Explosive tearing mode reconnection in relativistic plasmas

From Black Holes to Cosmic Rays : when plasmas go wild
A TRIBUTE TO GUY PELLETIER 14 -18 October 2013
Explosive tearing mode reconnec3on in rela3vis3c plasmas: applica3on to the Crab flares Hubert Baty, Observatoire de Strasbourg [email protected] Collaborators: Jérôme Pétri, Observatoire de Strasbourg, France Seiji Zenitani, National Observatory of Japan, Tokyo H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Plan •  Mo3va3on: Gamma-­‐ray flares in the Crab Nebula •  Models for the Crab flares •  Our study: Explosive reconnec3on of double tearing mode in rela3vis3c plasmas •  Conclusion/Applica3on to the Crab flares H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Mo3va3on (1) •  The Crab pulsar : pulsed radio emission ! 30 pulses/s -­‐> Spin Period : P = 33 ms Spin decrease = 10-­‐12.4 s/s Surface B = 4×1012 Gauss Radius = 10 km Lighthouse effect HST + Spitzer + Chandra (Crab nebula: visible + IR + X) Neutron star born a\er a supernova explosion in 1054 ! H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Radio waves: Synchrotron radia3on ! (rela'vis'c electrons) Mo3va3on (2) •  Crab nebula : (before 2007) steady emission -­‐> standard candle for gamma-­‐ray astronomy ! The Crab nebula is bright at all accessible wavelengths H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Mo3va3on (2) •  Crab nebula : (before 2007) steady emission -­‐> standard candle for gamma-­‐ray astronomy ! The Crab nebula is bright at all accessible wavelengths ⇒  The pulsar powers/energizes the nebula emission (Synchrotron + inverse-­‐Compton) Loss of rota3onal energy: -­‐> Magnetosphere Plasma pairs crea3on -­‐> cold ultra-­‐rela3vis3c wind inside the termina3on shock (with a frozen-­‐in magne3c field) -­‐> Synchrotron nebula Accelerated and radia3ng pairs See Review Kirk et al. 2009 Light cylinder RL ~ 10-9 pc
Termin. shock RTS ~ 0.1 pc
~ 1 pc
Aharonian & Bogovalov 2003 H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 quantities (in the observer frame): the position of the peak photon energy, Ep /
B, the peak emitted power ⌫F /
the rise time ⌧1 = l/(c ), and the cooling time ⌧2 = 8.9 ⇥ 103 /[(B/Gauss)2 ⇤2 ], assuming = 1 (see text).
K/↵ l B
,
Mo3va3on (3) •  Crab wind/nebula : gamma-­‐ray flares (energy > 100 Mev) VARIABLE GAMMA-­‐RAY EMISSION Gamma-­‐ray light curve from the Crab nebula : short and powerful flares ! The gamma-­‐ray space telescopes Fermi and Agile -­‐> E. Striani et al. ApJ 2013 -­‐ Dura3on: 1 day -­‐Frequency : a few events /year -­‐ Rise 3me: 10 hours ! Mo3va3on : 106 P and a few 100 3mes smaller than the nebula’s dynamical 3me-­‐scale ! Fig. 1.— Gamma-ray lightcurves above 100 MeV (12-hr time bins) from the Crab (pulsar plus Nebula) detected by AGILE and FermiLAT. From top to bottom, the Sept. - Oct. 2007 event (AGILE data), the Feb. 2009 event (Fermi-LAT data), the Sept. 2010 event
(Fermi-LAT data), and the Apr. 2011 event (Fermi-LAT data).
H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Plan •  Mo3va3on: Gamma-­‐ray flares in the Crab Nebula •  Models for the Crab flares •  Our study: Explosive reconnec3on of double tearing mode in rela3vis3c plasmas •  Conclusion/Applica3on to the Crab flares H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Models for the Crab flares (1) •  Different models proposed for the Crab flares : –4–
(1)  Large-­‐scale changes in the steadily wind flow + rela3vis3c beaming amplifica3on Komissarov & Lyu3kov (2011), Lyu3kov et al. (2011) Flaring region in the downstream region of an oblique shock The 3me-­‐scale determined by a change in the shock normal direc3on due to waves/instabili3es ! => post-­‐shock velocity can sweep across the line of sight and then create a flare in Doppler boosted emission ! Fig. 1.— Interpretation of the inner knot as a highly oblique part of the termination shock (Komissarov & Lyutikov 2011).
