Numerical Relativistic Hydrodynamics and Magnetohydrodynamics, and Extragalactic Jets
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Numerical Relativistic Hydrodynamics and Magnetohydrodynamics, and Extragalactic Jets. José Mª Martí. Departamento de Astronomía y Astrofísica Universidad de Valencia (Spain). Astrophysical Fluid Dynamics, April 4-9, 2005. Outline of the talk. Motivation. Extragalactic jets

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Numerical Relativistic Hydrodynamics and Magnetohydrodynamics, and Extragalactic Jets

José Mª Martí

Departamento de Astronomía y Astrofísica

Universidad de Valencia (Spain)

Astrophysical Fluid Dynamics, April 4-9, 2005

Outline of the talk

  • Motivation. Extragalactic jets

    • Discovery and observational status

    • Standard model

    • The jet/outflow family

    • Extragalactic jets: open questions

  • General relativistic hydrodynamics

    • RHD eqs. as a hyperbolic system of conservation laws

    • Numerical methods: AV, HRSC, …

  • Simulations of extragalactic jets

    • Large scale jets

    • Compact jets

    • Jet formation mechanisms

  • Relativistic Magnetohydrodynamics

  • Summary, conclusions, …

Cygnus A, VLA, Perley et al. 1984

Motivation. Extragalactic jets

General relativity and relativistic hydrodynamics play a major role in the description of many astrophysical scenarios. Extragalactic jets (present in radio-loud AGNs) are a paradigmatic example.

1918: Discovery of the optical jet in M87

1940-50: Radio source Virgo A associated with M87 and its jet

1970-80: Models of continuous supply to interpret double lobed extragalactic radio sources (obtained with aperture synthesis techniques)

M87, HST

Scheuer 1974

Cygnus A, Cambridge 5km array

Hardgrave & Ryle 1974

Blandford & Rees 1974

1980-…: with the advent of VLA, routine detection of jets

Very Large Array

Socorro, NM

Extragalactic jets: Observational status I

Nowadays, jets are a common ingredient of radio-loud AGNs detected and imaged at very different spatial scales with different arrays (kpc scales: VLA, Merlin; pc scales: VLBA, VLBI, VSOP)


Extragalactic jets: Observational status II

Kpc scales: Morphological dichotomy based on jet power

Pc scales: Superluminal motion, one-sidedness

Subpc scale: Collimation

Cyg A, VLA (6cm), Carilli et al. 1996

3C120, VLBA

Gómez et al. 2000


Junor et al. 1999

3C31, VLA (20cm), 1999

Extragalactic jets: Observational status III

Radio polarization: information about the topology of magnetic fields at both pc and kpc scales

1055+018, Attridge et al. 1999

Hints on jet internal structure: shocks,

shear layers

3C120: X-ray dips/component ejection correlations

3C31, Laing & Bridle 2002 (ticks show the orientation of magnetic field

Variability: time scales and correlations between variations in different continuum/line components

Nature and location of the different components at the central regions of AGNs (e.g., narrow and broad line regions) and on their interdependencies (e.g., jet/disc connection)

Marscher et al. 2002

Extragalactic jets: Observational status IV

Imaging of the central regions of AGNs: anatomy (narrow line region, obscuring torus) and dynamics (rotation) of the central engine



Hubble Space Telescope

Multiwavelength imaging of the jets and spectra: emission models, spectral evolution of the high

energy electron distribution, reacceleration sites, …

3C273, radio (MERLIN), optical (HST) and X-ray (CHANDRA) with optical contours; Marshall et al. 2001

Bertsch (NASA/GSFC)

Extragalactic jets: the standard model

  • The production of jets is connected with the process of accretion on supermassive black holes at the core of AGNs (see, e.g., Celotti & Blandford 2000)

    • Hydromagnetic acceleration of disc wind in a BH magnetosphere (Blandford-Payne mechanism)

    • Extraction of rotational energy from Kerr BH by magnetic processes (Blandford-Znajek mechanism, magnetic Penrose process)

  • Emission: synchrotron (responsible of the emission from radio to X-rays) and inverse Compton (g-ray emission) from a relativistic (e+/e-, ep) jet (e.g., Ghisellini et al. 1998). Target photons for the IC process:

