Transcript Problems in MHD Reconnection ?? (Cambridge, Aug 3, 2004) Eric Priest St Andrews.

Problems in MHD Reconnection ??
(Cambridge, Aug 3, 2004)
Eric Priest
St Andrews
CONTENTS
1. Introduction
2. 2D Reconnection
3. 3D Reconnection
4. [Solar Flares]
5. Coronal Heating
1. INTRODUCTION
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 Reconnection is a fundamental process in a plasma:
 Changes the topology
 Converts magnetic energy to heat/K.E
 Accelerates fast particles
 In solar system --> dynamic processes:
Magnetosphere
Reconnection -- at magnetopause (FTE’s)
& in tail (substorms) [Birn]
Solar Corona
Reconnection key role in
Solar flares, CME’s [Forbes] +
Coronal heating
Induction Equation
B
2
   (v  B)   B + ......
t
[Drake, Hesse, Pritchett]
 B changes due to transport + diffusion
 Rm>>1 in most of Universe -->
B frozen to plasma -- keeps its energy
Except SINGULARITIES -- B & j & E large
Heat, particle accelern
Current Sheets how form ?
 Driven by motions
 At null points
 Along separatrices
 Occur spontaneously
 By resistive instability in sheared field
 By eruptive instability or nonequilibrium
 In many cases shown in 2D but ?? in 3D
2. 2D RECONNECTION

In 2D takes place only at an X-Point
-- Current very large
-- Strong dissipation allows field-lines to break
/ change connectivity

In 2D theory well developed :
* (i) Slow Sweet-Parker Reconnection (1958)
* (ii) Fast Petschek Reconnection (1964)
* (iii) Many other fast regimes -- depend on b.c.'s
 Almost-Uniform (1986)
 Nonuniform (1992)
SweetParker
(1958)
Simple
current sheet
- uniform
Mass conservat ion
: L vi  l vo
inflow
Advection/ diffusion: v i   / l
Accelerat e along sheet:
vo  vA
vi
1
Recon. Rate M i 
 1/ 2
v Ai Rmi
Rmi 
L vA

,
Petschek (1964)
 SP sheet small
- bifurcates
Slow shocks
- most of
energy
 Reconnection
speed ve -any rate up to
maximum
ve

Me  
 0.1
v A 8 log Rm e
?? Effect of Boundary Conditions on Steady
Reconnection
NB - lessons:
1. Bc’s are crucial
2. Local behaviour is universal Sweet-Parker layer
3. Global ideal environment depends on bc’s
4. Reconnection rate the rate at which you drive it
5. Maximum rate depends on bc’s
Newer Generation of Fast Regimes
 Depend on b.c.’s
Almost uniform
Nonuniform
 Petschek is one particular case -
can occur if  enhanced in diff. region
 Theory agrees w numerical expts if bc’s same
Nature of inflow affects regime
Converging
Diverging
-> Flux Pileup regime
Same scale as SP,
f
but different f,
Me 
1/ 2
Rme
different inflow
 Collless models w. Hall effect (GEM, Birn, Drake) ->
fast reconnection - rate = 0.1 vA
2D - Questions ?
 2D mostly understood
 But -- ? effect of outflow bc’s -- fast-mode MHD characteristic
-- effect of environment
 Is nonlinear development of tmi understood ??
 Linking variety of external regions to collisionless
diffusion region ?? [Drake, Hesse, Pritchett, Bhattee]
3. 3D RECONNECTION
Many New Features
(i) Structure of Null Point
Simplest
B = (x, y, -2z)
2 families of field lines
through null point:
Spine Field Line
Fan Surface
Most generally, near a Null
(Neukirch…)
Bx = x + (q-J) y/2,
By = (q+J) x/2 + p y,
Bz = j y - (p+1) z,
in terms of parameters p, q, J (spine), j (fan)
J --> twist in fan,
j --> angle spine/fan
(ii) Topology of
Fields - Complex
In 2D -Separatrix curves
In 3D -Separatrix surfaces
-- intersect in Separator
In 2D, reconnection at X
transfers flux from one
2D region to another.
In 3D, reconnection at
separator
transfers flux from one
3D region to another.
? Reveal structure of complex field
? plot a few arbitrary B lines
E.g.
2 unbalanced sources
SKELETON -- set of nulls, separatrices -from fans
2 Unbalanced
Sources
Skeleton:
null + spine + fan
(separatrix dome)
Three-Source Topologies
Simplest configuration w. separator:
Sources, nulls, fans -> separator
Looking Down on Structure
Bifurcations from one state to another
Movie of Bifurcations
Separate -Touching -Enclosed
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Higher-Order Behaviour
Multiple
separators
Coronal
null points
[? more realistic models corona: Longcope,
Maclean]
(iii) 3D Reconnection
Can occur
at a null point (antiparallel merging ??)
or in absence of null (component merging ??)
At Null -- 3 Types of
Reconnection:
Spine reconnection
Fan reconnection
[Pontin, Hornig]
Separator reconnection
[Longcope, Galsgaard]
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Spine Reconnection
Assume kinematic, steady,
ideal. Impose B = (x, y, -2z)
Solve E + v x B = 0 and
curl E = 0 for v and E.
--> E = grad F
B.grad F = 0, v = ExB/B2
Impose continuous flow on
lateral boundary across fan
-> Singularity at Spine
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Fan Reconnection
(kinematic)
Impose continuous flow
on top/bottom boundary
across spine
[? Resolve singularities,
? Properties:
Pontin, Hornig, Galsgaard]
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Separator
Reconnection
(Longcope)
Numerical:
Galsgaard &
Parnell
In Absence of Null
Qualitative model generalise Sweet Parker.
2 Tubes inclined at  :
Reconnection Rate (local):
v
1
1/ 2
A