Since the reflected waves intensity depends on the relative phase of the incoming waves and
the corrugation waves, the corresponding perturbations of the downstream medium will include an
entropy wave and one cannot use the barotropic equation of state; this complicates the problem
considerably (Courant & Friedrichs 1948, §73). Still, the salient features of the interaction can
be derived in the limit of small amplitude of corrugation, in which case the entropy wave is weak
(Courant & Friedrichs 1948, §73). In the isentropic limit, the shock acts as a partially reflecting
surface (Landau & Lifshitz 1959, §91), with the angle of reflection not equal the incidence angle. The
incident and the reflected wave then form an interference pattern in the shocked fluid. Propagation
of the shock through this interference pattern then induces weaker shock corrugations.
As a simpler estimate we next consider the post-shock flow from a given corrugated shape of
the shock, without calculating the dependence of the corrugation on the amplitude of the perturbing
wave and neglecting the the entropy wave. General conditions at relativistic oblique perpendicular
H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Models for the Crab flares (2) •  Different models proposed for the Crab flares : (2) Magne3c reconnec3on (nebula, wind, …) Flare observed (beamed emission) -­‐ topological rearrangement of magne3c field lines -­‐> electrons accelerated by the reconnec3on electric field or/and by internal shocks bulk ouolow ! -­‐ Sta3s3cal model -­‐> random ac3va3on of rela3vis3c minijets reproduce aspects of the flaring events ! Clausen-­‐Brown &
L
yu3kov (
2012) (par3cles => synchrotron radia3on) But the basic reconnec3on mechanism is not included ! rising 3me ? H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Models for the Crab flares (2) •  Different models proposed for the Crab flares : (2) Magne3c reconnec3on (nebula, wind, …) Flare observed (beamed emission) -­‐ topological rearrangement of magne3c field lines -­‐> electrons accelerated by the reconnec3on electric field or/and by internal shocks bulk ouolow ! Clausen-­‐Brown & Lyu3kov (2012) The answer, my friend, is blowin’ in the wind ! H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Models for the Crab flares (3) The striped wind structure
•  Different models for the Crab flares : wind magne3c reconnec3on Asymptotic MHD solution: oblique rotator (Bogovalov 1999)
Jérôme
Pétri
Basic
features
Radio
emission
Gamma-ray
pulsars
20
15
15
Χ
10
Ζ
5
y
0
-5
-10
-10
-15
-15
-20
-10
0
r
10
Emission
mechanisms
Ω: rotation axis
Meridional plane Polar cap
χ: inclination of magnetic axis
(Kirk & Lyubarsky Magnétosphère
01)
Outer gap
Magnetosphere
0
-5
-30
Theory
!
5
Introduction
Discovery
20
10
z
Theory of
pulsar
magnetosphere
20
30
rotator Bogovalov (1999) ζ: Oblique inclination of
line of sight.
The wind
40
-15
-10
-5
0
x
5
Model
Synchrotron
emission
Crab pulsar
Inverse
Compton
emission
Vela
Geminga
H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 15
20
Equatorial plane Current sheets Magne3c structure of the rela3vis3c Properties striped wind is naturally subject to the tearing mode (a resis3ve nstability) ! assumes ionly
a Bϕ component
decreasing like 1/r
an exact analytical expression for Bϕ is known
-­‐> possible explana3on for the sigma-­‐problem independent of the!magnetospheric
structure inside the light cylinder
6
(loss of magne3za3on -­‐> B varies from 10 Gmagnetic
at light cylinder discontinuous
polarity
reversal.
