MHD instabilities and
Magnetic Reconnection
Gunnar Hornig
STFC Advanced Summer School 2014
Contents
• Motivation
• Stability and Instability
• Kink Instability
• Tearing Instability
• Reconnection: History, Literature, Occurrence
• 2D Reconnection (MHD)
• 2D Collisionless Reconnection
• 3D Reconnection
• Problems
Motivation
Brief Article
One of the key questions of solar physics (plasma astrophysics) is how the magnetic field
interacts with the plasma (e.g. dynamo The
problem
or coronal heating problem).
Author
This includes the exchange of energy and momentum between the magnetic field, and the
plasma. The latter consists of momentum/energy
bulk velocity and the random
September of
2,the
2014
(thermal) velocity.
For a high temperature plasma (e.g. the solar corona) the exchange of energy between the
magnetic field and the thermal energy seems to be small due to the low value of the resistivity
η ~ Τ -3/2 .
Brief Article@B
d
+ Lv B = 0; v(G(x, t)) =
@t
dt
The Author
n ⇥ A|@C = n ⇥ Ap |@C
✓
◆
2
1
September 2, 2014@ B
2
+r·
E ⇥ B = J · E Z= v · j ⇥ B ⌘j
Brief@tArticle
HR = (A + Ap ) · (B Bp ) d3 x
2µ0
µ0
Author
dThemagnetic
energy
B = 0; v(G(x, t)) = G(x, t); G(x,
0) = x
dt
September 2, 2014
✓
◆
1
r·
E ⇥ B = J · E = v · j ⇥ B ⌘j2
µ0
ez ⇥G(x,
A|BC =
ez ⇥=
Apx
TC = 0)
t);ez ⇥ A|
G(x,
thermal energy
@B
d
+ LMagnetic
t)) = G(x, t); G(x, 0) = x
energy
v B = 0; ⇤ v(G(x,
@t B(x, 0) = G B(x, t)
dt
✓
◆
@ B2
1
+r·
E ⇥ B = J · E = v · j ⇥ B ⌘j2
@t 2µ0) Fb = G(F
µ0 a , t)
@B
B(x, 0) = G⇤ B(x, t)
r ⇥ (v ⇥ B) = 0
r
2
B(x, 0) = G⇤ B(x, t)
Poynting flux
work done
Apby=Lorentz
B0 e force
T =
) Fb = G(Fa , t)
X
index(xp0 ) = T@D
xP
0
@B bulk kinetic energy
r ⇥ (v ⇥ B) = 0
@t
µv v
Ohmic heating
Motivation
Both, the resistive and the viscous heating terms are small unless there are strong gradients in
the magnetic field (electric currents) or the velocity field.
!
Also if strong gradients in the fields develop the assumptions for the derivation of these terms
breaks down (e.g. macroscopic length scale >> mean free path) and the coefficients, and
possibly also the form of the terms, may become anomalous.
!
Processes which generate strong gradients are typically instabilities. These instabilities in turn
lead to magnetic reconnection.
!
Stabilityand
and
Instability
Stability
Instability
3.
3.1.In Linear
stability
all four cases
the ball is in equilibrium, but
is globally
stable;that
all quantities are of the form
Linear a)
stability:
you assume
b) is an unstable equilibrium (but stable equilibria are close by);
X(x, t) = X(no
+ X1 (k,equilibrium
!) exp(i(k · xexists)
!t))
0 (x) stable
c) is an unstable equilibrium
is a stable equilibrium (for small perturbations)
Here Xd)
0 is the equilibrium quantity and X1 is a small quantity so that all terms quadratic in X1 can
be neglected. Solving the MHD equations using such an Ansatz yields a dispersion relation
For large perturbations b) behaves like a) and d) like c).
! = !(k),
!
where ! is a complex function of the wave modes k.
MHD stability: Take an MHD equilibrium (e.g. pressure equilibrium, force-free
equilibrium,
and perturb
Re(!):
Oscillating ..)
solutions
(waves)it. Does the amplitude of the perturbation increase (instability) or
decrease (stability)?
Im(!): Damping Im(!)  0 (stability) or exponential growth, Im(!) > 0 (instability)
!
Complication: Other than in the cartoon the space of possible perturbations is infinite
dimensional. Typically only a subspace of possible perturbations is tested. This allows only to
prove instability but not stability. !
3.Linear
Stability
and Instability
Stability
3.
Stability and Instability
Linear
stability
3.1. 3.1.
Linear
stability
stability:
assumethat
that all
all quantities
areare
of the
form form
LinearLinear
stability:
youyou
assume
quantities
of the
X(x, t) = X0 (x) + X1 (k, !) exp(i(k · x
!t))
X(x, t) = X0 (x) + X1 (k, !) exp(i(k · x
!t))
Here X0 is the equilibrium quantity and X1 is a small quantity so that all terms quadratic in X1 can
be0 neglected.
Solving the MHD
equations
an Ansatz
yields so
a dispersion
Here X
is the equilibrium
quantity
and using
X1 issuch
a small
quantity
that all relation
terms quadratic in X1 can
be neglected. Solving the MHD equations using such an Ansatz yields a dispersion relation
! = !(k),
where ! is a complex function of the wave modes!k.= !(k),
Oscillatingfunction
solutions (waves)
where !Re(!):
is a complex
of the wave modes k.
