# MHD jet collimation - Torino 2007 JETSET School

Urban and Civil

Nov 15, 2013 (4 years and 5 months ago)

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MHD jet collimation  the role of
magnetic diffusivity &
disk magnetic flux profile
Christian Fendt
Max Planck Institute f.
Astronomy, Heidelberg

Magnetohydrodynamic jets

Numerical simulations:

- disk magnetic field profile (Fendt 2006)
- magn.diffusive jets (Fendt & Cemeljic 2002)
JETSET School and Workshop
Sauze d'Oulx, Torino, Italy
January 8-13, 2007

NsSSS

collimation
and
acceleration
of a
disk wind into a jet ?

ejection
of disk
material
into wind?

accretion
of matter?
generation of
magnetic field
?

jet
propagation
/
interaction
with
ambient medium
-->
Jets
are collimated
disk winds,
launched / accelerated / collimated
by
magnetic forces
Standard model
of jet formation:
--> 4 basic questions
of jet theory:
Astrophysical jets:
Standard model
Astrophysical jets:
Magnetohydrodynamics (MHD)

MHD concept:
ionized, neutral
fluid:

average quantities:

Ideal MHD:

frozen-in field lines:

MHD

Lorentz force:

MHD equations
(to be solved numerically):
Axisymmetric jets -->
magnetic flux surfaces:
Lorentz force components:

--> projected on Ψ:
--> (de/)
accelerating:
--> (de-)
collimating:
Self-collimation
by tension of toroidal field
MHD jet formation:
Acceleration/collimation
Magneto-centrifugal acceleration:

(Blandford & Payne 1982)

-> magnetic
field
lines
corotate
with
disk

( beads on a wire )
-> if field line
inclination

< 60 deg
:
->
unstable
equilibrium,
initial
acceleration
outwards
Self-collimation of MHD jets:

kinetic > magnetic energy:
-> poloidal field wound-up by
inertial
forces
-> collimation
by toroidal field tension
->
MHD
acceleration (Lorentz forces)
MHD jet self-collimation
t=0
t=400
t=200

ρ (r,z) Bp (r,z)
Numerical proof of MHD self-collimation:
Ustyugova etal. 1996; Ouyed & Pudritz 1997

Model assumptions:
--> ideal MHD
--> Keplerian disk as
boundary condition
-->
disk magnetosphere,
Keplerian footpoints
-->
mass injection
polytropic gas + turb. Alfvenic pressure
-->
numerical stability ->

follow
evolution over 1000s of rotation preriods
Motivation for further modifications (Fendt et al ...):
--> investigate
magnetically diffusive jets:
jets are launched in turbulent accretion disk
--> turbulence pattern may be launched into the jet as well
--> investigate different disk magnetic field / mass
flux profiles:
--> nothing is known so far about the disk magnetic field / origin of jet magnetic field
--> different disks produce different jets (?)
Simulations of jet
acceleration
and
collimation:
--> solving time-dependent
MHD equations
-->
Zeus-3D
MHD code:
- magnetic flux conservative
- suited for supermagnetosonic velocities
- extended for
magnetic diffusivity
(Fendt & Cemeljic '02)
Model
assumptions:
--> disk as
boundary condition
(Ouyed&Pudritz '97):
-->
disk magnetosphere,
Keplerian rotation
-->
mass injection
--> gas + turb. Alfvenic pressure:
-->
turbulent magnetic diffusivity
--> self-consistent toy model:
MHD jet collimation:
Jets from accretion disks
P
=
P
A

P
G
P
A

T
initial condition:

-->

hydrostatic

pressure/density distribution
-->

force-free

magnetic field
==>
long-term simulation
> 1000 periods
Model
parameter:
--> mass flux, magnetic flux, gas pressure (polytropic)
--> grid size, axisymmetry
ρ (r,z; t=0)
Bp (r,z; t=0)

z
r
Disk jet formation
(Fendt & Cemeljic 2002, Fendt 2005):

--> magnetohydrodynamic

jet
self-collimation

--> , super-magnetosonic
--> (quasi-)

state reached (inner jet)
--> interrelation
turbulent diffusivity
<-->
collimation
degree (critical value η ∼ 0.3)

i
=
100,

p
=

=
1,

T
=
0.03,v
inj
r
=
10
3
v
K
r
,

inj
=
100

cor
,r
max
=
40,z
max
=
160
v
jet
~1.5v
Kep
MHD jet collimation:
Jets from accretion disks

Toy model for jet
magnetic diffusivity:
Idea:
disk turbulence pattern propagate into outflow
--> jet is turbulent ( turbulent diffusitivity )
--> jet turbulent magnetically diffusive

Toy model approach:

jets
are
''cold''
(low gas pressure) --> support by
turbulent Alfvenic pressure
(O & P 1997):

--> turbulent magnetic diffusivity:

--> polytropic
gas law:
-->

--> weak dependence : ( here for simplicity L ~ const. ! )

--> normalized mean diffusivity
(parameters: ):
MHD jet collimation:
Turbulent magn. diffusivity
P=P
A
P
G
,v
T
v
A
,
T
c
s
2
/v
T
2
=0.03

T

m
vL
=
m
v
T
L,

1
c
s
2
=P/
P
A
=
P
G
/
T
v
T
2
=

T
P

T
~
1/3
for=5/3

T
=
0.015

m
0.1
L
1.0
,
average

0.01

i
=100,
T
=0.03
Propagation of bow shock

(Fendt & Cemeljic 2002):
Simulations with
different η ,
same set-up otherwise:
-> same mass flux injected
-> same magnetic flux

-> bow shock
of diffusive

jets
propagates slower

( η = 0, 0.01, 0.1 )
( t = 400 rotations )

-> interpretation:

de-collimation of
mass flow
->
substructure
smoothens
MHD jet collimation:
Magn. diffusive jets


MHD jet collimation:
Magn. diffusive jets
Magnetic field structure

(Fendt & Cemeljic 2002):
Poloidal field lines
(different η , same setup otherwise)

t = 250, 300, 350, 400, η = 0.1

--> (quasi-) stationary state
of
poloidal field, in spite of diffusion

-
-> interpretation:
magnetic field ''replenished'' by disk
Poynting flux

--> small η < 0.1 : slight
field de-collimation,
not visible in velocity (!)