Conclusion
radius to few mG at the termina3on shock !) Structure
10
Models for the Crab flares (4) unner, we can study pair•  Different models for the Crab flares : the rela3vis3c tearing mode in 3D
Current Sheet
Reconnection
ring
!
inflow
!" Initial Relativistic Harris sheet :
i
ast &
kink
dt" = 0.223
1/2
Main esult -­‐> =30.5me scale xof the #/d =r0.707,
$ /"
z reconnec3on = (TA TD) (same as in classical MHD) length)
(dn /nis e=0.3inertial
tearing
(background density)
y
Cell
size
~
0.195
#
Problems: 3me-­‐scale not so fast with Lundquist number S1/2 scaling , simula3ons use S < 103 !" Periodic BC
>>>1 are expected ! !"while no initial Sperturbation
!" no
fieldbulk ouolow is also very sub-­‐rela3vis3c ! -­‐This is not sudden/explosive ! guide
-­‐ the A more energe3c/explosive, fast, and sudden reconnec3on mechanism is required ! +
i
affect
Resis've Magnetodynamic model (MHD + limit of vanishing rest mass density and maHer pressure) Komissarov et al. MNRAS 2007 ci
e:
outflow
Diffusion Region
(X-line)
Magnetic
Islands
me-
!
B
bg
pi
ci
0
D
TA = Alfvén crossing 3me scale TD = resis3ve 3me scale S -­‐> Inverse normalized resis3vity B. J. Albright, K. J. Bowers, J. Margulies, Phys. Rev. Lett., 101, 125001-1 (2008)
H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Plan •  Mo3va3on: Gamma-­‐ray flares in the Crab Nebula •  Models for the Crab flares •  Our study: Explosive reconnec3on of double tearing mode in rela3vis3c plasmas •  Conclusion/Applica3on to the Crab flares H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Explosive reconnec3on in rela3vis3c plasmas (1) Our study -­‐> Baty et al. 2013 MNRAS to appear •  A resis3ve rela3vis3c MHD (RRMHD) code is used RRMHD equations
Continuity
Momentum
Energy
Maxwell eqs.
Virtual potentials to fix div B, E
(Munz ’00, Dedner ‘02)
Charge conservation
Ohm’s law
(1) Isotropic MHD fluid
four velocity
momentum
(2) Simplest Ohm’s law
energy
enthalpy
Shock capturing discre3sa3on: (finite-­‐volume method) -­‐Time-­‐split HLL scheme -­‐Resis'vity by relaxa'on scheme -­‐Dedner scheme to clean the non-­‐
divergence free of B ! See Komissarov 2007 Zenitani et al. 2010 η : resis3vity H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 solenoidal condition (Dedner et al. 2002). For more details
equations
are solved
by a isvariant
of to
Komissarov
about
the code,
the reader
referred
Zenitani et(2007)
al. (2010)
time-split HLL method, where the sti↵ non-ideal terms are
and references therein. The light speed is conveniently nortreated using a relaxation scheme. A hyperbolic divergence
malized
toNRAS unity,
c =
We instart
with an initial double
Baty et al. M
2013 1.used
cleaning
method
is also
order to deal with the
Harris-sheet-like
configuration
solenoidal condition
(Dedner et al. 2002). For more details
Explosive reconnec3on in rela3vis3c plasmas (2) • 
1.5
1
0.5
about
reader is referred to Zenitani et al. (2010)
~ =the
B
B0code,
(1 + the
tanh((y
y0 )/l) tanh((y + y0 )/l)~
x,
(1)
and
references
therein.
The
light
speed
is
conveniently
norA numerical study: a 2D double Harris current sheet (in MHD malized
c = 1. We magnetic
start with field
an initial
double The
where
B0toisunity,
the maximum
amplitude.
equilibrium) Harris-sheet-like
configuration
initial
temperature
T is assumed to be uniform, and the
sqrt(2)*(1+tanh(x-4)-tanh(x+4))
Bx ~ =density
plasma
the expression
B
B0 (1 + variation
tanh((y yfollows
tanh((y
+ y0 )/l)~x,
0 )/l)
(1)
2
2
the maximum
field amplitude.
The (2)
⇢where
= ⇢bB+0 ⇢is0 (cosh
((y ymagnetic
((y + y0 )/l)).