    • Self Compton: synchrotron photons

    • External Compton: disc, BLR, dusty torus

  • Jets are relativistic, as indicated by:

    • Superluminal motions at pc scales (due to the finite speed of propagation of light)

    • One-sidedness of pc scale jets and brigthness asymmetries between jets and counterjets at kpc scales (due to Doppler boosting of the emitted radiation)

Jets: Relativistic collimated ejections of thermal (e+/e-, ep) plasma + ultrarelativistic electrons/positrons + magnetic fields + radiation, generated in the vicinity of SMBH


The jet/outflow family

  • X-ray binaries:

  • 20%

  • vjet < 0.9c

  • Ljet = 1038- 1040 erg/s

  • Size = 10-3 – 10-1 pc

  • Collimation: few degrees

  • Central engine: stellar BH/NS + disc

  • Active galactic nuclei:

  • 10% AGN (radio-loud AGN)

  • vjet = 0.995c

  • Ljet = 1043- 1048 erg/s

  • Size = 0.1-1 Mpc

  • Collimation: few degrees

  • Central engine: SMBH + disc

  • Gamma-ray bursts:

  • vjet/wind = 0.99995-0.999999995c

  • LGRB = 1052 erg/s (T = 1s)

  • Size = 1 pc (late afterglow evolution)

  • Collimation: few tens of degrees

  • Central engine: stellar BH + torus

  • Young stellar objects:

  • vjet = 10-3 c

  • Ljet = 1035 erg/s

  • Size = 10-3 – 10-2 pc

  • Collimation: few degrees

  • Central engine: YSO + inflow

And more… (pulsars, proto-planetary nebulae, cataclismic variables,…)

Extragalactic jets: open questions

Jet formation: hydromagnetic acceleration of disc wind ?

extraction of rotational energy from rotating BH

Influence of radiative acceleration

Influence of hydrodynamic acceleration

Origin of the poloidal magnetic field

Connection with jet composition: ep, e+e-

Acceleration: present numerical simulations fail to generate highly relativistic, steady jets

(several arguments point to Lorentz factor few-20; IDV: Lorentz factor 100)

Jet composition

Origin of the ultrarelativistic particle distribution

Nature of the radio components: relativistic shocks ?

instabilities ?

Kiloparsec scale parsec scale miliparsec scale

Structure and kinematics of jets; magnetic field topology; role of the magnetic fields in the jet dynamics and emission

Stability on large scales

FRI/FRII morphological dichotomy: environment? Jet power? Composition? Formation

mechanism? Accretion regime? Magnetic field?

KH instabilities?

Role in galaxy and cluster evolution: heating

RHD equations for a perfect fluid

The fluid is characterized by a four-velocityum and a stress-energy tensorTmn. The space-time in which the fluid evolves is characterized by a metric tensor whose components are gmn.

  • The equations of relativistic hydrodynamics are conservation laws:

    • Conservation of mass:

    • r: proper rest-mass density

    • Conservation of energy and momentum:

    • For a perfect fluid:

    • e: specific internal energy; p: pressure

    • Equation of state:

  • Approaches:

    • Full GRHD: Components of the metric as a function of coordinates from Einstein eqs.

    • ( ) (Numerical Relativity; consistent scenarios of jet/GRB production)

    • Test GRHD: Fluid evolving in a given space-time (simplified scenarios of jet formation)

    • SRHD: Flat space-time (propagation of jets at parsec and kiloparsec scales)

Relativistic hydrodynamics: SRHD equations

Relativistic hydrodynamics: hyperbolicity

RHD equations form a non-linear, hyperbolic system of conservation laws for causal EoS (e.g., Anile 1989)

  • For hyperbolic systems,

    • The Jacobians of the vectors of fluxes, , have real eigenvalues and a complete set of eigenvectors.

    • Information about the solution propagates at finite velocities (characteristic speeds) given by the eigenvalues of the Jacobians.

    • Hence, if the solution is known (in some spatial domain) at some given time, this fact can be used to advance the solution to some later time (initial value problem).

    • However, in general, it is not possible to derive the exact solution for this problem. Instead, one has to rely on numerical methods which provide an approximate to the solution.