Poloidal fileld lines
η = 0, 0.1, t = 400
η = 0, 0.1, t = 400
Velocity & field strength
(Fendt & Cemeljic 2002):
--> (simulations with different η , same mass flux & magnetic flux)

--> velocity increase
with increasing η (slice along z at r=15, t= 400),
field strength decrease

ad hoc interpretation: two competing mechanisms
(1)
diffusivity weakens magnetic field -
-> weakens Lorentz force ??
(2) less collimation / inclination --> more efficient
magneto-centrifugal acceleration
MHD jet collimation:
Magn. diffusive jets
Diffusivity & Lorentz force

(Fendt & Cemeljic 2002):

--> Lorentz force increases
with
increasing η
( slice along z at r=15, t= 400 )

--> note: F ~ j x B ~ (V x B) x B

--> F_||
increases
--> v_|| increases

--> F_φ
increases
--> v_φ increases

--> perpendicular F
is directed
inwards (collimating) :
--> balances increased
centrifugal force
MHD jet collimation:
Magn. diffusive jets

Collimation & mass flux
(Fendt & Cemeljic 2002):
( different η , same mass flow injected, magnetic flux )
--
mass flux
in r,z-direction
( normalized mass flow rate injected ~ 1.5 )
-->
mass flux ratio
in z/r-direction defines
degree of collimation

--> examples:
mass flow rate evolution
for

η = 0, 0.1, 0.5

--> at quasi-stationary state:
-> critical magn. diffusivity
concerning jet

self-collimation:

( for chosen numerical setup )

--> turbulent outflows less collimated ! (?)

crit
0.3
d
dt
M
r,z
=

0
out
2v
r,z

T
d
dt
M
r
d
dt
M
z
0.00.41.4
0.51.050.45
MHD jet collimation:
Magn. diffusive jets
solid: inflow; dashed: z_max=60; dotted: r_max=20
Collimation & disk magnetic flux:

Disk magnetic
field profile:

-> construct initial condition !!

Disk wind
magnetization:

Degree of collimation:

(mass flux in axial and lateral direction)

Box size:
(150x300) R_i <--> (7x14) AU

Parameter runs:
µ, | B|, dM(r)/dt

--> collimation, velocity profiles,
asymptotic speed, stability
(in stationary regime)
B
p
~r

µ = 1.5

µ = 1.25

µ = 1.25

µ = 1.25

µ = 1.0
MHD jet collimation:
Disk magnetic flux profile
Collimation & disk
magnetic flux:
(Fendt 2005, astro-ph)
- systematic parameter
study:

µ = 0.2 ... 2.0
- one year of CPU time
(workstation)

B
p
~r

MHD jet collimation:
Disk magnetic
flux profile

Collimation & disk magnetic flux
(Fendt 2006):

--> flat profile (B, σ) --> good
collimation
--> axial
instabilities
for too flat profile (not stationary state)

B
p
~r

,
0
~r

,
0
~r


MHD jet collimation:
Disk magnetic flux profile

Observations

(DG Tau):
- 0.''05 ~ 7 AU
-
jet rotation
indicated ~ 12 km/s max
- fast (400km/s) & slow (100 km/s) component

Simulations
(7x14 AU grid):

-
flat profile µ
~ 0.5 for DG Tau indicated
(fits both poloidal and toroidal jet velocity)
- rotational velocity

- only
slow component
could be reproduced

MHD jet collimation:
Rotation & disk magn. field profile
µ = 1.0
µ = 0.5
(Bacciotti et al. 2002)
.v3(r)
.v3(r)
Two (new) aspects of MHD jet formation:
(0) Procedure:
--> axisymmetric
MHD simulations: jet formation & acceleration from
Keplerian disks,
disk-to-jet mass-flux
prescribed (see OP 1997)

--> collimation degree ζ
defined by mass flux in axial / lateral direction

--> ideal MHD ZEUS code extended and tested for
magn. diffusivity
(1)

Magn. diffusive jets
and collimation

(A&A 395, 2002)
-->
jet turbulence
launched by turbulent disks;
--> toy model for magnetic diffusivity:
--> magn. diffusive jets
-
propagate slower
(bow shock), move faster (matter)
- may reach
stationary state
(energetically supported by disk Poynting flux)
- are
less collimated
(as measured by mass flux),
critical diffusivity
for collimation

--> hypothesis:
highly turbulent disks cannot launch collimated jets
(2) Jets with different
magnetic field & mass flux profile
(ApJ 651, 2006)

--> parameter study for wide range of power law profiles
--> unique relation between disk wind
magnetisation σ
and degree of
collimation ζ
--> better collimation for
flat magn. field profile
/ disk wind magnetization profile
--> implications origin of jet magnetic field ?? disk field ?, disk dynamo, advection ??
--> application:
jet rotation
vφ decreases with radius --> flat disk flux profile for DG Tau jet(?)
MHD jet collimation
Summary
B
p
~r

P
A

T