0 )/l) + cosh
initial temperature T is assumed to be uniform, and the
This
corresponds
to a structure
in equilibrium
plasma
density variation
follows the
expression as the thermal
pressure
is taken to be , P y= = ⇢.
amplitude of the magl = 1 (half-­‐thickness) 4 The
(half-­‐separa'on) ⇢ = ⇢b + ⇢0 (cosh 2 ((y y002)/l) + cosh 2 ((y + y0 )/l)). (2)
netic
field is bset
so that
= -­‐1.
our
Numerical oundary Ly= B105 /2
and 15 Throughout
(free ouTlow BC) study,
This
corresponds
a structure
equilibrium
thermal
the
sheet to
thickness
is in
taken
to be las
=the
1 and
the half current
pressure
isbetween
takenBto2 the
be Ptwo
= =⇢.
The
amplitude
magseparation
current
layers
is of
y0 the
= 4.
We also
Normaliza'on 1
(
uniform t
emperature), 0 = 2 , T
2
set
so that
ourdensity
study, val0 /2 =
fixnetic
specific
heat
at
initial
Γthe
= field
4/3 (isspecific hratio
eat B
ra'o) ,=
ρ1.04/3.
=Throughout
1 .The
s
y theare
current
sheet thickness
is taken
l = 1 and
halfues
normalized,
by using
⇢0 =to1,bewhilst
thethe
background
separation between the two current layers is y0 = 4. We also
⇢b varies
from case to
case. Typically,
we choose 4 val-> ρb is varied in the range [2, 1/3] =value
> different magne3za3on parameter σ between fix the specific heat ratio at = 4/3. The initial density values
this
background
⇢b background
= 2, 1,! 2/3
0.2 and 1.2 or different Alfvén speed CAfor
are
between 0.45 cby
and 0density,
.74 -­‐> 1,namely
mwhilst
ildly rthe
ela3vis3c and
ues
normalized,
using
⇢0c=
leading
tofrom
4 values
theTypically,
corresponding
background
1/3,
value
⇢b varies
case tofor
case.
we choose
4 val1/2
Alfvén
speed
such
that
c
=
0.447,
0.535,
0.612,
and
CA = (σ/(σ+1)) c = light speed ues for this background density,
A
namely ⇢b = 2, 1, 2/3 and0.739.
The
is defined
by corresponding
c2A = B02 /(⇢hbackground
+ B02 ), where h
1/3,Alfvén
leadingspeed
to 4 values
for the
-­‐> The resis3vity is varied with the isLundquist number S = l cch
/η r0.535,
ange 0.612,
[50 : 4and
00] 0.739.
Alfvén
speed
such
that
0.447,
the specific
enthalpy,
=in 1 t+he P/⇢(
1).
Note
that, the
A
2
2
2
The Alfvén speed
is defined
B0 magnetization
), where h
A = B0 /(⇢h
background
density
is alsoby
a cmeasure
of +
the
t of the flow velocity overlaid with magisthe
h
= 1 + P/⇢( parameter
1). Note that,
the the
H. of
Baty -­‐the
Les system,
Hspecific
ouches 14 enthalpy,
-­‐ 18 the
octobre 2013 as
magnetization
follows
i↵erent times, from left to right and up
background density is also a1/2
of the magnetization
relation:
cA = ( /( + 1)) measure
(Del Zanna
et al. 2007). Conmponent of the flow velocity overlaid with mag0
-0.5
-1
-1.5
-15
-10
-5
0
5
10
15
Explosive reconnec3on in rela3vis3c plasmas (3) -­‐We add a divergence-­‐free small perturba3on a Lx = 24 -­‐> wavelength periodic S = 200 ρb = 1 Phase 1: Linear evolu3on of a double tearing mode DTM (an3-­‐symmetric mode) Phase 2: Satura3on (Rutherford regime) O-­‐points Magne3c islands (end of Phase 2) UX + magne3c field lines Ux: x comp. of four velocity vector H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Explosive reconnec3on in rela3vis3c plasmas (3) -­‐We add a divergence-­‐free small perturba3on •  : a b c d Lx = 24 -­‐> wavelength periodic S = 200 ρb = 1 Ux/c (l/c) Phase 1: Linear evolu3on of a double tearing mode DTM (an3-­‐symmetric mode) Phase 2: Satura3on (Rutherford regime) Phase 3: Non-­‐linear secondary instability (coalescence) with fast growth ! UX + magne3c field lines Phase 4: Relaxa3on to Final state (the ini3al Ux: x comp. of four velocity vector magne3c field is released in heat and bulk flow) H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Explosive reconnec3on in rela3vis3c plasmas (4) •  Results : -­‐In agreement with classical MHD with applica3on to Tokamaks ! See Janvier et al. 2011, Zhang & Ma 2011 0.1
Ux/c L = 40 Maximum Vx