    • Moreover, these numerical methods must be able to handle discontinuous solutions, which are inherent to non-linear hyperbolic systems.

Relativistic hydrodynamics: characteristic structure I

Donat et al. 1998; Martí et al. 1991, Eulderink 1993, Font et al. 1994

Relativistic hydrodynamics: characteristic structure II

Numerical relativistic hydrodynamics: AV methods I

May & White’s code (May & White 1966, 1967)

Lagrangian (1D) finite difference scheme + artificial viscosity (AV) for spherically symmetric, relativistic, stellar collapse

Space-time metric:

m: total (barionic) rest-mass up to radius R

Mass, energy and momentum conservation:

Einstein equations:


Richtmyer & von Neumann’s (1950) AV

AV produces dissipation at shocks that reduces post-shock oscillations

Numerical relativistic hydrodynamics: AV methods II

Wilson’s approach (Wilson 1972, 1979)

Eulerian 2D finite difference scheme + artificial viscosity (AV) for test GRHD (3+1formalism)

Mass conservation:

Momentum conservation:

Energy conservation:


  • Extension to Full GRHD: Smarr & Wilson code (Wilson 1979): Wilson’s formulation built on a vacuum numerical relativity code for the head-on collision of two black holes (Smarr 1975)

  • Smarr & Wilson’s approach allowed to simulate complex relativistic scenarios for the first time:

    • Axisymmetric stellar core collapse(Wilson 1979, Dykema 1980, Nakamura et al. 1980, Bardeen & Piran 1983, Evans 1984, …)

    • Issues on Numerical Cosmology: inflation, primordial nucleosynthesis, microwave anisotropy, evolution of primordial gravity waves,…(Centrella & Wilson 1983, 1984, …)

    • Accretion onto compact objects(Hawley et al. 1984a,b, …)

    • Heavy ion collisions(Wilson & Mathews 1989, …)

Numerical relativistic hydrodynamics: AV methods III

Wilson’s approach. Numerical method (Wilson 1972, 1979; see the recent book by Wilson & Mathews 2003)

Eulerian 2D finite difference scheme + artificial viscosity (AV) for test GRHD (3+1formalism)

Model equation in 1D: (transport equation with source terms)


(Donnor cell or Lelevier’s method; upwind, first order)

Newer developments: Consistent AV

artificial viscosity as a bulk scalar viscosity (Norman & Winkler 1986):

Non-consistent AV: artificial viscosity added to the pressure in some terms of the hydro eqs.

Large innacuracies in mildly relativistic flows

Very accurate; increase of coupling

see LeVeque’s 1992 book

High-Resolution Shock-Capturing methods

HRSC methods deal with hyperbolic systems of conservation laws. Model equation in 1D:

HRSC methods: finite difference (volume) schemes in conservation form. Integrating the PDE over a finite space-time domain :

Lax-Wendroff theorem (LW 1960): conservation form ensures the convergence of the solution under grid refinement to one of the weak solutions of the original system of equations

Numerical fluxes:

  • exact or approximate Riemann solvers (Godunov-type methods)

  • standard finite-difference methods + local conservative dissipation terms (e.g., symmetric schemes)

  • High-order of accuracy:

    • Total variation stable (TVD, TVB) algorithms

Stability (no spurious oscillations)

and convergence

  • Standard approach: Conservative monotonic polynomials as interpolant functions within zones (slope limiter methods; also flux limiter methods)

  • MINMOD (van Leer 1977; second order)

  • PPM (Colella & Woodward 1984; third order)

  • ENO (Harten et al. 1987)

See Martí & Müller, Numerical Hydrodynamics in Special Relativity,Living Reviews in Relativity,

HRSC methods in Relativistic Hydrodynamics

  • Based on Riemann solvers (upwind methods):

    • Linearized solvers: based on local linearizations of the Jacobian matrices of the vector of fluxes

      • Roe-type Riemann solvers(Roe-Eulderink: Eulderink 1993; LCA: Martí et al. 1991)

      • Falle-Komissarov(Falle & Komissarov 1996): based on a primitive-variable formulation of the eqs.

