Akira Mizuta, Shoichi Yamada and Hideaki Takabe- Propagation and Dynamics of Relativistic Jets

8/3/2019 Akira Mizuta, Shoichi Yamada and Hideaki Takabe- Propagation and Dynamics of Relativistic Jets

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PROPAGATION AND DYNAMICS OF RELATIVISTIC JETS

Akira Mizuta,1

Shoichi Yamada,2

and Hideaki Takabe1

Received 2003 September 21; accepted 2004 January 14

ABSTRACT

We investigate the dynamics and morphology of jets propagating into the interstellar medium using two-dimensional relativistic hydrodynamic simulations. The calculations are performed assuming axisymmetric ge-ometry and follow jet propagation over a long distance. The jets are assumed to be light, with the density ratiobetween the beam to the ambient gas much less than unity. We examine the mechanism for the appearance ofvortices at the head of jets in the hot spot. Such vortices are known as a trigger of a deceleration phase, whichappears after a short phase in which the jet propagation follows the results from one-dimensional analysis. Wefind that an oblique shock at the boundary rim near the end of the beam strongly affects the flow structure in andaround the hot spot. Weakly shocked gas passes through this oblique shock and becomes a trigger for thegeneration of vortices. We also find the parameter dependence of these effects for the propagation and dynamicsof the jets. The jet with slower propagation velocity is weakly pinched, has large vortices, and shows verycomplex structure at the head of the jets and extended synchrotron emissivity.

Subject headings: galaxies: jets hydrodynamics methods: numerical relativity shock waves

1. INTRODUCTION

It is widely known that there are three classes of highlycollimated and supersonic jets from dense central objectswith accretion disks, which depend on the central object, protostars, binary stars, or active galactic nuclei (AGNs).AGN jets are the largest scale phenomena, and the velocity ofthe jet beam is highly relativistic, at least close to the centralobject (see, e.g., Pearson et al. 1981; Biretta, Sparks, &Macchetto 1999). The jet, which originates near an accretiondisk that surrounds an AGN, can propagate over a longdistance, up to a few Mpc, while remaining well collimated.

There are two shocks at the end of the jet. One is a bowshock (or a forward shock), which accelerates the ambientgas. The other is a terminal Mach shock (or a reverse shock)at which the beam ends. At the terminal Mach shock, non-thermal particles are accelerated and emit photons throughsynchrotron radiation and inverse Compton scattering. Thegas that crosses the terminal Mach shock into a hot spot ishot and pressurized, and expands laterally, enveloping the beam with the shocked ambient gas, creating a so-calledcocoon structure. At the contact discontinuity between theambient gas and the jet in the cocoon, Kelvin-Helmholtzinstabilities develop.

Cygnus A is a suitable object in which to see these features,because it is one of the closest radio galaxies, and the beam

propagates perpendicularly to the line of sight. From obser-vations, its size is about 120 kpc, the beam velocity is $0.41c, and the hot spots ram pressure advance speed is 0.03c(Carilli et al. 1998), where c is the speed of light. For moredetails of AGN jets, see, for example, Carilli & Barthel(1996), Ferrari (1998), Livio (1999), and references therein.

Other active regions in jets are knots and secondary hotspots. Some knots with nonthermal emission can be seensporadically along the straight beam flow. The emissivity

between the knots varies even if they belong to the same jet.Recent high-resolution observations of AGN jets show finestructure of knots in the beam flow for up to several tens ofkiloparsecs in M87 (Biretta et al. 1999; Marshall et al. 2002;Wilson & Yang 2002), 3C 273 (Bahcall et al. 1995; Marshallet al. 2001; Sambruna et al. 2002; Jester et al. 2002),Centaurus A (Kraft et al. 2002), 3C 303 (Kataoka et al. 2003),and others (Sambruna et al. 2001). Knots on subparsec orparsec scales are thought to be from intermittent flows fromthe central source, and knots in blazars are due to ex-ceptionally strong shocks that are caused by the collisions of

internal shocks (Spada et al. 2001; Bicknell & Wagner2002). Knots on kiloparsec or larger scales are not wellunderstood. The high-energy particles that are accelerated on parsec scales may retain their energy for a long time. Insome jets, one observes secondary hot spots adjacent to theprimary hot spot at the head of the jet. The emissivity of thesecondary hot spot can be as high as that of the primary hotspot. It is thought that the gas in the secondary hot spot isseparating from the primary hot spot. For example, a brightsecondary hot spot is observed at both the eastern andwestern sides of the Cygnus A jet (Wilson, Young, &Shopbell 2000). The reason for these multiple hot spots isnot well understood.

Recent observations of AGN jets provide information not

only about the features of large-scale jets but also about thoseof very small scale jets such as in compact symmetricobjects (CSOs), which are believed from their morphologyto be the first stage of AGN jets. CSOs show two-sidedemissivity within the central few kpc of the parent galaxy core.The physical properties of the hot spots and lobes at both sideshave been measured (Owsianik & Conway 1998; Polatidis &Conway 2003; Giroletti et al. 2003). CSOs are only a fewthousand years old, and the propagation velocity of the hotspot is a few tens of percent of the speed of light. CSOs maytell us about the physical conditions during the earliest phasesof jets.

Analytical studies and numerical simulations of the morphology and the dynamics of jets have been performed for

the past thirty years. Blandford & Rees (1974) and Scheuer

1 Institute of Laser Engineering, Osaka University, 2-6 Yamada-Oka, Suita,Osaka, 565-0871, Japan; [email protected].

2 Department of Physics, Waseda University, Okubo, Shinjuku, Tokyo,

169-8555, Japan.