Im(!): Damping Im(!)  0 (stability) or exponential growth, Im(!) > 0 (instability)
Re(!): Oscillating solutions (waves)
Im(!): Damping Im(!)  0 (stability) or exponential growth, Im(!) > 0 (instability)
Non-Linear Stability
3.
Stability and Instability
Linear stability
3.2.3.1.Non-linear
stability: Energy principle
Linear stability: you assume that all quantities are of the form
Define the total energy of a closed system:
X(x, t) = X0 (x) + X1 (k, !) exp(i(k · x
!t))
(t)X1=is a smallT quantity
(t)
V (t)quadratic in X1 can
Here X0 is the equilibrium quantityW
and
so +
that all terms
|{z}relation
be neglected. Solving the MHD equations using such |{z}
an Ansatz yields a dispersion
Kinetic energy
! = !(k),
potential energy
In a static equilibrium at t = 0: T (0) = 0, V (0) = V0 and W0 = V0
where ! is a complex function of the wave modes k.
Oscillating
solutions (waves)
If we Re(!):
disturb
the equilibrium
W0 ! W = W0 +
W.
Im(!): Damping Im(!)  0 (stability) or exponential growth, Im(!) > 0 (instability)
Define stability: T (t) 
W for any disturbance .
Ideal instability: kink instability
3.3.
Ideal instability: Kink instability
An ideal instability of flux tubes which occurs at a certain threshold of twist in the flux tube. The
threshold for the onset of the instability depends on the details of the considered equilibrium, in particular on the radial profile of the twist, and on the aspect ratio, l/a, where a is the characteristic radius of
the configuration (the minor radius in a toroidal system). For a straight, cylindrically symmetric flux
tube with fixed (line-tied) ends and uniform twist (the Gold and Hoyle (1960) force-free equilibrium),
which is the simplest possible model of a coronal loop, the threshold was numerically determined (Hood
and Priest, 1981) to be 2.49 ⇡.
From T. Török and B. Kliem.
3.4.
Resistiveinstability:
instability: Tearing
instability
Resistive
tearing
instability
Initial equilibrium: One dimensional current sheet (Harris sheet)
B(z) = B0 tanh(z)ex ) j =
B0
µ
cosh(z)
2
ey .
The equilibrium is unstable to long wave length resistive modes.
) Change of topology (not possible for an ideal instability)
) Release of energy stored in the magnetic configuration
) Reconnection (see talk by A. Hood).
-3/5
Growth rate of the instability: γ ~ S
where S = VA a / η.
Tearing instability
Courtesy of M Hesse
Occurrence of tearing instability in a less symmetric set up:
Shearing of an 2D x-type configuration generates a thin current sheet which becomes
unstable.
Magnetic Reconnection
A very short history of magnetic reconnection
• 1946 Giovanelli suggested reconnection as a mechanism for particle acceleration
in solar flares. !
• 1957-58 Sweet and Parker developed the first quantitative model • 1961 Dungey investigated the interaction between a dipole (magnetosphere) and the
surrounding (interplanetary) magnetic field which included reconnection. • 1964 Petschek proposed another model in order to overcome the slow reconnection
rate of the Sweet-Parker model • 1974 J.B. Taylor explains the turbulent relaxation of a Reversed Field Pinch by a cascade
of reconnection processes which preserve only the total helicity.
• 1975 Kadomtsev explains the sawtooth crash (m=1,n=1 mode) in a tokamak by
reconnection • 1988 Schindler et al. introduced the concept of generalised magnetic reconnection in
three dimensions
Literature:
• several thousands publications which refer to magnetic reconnection !
• Biskamp, D., Magnetic Reconnection in Plasmas, CUP 2000 !
• Priest and Forbes, Magnetic Reconnection, CUP 2000 !
• Reconnection of Magnetic Fields, Editors: J. Birn and E. R. Priest, CUP
2007
Reconnection in other systems
•Hydrodynamics: Reconnection of vorticity (“crosslinking ” or “cut and
connect” of vortex tubes) , e.g. [Kida, S., and M. Takaoka, Vortex
Reconnection, Annu. Rev. Fluid Mech. 26, 169 (1994)] !
•Superfluids: Reconnection of quantized vortex elements [e.g. Koplik, J. and
Levine, H., Vortex reconnection in superfluid helium, Phys. Rev. Letters 71,
1375, (1993)]
Movie by O.N. Boratav
and R.B. Pelz
Reconnection in other systems
• Cosmic Strings: Reconnection of topological defects [e.g. Shellard, E.P.S.,
Cosmic String interactions, Nucl. Phys. B 282, 624, (1987)] !
• Liquid Crystals: Reconnection of topological defects [e.g. Chuang, I.,
Durrer, R., Turok, N., and Yurke, B., Cosmology in the laboratory: Defect
Dynamics in Liquid Crystals, Science 251, 1336, (1991)]!
• Knot theory: Surgery of framed knots [e.g. Kauffman, L.H. (1991)Knots and
Physics, World Scientific, London] !
