alfio bonanno

MHD instabili,es of current-­‐carrying jets Alfio Bonanno – INAF Catania – Vadim Urpin – Ioffe Ins. St. Peterbsurg Based on A&A, 525, 100, 2011 Outline of the talk • 
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Mo,va,on Review of main results Linear stability analysis results -­‐ Numerical simula,ons Conclusions Mo,va,ons —  The magnetic field likely plays a key role in formation and propagation of astrophysical jet —  Polarization observations (see Cawthorne et.al 1993, Leppanen, Zenus&Diamond 1995,Gabuzda et al,2004,Kharb et al.2008) imply higly organized magnetic field in jets —  Recent observation of a radio jet in the galaxy CGCG 049-­‐033 reveals the presence of a predominantly toroidal field (Bagchi, 08) —  it could be generated by the dynamo action or generated by hydrodynamic motions. (see, e.g., Eichler 1993,Appl 1996,Appl et.al 2000,Lety et al.2000,Baty&Keppens 2002) Current driven instability of superthermal fields: axial field+ diff. profiles of azimuthal(comp.) Appl et al. 2000 " 
Non-­‐linear develop. of instability modify the magnetic structure with current density in the inner part Lery at el.2000 " 
The dominant instabilities with toroidal field are kink(m=1) and pinch(m=0) Begelman 1998 " 
In general we would also like to understand the stably stra,fied stellar interiors ! (see V.Duez ) Proto-neutron star(PNS) –The features of PNS -
・ strong instabilities driven by lepton number gradients and T gradients
t ∼ 40 s
" ! 1000 rad/sec
(This is ten times as fast as young pulsar with radio SNR（spin period ∼10msec）)
・The features of pulsar suggested by observation
Magnetic field of young pulsar
12!13
B " 10
G
It is suggested that PNS is accompanied with magnetic fields
n  Possibility
of dynamo action (AB, L.Rezzolla, V.Urpin,
2003)
n  Strong
n  What
fields are generated with Bp/Bphi = O(1)
happens after diff rotation has disappeared?
Crab nebula
（Typical pulsar）
Consider the comoving frame of the jet. Ideal MHD, incompressible There are essentially two types of instability:
・ axisymmetric m=0, non-axisymmetric m=1,2,3,…
n  Sufficient
condition for stability for a non
stratified medium for m=0:
Pinch-type instability for m=0
dln B#
"=
<1
dln s
Sufficient condition for stability for a non stratified
medium m=1 (also necessary, Tayler 1957, 1973)
n 
!
dln B#
"=
< $1/2
dln s
Kink-type instability for m=1
Instabilities of combined toroidal and poloidal fields:
n Necessary
condition for INSTABILITY
for a non stratified medium for m=0
(Howard and Gupta, 1962):
dln B#
"=
> $1
dln s
Relevant questions: what are the SUFFICIENT
conditions for INSTABILITY / STABILITY in general?
n 
!
Knobloch, 1992; Knoboloch & Dubrulle 1993 Kink-type instability for m=1
Braithwaithe & Nordlund 1996, Kitchatinov Ruediger 2007,
2008, Rudiger, Kitchatinov, Gellert 2009
Axially symmetric basic state and axially-­‐symmetric perturba4ons Cylindrical Geometry, use the spectral method. Basic state characterized by a toroidal field depending on the cylindrical radius in the incompressible limit. We define: α=
∂ ln Bϕ
∂ ln s
Bz �= 0
q = kz s 1
m=0
Ideal incompressible MHD equilibrium ∂�v
∇P
1
� ×B
�
+ (�v · ∇)�v = −
+
(∇ × B)
∂t
ρ
4πρ
∇ · �v = 0
�
∂B
� =0
− ∇ × (�v × B)
∂t
� =0
∇·B
BASIC STATE: 1
� ×B
�
∇P =
(∇ × B)
4π
Axisymmetric perturba,ons !
"
#
v1s
$
2ω2 ω2
4k
−k2v1s + 2 A 2B v1s
(σ + ωA)
"
#
v1s
2 2 ∂v1s
2 2
+
−2k ωB (1 − α)v1s = − δωA
s
∂s
s
d dv1s
2
2
+
(σ + ωA)
ds ds
s
where
∂ ln Bϕ
∂ ln Bz
kBz
Bϕ
, ωB = √
, α=
, δ=
ωA = √
4πρ
s 4πρ
∂ ln s
∂ ln s
In the case δ = 0 (Bz =const), one recovers the
equation derived by Acheson (1973) and Knobloch
(1992).
Analy,c solu,on: d2 v
1 dv1s
1s
+
+
2
dξ
ξ dξ
!
Bϕ ∝ s, Bz = const
1
1− 2
ξ
"
v1s = 0
where
√
ξ = ks F ,
2
4ωA ωB
F =
−1
2
2
2
(σ + ωA )
Then,
2 ± 2√µω ω
σ 2 = −ωA
A B
One of the roots is positive if the toroidal field satisfies
the inequality
Bϕ (s1 ) > Bz
!
ks1
√
2 µ
"
where Bϕ (s1 ) is the strength of the toroidal field at
the inner boundary and µ is a number of the order
unity ∝ k 2 . This is a new type of MHD wave.