804

The Astrophysical Journal, 606:804818, 2004 May 10

# 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.


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(1974) discussed the structure of jets with a theoretical rela-tivistic beam model. Some of the early numerical work wasdone by Norman et al. (1982). Although these were nonrela-tivistic simulations, they showed the main structure of jets,that is, two shocks at the head of the jet and the cocoon. Thedifficulty of numerical relativistic hydrodynamics has delayedthe investigation of the relativistic effects on the morphologyand the dynamics of jets. Only in the past ten years have stablecodes, with or without external magnetic fields, been devel-oped for the ultrarelativistic regime (Eulderink & Mellema1995; Duncan & Hughes 1994; Font et al. 1994; Koide, Nishikawa, & Mutel 1996; Komissarov 1999a; Aloy et al.1999b; Hughes, Miller, & Duncan 2002), which allowed in-vestigation of the formation, collimation, and propagation ofjets (Rosen et al. 1999; Koide et al. 2002).

Early simulations have been done by van Putten (1993,1996), Mart et al. (1994, 1995), and Duncan & Hughes (1994).Mart et al. (1997) and Aloy et al. (1999a) performed long-term simulations with two-dimensional and three-dimensionalrelativistic hydrodynamic codes to study the morphology of jets. Propagation of relativistic jets in an external magnetic

field were computed by Komissarov (1999b). These studiesverified that the morphology of jets depends on a number ofdimensionless parameters: the density ratio of the jet beam tothe ambient gas ( b=a), the Lorentz factor of the beam,and the Mach number of the beam (Mb vb=cs;b). Here thesubscripts b and a stand for beam and ambient gas,respectively. In most of these simulations, a pressure-matched jet is assumed, with the pressure ratio K pb=pa 1. Mart,Muller, & Ibanez (1998) performed long-term simulations of jets and found a deceleration phase of the propagating jets.Recently, Scheck et al. (2002) pointed out that the propa-gation velocity v1Dj , which is derived from one-dimensionalmomentum balance at the rest frame of the contact disconti-nuity, is very useful for understanding the propagation of the

jet into the ambient gas. They assumed a constant velocityv

1Dj 0:2c and fixed the kinetic luminosity of the beam to be

Lkin 1046 ergs s1 in their models. These are the first nu-merical simulations, using a relativistic hydrodynamic code,that assume the propagation efficiency v1Dj =vb to be less than0.5 (cf. most numerical simulations done so far assume theefficiency is $0.51). Thus a wider range of parameter spaceof v1Dj must be studied if we agree with recent observationsthat CSOs are young AGN jets. Scheck et al. (2002) also in-vestigated the dependence of jet dynamics and morphologyon their composition, namely, pure electron-positron, electron- proton, or a mixture using a generic equation of state.They varied the density ratio (), the pressure ratio (K), theMach number (Mb), and the fraction of protons, electrons, and

positrons in their three models. The main difference betweenthe various models is seen in the radiative properties of the jets. The deceleration phase is also confirmed and shown tobegin with the generation of a large vortex at the head of the jet.This seems consistent with observations that jets have a de-celeration phase, because CSOs have a propagation speed ofa few tens of percent of the speed of light, while large-scale jets, such as Cygnus A, have a velocity only a few percent ofthe speed of light. However, the physics of the generation ofvortices is not well understood.

A first attempt to compare the vortex structures in numeri-cal simulations of the jets with observational data was made by Saxton et al. (2002b) and Saxton, Bicknell, & Sutherland(2002a), who discussed the shock structures at the head of the

jet of Pictor A and rings in the cocoon of Hercules A. We note

that radio jets with such features are rare (Gizani & Leahy2003).

In this paper we examine the mechanism that causes suchvortices in some detail. We present three simulations withdifferent v1Dj as injected beam conditions. We pay special at-tention to the generation of vortices in the hot spot and theseparation of vortices from hot spots, which strongly affectsthe dynamics of jets. We show the mechanism of generation ofvortices at the head of the jet. We hope that this study will helpus to better understand the propagation of jets into the ambi-ent gas. This paper is organized as follows. In x 2, the basicequations are presented. In x 3, we show our three models forthe numerical simulations. The results and discussion are shownin x 4. The surface brightness of synchrotron emissivity and aresolution study are also shown. Conclusions are given in x 5.

2. BASIC EQUATIONS

2.1. Relativistic Hydrodynamic Equations

Relativistic flows, v $1, affect the morphology and dy-namics of jets. The relativistic hydrodynamic equations (Lan-dau & Lifshitz 1987) then have to be solved:

@(W)

@t 1

r

@r(Wvr)

@r @(Wvz)

@z 0; 1

@(hW2vr)

@t 1

r

@r(hWv2r p)@r

@(hW2vrvz)

@z p

r; 2

@(hW2vz)

@t 1

r

@r(hW2vrvz)

@r @(hW

2v

2z p)

@z 0; 3

@(hW2 p)@t

1r

@r(hW2vr)

@r @(hW

2vz)

@z 0; 4

where , p, vi , W, and h are the rest mass density, the pressure,the three-velocity component in the i direction, the Lorentzfactor (1 v2)1=2, and the specific enthalpy (1 p=), respectively. In this section we equate the velocity oflight to unity. We assume axisymmetry. The set of equations(1)(4) is not closed until the equation of state is given. In thispaper, we assume an ideal gas

p ( 1); 5where and are a constant specific heat ratio ( 5=3) andthe specific internal energy, respectively.

2.2. Propagation Velocity

A one-dimensional analysis is useful for estimating thepropagation velocity of the jet. Here the propagation velocity

vj is defined to be the velocity of the contact discontinuity atthe end of the jet in the rest frame of the ambient gas. For verylight jets (T1), in general, the propagation velocity is lessthan that of the beam, because the conversion of the largefraction of the beam kinetic energy to the thermal energyoccurs at the terminal Mach shock.