• Enzymology: Reconnection (also “recombination”) of strands of the DNA !
[e.g. Sumners D., Untangling DNA, Math. Intell. 12, 71–80 (1990)]
Occurrence of magnetic reconnection:
• Fusion devices, in particular Tokamaks. Sawtooth oscillations limit
confinement (see e.g Kadomtsev, 1987).
Reconnection Experiments: Swarthmore Spheromak Experiment (SSX),
Princeton Magnetic Reconnection Experiment (MRX), High Temperature
Pasma Center Tokyo (TS-3/4), Versatile Toroidal Facility (VTF) at MIT, ...
• Magnetospheres of planets
• Solar Flares, Coronal Mass Ejections
• Accretion disks around black holes !
• Magnetars (Soft Gamma Repeaters)
A large prominence eruption on the Sun
on June 7, 2011 (SDO)
What have these systems in common? • A primarily ideal, i.e. topology conserving dynamics. • A (spontaneous) local non-ideal process, which changes the topology. !
Characteristics associated with magnetic reconnection:
a) generation of an electric field which accelerates particles parallel to B b) dissipation of magnetic energy (direct heating) c) acceleration of plasma, i.e. it converts magnetic energy into bulk kinetic energy d) change of magnetic topology
Note a)-c) are not properties of magnetic reconnection alone, but occur also in other
plasma processes, in particular c) can occur even in an ideal plasma.
3.
Topological Conservation Laws
3.1.
Flux conservation
of reconnection
ideal evolution
tions for Role
magnetic
to occurin
E+v⇥B=0
⇤
⇤
B
⇤t
⇧⇥v⇥B=0
reconnection:
⇤t B
⌅
2
⇧ ⇤ (v ⇤ B) = 0
B · da = const. for a comoving surface C 2 ,
C (t)
!
⌅ Conservation
flux
Alfvén’s Theorem (1942):
The flux of
through
any comoving surface is conserved. ⌅ Conservation of field lines
Mathematically this is an application of the Lie-derivative theorem: The
⌅ Conservation of null points
magnetic field is transported (or Lie-dragged, Lie-transported, advected) by the
⌅ Conservation of knots and linkages of field lines
flow v. n of magnetic!flux and field line topology: No reconnection.
Topology conservation can only occur if the plasma is non-ideal.
Conservation of flux, E + v ⇥ B of
= field
N (e.g.
Conservation
lines N
= ⇥j)
⇤
B ⇧ ⇥ v ⇥ B = null
⇧ ⇥points
N
t) conserves the magnetic flux then there exists a transport velocity w s
⇤tConservation ofVice
versa: If B(x,
if N⇧=⇤ (w
v ⇥⇤BB)
+=
⇧ 0. Note that w is not unique.
⇤t B
⇤ Conservation of knots ⇤ B ⇧ ⇥ w ⇥ B = 0 for w = v
v
⇤t
and linkages of field lines ⇥B+⇧
and v smooth: No reconnection but slippage.
⇥B+⇧
Reconnection (3D).
⇥B+⇧
and! v singular: Reconnection (2D).
Kinematic MHD Equations
Role of
evolution
reconnection:
ertain applications
the ideal
plasma flow
v is knownin
or assumed
to be of a certain form (e.g. dynamo
y).
In this case we
only have
to find the magnetic field from the induction equation.
Kinematic
MHD
Equations
• Storage of magnetic energy!
⌥B flow v is known or assumed1to be of a certain form (e.g. dynamo
tain applications
the
plasma
• Generates small scales
⇤ (v(typically
⇤ B) =current⇤sheets)
( ⇤ B).
(21)
⌥t to find the magnetic field from
⇤µ the induction equation.
. In this case we only have
given v(x, t) this equation
⌥B determines the magnetic field
1 B(x, t) provided the field is given at an
⇤ (v ⇤ B) =
⇤
( ⇤ B).
(21)
time B(x, t0 ). Note that
if
the
initial
field
is
divergence-free
then
the
induction
equation
ensures
⌥t
⇤µ
he magnetic field is divergence-free for all later times, hence (20) is nothing more than an initial
iven for
v(x,
t) thisthe
equation
determines
theAmagnetic
B(x,
provided
the field issolution.
given at an
tion
solving
induction
equation.
solution field
of (21)
is t)
called
a “kinematic”
time B(x,
Note thatdiffusivity
if the initial field is divergence-free then the induction equation ensures
0 ). magnetic
ming
that tthe
1
e magnetic field is divergence-free for all later times,
hence (20) is nothing more than an initial
=
,
(22)
on for solving the induction equation. A solution
µ⇤ of (21) is called a “kinematic” solution.
ng that the magnetic diffusivity
stant (generally depends on T ), (21) can be rewritten
using
⇤ ( ⇤ B) = B
( · B) as
1
=
,
(22)
µ⇤
⌥B
⌅
⇤ (v ⇤ B) =
B
(23)
R
=10
tant (generally dependsRmon=10000
T ),
(21)
can
be
rewritten
using
⇤
(
⇤
B)
=
B
(
·
B)
as
⇤
⇥
⌅
m
⇤
⇥
⌅
⌥t
di⇥usion term
advection term
⌥B
⇤ (vas⇤ the
B) ratio
= of the advection
B
(23)
e, Rm , the magnetic ⌅
Reynolds
number,
and
di⇥usion
terms
in the
⇤
⇥
⌅
⇤
⇥
⌅
⌥t
tion equation.