1) Effect of varying the wavelength (or Lx) -­‐> secondary growth (nonlinear instability): a cri3cal minimum wavelength > 6 3mes y0 ! 1
x
0.01
Lx= 30 Lx = 24 0.001
Lx = 20 0.0001
0
200 400 600 800 1000 1200 1400 1600 1800 2000
Time
Explana3on: structural instability is nonlinearly driven ! 2 isolated single tearing mode (saturated) -­‐> The bulk ouolow velocity is mildly rela3vis3c (~ 0.5 C) >> single tearing mode velocity ! H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Explosive reconnec3on in rela3vis3c plasmas (5) •  Results : S = 200 S = 100 0.1
Ux/c Maximum Vx
2) Effect of varying the resis3vity or S -­‐> Satura3on phase 3me-­‐scale as S 1 -­‐> Secondary instabilty phase is explosive and fast as it scales as S 0.2-­‐0.3 (even faster in classical MHD -­‐> nearly independant of S) S = 50 S = 400 0.01
0.001
0
H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 500
1000
Time
1500
2000
Case Lx= 24 Explosive reconnec3on in rela3vis3c plasmas (5) •  Results : S = 200 S = 100 0.1
2) Effect of varying the magne3za3on via ρb -­‐> Weak scaling dependance as σ 0.3 Maximum Vx
Ux/c S = 50 S = 400 0.01
0.001
0
500
1000
1500
Time
2000
Case Lx= 24 0.1
Ux/c Maximum Vx
2) Effect of varying the resis3vity or S -­‐> Satura3on phase 3me-­‐scale as S 1 -­‐> Secondary instabilty phase is explosive and fast as it scales as S 0.2-­‐0.3 (even faster in classical MHD -­‐> nearly independant of S) σ = 1.2 σ = 0.2 0.01
1000
H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 1200
1400
Time
1600
1800
2000
Plan •  Mo3va3on: Gamma-­‐ray flares in the Crab Nebula •  Models for the Crab flares •  Our study: Explosive reconnec3on of double tearing mode in rela3vis3c plasmas •  Conclusion/Applica3on to the Crab flares H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Conclusion (1) Our study -­‐> Baty et al. MNRAS 2013 tructure du vent
e modèle
pplication au pulsar du Crabe
fluence des paramètres
•  Applica3on to Crab flares : eur oblique (Bogovalov 1999)
20
15
10
y
5
0
-5
-10
-15
30
40
-15
-10
-5
0
x
5
10
15
20
roissant en 1/rR;ts ~ 108 rL est connue ; (rL : light cylinder radius) érique à l’intérieur du cylindre lumière ;
-­‐Proposed mechanism: DTM reconnec3on event in the wind structure (where λ > ~ 6 y0) : -­‐> radially sheared rela3vis3c ouolow => Poten3al source for internal shocks and beamed emission ! -­‐Time scale seen by distant observer (rela3vis3c correc3on): ∆T ≈ 1500 γ2 P/16 (y0 is es3mated from λ ~ 2 π rL ) rising 3me of about 10 hrs for ∆T => γ = 50-­‐100 -­‐> value expected rela3vely close to the light-­‐cylinder -­‐Energe3cs : magne3c energy released in the localized region during 1 day (typical flare 1034 J is needed) ⇒  a local B magnitude of order 100 T ! -­‐> value expected at ~ 50 rL H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Conclusion (2) •  Summary : -­‐The DTM is a be{er ‘reconnec3on’ candidate for powerful gamma-­‐ray flares than simple tearing mode -­‐> explosive evolu3on with fast 3me-­‐scale ! -­‐> more energe3c with rela3vis3c bulk ouolow ! -­‐Higher S and sigma (more realis3c) values need to be inves3gated ! ⇒  Need for be{er (robustness) numerical schemes achieving theses values: Palenzuela (2009), Dumbser & Zano} (2009) for discon3nuous Galerkin discre3za3on. -­‐MHD model -­‐> PIC model/simula3ons (the accelera3on of par3cles is naturally included and ‘easy’ modelling of the synchrotron emission, see Ceru} et al. (2012) -­‐> talk tomorrow) H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Conclusion (3) •  Summary : -­‐More complex/realis3c configura3ons must be inves3gated; see recent results on the plasmoid instability in classical plasmas ! Plasmoid Chains
Samtaney PRL Sept. 2009
S = 104
S = 105
S = L VA/η
S = 106
S = 107
• Sweet-Parker current
sheets
with
> 100 aspect
ratio
are unstable
*Sweet-­‐Parker (
S-­‐P) c
urrent s
heets w
ith a
spect ra3o > 100 are unstable to forma3on to formation of plasmoid (magnetic-island) chains.