2.2.1. Nonrelativistic Case

If the velocity and temperature of the gas are in the non-relativistic regime, it is sufficient to consider the Euler equa-tion. Assuming momentum balance in the rest frame of con-tact discontinuity,

Sb b(vb

v1Dj )

2

pbh i Sa a(v

1Dj )

2

pah i; 6

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where Sb and Sa are the cross section of the beam and the hotspot, and v1Dj is the propagation velocity derived from one-dimensional analysis. Solving equation (6) for v1Dj , we obtain

v1Dj

1

A 1

(Avb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A22v2b (A 1) Av2b (AK 1)c2s;a

" #vuut ); 7where cs is the sound speed and A is the cross section ratio(A Sb=Sa). If we assume A 1 and neglect the term thatincludes K(pb=pa) in equation (6), then this equationreduces to

v1Dj

ffiffiffi

pffiffiffi

p 1 vb: 8

This is the same as the equation derived by Norman, Winkler,& Smarr (1983) for the pressure-matched jet (K

1). If the

density ratio is much less than unity, in the case of a lightjet, the propagation efficiency, vj=vb, is much less than unity.

2.2.2. Relativistic Case

For the relativistic case, assuming again one-dimensionalmomentum balance between the beam and the ambient gas inthe rest frame of the contact discontinuity at the head of thejet,

Sb bhbW2

j W2b (vb v1Dj )2 pb

h i Sa ahaW2j ( v1Dj )2 pa

h i; 9

where Wj

1

(v1Dj

)2

1=2

. This equation leads to

v1Dj

1

AR 1

(ARvb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA22R v

2b (AR 1) ARv2b

(AK1)c 2saW2j

" #vuut ); 10with R RW2b ; R

bhbaha

: 11

In general, the sound speed of interstellar matter is muchsmaller than the velocity of the beam (cs;aTvb), so we canneglect the term including K. As a result, equation (10)becomes

v1Dj

ffiffiffiffiffiffiffiffiAR

p1 ffiffiffiffiffiffiffiffiARp vb: 12

This is equal to the equation derived by Mart et al. (1997) forthe pressure-matched jet (K 1), if we assume A 1. Schecket al. (2002) showed that the time evolution of the actual propagation velocity is almost the same in their models forfixed v1Dj and A 1, in spite of the large variation of thespecific internal energy of the beam during the first phase.There are several ways to explain the deceleration of jetsbetween subparsec and megaparsec scales. If the density ratio

varies, should become smaller than that in the earlier

phase. The Lorentz factor should also become smaller. Thevelocity at the head of the beam should be relativistic to produce strong emissivity at the hot spot. The effect of theLorentz factor seems small. Another possibility is variation ofthe density profile of the ambient gas. For example, it isnatural to assume that the ambient gas decreases in density asfunction of distance from the AGN. In the models of Schecket al. (2002), the multidimensional effect, namely decreasingthe cross section ratio, becomes very important for the dy-namics, and the jet decelerates gradually. Scheck et al. (2002)compared the time evolution of the dynamics with the ex-tended Begelman-Cioffi model. The original model derivedby Begelman & Cioffi (1989) assumes a constant propagationvelocity, and Scheck et al. (2002) extended the assumption toa power law.

3. NUMERICAL CONDITIONS

We solve the relativistic hydrodynamic equations (1)(5)with a two-dimensional relativistic hydrodynamic code re-cently developed by one of the authors (A. M.). The detailednumerical method and test calculations of the code are given

in the Appendix.We use a two-dimensional cylindrical computational region

r; z. The grid size in both rand zdirections is uniform, namelyr z const. We assume that the ambient gas is homo-geneous initially. Calculations including clouds in the ambientgas have been made by de Gouveia Dal Pino (1999), Zhang,Koide, & Sakai (1999), and Hughes et al. (2002). A relativisticbeam flow (vb), which is parallel to the z-axis, is continuouslyinjected from one side of the computational region (z 0). Theinner 10 grid points from the symmetry axis are used for this.The radius of the injected beam, Rb 10r, is used as ascaling unit in this study. The computational region covers50Rb(r) ; 180Rb(z), which corresponds to a 500 ; 1800 grid.The radius of the beam near the central engine is unknown, and

we assume a plausible value of 1Rb 0:5 kpc, which corre-sponds to 25 kpc ; 90 kpc for the computational region. A freeoutflow condition is employed at the outer boundaries forr 50Rb and z 180Rb. A reflection boundary is imposed onthe symmetry axis, as well atz 0 with r > Rb. The boundarycondition at z 0 is crucial for dynamics and the outer shapeof the jets. The free boundary condition permits the gas toescape at the back side (Saxton et al. 2002b). According toobservations, jets have counterjets that propagate in theopposite direction. We assume the reflective boundary condi-tion so that our calculations begin near the central engine.

We examine three injection beam conditions (Table 1) withdifferentv1Dj . The description ofv

1Dj is given in equation (12);

here A

1 is assumed. The beam velocity is fixed at vb

0:99c (the corresponding Lorentz factor is Wb 7:1). TheTABLE 1

Numerical Conditions of Models

JB02 JB03 JB04

b/a......................... 1.28 ; 103 3.76 ; 103 9.15 ; 103Mb vb/cb ...................... 6.0 6.0 6.0b .................................... 2.55 ; 10

2 2.55 ; 102 2.55 ; 102..................................... 5/3 5/3 5/3

Kpb/pa ........................ 10 33 100Wb(vb). ... .. .. ... ... ... ... .. .. ... .. 7.1(0.99) 7.1(0.99) 7.1(0.99)

v1D

j =c............................... 0.2 0.3 0.4Expiration Time Rb/c ..... 1800 1200 600

MIZUTA, YAMADA, & TAKABE806 Vol. 606


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Mach number of the beam is also fixed at Mb 6:0. Weassume that the beams do not have highly relativistic tem-peratures. The incident beam velocity vb is fixed. As a result,v