di⇥usion term
advection term
| ⇤as(vthe
⇤ B)|
B/l0advection
l0 v0 and di⇥usion terms in the
Rm , the magnetic Reynolds number,
ratio ofv0the
Rm =
=
,
(24)
=
2
2
on equation.
| ⌃ B|
B/l0
| ⇤ (vdetermining
⇤ B)|
v0 B/l
l0 v0
an important dimensionless parameter
the0 importance
of advection w.r.t. di⇥usion
Role of ideal evolution in reconnection:
• Under ideal evolution a non-trivial topology can store energy
trivial topology:
ideal relaxation can
reduce the magnetic
energy to zero.
non-trivial topology:
(linkage of magnetic flux)
ideal relaxation cannot
reduce the magnetic
energy to zero.
ng of flux tubes [?]:
• The fields B and w are locally tangential to a plane perpendicular to Ẽ.
• The fields B and w are locally tangential to a plane perpendicular to Ẽ.
Magnetic
Helicity:
lk(T , T )
,
• Reconnection can occur only at null points of the magnetic field.
i
j
i can
j occur only at null points of the magnetic field.
• Reconnection
• The source term of the helicity density Ẽ · B vanishes.
The
homogeneous
Maxwell’s
equation
yield
a balance equation for the helicity
•
The
source
term
of
the
helicity
density
Ẽ
·
B
vanishes.
Reminder on magnetic helicity: The homogeneous Maxwell’s equation yield a balance equation for the
density:
helicity
density:
e
linking
numberhelicity:
of⌅ theThe homogeneous Maxwell’s equation yield a balance equation for th
Reminder on magnetic
A
· B⇧
+⌥ · ( B + E ⇤ A) =
2E · B
⌅
⇤⇥
⌅
⇤⇥
⇧
⌅
⇤⇥ ⇧
helicity
density:
magnetic fluxes i⌅t and
hel. current
hel. source
⌅hel. density
A
· B⇧ +⌥ · ( B + E ⇤ A) =
2E · B
⌅
⇤⇥
⌅
⇤⇥
⇧
⌅
⇤⇥ ⇧
Integrating over are volume ⌅t
yields an expression for the total helicity
hel. density
hel. current
hel. source
Integrating over a closed
volume
(no
helicity
current
across
the boundary) yields an
d
3
3
Aexpression
· B d x = for
2 the
E·B
d x helicity
,
c)
ntegrating
over are volume
yields
an
total
expression
for the total
helicity:
dt V
V
was generalized by [?] ford the generic
case where3 field lines are not close
3
A·B d x= 2
E·B d x ,
dt V
umbers.
V
age of
The helicity integral (total helicity) is a measure for the linkage of magnetic !
sub-flux
tubes:
flux in the domain. It is invariant under an ideal dynamics.
Role of ideal evolution in reconnection:
Allows to build up magnetic energy in the field Allows to store magnetic energy in complex magnetic topology
.
2D Reconnection
2D• 2D
or E
· B = 0-Reconnection
-> B= Bx(x,y) ex + By(x,y) ey; N= η J = Jz(x,y) ez
!
E + v ⇤ B = N; N = v ⇤ B
!
⇧ E + w ⇤ B = 0 with w = v
v
!
E⇤B
!
w =
, ⇧ E · B = 0; E · w = 0
2
B
!
flux conserving
flow w exists
of points where B = 0
occursand
onlyitatisansmooth
x-pointwith
of theexception
magnetic field.
• The reconnection
ut N ⌃= •0.The source term of the magnetic helicity density vanishes, E⋄B=0.
• One can map the plasma flow v to an ideal flow w which transports the magnetic field and
at the
x-point.
• Thebecomes
fields Bsingular
and w are
locally
tangential to a plane perpendicular to Ẽ.
Note the process is only locally stationary, globally the field is always time-dependent.
•
• Reconnection can occur only at null points of the magnetic field.
2.
Non-Ideal
Evolution
at an X-point
2D
Reconnection
a)
b)
Manifestation
or
creation
(b) ofinflux
at isanofO-point
Since
generic
null
B is linear
x, w
the type −x/x2 along the inflow direction,
ructure
offor
wofaatloss
an(a)
X-point:
2
ull B hence
is linear
in
x,
w
is
of
the
type
x/x
. It hasThus
a 1/xa singularity
at the
origin. flux is transported
it has a 1/x singularity at the origin.
cross-section
of magnetic
is transported
a finite
onto the null-line.
in a finite in
time
ontotime
the null-line.