      • Marquina Flux Formula(Donat & Marquina 96)

    • Solvers relying on the exact solution of the Riemann problem (Martí & Müller 1994; Pons, Martí & Müller 2000)

      • rPPM(Martí & Müller 1996)

      • Random choice method(Wen et al. 1997)

      • Two-shock approximation(Balsara 1994; Dai & Woodward 1997)


  • Symmetric TVD, ENO schemes with nonlinear numerical dissipation:

    • LW scheme with conservative TVD dissipation terms (Koide at al. 1996)

    • Del Zanna & Bucciantini 2002: Third-order ENO reconstruction algorithm + spectral-decomposition-avoiding RS (LF, HLL)

    • NOCD (Anninos & Fragile 2002)

Exact Riemann solver in RHD I

(Martí & Müller 1994; Pons, Martí & Müller 2000)

Riemann Problem: IVP with initial discontinuous data L, R

W: shock / rarefaction (self-similar expansion); C: contact discontinuity

1. The compressive character of shock waves allows as to discriminate between shocks (S) and rarefaction waves (R):

pressure ahead and behind the wave

2. The functions Wg, Wfallow one to determine the functions

and , respectively.

RS(p) / SS(p): family of all states which can be connected through a rarefaction / shock with a given state S ahead the wave.

3. The pressure p* and the velocity vx* in the intermediate states are then given by the condition across the contact discontinuity

Intrinsic relativistic effects: Riemann solution depends on tangential velocities

solution for pure normal flow

Solution of the Riemann problem in the pressure – normal flow velocity diagram

Exact Riemann solver in RHD II

(Martí & Müller 1994,

Pons, Martí & Müller 2000)

The wave-pattern (RR, SR, SS) of the solution can be predicted in terms of the relativistic invariant relative velocity between the initial left and right states (Rezzolla & Zanotti 2001,Rezzolla, Zanotti & Pons 2002)

changing initial tangential speeds…


Relative velocity between L, R initial states




Algorithms for Numerical RHD I: upwind HRSC methods

Algorithms for Numerical RHD II: central HRSC methods

Harten-Lax-van Leer Flux

Algorithms for Numerical RHD III: other approaches

Pons et al. 1998

Special relativistic Riemann solvers in GR

According to the Equivalence Principle, physical laws in a local inertial frame* of a curved spacetime have the same form as in Special Relativity

* free falling frames

The solution of the Riemann problem in GR coincides (locally) with the one in SR if described in a LIF

  • 1. Perform at each numerical interface, , a coordinate transformation to locally Minkowskian

  • coordinates, , according to

  • where are the coordinates of the center of the interface and is given by

  • where is the orthonormal basis attached to , with orthogonal to .

  • 2. Set up the Riemann problem at the two sides of by transforming the velocity

  • components to the new basis, .

  • 3. Solve the Riemann problem as in special relativity and compute numerical fluxes

  • 4. Transform numerical fluxes to the coordinate basis, .

This procedure has been used with success in numerical GRMHD and force-free degenerate electrodynamics in the context of jet formation scenarios (Komissarov 2001, 2004, 2005)

Simulations of relativistic jets: Kiloparsec scale jets I

Hydrodynamical non-relativistic simulations(Rayburn 1977; Norman et al. 1982) verified the basic jet model for classical radio sources (Blandford & Rees 1974; Scheuer 1974).

Two parameters control the morphology and dynamics of jets: the beam to external density ratio and the internal beam Mach number

Morphology and dynamics governed by interaction with the external (intergalactic) medium. The simulations have allowed to identify the structural components of radio jets

First relativistic simulations: van Putten 1993, Martí et al. 1994, 1995, 1997; Duncan & Hughes 1994

Relativistic, hot jet models

Relativistic, cold jet models

Density + velocity field vectors

“featureless” jet + thin cocoons without backflow + stable terminal shock: naked quasar jets (e.g., 3C273)

“knotty” jet + extended cocoon + dynamical working surface: FRII radio galaxies and lobe dominated quasars (e.g., Cyg A)

Simulations of relativistic jets: Kiloparsec scale jets II

  • 3D simulations (Nishikawa et al. 1997, 1998; Aloy et al. 1999; Hughes et al. 2002, …):

    • Simulations too short (mean jet advance speed too high; poorly developed cocoons)

    • Two-component jet structure: fast (LoF 7) inner jet + slower (LoF 1.7) shear layer with high specific internal energy

8.6 Mcells

3.6 Mcells

Aloy et al. 1999

  • Long term evolution and jet composition (Scheck et al. 2002):

    • Evolution followed up to 6 106 y (10% of a realistic lifetime).