v i  1/ 2 [2 sin 2  ]
Rmi
Varies with  - max when antiparl
Numerical expts:
(i) Sheet can fragment
(ii) Role of magnetic helicity
Numerical Expt (Linton & Priest)
3D pseudospectral code,
2563 modes.
Impose initial
stagn-pt flow
v = vA/30
Rm = 5600
Isosurfaces of B2:
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B-Lines
for 1
Tube
Colour
shows
locations of
strong Ep
stronger Ep
Final twist 
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Features
 Reconnection fragments (cf Parnell & Galsgaard)
 Complex twisting/ braiding created
 Approx conservation of magnetic helicity:
Initial mutual helicity = final self helicity
 2
2
F 2
F
2
  
 Higher Rm -> more reconnection locations & braiding

? keep
 as tubes / add twist: Linton
(iv) Nature of B-line velocities (w)
[Hornig, Pontin]
(E + wB = 0)
 Outside diffusion
region (D), v = w
In 2D
 Inside D, w exists
everywhere except at Xpoint.
 B-lines change
connections at X
 Flux tubes rejoin
perfectly
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In 3D : w does not exist for an
isolated diffusion region (D)

i.e.,

no solution for w to
E + wB = 0

fieldlines continually
change their connections in D
(1,2,3 different B-lines)

flux tubes split, flip and in
general do not rejoin
perfectly !
Locally 3D Example
Tubes
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split
&
flip
Fully 3D Example
Tubes split & flip - but
don’t rejoin
perfectly
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3D - Questions ?
 Topology - nature of complex coronal fields ?
[Longcope, Maclean]
 Spine, fan, separator reconnection - models ??
[Galsgaard, Hornig, Pontin]
 Non-null reconnection - details ??
[Linton]
 Basic features 3D reconnection such as nature w ?
[Hornig, Pontin]
4. FLARE - OVERALL PICTURE
Magnetic tube twisted - erupts magnetic catastrophe/instability
drives reconnection
Reconnection heats loops/ribbons
- rise /
separate
[Forbes]
5. HOW is CORONA HEATED ?
Bright Pts,
Loops,
Holes
Reconnection
likely
Reconnection can Heat Corona:
(i) Drive Simple Recon. at Null by photc. motions
--> X-ray bright point (Parnell)
(ii) Binary Reconnection -- motion of 2 sources
(iii) Separator Reconnection -- complex B
(iv) Braiding
(v) Coronal Tectonics
(ii) Binary Reconnection
(P and Longcope)
Many magnetic sources in solar surface
 Relative motion of 2 sources -- "binary" interaction
 Suppose unbalanced and connected --> Skeleton
 Move sources --> "Binary" Reconnection
 Flux constant - - but individual B-lines reconnect
Cartoon Movie (Binary Recon.)
Potential B
Rotate one
source
about
another
(iii) Separator Reconnection
[Longcope, Galsgaard]
 Relative motion of 2 sources in solar surface
 Initially unconnected
Initial state of numerical expt. (Galsgaard & Parnell)
Comput. Expt. (Parnell / Galsgaard
Magnetic field
lines -- red
and yellow
Strong
current
Velocity
isosurface
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(iv) Braiding
Parker’s Model
Initial B uniform / motions braiding
Numerical Experiment (Galsgaard)
Current sheets grow --> turb. recon.
Current Fluctuations
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Heating localised in space -Impulsive in time
(v) CORONAL TECTONICS
? Effect on Coronal Heating of
“Magnetic Carpet”
* (I) Magnetic sources in surface are
concentrated
* (II) Flux Sources Highly Dynamic
Magnetogram movie (white +ve , black -ve)
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