T
T =
dt =
0
0
dt/dx dx =
x0
0
x0
1/wx dx ⇤ x20 /2
This gives the impression that the flux is “cut” and reconnected at the z-axis. Rate of flux annihilation:
d
rec
=
d
dt
B·nda =
Ez dz
(=
wr Bphi dz)
Reconnection
gain the transporting flow has a 1/x singularity ⇥ The flux (a cross section) is transported in
nite time onto the null-line. But this time the singularity is of X-type. Simultaneously the null lin
the source of flux leaving the axis along the other direction. ⇥ No loss of flux but a re-connection
dt
Rate of reconnected flux:
Rate of reconnected flux:
d
rec
dt
d
=
dt
B·nda =
Ez dz
(=
wx By dz)
2D Reconnection:
First simple models of two-dimensional steady state reconnection have been set up by Sweet
and Parker (1957/8) and later Petcheck (1964) on the basis of conservation of mass and flux
in the framework of resistive MHD. 6.3.
Reconnection rate (2D)
• The
reconnection
rate
defined
for a two-dimensional
model plasma
of steadyvelocity
state reconThey
provide
a scaling
ofwas
the originally
Alfvén Mach
number
(the ratio of inflow
to
nection
(Sweet-Parker
the Alfvénnumber.
Mach number, i.e. the ratio of inflow plasma
inflow
Alfvén
velocity) inmodel,
terms 1957)
of theasLundquist
velocity to inflow Alfvén velocity:
⇥
Le is the global scale length, vAe = Be / ⇥µ0 is the Alfvén speed in the inflow region
MAe
ve
E
absolute reconnection rate
=
=
=
= relative reconnection rate,
vAe
vAe Be
maximum inflow of flux
• The electric field at the x-point is an absolute measure of the reconnected flux.
d2 rec
= Ez
dt dz
• Both definitions are equivalent if the magnetic field does not vary along the inflow channel.
2D Reconnection:
The models provide a scaling of the Alfvén Mach number in terms of the Lundquist
number:
én Mach number in terms of the Lundquist number (magnetic Reynolds number) :
S = µ0 Le vAe /
MAe
⇤
ve
E
=
=
=
vAe
vAe Be ⇥
⌅1
S
8 ln S
“Sweet-Parker”
“Petschek”
S=108
=
S=108
=
10
4
2 · 10
2
12) the
Due
to the
highorder
Lundquist
numbers
in the
Solar
Corona
(108- 10
scaling
of magnetic
the
nnection
rates
of the
of MAe
= 0.1 yield
time
scales
for global
energy
release
and
reconnection
rate with S
is important.!
nfiguration
that are consistent
with
those seen in solar flares and magnetospheric substorms (Miller
, 1997).!
Reconnection rates of the order of MAe = 0.1 yield time scales for global energy
reconnection:
release and magnetic reconfiguration that are consistent with those seen in solar
lim (⇥ 1/ ln(S))
flares and magnetospheric substorms
(Miller et al., 1997). S⇥⇤
In this respect Petschek’s model was a significant improvement to Sweet &Parker as
it provides higher reconnection rates.!
The term “fast reconnection” is used to describe reconnection rates which!
scale like Petschek reconnection or faster.
2D Reconnection: Collisionless
6.4.
Collisionless: There are not enough collisions between electrons and ions to explain the
6.4. Collisionless reconnection
reconnective
electric
field at the x-point as caused by resistivity (E ≠ η j).
Collisionless
reconnection
Generalised Ohm’s law:
Generalised
Ohm’sOhm’s
law: law:
Generalised
me
n e2
⌃j
+ ⌃ · (j v + v j
⌃t
1
jj
ne
⇥
=E+v⇤B
me
n 1e2
ne
⌃j
+ ⌃ · (j v + v j
⌃t
1
j⇤B+
ne
⌃ · Pe
1
jj
ne
j
⇥
=E
• Scale on which the hall term dominates over t
inertial <
length:
de=c/ωion
pe inertial lengt
• Scale on which the hall term dominates over the v•Electron
⇤
B
term:
d
=
c/⇤
i electron
pi
Scale on which
the
inertia dominat
⇤
! c/⇤pe ,(⇤pe = 4⇥ne2 /me ). In the solar coron
• Scale on which
⇤ the electron inertia dominates the
! hall term: electron inertial length, de
c/⇤pe ,(⇤pe = 4⇥ne2 /me ). In the solar corona de ⇧ 1m, in the magnetotail de ⇧ 1km!
Schematic of the multiscale structure of the
dissipation region during anti-parallel
reconnection. Electron (ion) dissipation
region in white (gray) with scale size c/ωpe
(c/ωpi). Electron (ion) flows in long (short)
dashed lines. Inplane currents marked with
solid dark lines and associated out-of-plane
magnetic quadrupole field in gray (From
Drake & Shay, Collisionless Reconnection
in ``Reconnection of magnetic fields'', CUP
2007)
2D Reconnection: Collisionless
Data from a PIC simulation of anti-parallel
reconnection with mi/me =100, Ti/Tp = 12.0
and c = 20.0 showing:
(a) the current Jz and in-plane magnetic!
field lines; (b) the self-generated out-of-plane field Bz (c) the ion outflow velocity vx; (d) the electron outflow velocity vxe; and (e) the Hall electric field Ey. Noticeable are the distinct spatial scales of
the electron and ion motion, the substantial
value of Bz which is the signature of the
standing whistler and the strong Hall
electric field Ey, which maps the magnetic
separatrix. The overall reconnection
geometry reflects the open outflow model of
Petschek rather than the elongated current
layers of Sweet-Parker (From Drake &
Shay, Collisionless Reconnection in
``Reconnection of magnetic fields'', CUP
2006).