    • Realistic EoS (mixture of e-, e+, p)

    • Long term evolution consistent with that inferred for powerful radio sources

    • Relativistic speeds up to kpc scales

    • Neither important morphological nor evolutionary differences related with the plasma composition

(Nishikawa et al. 1997, 1998; Komissarov 1999; Leismann et al. 2005)

Simulations of relativistic jets: RMHD sims of kpc jets

Relativistic jet propagation along aligned and oblique magnetic fields (Nishikawa et al. 1997, 1998)

  • Relativistic jets carrying toroidal magnetic fields (Komissarov 1999):

    • Beams are pinched

    • Large nose cones (already discovered in classical MHD simulations) develop in the case of jets with Poynting flux

    • Low Poynting flux jets may develop magnetically confining cocoons (large scale jet confinement by dynamically important magnetic fields)

  • Models with poloidal magnetic fields (Leismann et al. 2005):

    • The magnetic tension along the jet affects the structure and dynamics of the flow.

    • Comparison with models with toroidal magnetic fields:

      • The magnetic field is almost evacuated form the cocoon. Cocoons are smoother.

      • - Oblique shocks in the beam are weaker.

Relativistic perturbation

Pressure-matched jet

steady jet

Overpressured jet

standing shocks

standing shocks

Shok-in-jet model: steady relativistic jet with finite opening angle + small perturbation (Gómez et al. 1996, 1997; Komissarov & Falle 1996, 1997)

Simulations of relativistic jets: pc-scale jets and superluminal radio sources

Radio emission (synchrotron; overpressured jet)

  • Convolved maps (typical VLBI resolution; contours): core-jet structure with superluminal (8.6c) component

  • Unconvolved maps (grey scale):

    • Steady components associated to recollimation shocks

    • dragging of components accompanied by an increase in flux

    • correct identification of components (left panel) based on the analysis of hydrodynamical quantities in the observer’s frame

3D hydro+emission sims of relativistic precessing jets (including light travel time delays): Aloy et al. 2003

In order to compare with observations, simulations of parsec scale jets must account for relativistic effects (light aberration, Doppler shift, light travel time delays) in the emission

Relativistic hydrodynamics and emission models

  • Basic hydro/emission coupling (only synchrotron emission considered so far! Gómez et al. 1995, 1997; Mioduszewski et al. 1997; Komissarov and Falle 1997):

    • Dynamics governed by the thermal (hydrodynamic) population

    • Particle and energy densities of the radiating (non-thermal) and hydrodynamic populations proportional (valid for adiabatic processes

    • (Dynamically negligible) ad-hoc magnetic field with the energy density proportional to fluid energy density

    • Integration of the radiative transfer equations in the observer’s frame for the Stokes parameters along the line of sight

      • Time delays: emission ( ) and absortion coefficients ( ) computed at retarded times

      • Doppler boosting (aberration + Doppler shift):

  • Further improvements:

    • Compute relativistic electron transport during the jet evolution to acount for adiabatic and radiative losses and particle accelerations of the non-thermal population (e.g., Jones et al. 1999, non-relativistic MHD sims.)

    • Include inverse Compton (scattering process!)