1Dj depends only on the density ratio . Scheck et al. (2002)

adopted v1Dj 0:2c as the injected beam conditions for theircalculations, which is in the range of acceptable propagationvelocities based on observations of CSOs. We explore theregion of the propagation velocity v1Dj from 0.2c to 0.4c and

investigate the dependence of the difference ofv

1D

j caused bydifferent . With increasing v1Dj , varies about a factor of 10,from 1:28 ; 103 to 9:15 ; 103. We label these cases JB02,JB03, and JB04, and the corresponding v1Dj are 0.2c, 0.3c, and0.4c, respectively. Although there is only about 1 order ofmagnitude difference in the density ratio for all beams, dou-bling v1Dj has a big effect on the dynamics of jets and theirlong-term propagation. Because the spatial scale is fixed, weadopt different expiration times for each model; see Table 1.

The initial rest mass density of ambient gas a is unity forall models. The pressure ratio K is chosen to be K 10 100,so that the ambient gas is approximately similar in all cases.As a result, the temperature of the ambient gas for all injectedbeams is a few eV, if we assume the ambient matter is pure

hydrogen gas. Because we choose similar values of v1D

j to

those of Scheck et al. (2002), other parameters (, K, Mb) arealso very similar.

4. RESULTS AND DISCUSSION

4.1. Morphology And Dynamics

From our simulations, we find that the overall morphologyand dynamics of the jets are similar to that discussed in pre-

vious work. Figures 1, 2, and 3 show snapshots of rest massdensity, pressure, and Lorentz factor for three models nearthe end of the simulations, t 1770Rb=c (model JB02), t1140Rb=c (model JB03), and t 570Rb=c (model JB04). Sincethese jets have different beam conditions, the end times arealso different, as discussed in x 3. However, this is not onlydue to different v1Dj values but also because of differences inthe deceleration phase of the jets, which is caused by thegeneration and separation of vortices at the head of the jet,as we discuss later. At first glance, the outer shapes of the jetsare quite different. Model JB02 shows a conical outer shape,which is very similar to the results found by Scheck et al.(2002). This is not seen in the other two models, JB04 andJB03. All beams remain collimated from z 0, where thebeam is injected into the computational region, to the head of

Fig. 1.Contours of rest mass density (top), pressure (middle), and Lorentzfactor (bottom) of model JB02 at the end of the simulation ( t 1770Rb=c). Fig. 2.Contours of rest mass density (top), pressure (middle), and Lorentzfactor (bottom) of model JB03 at the end of the simulation ( t 1140Rb=c).

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the jets. It is also important to note that the beam radius doesnot increase monotonically from the source to the head of thejet. At the head of the jet, the radius of the beam is $3 5Rb.The opening angle is very small, $12, for all models. HighLorentz factors exist only along the z-axis in the beamflow, but they do vary. That means that a large part of theinjected kinetic energy is transported from the central engineto the head of the jet along the beam flow. At the end of thebeam, a strong terminal Mach shock can be seen. One ofthe most active points is called a hot spot, into whichshocked beam gas enters through the terminal Mach shock atthe head of the jet. The pressure in the hot spot is very high,because of the energy dissipation at the terminal Mach shock,and is matched by ambient gas compressed at the bow shock.Moderate Lorentz factors (W $ 2) exist in the cocoon. Theseare due to back flows that begin at a hot spot and flow back inthe center of the cocoon parallel to the beam flow. This mildlyrelativistic back flow seems a little strange, because the headis proceeding very slowly (P0.2c), and the expanding velocityfrom the hot spot is as much as 0.5c in the comoving frame(maximum sound speed). It should be noted that this mildlyrelativistic backflow is longer for slower jets. Thus some

acceleration mechanism or other effects are necessary. Wediscuss this in x 4.2. This back flow creates a shear flow,and the contact surface becomes unstable because of Kelvin-Helmholtz instabilities. The surface between the back flowand the shocked ambient gas flow also becomes unstable andcauses the appearance of vortices in the larger cocoon.

To understand the difference of these models after long-term propagation, we must study the dynamics of the jets.Figure 4 shows the time evolution of the positions of thecontact discontinuity, the bow shock (forward shock), and theterminal Mach shock (reverse shock) at the head of the jet foreach model. Three straight lines that correspond to the lines ofone-dimensional analysis are also shown for comparison. It isdifficult to define the positions of these surfaces because of thecomplex structure at the head of the jets. In this paper weshow the positions atr 0, namely along the z-axis. It is hardto identify the contact discontinuity between shocked ambientgas and shocked beam gas, especially in later phases, becauseof mixing of the shocked beam with the ambient gas by thegeneration of vortices. The contact discontinuity is defined asthe boundary where the shocked ambient gas has half the

maximum density. We plot the points where the Lorentz factorbecomes 2 at the head of the jet. The two phases indicated byMart et al. (1998) and Scheck et al. (2002) are also seen in ourresults. During the first phase, all of the slopes are constant.Model JB02 has a propagation velocity $0.2c, and the one-dimensional analysis discussed in x 2.2 is in good agreementwith our result. For JB03 and JB04, the propagation velocityis a little faster than that of our one-dimensional estimates.During the first phase, the surfaces are very close to eachother. On the other hand, in the second phase, the surfacesseparate and approach each other repeatedly. The propagationvelocity is no longer constant, and the jet is deceleratinggradually. Some vortices grow at the head of the jet and sep-arate toward the back side. This affects the jet dynamics.

Figure 5 shows snapshots of the velocity (v

2

r v2

z)1/2

during the earlier phase of each models. Although its path isnot as straight as that of the beam, the back flow does remainparallel to the beam flow. The back flow velocities are veryFig. 3.Contours of rest mass density (top), pressure (middle), and Lorentz

factor (bottom) of model JB03 at the end of the simulation ( t 570Rb=c).

