shows that region of the computational domain 0 < x < 3, therefore the actualnumber of cells in the plot is 360× 120. The result compares well with that in the paperof Toth and Odstrcil (1996) and shows sharply resolved discontinuities, as seen on boththe main shock and the reflected shock, which is expected for a shock-capturing algorithmlike TVDLF.

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4.2 Magnetised accretion / ejection structure

4.2.1 Reproduction of jet simulation

The VAC simulation of a magnetised accretion disk launching a self-collimating jet isshown as the middle panel in figure 6. This simulation has a resolution of 70× 100 in therange 0 < R < 40 and 0 < Z < 80 – i.e. one quarter of the image. The colour relatesto logarithmic density as explained in the caption. The snapshot is taken at T = 10 andshows the jet as a hollow structure. Some sample magnetic field lines are shown in blackand the sink region is visible as the black square at centre.

4.2.2 Parameter study

With the model working in VAC, I decided to experiment with some of the parametersgoverning the dynamics. Presented in this section are the results of varying the plasmabeta parameter β, and also the initial accretion flow speed ms.

Figure 6 shows the results of changing the initial plasma beta parameter from the jet-launching value of β = 3.33 throughout the disk (central panel - the ‘control’) to β = 100in the left panel and β = 0.6 in the right. As the thermal pressure is the same in eachsimulation, this effectively changes the ‘stiffness’ of the magnetic field lines and I shallnow give a qualitative account of the effects of such a change.

Starting with the left panel of figure 6 and comparing with the jet simulation in thecentre, it is immediately obvious that changing the plasma beta parameter can drasticallyaffect the dynamics of the system. No jet is launched, but a plasma outflow is observed.Looking more closely at the left panel and the central panel we notice that the magneticfield structure is very different. This is to be expected; with a high plasma-β, the magneticfield is more easily advected with the plasma motion which, as it is accreting, drags thefield lines towards the central object. The high β also helps explain the outflow as thedisk is no longer in the equilibrium discussed in section 2.2. The thermal pressure will begreater than the vertical magnetic force trying to hold the disk flat with the result thatthe disk vents material. This outflow differs from the jet in that it is not driven initially bythe magnetic field and also the magnetic field is not strong enough to collimate materialand so it is more like a thermally driven wind. Snapshots showing streamlines of plasmaflow in this simulation are shown in figure 7. These show that the wind is fairly uniformfor a time, especially at the resolution of 70× 100 used for the left panel. However, aftera longer time scale and at the higher resolution 100×200 (right panel), it seems the windis perhaps a transient event caused by the accretion disk reordering itself as it comes to anew equilibrium between magnetic forces and thermal pressure. I believe that the largevortices in the lower left corner of each simulation may be due to problems with the sinkboundary condition but I do not think that they affect the general qualitative conclusionsreached here.

The right panel in figure 6 shows the results of a decreased plasma beta. Comparison withthe jet launching simulation shows that, as expected, the magnetic field is not advectedas much by the plasma in the accretion disk. I would have anticipated this simulation

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Figure 6: The results from an investigation into the effect of different values of plasma beta on jet launching from amagnetised accretion disk at T = 10. The values of β are: (left panel) β = 100, (centre panel) β = 3.33 and (right panel)β = 0.6. The images therefore show the effect of ‘stiffening’ of the magnetic field lines from left to right. The black linesshow selected magnetic field lines and the colour fill represents logarithmic density with white corresponding to ρ = 1 anddark blue / purple to ρ = 10−5. The resolution for each simulation (one quarter of each panel) is 70 × 100.

to exhibit a thinner disk than in the high-β example as I expected that the thermalpressure would now be weaker compared to the magnetic pressure. The force exertedby the magnetic pressure depends on the gradient however, and it can be seen from thefigure that the field does not change as much as in the other simulations. Thereforethe thermal pressure gradient causes the disk to inflate until a new equilibrium is found,causing some mass loss in the process.

The parameter ms determines the initial radial velocity of the accreting matter in thesimulation. For the jet launching case shown in section 4.2.1, ms = 0.15. I varied this tohave both no initial inward velocity ms = 0, and also to have a very rapid initial in-fallms = 0.75. As there were no qualitative differences in the three cases, I conclude thatthis parameter is not significant to jet launching in this model.

5 Results of AMRVAC simulations

5.1 Double Mach reflection problem revisited

The result of an AMRVAC simulation of the hydrodynamical problem of a Mach 10 shockreflecting from a wedge at an incidence angle of 60, is shown in figure 8. The plot showsdensity contours over the range 1.4 < ρ < 21.9. The coloured rectangles show the level

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Figure 7: Poloidal velocity streamlines at time T = 20 at a resolution of 70 × 100 (left). Right panel shows the poloidalvelocity streamlines at an increased resolution of 100 × 200 for a slightly larger computational domain and later time(T = 30). Both simulations have the same parameters as that in the left panel in figure 6.

of refinement throughout the computational domain with white representing the coarsestlevel and the darkest grey representing the highest resolution regions. The base resolutionhere is 80 × 20 with three levels of refinement. At each refinement level, the resolutionis doubled producing a maximum resolution of 640× 160. The highest resolution is onlyrequired at those regions with high gradients of the physical properties – shock frontsin this case. So, even though the simulation has a higher effective resolution than thecorresponding VAC simulation, it executed in a fraction of the time.

