Download The centrifugal term is nearly equal to zero.

(1/36)
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
(2/36)
Introduction (1)
MHD acceleration mechanism: a model for astrophysical jets
There 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).
(3/36)
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), KelvinHelmholtz(K-H), current-driven(CD) instabilities (Kadomtsev
1966; Bateman 1980; Freidberg 1982).
(4/36)
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.
(5/36)
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.
(6/36)
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
(7/36)
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.
(8/36)
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.
(9/36)
MHD equations and Code
Two-step LaxWendroff Scheme
+artificial viscosity
The numbers of the
grid points:
Add the term, -∇(p0) for initial
hydrostatic equilibrium.
261x261x729
(10/36)
Initial Conditions
A current- (and therefore force-) free magnetic field.
ρ∝|B|α
α=2: Alfven speed is constant
Α<2: Alfven speed is decreasing with
with distance.
p∝ρΓ (Γ: polytropic index; =5/3)
Plasma-beta is 0.01 at the origin.
Simulation domain Ring current
Time is normalized by τA0
(Alfven crossing time)
(11/36)
Boundary Conditions
Free boundary
Free boundary
Free boundary
Z=0.0
Z=Zmin
Physical variables except the
magnetic field are damped.
(12/36)
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
(13/36)
The Four Models (2)
Alfven velocity is constant, but
sound speed is decreasing.
(14/36)
The Four Models (3)
(15/36)
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).
(16/36)
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.
(17/36)
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～ VA).
(18/36)
Early jet evolution (4)
Model B-2
The authors identify CD as
defining the rest frame of the jet
flow.
(19/36)
Intermediate jet evolution
R-S
More
twisted
F-S
Less
twisted
F-F
More
twisted
(20/36)
Nature of PFD jets as currentcarrying jets (1)
The “return current density” Jrc
The “forward jet current density” Jjc
The force-free parameter λｆｆ (=J･B/|J||B|) λ
ｆｆ=1 or –1: force-free
(21/36)
Nature of PFD jets as currentcarrying 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
(22/36)
Final evolutionary stages (1)
Vz is still sub-Alfvenic, that is the highly PFD jet remains
Poynting flux dominated.
(23/36)
Final evolutionary stages (2)
(24/36)
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.
Nonlinear growth of CD
instabilities (1)
Calculate the power spectrum.
Define the jet current.
(25/36)
Nonlinear growth of CD
instabilities (2)
(26/36)
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.
Nonlinear growth of CD
instabilities (3)
(27/36)
Classical KruskalShafranov (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).
Nonlinear growth of CD
instabilities (4)
(28/36)
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)
Nonlinear growth of CD
instabilities (5)
Thinking the balance of VΦ2/r - BΦ2/r/ρ - ∇pm
(29/36)
Nonlinear growth of CD
instabilities (6)
(30/36)
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.
Nonlinear growth of CD
instabilities (7)
(31/36)
How and where the classical K-S criterion is violated?
Decrease of Bz in the
region 0.3<r<1.0 makes
the situation of Φ> Φcrit.
Concentration of magnetic flux to near
the central axis due to the pinch
Nonlinear growth of CD
instabilities (8)
(32/36)
How and where the classical K-S criterion is violated?
Behind the R-S shock
wave, the azimuthal
F-F velocity becomes nearly
equal to zero.
→The pinch effect
becomes strong.
→The classical K-S
criterion is violated.
R-S
F-S
Suppression of MHD KH (33/36)
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)
Suppression of MHD KH (34/36)
Instabilities due to the External
Magnetized Winds (2)
(35/36)
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).
(36/36)
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.