Masanori Nakamura and David L. Meier Astro-ph/0406405

Poynting Flux Dominated Jets in Decreasing Density Atmospheres.

I. The Non-relativistic Current-driven Kink Instability and the Formation of “Wiggled”

Structures

Masanori Nakamura and David L. Meier

Astro-ph/0406405

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Introduction (1)MHD acceleration mechanism: a model for astrophysical jetsThere has been a growing recognition in recent years that the influence of strong magnetic fields within the jet may extend beyond the central engine into the region where the jet freely propagates.

This is particularly evident in observations of jets in AGNs, QSOs, winds from pulsars, and γ-ray burst sources (e.g., Perley, Bridle, & Willis 1984; Conway & Murphy 1993; Hester et al. 2002; Coburn & Boggs 2003).

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Introduction (2)Jets with a strong toroidal field encircling the collimated flow: “current-carrying” or “Poynting flux dominated” (PFD) jets.

Rotation of the helical field drives a torsional Alfven wave (TAW) forward in the direction of the jet flow.

TAW carries electromagnetic energy and accelerate the plasma.A cylindrical plasma column with helical magnetic configuration is subject to MHD instabilities.

Classification of instabilities: pressure-driven(PD), Kelvin-Helmholtz(K-H), current-driven(CD) instabilities (Kadomtsev 1966; Bateman 1980; Freidberg 1982).

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Introduction (3)PD instabilities: considered not to be very important for supersonic jet.

K-H instabilities: important at the shearing boundary between the jet and the external medium, particularly in kinetic flux dominated (KFD) jets.

On the other hand, PFD jets should be especially susceptible to CD instabilities because of the presence of the strong axial electric current.→ The study of CD instabilities on jet flow is important.

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Introduction (4)Purpose of this paper: numerical investigation of the nonlinear development of CD instabilities in PFD jets, especially the CD kink (m=1) mode.

Previous study (Nakamura, Uchida, & Hirose 2001) : simplified atmospheric conditions.

This study : more realistic atmospheric situations, including density, pressure, magnetic field, and temperature gradients in the ambient medium.

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Basic Astrophysical Model (1)

Nakamura et al. 2001 A collimated outflow

(V>Vk) has been established, but that flow still is dominated by magnetic forces and is not yet a super-(fast magneto) sonic velocity (EEM >

Ek).Cs<<VA

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Basic Astrophysical Model (2)

Near the central engine: a rotating, magnetic structure can be created (Blandford & Znajek 1977; Blandford & Payne 1982; Koide et al. 2002).

How about the physical connection between this central region and the sub-parsec region? ← not yet well understood.

However, it is reasonable to suppose some connection.

Assume a rotational structure.

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Basic Astrophysical Model (3)

Assumption of a large scale poloidal magnetic field in the ambient medium. The origin is not yet fully understood.

← But, there are observational suggestions of both synchrotron emission and Faraday rotation.

The magnetic field assumed here might be

the primordial inter-stellar field (Kulsrud & Andersen 1992; the galactic field must have a primordial origin.) or the central part of the amplified field by a galactic turbulent dynamo process (Kronberg 1994; Han & Wielebinski 2002, and reference therein) or a field carried out from the central engin by a low-velocity magnetized disk wind.

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MHD equations and Code

Two-step Lax-Wendroff Scheme

+artificial viscosityThe numbers of the grid points:

261x261x729Add the term, -∇(p0) for initial hydrostatic equilibrium.

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Initial ConditionsA current- (and therefore force-) free magnetic field.

Ring current

Simulation domain

Time is normalized by τA0 (Alfven crossing time)

ρ∝|B|α

α=2: Alfven speed is constantΑ<2: Alfven speed is decreasing with with distance.

p∝ ρΓ (Γ: polytropic index; =5/3)

Plasma-beta is 0.01 at the origin.

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Boundary Conditions

Z=0.0

Z=Zmi

n

Free boundary

Free

bou

nd

ary

Free

bou

nd

ary

Physical variables except the magnetic field are damped.

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The Four Models (1)

Two models for ambient medium

A: shallow-atmosphere model (α=1)

B: steep-atmosphere model (α=2)

1: Highly PFD jets (FExB/Ftot ～ 0.9)

2. Mildly PFD jets (FExB/Ftot ～ 0.6)

Four models of A-1, A-2, B-1, and B-2

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The Four Models (2)Alfven velocity is constant, but sound speed is decreasing.

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The Four Models (3)

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Early jet evolution (1)Model A-1

The F-F compressive wave decelerates and steepens into a fast-mode MHD bow shock due to a gradual decreasing ambient VA (check the jump of Vz, becoming super-fast magneto sonic).