5.2 Jet launching with AMRVAC

The best result of a jet being launched from an accretion disk, simulated using AMRVAC,is shown in figure 9. The snapshots show the logarithmic density of the system at T = 1,T = 20.3 and T = 50. The density has been colour coded and is explained in the captionto the figure. Immediately apparent in the images, although most noticeable in the leftpanel, is a vertical disturbance in the density which propagates through the system fromleft to right and interferes to some extent with the outer part of the accretion disk. Thisis not a significant effect as it is of the order of ρ = 10−5, but its cause was found

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Figure 8: Density profile from an AMRVAC solution to the problem of a Mach 10 shock reflecting from a wedge [13]. Thebase resolution for this “lambda point” is 80× 20 with 3 levels of refinement, each with twice the resolution of the previouslevel, making a maximum resolution (the darkest grey rectangles) of 640 × 160. Axes show dimensionless coordinates.

nevertheless. It is due to a new definition of the initial magnetic field as found in Casseand Keppens (2004). This definition includes the term

√βP

(1 + R2),

in the vertical component of the magnetic field which avoids a vanishing magnetic fieldin that direction at the jet axis. This term generates a small magnetic pressure at thefirst iteration and the disturbance is some type of MHD wave propagating through thecomputational domain as the magnetic field comes to equilibrium. Despite this minorpoint, the simulation shown shows a jet reaching a greater stage of evolution than withthe VAC simulation, on a domain which is slightly more than double the size. Thissimulation was completed within a single day, rather than the week or so it would havetaken without AMR.

This simulation includes a sink region which functions in much the same way as inthe VAC run. An important point however, was that even at an effective resolution of200×200, the inner part of the accretion disk which interacts with the sink region is onlyone cell high for a domain which is 80× 200. This means that a single cell must simulateboth accretion and ejection of the plasma. The obvious solution was to raise either thenumber of nested grids in the AMR routine, the base resolution, or both. As it turns out,this was not possible. It seems that once the resolution became close to resolving thesink region, the simulation would halt either immediately or very early in the advance.

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Figure 9: Snapshots of colour-coded logarithmic density (white: ρ = 1, black: ρ = 10−6) for an AMRVAC simulation ofa magnetised accretion disk launching a jet. The rectangles denote regions of grid refinement of which there are 3 levels.The base resolution is 50×50 and this is doubled at each refinement level, giving a maximum resolution of 200×200. Notethe disturbance in the density (most noticeable in the first snapshot) which propagates from left to right and is due to theway the initial magnetic field is defined (see text).

Further investigation lead me to believe that the problem was created by the refinementmethod as, with AMR turned on, the code would halt when the refinement level reachedsome critical resolution which seemed linked to the size of the sink region. Conversely,the code would complete with no refinement, but with the resolution set to the ‘criticalresolution’. After research of this problem with Rony Keppens and Fabien Casse however,it seems that even though the code will run with no refinement in AMRVAC, thereexist problems with the limiting values of some of the plasma properties. For example,although density and pressure were supposedly limited via the code to always be greaterthan certain small values, they were frequently found to ignore this limit and in the caseof the pressure were sometimes negative! Also found were extremely high and unphysicalvalues for temperature and this is believed to be linked to this artificial minimum limiton pressure. Most likely, the problem stems from the difference in the way VAC andAMRVAC treat the sink region. In VAC, it is implemented via the boundary conditions,whereas in AMRVAC it is actually part of the computational domain as it is currentlytreated as a source term to the MHD equations.

It is regrettable that this problem has not yet been solved. I am confident however, thatAMRVAC can be used to provide new insight into the dynamics of jets of the kind shownhere.

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6 Conclusion and outlook

In this report, we have seen that numerical integration codes are a useful tool in the studyof physical systems described by hydrodynamics and magnetohydrodynamics, in whichanalytical solutions are not easily obtained. VAC was introduced as a very flexible code,capable of solving a wide range of problems from a simpler hydrodynamics problem, up toa full MHD treatment of an important astrophysical object – an accretion disk producingbipolar outflows. After running simulations in both VAC and AMRVAC, I am convincedthat adaptive mesh refinement is a welcome addition to VAC, as it allows both betterresolution and greater domain size with much more efficient use of computer time.

We turned the tool of AMRVAC on the study of astrophysical jets launched from accretiondisks and found, perhaps, some teething problems. If these problems are resolved, thereare many further studies which can done with the jet model presented here. For example,in the small computational domains used so far, it is not clear what the structure of the jetwill be once it develops further and leaves the region of the accretion disk. The observedjets remain highly collimated over great distances when compared to their parent disksand obviously it is desired that the modelled jets do the same.

It appears from even the VAC simulations that the jet assumes a ‘quasi-stationary’ stateafter very short dynamical time-scales. One study could test if this really is a stable state.Does the jet only exist for a short time? Is it resistant to perturbations in the inflowof matter from the outer accretion disk? Many observed astrophysical jets exhibit knotsand other density structure. Is this caused purely by interaction with the surroundingmedium or does the flow of matter to the base of the jet dominate?

It is obvious that there are many unanswered questions in this topic, but perhaps thiswill be addressed with codes like AMRVAC very soon.

7 Acknowledgements

I would like to thank everyone at FOM-Rijnhuizen who made my stay there a veryenjoyable one. Special thanks go however, to Dr. Fabien Casse and Dr. Rony Keppenswithout whose supervision and previous work in this area, I could not have accomplishedthis project.

This project was undertaken with funding from the EU network PLATON, supportedby the European Community’s Human Potential Programme under contract HPRN-CT-2000-00153.

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