Sequence is repeated 4 times
Flux emerges ... cancels
Reprocessed very quickly (14 hrs !!!)
? Effect of structure/motion of carpet on Heating
Life of Magnetic Flux
in Surface

(a) 90% flux in Quiet Sun
emerges as ephemeral regions
 (b) Each pole migrates to
boundary, fragments
-->
10 "network elements" (3x1018 Mx)

(c) -- move along boundary
-- cancel
From observed
magnetograms
- construct
coronal field
lines
- statistical properties:
most close low down
- each source
connected to 8 others
Time for all field
lines to reconnect
only 1.5 hours
(Close, Parnell, Priest):
Coronal Tectonics Model
(Priest, Heyvaerts & Title)
 Each "Loop" --> surface in many sources
 Flux from each
source topology
distinct -Separated by
separatrix surfaces
 As sources move, coronal fields slip ("Tectonics")
--> J sheets on separatrices & separators
--> Reconnect --> Heat
 Corona filled w. myriads of separatrix/
separator J sheets, heating impulsively
Fundamental Flux Units



not Network Elements
Intense tubes (B -- 1200 G, 100 km, 3 x 1017 Mx)
Each network element -- 10 intense tubes
Single ephemeral
region (XBP) -100 sources
800 seprs, 1600 sepces

Each TRACE
Loop -10 finer loops
80 seprs, 160 sepces
Theory
 Parker -- uniform B -- 2 planes -- complex motions
 Tectonics -- array tubes (sources) -- simple motions
(a) 2.5 D Model
 Calculate equilibria -Move sources -->
Find new f-f equilibria
 --> Current sheets
and heating
3 D Model
Demonstrate
sheet
formation
Estimate
heating
Preliminary
numerical expt.
(Galsgaard,
Mellor …)

Results
Heating uniform along separatrix
Elementary (sub-telc) tube heated uniformly

But 95% photc. flux closes low down in carpet
-- remaining 5% forms large-scale connections
 --> Carpet heated more than large-scale corona

So unresolved observations of coronal loops
--> Enhanced heat near feet in carpet
--> Upper parts large-scale loops heated uniformly
& less strongly
6. CONCLUSIONS
 2D recon - many fast regimes - depend on nature inflow
 3D - can occur with or without nulls
- several regimes (spine, fan, separator)
- sheet can fragment - role of twist/braiding
- concept of single field-line vely replaced
- field lines continually change connections in D
- tubes split, flip, don’t rejoin perfectly

Reconnection on Sun crucial role * Solar flares
* Coronal heating
?? Extra Questions ??
 ? Threshold / conditions for onset of reconnection
 ? Occur equally easily at nulls or without
 ? Determines where non-null recon. occurs
 ? Rate and partition of energy
 ? Role of microscopic processes
 ? How does reconnection accelerate particles cf DC electric fields, stochastic accn, shocks
PS-Example from SOHO (EIT - 1.5 MK)
 Eruption
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 Inflow to
reconnection
site
 Rising loops
that have cooled
(Yokoyama)
Example from TRACE
 Eruption
 Rising loops
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 Overlying
current sheet
(30 MK) with
downflowing
plasma
Reconnection proceeds
New loops
form
Old loops
cool
PS-B-Lines for 1 Tube
PS-Cause
of Eruption
?
Magnetic
Catastrophe
2.5 D
Model
Numeric
al Model
Suggestive
of
Catastrophe
PS- Reconn - Elegant Explanation for many
Recent Space Observations
Yohkoh
 Hottest loops are cusps or interacting loops
 X-ray jets - accelerated by reconnection
SOHO
 X-ray bright points (NIXT, EIT, TRACE)
 Magnetic carpet (MDI)
 Explosive events (SUMER)
 Nanoflares (EIT, TRACE, CDS)
TRACE Loop
Reaches to
surface in
many
footpoints.
Separatrices
& Separators
form web in
corona
Corona - Myriads Different Loops
Each flux element --> many neighbours
But in practice each source has 8 connections