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Early jet evolution (2)Model A-2

There is the contact discontinuity (CD) between R-S and F-S.

CD ～ R-S: decreasing and heating magnetized jet itself.

CD ～ F-S: compressing and heating the ambient medium.

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Early jet evolution (3)Model B-1

Only a very low amplitude F-F compressive wave front can be seen due to a constant ambient VA (if Cs<<VA, VF ～ V

A).

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Early jet evolution (4)Model B-2

The authors identify CD as defining the rest frame of the jet flow.

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Intermediate jet evolution

R-S F-S F-F

More twisted

Less twisted

More twisted

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Nature of PFD jets as current-carrying jets (1)

The “forward jet current density” Jjc

The “return current density” Jrc

The force-free parameter λ ｆｆ (=J･ B/|J||B|) λ ｆｆ =1 or –1: force-free

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Nature of PFD jets as current-carrying jets (2)

The distribution of the force-free parameter and the current density. Jet is force-free in almost all region.

Large difference between highly and mildly PFD jet (related to accumulation of Bφ)

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Final evolutionary stages (1)

Vz is still sub-Alfvenic, that is the highly PFD jet remains Poynting flux dominated.

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Final evolutionary stages (2)

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Final evolutionary stages (3)

The mildly PFD jet → (mildly) kinetic energy flux dominated (KFD) jet.

The highly PFD jet with the steep atmosphere → an equipartition state between the kinetic and Poynting fluxes.

Jets propagating in the trans-Alfvenic region before they become kinetic energy dominated, can be deformed into wiggled structures.

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Nonlinear growth of CD instabilities (1)

Calculate the power spectrum.

Define the jet current.

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Nonlinear growth of CD instabilities (2)

Calculate power spectrum of k→0.

Pure CD modes can develop typical on the Alfven crossing time scale (Begelman 1998, etc.)This is consistent with the result in this paper.

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Nonlinear growth of CD instabilities (3)

Classical Kruskal-Shafranov (K-S) criterion.

Originally Φcrit is set equal to 2π.

Φ> Φcrit and Alfven mach number for azimuthal velocity is not so much large (nearly equal to 0).

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Nonlinear growth of CD instabilities (4)

Despite Φ> Φcrit, the jet is stable because of relatively large MAΦ.

This is consistent with the linear theory (Appl 1996, Appl, Lery, & Baty 2000)

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Nonlinear growth of CD instabilities (5)

Thinking the balance of VΦ2/r - BΦ

2/r/ρ - ∇pm

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Nonlinear growth of CD instabilities (6)

Models, B-1 and B-2: the sum of the first and second term is nearly equal to 0. ∇pm is also nearly equal to 0.Model A-1: The magnetic pinch term is a bit grater than the centrifugal term (azimuthal velocity is sub-Alfvenic), but the difference is vanished by the gradient of the magnetic pressure.

Model A-2: The centrifugal term is nearly equal to zero. 　→ The promotion of the pinch. → The magnetic pinch term and the magnetic pressure are asymmetric with respect to the jet axis due to the kink instabilities.

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Nonlinear growth of CD instabilities (7)

How and where the classical K-S criterion is violated?

Concentration of magnetic flux to near the central axis due to the pinch

Decrease of Bz in the region 0.3<r<1.0 makes the situation of Φ> Φcrit.

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Nonlinear growth of CD instabilities (8)

How and where the classical K-S criterion is violated?

F-F

F-SR-S

Behind the R-S shock wave, the azimuthal velocity becomes nearly equal to zero.→ The pinch effect becomes strong.→ The classical K-S criterion is violated.

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Suppression of MHD KH Instabilities due to the

External Magnetized Winds (1)

The definition of winds in this paper is current-free flow (the flow between the jet current and the return current).The condition for the stability is

; surface Alfven speed

Hardee & Rosen (2002)

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Suppression of MHD KH Instabilities due to the

External Magnetized Winds (2)

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Summary (1)

Due to the centrifugal effect, rotating jets can be stabilized against the CD kink mode beyond the point predicted by the classical K-S criterion.

Non-rotating jets will be subject to the CD kink mode when the classical K-S criterion is violated.

The driving force of the CD kink mode is an asymmetrically distribution of hoop-stress (the magnetic tension force). This is caused by a sudden decrease of jet rotation and a concentration of the poloidal magnetic flux toward the central axis (which is related to a reverse slow-mode MHD shock wave).

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Summary (2)

The CD kink mode grows on time scales of order of the local Alfven crossing time.

The MHD KH instability is suppressed even when flows become super-Alfvenic.

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