Reconnection geometry:
2D: B(x,y) and v(x,y)
are in one plane. The
diffusion region (gray
area) is bounded in
the plane.
2.5D: B and v are
functions of (x,y) but B
has a component out
of the plane. The
diffusion region is
bounded in (x,y) but
not in z.
3D: B and v are
functions of (x,y,z).
Also the diffusion
region is bounded in
all 3 dimensions
Historically most work on reconnection has been done using the (much more
simple) 2D and 2.5D case. However, while this approximation is justified for
Tokamaks and other technical plasmas which have an inbuilt symmetry, in
astrophysical plasmas reconnection is inherently 3D.
Reconnection
canflux
onlyand
occur
thetopology:
plasma isNo
non-ideal.
⇤ Conservation
of magnetic
fieldifline
reconnection.
⇤ Conservation of magnetic flux and field line topology: No reconnection.
Reconnection can only occur if the plasma is non-ideal.
E + vis⇥
Breconnection.
= N (e.g. N = ⇥j
Conservation
of
magnetic
flux and
field
line plasma
topology:
No
c flux andReconnection
field⇤line
topology:
No
reconnection.
vs
Slippage
: if the
Reconnection
can only
occur
non-ideal.
⇤
⇧
vNo
⇥(e.g.
B
=N =
⇧ ⇥j)
⇥N
Reconnection
only occur
thefield
plasma
isB
non-ideal.
⇤ Conservation
of can
magnetic
flux ifand
topology:
reconnection.
E + line
v⇤t⇥
B
= ⇥N
ur if the plasma
is non-ideal.
xample where
the transport
velocity
the
magnetic
flux wis differs
from
velocity
v⇥j)
in +
a
• Reconnection
can only
occur of
if the
dynamics
non-ideal.
E
+v⇥
B the
= plasma
Nif (e.g.
⇤plasma
N
= Nv=⇥
B
Bplasma
⇧region
⇥ is
v⇥
B plasma
= ⇧ is
⇥ ideal
N (v = w). The blue and
on-ideal region
(gray)
while
outside
the
non-ideal
the
Reconnection
can
only
occur
if
the
non-ideal.
!
⇤t
E +⇤⇤
v ⇥ B = N (e.g. N = ⇥j)
⇧
vw⇥⇥B
⇧0⇥ N
⇤
B blue
⇧⇥
⇥cross
B===
for
w =inv the
d cross sections
with(e.g.
the plasma
velocity. The
section
always
remains
E + v are
⇥ Bmoving
= N
N = ⇥j)
if
N
v
⇥
B
+
⇧
⇤t
⇤
⇤t
eal ⇤domain whileOhms
the red
the ⇧
non-ideal
B
⇥ vinduction
⇥ Bregion.
= equation
⇧⇥N
lawcross section crosses
⇤
if N = v ⇥ B + ⇧
⇤t E⇧+⇥vw
B
⇧
⇥
v
⇥
B
=
⇧
⇥
N
⇥
B
=
N
(e.g.
N
=
⇤
B
⇥
B
=
0
for
w
=
v⇥j) v
xamples
which satisfy ⇥t B
⇤ (w
⇤ B)
0: and v smooth: No reconnection
a) N⌥=
v⇥
B +=⇧
but slippage.
⇤t
if
N
=
v
⇥
B
+
⇧
⇤
⇤t
⇤ B ⇧ ⇥ w ⇥ B = 0 for w = v
v
⇤
if
N
=
v
⇥
B
+
⇧
⇤
• Ideal• plasma
dynamics:
For
a) N = vb)
B
⇧ and
⇥
v⇥⇤t
⇥
= = ⇧0⇥Reconnection
N
N+=⇧ v and
⇥B
+
⇧
vB
singular:
(2D).
⇤
B
⇧
w
⇥
B
for
w
=
v
v
⇥
B
v
smooth:
No
reconnection
but
slippage.
⇤t
⇤
⇤t= 0
E
+
v
⇤
B
⇤ B ⇧ ⇥ w ⇥ B = a)0 Nfor
=Bv+ ⇧ v and v smooth:if NoNreconnection
= vvw
⇥B
but
slippage.
=
v
⇥
B
+
⇧
c)
N
=
⌅
⇥
+
⇧
Reconnection
(3D).
⇤t
b)
N
=
v
⇥
B
+
⇧
and
v
singular:
Reconnection
(2D).
the evolution
ideal.
called
slippage
(ofreconnection
plasma w.r.t.but
the slippage.
field). a) Nis=still
v⇥
B +This
⇧ is
and
v smooth:
1 No
⇤B Flux
E=+velocity.
v
⇤
=⇧ transport
⌥P
(n) ⇧ are
w=
v
evelocities
w
is
called
a
flux
transport
non-unique.
b)
N
v
⇥
B
+
and
v
singular:
Reconnection
(2D).