Fig. 4.Time evolution of the position of the bow shock, the contactdiscontinuity, and the terminal Mach shock for models JB02, JB03, and JB04.Analytic lines (solid) are written for comparison with each model.



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similar for all models ($0.30.4c). Because the propagationvelocity is less than that of the back flow for the JB02 model,most of it reaches the boundary, after which some of the gasexpands in a lateral direction, and other gas forms a new flow

between the beam and the back flow. This new flow also can

be seen in the JB03 model near the boundary, but it does notreach the head of the jet like JB02, because the propagationvelocity is faster. The beam flow and back flow lie side byside near the head of the jets in JB03 and JB04.

In later phases, this third flow, which is not a beam flowbut has the same propagating direction, exists in all models because of vortices generated in the cocoon by Kelvin-Helmholtz instabilities. At this stage the back flow becomesvery complex. How much gas reaches the boundary at r 0affects the expansion into lateral directions for each model.Slower propagation velocities result in conelike outer shapesof the jets. The shocked ambient gas forms a shell-like struc-ture like a football, resembling the X-ray emission Cygnus Aobserved by Chandra (Wilson et al. 2000), which is consistentwith our results, because the present propagation velocity ofCygnus A is very small ($0.03c). A long time may havepassed after the deceleration phase began in the Cygnus A jet.In contrast, models JB03 and JB04, which have propagationvelocities faster than that of the back flow in the earlier phase,do not have a conical shape. These jets are decelerating. Theeffect of a decreasing propagation velocity can be seen in

JB03 near the head of the jet, which shows a conical shape.4.2. Vortex Formation in Hot Spots

We discussed the generation and separation of large vorticesat the head of the jet, which occurs repeatedly during thesecond phase. Such processes strongly affect the dynamics ofjets, as also found by Scheck et al. (2002), and are also con-sistent with observations of the high-speed CSO sources andslower large-scale jets. The reason for this deceleration seemsto be the formation of vortices. Where and how are thesevortices created? The possibility of hydrodynamic instabilitiesat the head of jets has been discussed before. Norman et al.(1982) studied Rayleigh-Taylor instabilities at the contactdiscontinuity. Recently, Krause (2003) explored the Kelvin-

Helmholtz instability between the flow from the hot spot andthe shocked ambient gas.To understand the generation of these vortices, we need to

focus on the flow structure in and around the hot spot. Thevortices appear in the hot spot, and they originate by the flowthrough an oblique shock at the end of the beam.

It is known that oblique shocks appear within the beam,even before the end, when the beam expands in lateral di-rections. This re-establishes pressure equilibrium between the beam and its surroundings and keeps it confined (Fig. 6a).When the beam is pinched for some reason, the gas tends toexpand because of increased pressure caused by the compression (Fig. 6b). If the pressure outside is high enough toconfine the beam, an oblique shock appears to prevent it from

expansion (Fig. 6c). The radius of the beam after reconfine-ment depends on the pressure outside of the beam and how thebeam is pinched.

Figure 7 shows the rest mass density, pressure, and Lorentzfactor profile along the z-axis of model JB02 at t 300, 600,900, 1200, and 1500Rb=c. The beam is confined by the pressure of the cocoon and the shocked ambient gas. How-ever, the surface of the beam is not stable. In our calculations,the first oblique shock appears where the beam is injected.Such shocks appear irregularly in the beam. Since theseshocks have different speeds, a slower shock is caught up by afaster shock. This might explain the high-emissivity knots inthe jets. When an oblique shock appears at the end of thebeam, most of the gas passes a terminal Mach shock with large

dissipation. However, some gas passes through the oblique

Fig. 5.Absolute velocity (r > 0) and log scale rest mass density contour(r < 0)of modelsJB02 (top),JB03 (middle),and JB04(bottom) in early phases,at t 107:5Rb=c (JB02), t 50:0Rb=c (JB03), and t 35:0Rb=c (JB04).



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shocks on the side (Figs. 8a and 8b). If the angle is small, theoblique shock is very weak. The loss of kinetic energy should be small, but the pressure becomes as high as that in the

cocoon. Then a mildly relativistic back flow begins. The effectof these oblique shocks can be seen in slower jets, which iswhy the mildly relativistic flow is longer for slower jets. Sucha fast velocity flow through the oblique shock can propagatefurther in lateral directions than slower gas from the hot spotbefore it propagates backward. The flow path becomes thena circular, arclike vortex. This vortex can sometimes reachthe beam flow, which triggers instabilities at the beam surfaceand internal oblique shocks. The surface of the beam becomesmore unstable and has oblique shocks there. The obliqueshocked flow has an important effect on the gas in the hot spot,since it blocks further outflow. In the hot spot, the velocity isnot constant. The nearer the gas is to a corner that is a cross-ing point between a contact discontinuity and the z-axis, the

slower the velocity. Although the outflow from the hot spot isblocked, the beam flow continues to enter the hot spot. Thisthen drives a clockwise rotation vortex in the hot spot (Figs. 8cand 8e). This vortex is very important for the dynamics, be-cause the increasing radius of vortices causes an increase ofthe beam cross section. As a result, the beam decelerates (seeeq. 12]). The vortex can grow until the radius becomes abouttwice that of the beam. The slower a jet is, the larger thevortex it can support, because the ram pressure of the shockedambient gas, which is going backward and pushing the vortex,is low. When a vortex grows, the jet decelerates. On the otherhand, when a vortex separates (Figs. 8fand 8g), the jet accel-erates slightly. However, the next vortex soon grows. InFigure 4, the size of the hot spot, namely, the distance be-

tween a contact discontinuity and a terminal Mach shock,

oscillates with increasing amplitude. This occurs in all modelsbut is strongest in the slower jets. The increasing oscillationamplitude increases the growth time of a vortex. A long growthtime causes a larger radius of the vortex. The dynamics is arepetition of these processes during the second phase.