    • Include emission back reaction on the flow (important at high frequencies)

Simulations of superluminal sources: interpreting the observations with the hydrodynamical shock-in-jet model

Isolated(3C279, Wehrle et al. 2001) and regularly spaced stationary components(0836+710, Krichbaum et al. 1990; 0735+178, Gabuzda et al. 1994; M87, Junor & Biretta 1995; 3C371, Gómez & Marscher 2000)

Variations in the apparent motion and light curves of components(3C345, 0836+71, 3C454.3, 3C273, Zensus et al. 1995; 4C39.25, Alberdi et al. 1993; 3C263, Hough et al. 1996)

Coexistence of sub and superluminal components(4C39.25, Alberdi et al. 1993; 1606+106, Piner & Kingham 1998) and differences between pattern and bulk Lorentz factors(Mrk 421, Piner et al. 1999)

Dragging of components(0735+178, Gabuzda et al. 1994; 3C120, Gómez et al. 1998; 3C279, Wehrle et al. 1997)

Trailing components(3C120, Gómez et al. 1998, 2001; Cen A, Tingay et al. 2001)

Pop-up components (PKS0420-014, Zhou et al. 2000)


Gómez et al. 1998


Piner & Kingham 1998


Wehrle et al. 1997


Gómez & Marscher 2000

Gabuzda et al.1994



Hough et al. 1996

KH stability analysis is currently used to probe the physical conditions in extragalactic jets

Kelvin-Helmholtz instabilities and extragalactic jets

  • Linear KH stability theory:

    • Production of radio components

    • Interpretation of structures (bends, knots) as signatures of pinch/helical modes

  • Non-linear regime:

    • Overall stability and jet disruption

    • Shear layer formation and generation of transversal structure

    • FRI/FRII morphological jet dichotomy

Interpretation of parsec scale jets

Wavelike helical structures with differentially moving and stationary features can be produced by precession and wave-wave interactions (Hardee 2000, 2001)

[used to constrain the physical conditions in the inner jet of 3C120 (Hardee 2003, Hardee et al. 2005)]

The 3C273 case

1. Emission across the jet resolved double helix inside the jet

2. Five sinusoidal modes are required to fit the double helix

3. The sinusoidal modes are then identified with instability modes (elliptical/helical body/surface modes) at their respective resonant wavelengths from which physical jet conditions are derived:

Lorentz factor: 2.1 0.4; Mach number: 3.51.4

Density ratio: 0.023 0.012; Jet sound speed: 0.53 0.16

Perucho et al. 2004a,b

Perucho et al. 2005

  • Goals of the study:

    • Relativistic effects on the stability of jets

    • Transition from linear to nonlinear stability

Kelvin-Helmholtz instabilities: non-linear regime

  • Dynamics of jet disruption

  • Long term evolution (jet disruption, shear layer formation, …)

  • Numerical simulations:

    • Planar (2D) symmetric (pinch) / antysymmetric (helical) modes

    • Resolution: 400 zones/Rj (transversal) x 16 zones/Rj (longitudinal)

  • Initial conditions: steady jet + small amplitude perturbation (first body mode)

  • Temporal approach

Lorentz factor 5 model

Initial model

Linear phase

Non-linear evolution

Evolutionary phases:


Pressure maximum




Forces acting on

plasma particles

along B lines

particles tied

to B lines

(MHD approx.)

B line anchored

to the disk

Plasma acceleration in BP mechanism

Simulations of jet formation (accretion/outflow, acceleration, collimation)

Blandford-Payne mechanism: hydromagnetic acceleration of disk wind in a BH magnetosphere (barion loaded jet; GRMHD)

Accretion/ejection from Keplerian (co-rotating/counter-rotating) cold disks around Schwarzschild (Koide et al. 1997), rapidly rotating Kerr BH (Koide et al. 2000)

Kerr BH

Koide et al. 2000

Magnetic Penrose process

Schwarzschild BH

Koide et al. 1997

Magnetic line twist extract energy from BH ergosphere

  • Extraction of rotational energy form Kerr BH by magnetic processes:

    • Blandford-Znajek mechanism: BH as a magnetized rotating conductor whose rotational energy can be efficiently extracted by means of magnetic torque (Poynting flux jet; force-free electrodynamics). Confirmed recently by Komissarov (2001 -FFDE-, 2004 -GRMHD, rarefied plasma-): formation of a UR particle, Poynting dominated wind

    • magnetic Penrose process: magnetic field lines accross the ergosphere twisted by frame dragging line twist propagates outwards as torsional Alfven wave train carrying e.m. energy total energy of the plasma near the hole decrease to negative values swallowing of this plasma by the hole reduces the BH rotational energy (Poynting flux jet; GRMHD). Koide et al. 2002, Koide 2003; controversial: Komissarov 2005 (magnetic Penrose process does not operate…)

Koide et al. 2002

Relativistic Magnetohydrodynamics

RMHD as hyperbolic system of conservation laws

In one spatial dimension, the system of RMHD can be written as a hyperbolic system of conservation laws for the unknowns (conserved variables) subject to the

constraint along the evolution.