⇤
B
⇧
⇥
w
⇥
B
=
0
for
w
=
v
v
e
n
d v smooth:c)No
reconnection
but
slippage.
Nb)⌅=N v=⇥vB⇥+B⇧
(3D).
+ ⇧Reconnection
and
v
singular:
Reconnection
(2D).
⇤t
E.g Hall MHD:
MHD: Reconnection
c)
N(2D).
⌅=+ ⇧v ⇥Reconnection
B + ⇧ Reconnection
(3D).
d • vHall
singular:
c)
N
=
⌅
v
⇥
B
(3D).
1
No reconnection but1slippage.
a) N = v ⇥ B + ⇧ and v smooth:
E+v⇤B=
j ⇤ B ⇧ w = ve = v
j
en
en
econnectionb)(3D).
N = v ⇥ B + ⇧ and v singular: Reconnection (2D).
• many other cases under constraints: e.g. 2D dynamics with B ⌃= 0:
⇥ occurs in an
the topology m
of (3D).
the field
changes.
If
this
• Forc) N ⌅= v ⇥ B + ⇧ Reconnection
⇥j
e
+ called
v ⇤ Breconnection.
= .... +
+ ⌥ ·globally
(vj + jv)
... diffusion.
isolated region only E
it is
If it occurs
it is +
called
2
ne
⇥t
tic flux and field line topology: No reconnection.
Reconnection:
cur if the plasma is non-ideal.
⇤
B
⇤t
⇤t
B
⇧ ⇥ w ⇥ B = 0 for w = v
a) N = v ⇥ B + ⇧ and v smooth: No reconnection but slippage.
b) N = v ⇥ B + ⇧ and v singular: Reconnection (2D).
E + v ⇥ B = N with
(e.g.c) N ⌅=
= ⇥j)
v⇥B+⇧
⇧⇥v⇥B = ⇧⇥N
⇤
implies either
Reconnection
(3D).
a) E⋄B ≠ 0 and (E⋄B
≠ B⋄∇Φ).
This
is 3D- (or B≠0 or E⋄B ≠ 0) reconnection or 2.5D (guide
if N
= v⇥B
+⇧
field) reconnection
⇤
⇤ B ⇧ ⇥ w ⇥ B = 0 for w = v
v
⇤t
!
b) E⋄B
and E≠0 where
This is 2D (or x-point or E⋄B =0) reconnection.
and v smooth:
No=0reconnection
but B=0.
slippage.
!
and v singular: Reconnection (2D).
Remark: Note that these are conditions on the evolution of the electromagnetic field - they are
Reconnection
not(3D).
restricted to MHD.
Two cases of reconnection: vanishing or non-vanishing helicity source term (E⋄B).
3D Reconnection: Stationary
E+ v x B = N; E⋄B= N⋄B ≠0; stationary: E= -∇Φ.
=
⇥
⌅ +v⇥B=R
N
⌅
⌅
R⇥ ds + (x0 , y0 ) ⇤
N
⇧ ⌅⇤ ⌃
⇧ ⌅⇤ ⌃
id
n id.
id.
+ vn id. ⇥ B = R
N
+ vid. ⇥ B = 0
n id
v = vn
id.
+ vid.
General property: No direct coupling between external
flow (MAe ) and absolute reconnection rate.
Non-ideal solutions shows counter rotating flows above
and below the reconnection region.
Basicbasic
non-ideal
solution issolutions
inherentare
3Dnotand
The
3D reconnection
wellnonperiodic ⇤ 3D is not a simple approximation of any
approximated
by 2 or 2.5D reconnection solutions.
2/2.5 D solution.
!
2.2. Reconnection rate 3D (Rate of reconnected flux)
3D Reconnection: Reconnection
rate
Proof of counter rotating flows:
E · dl =
0=
⇤
⇥
L
⇥
L
E · dl +
E · dl = 2
⇥
⇥
⇥
Er dr
R1
Er dr = 2
⇥
Er dr
R2
v Bz dr
The rate of ‘mismatching’ of flux is
given by the difference of the plasma
The rate of ‘mismatching’
flux is given
the di⇥erence
of the plasma ve
velocityof above
and by
below
the
and below the reconnection region. The rate at which flux crosses any ra
reconnection
tween the origin and
the boundaryregion.
of D is given by the potential di⇥eren
line, i.e
d
mag
dt
=
⇤
L
E · dl =
⇤
R
w
in
w
out
⇥
⇤ B · dr
3D Reconnection: Stationary
3.
Composite solutions
Superimpose an ideal solution (x-type flow):
Region I: Ideal evolution
Region II: Slippage
(separation bounded)
Region
III:
Reconnection
(separation exponential)
Flux through region III
= E · dl.
E⋄B ≠0 Reconnection: Stationary
3.1.
3.1. Ideal
Idealvs.
vs.reconnective
reconnectiveflows
flows
vidvid>>vnvnidid
3.2. Slow
Slowideal
idealflow
flow
3.2.
vvidid
vvnn idid
< vvnn
id <
vvid
id
id
E⋄B ≠0 Reconnection
!
3.3.