Most gas in this vortex has passed trough the terminalMach shock. The vortex also contains nonthermal particlesaccelerated by this shock. A separating vortex can be observedas an active region around the hot spot. The best candidatefor this region is the secondary hot spot discussed in x 1.The high-energy particles lose their energy quickly, so that aseparating vortex is a little fainter than the primary hot spot.

When a jet accelerates, the bow shock has a nose conelikeshape. On the other hand, the bow shock, which is deceler-ating, has a flat shape. This can be observed in the eastern lobeof Cygnus A, where the hot spot seems to be slightly ahead ofthe lobe, which may be because the vortex may just haveseparated and the jet is now in a brief accelerating phase.

Once a jet exhibits such oblique shocks and vortices, theflow structure around the hot spot becomes more complex.The effect of this turbulence is strong in the case of slowly

propagating jets, because the confinement effect of shockedambient gas is weak. The beam is pinched when a circulararclike flow from the oblique shock reaches the beam or aseparated vortex moves backward. The end of the beam tendsto expand and has more oblique shocks there. The beam issometimes constricted in the middle by such effects, and astrong shock appears. This shock may be related to knotsobserved in the beam.

During the later phase of model JB02, the hot spot becomesvery large and clumpy. As a result, the gas that passed theterminal Mach shock follows the edge of this clumpy gas.Some of it is mixed in at locations where the gas reaches theshocked ambient gas and back flow begins. This then alsoseparates from the head of the jet, and a normal hot spot

appears again. The large clumpy gas does not display a brighthot spot. It corresponds to a temporarily absent hot spotdiscussed by Saxton et al. (2002a) and can be a good candi-date for the absent hot spot in Hercules A.

4.3. Synchrotron Emissivity

Recently, some numerical simulations of jets show thesurface brightness of the synchrotron emissivity (Scheck et al.2002; Saxton et al. 2002a, 2002b; Aloy et al. 2003). We fol-low the analysis by Saxton et al. and compare our results withtheirs and observations.

We use an approximation that assumes the synchrotronemissivity is proportional to pB1 , where p is the pressure,B is the magnetic field, and is the spectral index. We use

the typical value 0:6. The magnetic pressure, $B2

, is as-sumed to be in equilibrium with the thermal pressure p. Thesurface brightness is derived from revolved two-dimensionalnumerical results. The emissivity is fp1:8, where fis the frac-tion of the gas that originates from the injected beam gas. Theadvection equation of fis transformed to a conservative formusing mass conservation equation and solved with hydrody-namic equations.

Figures 9 and 10 show the emissivity of the models JB02and JB04 on a log scale with 4 orders of magnitude from themaximum intensity (white is the brightest emissivity) is shownfor different several angles, 0, 15, 30, 45, 60, 75, and90, where is the angle between the z-axis and the line ofsight. JB02 has very extended emissivity from the head to

the root of the jet. On the contrary, JB04, which has less

Fig. 6.Schematic of appearance of an oblique shock in the beam (notscaled).



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complex structure, has emissivity only around the head. Thus,the appearance of vortices causes the extension of emissiveregions toward the back side. The brightest region corre-sponds to the hot spot. The other bright regions, which cor-respond to the oblique shocks in the beam, appear irregularly,which may correspond to the observed knots in the beam.The same emissivity is usually seen in the observations ofthe jets.

Both models have a ringlike emissivity near the head ofthe jet when the angle is $3075. This feature is alsoshown and compared with observations of Hercules A in

Saxton et al. (2002a, 2002b). Several rings in the lobe havedifferent radii and emissivities. The observations of HerculesA by Gizani & Leahy (2003) resemble our emissivity resultsfor JB02. These results indicate that the lobe of Hercules Ahas a complex structure with vortices.

4.4. Resolution Dependence

Finally, we discuss the resolution dependence of our re-sults. Calculations with higher resolution have finer struc-tures because how much numerical dissipation is included

depends on the resolution. In particular, for a problems in

Fig. 7.Plots of log scale rest-mass density ( solid line), log scale pressure (dot dashed line), and Lorentz factor (dashed line) profiles along the z-axis of modelJB02 at t 300, 600, 900, 1200, and 1500Rb=c.



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which vortices appear, such an effect is very important andmay affect the size of vortices and when they form. We per-formed calculations with resolutions 1.5 and 2 times higherfor the three cases discussed above, but with restricted scalesto save CPU time. In other words, 15 (1.5 times resolution) and20 (twice resolution) grid points are used for the inner beam.

Higher resolution calculations also show the same prop-erties as those discussed above; namely, two phases are

observed. The jets have the generation and separation vor-

tices at the head of the jet, and these effects are strong inthe slower injected beam models. Figure 11 shows the timeevolution of the positions of the bow shocks, contact surfaces,and terminal Mach shocks with different resolutions for thecondition of JB02. This figure is as same as Figure 4, but onlythe JB02 case is shown, and the scales are restricted. The jetswith higher resolution propagate a little slower than those inthe lower resolution calculations. This is caused by a large

separation of a terminal Mach disk from the contact surface in

Fig. 8.Series of profiles of growing and separating vortices. Each panel has the domain 20Rb ; 20Rb, and time evolves from (a) t 190Rb=c to (h) t 225Rb=cof model JB02. In each panel, the top shows the absolute velocity r > 0 and the bottom shows the rest mass density r < 0 for (a) a normal profile; (b) an obliqueshock appears at the rim of the end of the beam (see density jump from blue to green); ( c) (e) a vortex grows; ( f) (g) separation of the vortex from the head of the

jet to backward.