Anile & Pennisi 1987, Anile 1989 (see also van Putten 1991) have studied the characteristic structure of the equations (eigenvalues, right/left eigenvectors) in the space of covariant variables

Wavefront diagrams in the fluid rest frame (Jeffrey & Taniuti 1964)

  • There are seven physical waves:

    • Two Alfven waves,

    • Two fast magnetosonic waves,

    • Two slow magnetosonic waves,

    • One entropy wave,

(Deg II)

The wave propagation velocity depend on the relative orientation of the magnetic field, q

(Deg I)

orientation of the magnetic field

  • Physically: two or more wavespeeds become equal (compound waves)

  • Numerically: the spectral decomposition (needed in upwind HRSC methods) blows up

  • As in classical MHD there are two kinds of degeneracies:

    • Degeneracy I:

    • Degeneracy II:

Algorithms for Numerical RMHD


Compound wave





Deg I case
































Lorentz factor

Lorentz factor






Numerical RMHD: 1D Tests

Fast, slow shocks & rarefactions; Alfvén waves; shock tubes*; magnetic field divergence-free condition satisfied by construction

*still lacking an analytical solution!!

Antón et al. 2005 (also van Putten 1993, Balsara 2001, Del

Zanna et al. 2002, De Villiers & Hawley 2003, …)

Antón et al. 2005 (also Komissarov 1999, Del Zanna et al.

2002, De Villiers & Hawley 2003, …)




log r + B lines

log p

Lorentz factor




Numerical RMHD: 2D Tests

Cylindrical explosions:

Ambient (r >1.0): p = 3.e-5, r = 1.e-4 Homogeneous magnetic field, B = (Bx, 0, 0)

(Smooth) transition layer (0.8 < r < 1.0) Bx = 0.01, 0.1, 1.0 (b = 1.6, 1.6e2, 1.6e4)

Cylinder (r < 0.8): p = 1.0, r = 1.e-2

Antón et al. 2005 (proposed by Komissarov 1999; see also Del Zanna et al. 2002, b = 8.e2)

Rotors:Static ambient (r > 0.1): p = 1.0, r = 1.0, B = (Bx, 0, 0), Bx = 1.0 (b = 0.5)

Rotating disk (r < 0.1): p = 1.0, r = 10.0, w = 9.95 (Lorentz factor approx. 10)

Del Zanna et al. 2002

t = 0.4

0.35 (w) – 8.19 (b)

5.3e-3 (w) – 3.9 (b)

3.8e-4 (w) – 2.4 (b)

1.0 (w) – 1.79 (b)

Summary, conclusions,…

  • Extraordinary advance, in the last decade, in numerical methods for (ultra)relativistic hydrodynamics, specially with RS/Sym HRSC methods

  • Easy extension to (test) GRHD via local linearizations of the geometrical terms or local coordinate transformations (for RS HRSC methods; Pons et al. 1998)

  • Important advances in numericalRMHD(RS/Sym HRSC methods, also AV methods), however present numerical codes are less robust than in the purely hydro case

  • Big impact in extragalactic jet research, specially parsec scale jets, superluminal sources and jet

  • formation mechanisms

  • Important advances in the understanding of the morphology and dynamics of large scale relativistic jets

  • First simulations of superluminal sources (success of the relativistic shock-in-jet model). First steps in the combination of hydro + relativistic electron transport + radiation transfer codes

  • First simulations of relativistic jet formation

  • Numerical study of the non-linear regime of KH instabilities

  • Still lacking…

  • RMHD sims of pc and kpc jets to elucidate the configuration and role of magnetic fields at these scales

  • Jet formation mechanisms need further numerical study (no steady relativistic outflow yet found)

  • Consistent simulation of all the jet components (thermal matter, high energy particles, radiation and magnetic fields) and their mutual interaction.

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