Definitions
• Reconnection rate =
E dl along the field line where it is maximal
• External reconnection rate = voltage drop between the stagnation points of the
ideal flows = magnetic flux per unit time entering (or leaving) the reconnection
region
• Internal reconnection rate = sum of the voltage di⇥erence between the stagnation
points and the reconnection line = magnetic flux per unit time reconnecting within
the flux tube bounding the di⇥usion region only
d
rec
dt
=
⇥
E dl =
d
ext
rec
dt
+
d
int
rec
dt
E⋄B ≠0 Reconnection: Time-dependent
E⋄B ≠0 Reconnection
There are no pairwise reconnecting field lines. For processes bounded in
time reconnecting flux tubes exist which show a twist consistent with the
helicity production.
E⋄B ≠0 Reconnection: Null points
Process depends on the direction of the electric field at the null point. Two
basic cases: !
• Electric field aligned with the spine axis !
• Electric field tangent to the fan plane
Generic (hyperbolic) 3D null point. Fan:
blue, Spine: red.
Fan planes of two null points
intersect in a separator.
E⋄B ≠0 Reconnection: Null points
Spine aligned electric field (electric current)
Fan aligned electric field
Reconnection Cascades
Dundee braiding experiment:
BRIEF ARTICLE
Start with a force-free field
modelledTHE
on the
pigtail braid
AUTHOR
with total helicity H=0. Let the
field relax (resistive MHD)
!
! @t B r ⇥ (v ⇥ B) = ⌘ B
Taylor hypothesis: the end state
of a turbulent relaxation should
be a linear force-free field with
H=0 => B = 1 ez
!
Result: We found a non-linear
force-free field (with H=0) in
contradiction to Taylor’s
hypothesis. Wilmot-Smith, A.L., Pontin, D.I., Hornig,
G., A&A. 516 A5 (2010)
Pontin, D.I., Wilmot-Smith, A.L., Hornig,
G., Galsgaard, K, A&A, 525, A57 (2011)
Formation of a turbulent cascade of reconnection in a braided magnetic field:
3D Reconnection: Summary
• Three-dimensional reconnection is structurally different from 2 or 2.5 dimensional
reconnection. !
• The reconnection rate is given by the integral over the parallel electric field along
a certain field line. !
• The reconnection rate is not directly related to inflow or outflow velocities. !
• The flux undergoing reconnection is restricted to thin flux layers intersecting in the !
non-ideal region.!
• In particular we have no 1-1 correspondence of reconnecting field lines. Instead
we have flux surfaces which are mapped onto each other. !
• For a reconnection process which is limited in time, these flux surfaces are closed, !
i.e. they form perfectly reconnecting flux tubes. The counter rotating flows induce a !
twist in these tubes, which is consistent with the helicity production of the process.
Common Misconceptions about Reconnection:
•
•
•
“Magnetic reconnection heats the plasma” The Ohmic heating due to the
reconnection event itself is small due to the small size of dissipation
region and a low resistivity. The major part of the magnetic energy
released is converted into kinetic energy. The heating usually occurs
away from the reconnection region proper via a number of non-ideal
plasma processes (shocks, waves, adiabatic heating, viscous heating). !
“Magnetic reconnection occurs at an X-line” The notion of an X-line (a magnetic field line which has in a plane
perpendicular to it an X-type field) is a notion deduced from 2D (or 2.5D)
models. There is no distinguished “X-line” in a generic 3D magnetic field,
instead there are whole regions of magnetic flux which satisfy this
criterium. Moreover reconnection can also occur at an 0-line. !
“Magnetic reconnection occurs only along separators”
A separator is a field line formed by the intersection the fan planes of
two null points. It is, for instance, not present in Tokamaks but still we
have reconnection in these devices.!
Common Misconceptions about Reconnection:
!
•
“Magnetic reconnection is associated with fast (Alfvénic) flows” Mach numbers of 0.02 - 0.2 are typically found in stationary 2D
simulations. However, it is known that reconnection in reality is 3D and
time-dependent. There is no necessity for time-dependent processes to
have high flow velocities.!
!
!
!
!
!
!
Some open questions:
!
•
•
•
•
•
•
How does collisionless reconnection work in 3D configurations? !
Is there a “generic” dissipation mechanism? !
How common is magnetic reconnection? What is the spectrum of
reconnection? !
Is reconnection at magnetic null points spontaneous or driven? !
How does reconnection work in relativistic systems? !
Observational consequences?!
!
!
!
What is the problem?
• Complex three-dimensional geometry !
is the
Coupling
of problem?
micro (kinetic) physics and large scale dynamics (cross-scale
•What
coupling)
! three-dimensional geometry
• Complex
• Coupling of micro (kinetic) physics and large scale dynamics (cross-scale coupling)
• Spatial resolution in simulations is too low by many orders of magnitude. !
• Spatial resolution in simulations is too low by many orders of magnitude.
global length scale
108 m
9
=
=
10
electron-inertial length 10 1 m
global length scale
108 m
10
=
=
10
gyro effects
10 2 m
Typical
resolution of
of a
a 3D
(Typical
resolution
3D simulation:
simulation:300.
300)!
!
!
!