MIZUTA, YAMADA, & TAKABE812


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Fig. 9.Synchrotron emissivity of JB02 at t 1770Rb=c at angles 0, 15, 30, 45, 60, 75, and 90, where is the angle between the z-xis and the line ofsight. Counters are shown in log scale with 4 orders of magnitude from the maximum intensity (white is the highest emissivity) in each panel.

Fig. 10.Same as Fig. 9, but for JB04 at t 570Rb=c


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the early second phase, because the large separation allows theappearance of a large hot spot, which is seen in the later phaseof the calculation in JB02 with normal resolution, and a largecross section for the efficiency of the propagation.

5. CONCLUSION

Three numerical simulations of relativistic jets are shownin this paper. We pay special attention to the formation ofvortices and their parameter dependences. The propagationvelocity derived from one-dimensional momentum balanceis varied from 0.2c to 0.4c. These estimations for the velocityare based on recent observations of CSOs. We identify two phases based on long-term simulations, confirming previousresults. The propagation velocity during the first phase may be estimated using a one-dimensional analysis. The length between the bow shock and the contact discontinuity doesnot change significantly. On the other hand, the distance

between the terminal Mach shock and the contact disconti-nuity oscillates, and its amplitude increases gradually duringthe second phase. This effect is strongest in slow jets. Duringthe second phase, an oblique shock with a small angle at theend of the beam appears. This flow dramatically affects thedynamics, morphology, and emissivity of the jets in the second phase. Weakly shocked gas then forms a back flow withoutlarge energy dissipation. The velocity of this flow is faster thanthat of the expanding flow from the hot spot. The hot spot isblocked by this weakly shocked flow. As a result, a vortexformsin the hot spot. When a vortex grows, the increasing crosssection decelerates the jet. When a vortex separates from the jet, the jet accelerates. After this a new vortex appears, andthis process occurs repeatedly during the second phase. Thejet gradually decelerates. In future work, the effect of the gen-eration and separation of vortices at the head of the jet shouldbe investigated using three-dimensional numerical simulations.

The synchrotron emissivity of the jets are shown. Strongsynchrotron radiation emissivity appears at the hot spot. Theirregular emissivity on the z-axis appears and has some simi-larity to observed knots in the beam. In the slowest beam

model, very extended emissivity is shown from the head to theroot of the jet, and the ringlike structure of unusual emissivityin Hercules A is reproduced. In the fastest beam model, theemissivity can be seen only around the head of the jet andalong the beam. The separation of vortices from the head of thejet strongly affects the extension of the emissivity.

This work was carried out on NEC SX5, Cybermedia Centerand Institute of Laser Engineering, Osaka University. We ap-preciate computational administrators for technical support.

A. M. acknowledges support from the Japan Society forthe Promotion of Science (JSPS). A. M. would like to thank N. Ohnishi, H. Nagatomo, and K. Sawada for useful sugges-tions to the numerical methods. We also gratefully acknowl-edge J. M. Ibanez, S. Koide, M. Kino, and T. Yamasaki forhelpful discussions on AGN jets. We acknowledge commentson the emissivity by C. Saxton and G. Bicknell. We appreciateuseful comments and suggestions by W. van Breugel and theanonymous referee that have improved this manuscript.

APPENDIX

RELATIVISTIC HYDRODYNAMIC CODE

In this appendix, we describe the numerical method used in our relativistic hydrodynamic code and show some results oftypical test problems for relativistic hydrodynamic codes, namely, the shock tube problem and the strong reflection shock problem.

During the past 10 years, numerical methods to solve the relativistic or magneto-relativistic hydrodynamic equations have

advanced significantly (see a review paper by Ibanez & Mart 1999 and references therein). In particular, the method usingapproximate Riemann solvers provides good accuracy even if the flow includes strong shocks and high Lorentz factors. Recentlywe have developed a two-dimensional special relativistic code using approximate Riemann solvers, which are derived fromspectral composition of Jacobian matrices of special relativistic hydrodynamic equations. Either plane, cylindrical (r-z), orspherical (r-) geometry are assumed.

Recently other new methods also have been employed or proposed for relativistic hydrodynamic or magnetohydrodynamiccodes (Sokolov, Zhang, & Sakai 2001; Del Zanna & Bucciantini 2002; Anninos & Fragile 2003; Del Zanna, Bucciantini, &Londrillo 2003).

A1. NUMERICAL METHOD

The relativistic hydrodynamics equations are written in conservative form in each coordinate system: plane,

@u

@t

@f(u)

@x

@g(u)

@y 0;

A1

Fig. 11.Time evolution of the surfaces at the head of the jets (same as

Fig. 4, but for only JB02) using different resolutions for the calculation: A(normal case), B (inner jet beam 1.5 times normal), and C (2.0 times).



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cylindrical,

@u

@t 1

r

@ rf(u) @r

@g(u)@z

sc(u); A2

and spherical,

@u@t

1r2

@ r2f(u) @r

1rsin

@ sin g(u) @

ss(u); A3

where u is a vector of conserved quantities, fand g are Cux vectors, and sc and ss are source vectors. They are deBned as follows:

u (W; hW2v1; hW2v2; hW2 p W)T; A4f(u) (Wv1; hW2v1v1 p; hW2v1v2; hW2v1 Wv1)T; A5g(u) (Wv2; hW2v1v2; hW2v2v2 p; hW2v2 Wv2)T; A6

sc 0; pr

; 0; 0 T

; A7

ss 0; 2p hW2v

2v

2

r; hW

2v

1v

2 cotpr

; 0

T

: A8

We then discretize these equations for the numerical calculations,

un1i; j uni; j 1

ri; jri1=2; j fi1=2; j ri1=2; j fi1=2; j

t

ri gi; j1=2 gi; j1=2

t

zj sci; jt; A9

where uni; j stands for the average value ofu in the i,jth grid attn, andf and g stand for numerical Cux through the cell surface. Here,

we show the cylindrical coordinate case. Numerical Cuxes f and g at each cell surface are calculated by Marquinas Cux formuladerived from left and right eigenvectors and eigenvalues of Jacobian matrices of relativistic hydrodynamic equations (Donat &Marquina 1996; Donat et al. 1998). To obtain higher accuracy in spatial dimensions, we adopt the MUSCL method (van Leer 1977;van Leer 1979) for the reconstruction of the left and right state at each cell surface:

(qL)i1=2 qi 1

4(1 )()i (1 )()i

; A10

(qR)i1=2 qi1 1

4(1 )()i1 (1 )()i1

; A11

where q is the physical value for interpolation. In our code, values such as rest mass density, pressure, and velocity use thisreconstruction. The accuracy of the code for spatial dimensions is second order when a linear interpolation is adopted ( 1 or 0)and third order when a quadratic function is used for the interpolation ( 1=3). We use a minmod limiter to keep the totalvariation diminishing (TVD) condition, which prevents the development of numerical oscillations. Then and are deBned as

()i

minmod

qi1

qi; b(qi

qi1)

;

A12

()i minmodqi qi1; b(qi1 qi); A13minmod(a; b) sign(a) max 0; min (jaj; sign(a) b); A14

where b is a parameter that satisBes 1 b (3 )=(1 ). The results become diAusive with small b. In this study we use 1 and b 2. The time step is accurate to Brst order. Recovery of primitive values from conservative vector u is done by the Newton-Raphson method at each time step (Aloy et al. 1999b).

A2. TEST CALCULATIONS

We show two types of of one-dimensional relativistic hydrodynamics test calculations. The calculation is done with second orderfor space and Brst order for time. For both cases, 400 grid points are used in the direction of interest. Since we assume thedynamics is only one-dimensional, the velocity in the other direction is set to be zero initially. So, the velocity of this direction

remains zero as time evolves.



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A2.1. Shock Tube Problem

A shock tube problem is a kind of initial value problem. Two states are given between a discontinuity at t 0. This is areasonable problem, because the analytical solution is given by Thompson (1986) and Mart & Muller (1994). We use two initialconditions of this problem following Donat et al. (1998).

Problem 1 : SH1 (Plane)Left state (x < 0:5): L 10;pL 13:3; vL 0; L 5=3.Right state (x > 0:5): R 1;pR 1 ; 106; vR 0; R 5=3.

Problem 2 : SH2 (Plane)Left state (x < 0:5): L 1;pL 1000; vL 0; L 5=3.Right state (x > 0:5): R 1;pR 0:01; vR 0; R 5=3.

Figures 12 and 13 show the results of these problems at t 0:5 (SH1) and 0.35 (SH2). Analytical solutions are also shown forcomparison. Despite a jump more than 5 orders of magnitude in pressure at the initial discontinuity, a rarefaction fan whose front

proceeds toward the left with the sound velocity of the initial left side state, a contact discontinuity, and a shock are given withgood accuracy in each case, except at the density jump behind the shock in SH2.

A2.2. Reflection Shock Problem

A reCection shock problem is suitable for studying a Cow with a strong shock. Initially a relativistic homogeneous cold Cow(0; W0; 0 $ 0) reCects atx 0 from either a wall (plane geometry), a symmetric axis (cylindrical geometry), or a symmetric point

Fig. 12.Test calculation of shock tube problem (SH1). Rest mass density (), pressure (p), and velocity (v) are shown at t 0:5. Initial discontinuity is atx 0:5. Solid lines are analytic solutions.

Fig. 13.Same as Fig. 12, but for SH2 at t 0:35



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Fig. 14.Test calculation of reflection shock problem (REP; plane). Rest mass density ( ), pressure (p), and velocity (v) are shown at t 1:57. Solid lines areanalytic solutions.

Fig. 15.Same as Fig. 14, but for REC (cylindrical) at t 1:57

Fig. 16.Same as Fig. 14, but for RES (spherical) at t

1:57


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(spherical geometry). A strong shock then runs against the Cow with a high density and pressure jump. Analytical solutions of thisproblem are given by the Rankine-Hugoniot relation of relativistic hydrodynamics (Johnson & McKee 1971),

1 1

1 (W0 1) !

0; W0 1; v 0; for x < Vst A15

0 1 jv0jtx

; $ 0; v v0; for x > Vst; A16

where Vs is the shock velocity,

Vs W0 1W0 1

1=2( 1): A17

The geometry is plane ( 0), cylindrical ( 1), or spherical ( 2). The expression of rest mass density jump at the shockfront is divided in two parts, namely, a maximum density compression ratio in nonrelativistic hydrodynamics [( 1)=( 1)]and a term that includes a Lorentz factor:

Problem 3 : REP 0 1:0; 0 104; v0 0:999(W0 22); 4=3, plane;Problem 4 : REC 0 1:0; 0 104; v0 0:999(W0 22); 4=3, cylindrical;Problem 5 : RES 0 1:0; 0 104; v0 0:999(W0 22); 4=3, spherical.Figures 14, 15, and 16 show the numerical results for rest mass density, pressure, and velocity for each problem. Analytical

solutions are also given. We show the results att 1:57 when the shock propagates 0.5 from x 0. In all cases the shock front isreproduced with good accuracy. Because of a numerical problem, an oscillation appears at the front. At the boundary, the error iswithin 2% (REP), 6% (REC), and 5% (RES) in density. The geometric errors in the cylindrical and spherical cases are 1% (REC)and 2% (RES) in density, respectively.

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Akira Mizuta, Shoichi Yamada and Hideaki Takabe- Propagation and Dynamics of Relativistic Jets

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