of p• lasmoid (magne3c-­‐island) chains time-dependent ant
Magnetic reconnection
becomes inherently
potentially faster than S-P at S>104.
• Secondary sheets between islands can be unstable in turn, giving
*Magne3c reconnec3on becomes inherently 3me-­‐dependent ant poten3ally faster rise to a multilevel hierarchy of plasmoids (Uzdensky PRL Dec.
than 2010).
for the standard Tearing mode, at S > 104 : • Besides the
single b
x-line
wedged
structure,
this
multiple x-line
-­‐Secondary s
heets etween i
slands :
m
ul3level hierarchy of plasmoids -­‐> fast reconnec3on ! configuration can give fast reconnection.
P. Buratti Flaring CRAB 2012
-­‐> extension to rela3vis3c regime : see Takamoto 2013 H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 Conclusion (4) June 1, 2008
13:29
WSPC - Proceedings Trim Size: 11in x 8.5in
gmurphy
•  Thank you ! 1
3-d resistive MHD simulations of magnetic reconnection and the tearing mode instability in
current sheets
Guy also explored the tearing instability ! G. C. Murphy∗
806.0144v1 [astro-ph] 1 Jun 2008
Laboratoire d’Astrophysique de Grenoble, CNRS, Université Joseph Fourier, Grenoble, France
∗ E-mail: [email protected]
R. Ouyed
Department of Physics and Astronomy, University of Calgary,AB, Canada
E-mail: [email protected]
G. Pelletier
Laboratoire d’Astrophysique de Grenoble, CNRS, Université Joseph Fourier, Grenoble, France
E-mail: [email protected]
Magnetic reconnection plays a critical role in many astrophysical processes where high energy emission is observed,
e.g. particle acceleration, relativistic accretion powered outflows, pulsar winds and probably in dissipation of Poynting
flux in GRBs. The magnetic field acts as a reservoir of energy and can dissipate its energy to thermal and kinetic
energy via the tearing mode instability. We have performed 3d nonlinear MHD simulations of the tearing mode
instability in a current sheet. Results from a temporal stability analysis in both the linear regime and weakly nonlinear
(Rutherford) regime are compared to the numerical simulations. We observe magnetic island formation, island merging
and oscillation once the instability has saturated. The growth in the linear regime is exponential in agreement with
linear theory. In the second, Rutherford regime the island width grows linearly with time. We find that thermal energy
produced in the current sheet strongly dominates the kinetic energy. Finally preliminary analysis indicates a P(k)
4.8 power law for the power spectral density which suggests that the tearing mode vortices play a role in setting up
an energy cascade.
Interna'onal Journal of Modern Physics D, Volume 17, Issue 10, pp. 1715-­‐1721 (2008)
Keywords: MHD-Plasma physics-numerical simulations
H. Baty -­‐ Les Houches 14 -­‐ 18 octobre 2013 1. Introduction
and weakly nonlinear analyses of Furth et al. [4] and