Combined poloidal and toloroidal field in the general case We find numerical evidence that the critical value of the exponent is less than 1, close to -­‐0.1 αc � 0.1
Bonanno Urpin, 2008
q = kz s 1
Increasing the poloidal field strength Axially symmetric basic state and non-­‐axially symmetric perturba4ons Bz = const Bϕ = Bϕ0 ψ(s)
Bz
�=
Bϕ0
ψ(s) = xn e−p(x−1)
Current-­‐driven vs pressure driven ωB0
Bϕ0
= √
s1 4πρ
� = 0.1
σ
Γ=
ωB0
�=2
x = s/s1
Explicit symmetry breaking term: 1
∇p1
" +(∇× B)×
" B
" 1],
+
[(∇× B" 1)× B
σ"v1 = −
ρ
4πρ
(1)
∇ · "v1 = 0,
(2)
" 1 − (B" · ∇)"v1 + ("v1 · ∇)B" = 0,
σB
(3)
" 1 = 0.
∇·B
(4)
non-­‐linear eigenvalue problem !
"
#$
d 1 2
dv1s
v1s
2
2
(σ + ωA)
+
− kz2(σ 2 + ωA
)v1s
ds λ
ds
s
%
"
#
m(1
+
λ)
αλ
−2ωB kz2 ωB (1 − α) −
1−
(ωAz + 2mωB )
2
2
s λ
1+λ
$
2 ω2
4kz2ωA
mωAz
B v = 0,
− 2 2 v1s +
1s
2
2
s λ
λ(σ + ωA)
where
"
#
1
m
kz Bz
ωA = √
,
kz Bz + Bϕ , ωAz = √
4πρ
s
4πρ
∂ ln Bϕ
m2
Bϕ
ωB = √
, α=
, λ = 1 + 2 2.
s 4πρ
∂ ln s
s kz
non-­‐axisymmetric perturba,ons (p=0) Bz
�=
= 0.1
Bϕ
•  m=0,1,2,3…can have greater growth rate! •  They form a discrete spectrum, with the BC so that the perturba,on vanishes at the inner and outer boundary •  Does this trend saturate? Note the resonant character. Magne,c resonance condi,on 1 �
m �
ωA = √
kz Bz + Bϕ ≈ 0
s
4πρ
q ∼ −m/ε
Most dangerous instabilites are decided by the resonance q ∼ −m/ε
Eigenfunc,ons for various values of m Does a strong poloidal field stabilize the system? Bonanno and Urpin, 2011
Suydam’s criterion: necessary condi,on for stability sBz2
4π
�
1 dh
h ds
�2
dP
+8
>0
ds
No informa,on on growth rates, resonance width and eigenfunc,ons ! ε2
2
2
(1 − n + px) − ψ
8x
�
n+1
−p
x
�
>0
Do we see the resonance in numerical simula,ons? #$
+ " ! ( $u) = 0
#t
& #u
)
1
"( + (u $ %)u+ = ,%p +
[% . B] . B
' #t
*
4-
"B
= # $ [u $ B]
"t
" ! B = 0,
p=c !
s
−(s − s0 )2
B φ = b0
exp
s0
σ
2
S
Boundary conditions: periodic (z), reflection/outflow (r), periodic (phi)
Basic
! state:Lorentz Force balanced with P or gravity
Geometry: const./ring Bphi •  Cylindrical geometry •  Perfect conductor BC in the radial direc,on •  MHD equilibrium with constant Bphi, and a ‘ring’ Bphi z
z
vϕ
Bϕ
z
r
Br
r
r
r
Bz
Bϕ
ϕ
vz
Final state is current-free!
Hint that the almost 100 years old Le Chatelier principle can s,ll be useful! •  Changes in reactant or product concentra,ons is one type of stress on an equilibrium •  Other stresses are temperature, and pressure. •  The response of equilibria to these stresses is explained by Le Chatelier s principle: If an equilibrium in a system is upset, the system will tend to react in a direc,on that will reestablish equilibrium •  Thus we have: 1) Equilibrium, 2) Disturbance of equilibrium, 3) Shig to restore equilibrium •  Le Chatelier s principle predicts where an equilibrium will evolve s
−(s − s0 )2
B φ = b0
exp
s0
σ
Balance this basic state with either gradient of pressure, or external force (gravita,onal poten,al). �=0
In this case a mode with m=1 dominates the non-­‐linear evolu,on. s
−(s − s0 )2
B φ = b0
exp
s0
σ
Balance this basic state with either gradient of pressure, or external force (gravita,onal poten,al). �=2
In this case a mode with m=6 dominates the non-­‐linear evolu,on. Similar selec,on occurs for the ver,cal wavenumber! Kine,c energy evolu,on �=0
�=2
�=1
�=2→m=6
�=1→m=3
�=0→m=1
Magne,c energy evolu,on �=0
�=2
�=1
�=2→m=6
�=1→m=3
�=0→m=1
Conclusions •  The statement that a strong poloidal field stabilize the kink instability “is not even wrong! ” •  Numerical stability analysis of stable stra,fied stellar interior can miss important modes •  Role of the turbulent state need to be further analyzed (how much helicity is produced?)