Problems in MHD Reconnection ?? (Cambridge, Aug 3, 2004)

Eric PriestSt Andrews

CONTENTS

1. Introduction

2. 2D Reconnection

3. 3D Reconnection

4. [Solar Flares]

5. Coronal Heating

1. INTRODUCTION

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:

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Magnetosphere

Reconnection -- at magnetopause (FTE’s)& in tail (substorms) [Birn]

Solar Corona

Reconnection key role inSolar flares, CME’s [Forbes] +

Coronal heating

Induction Equation∂B

∂ t= ∇ × (v × B) + η∇2B + ......

B changes due to transport + diffusion

[Drake, Hesse, Pritchett]

Rm>>1 in most of Universe -->

B frozen to plasma -- keeps its energy

Except SINGULARITIES -- & j & E large ∇B

Heat, particle accelern

Current Sheets - how form ?

Driven by motions

At null points

Occur spontaneously

By resistive instability in sheared field

Along separatrices

By eruptive instability or nonequilibrium

In many cases shown in 2D but ?? in 3D

2. 2D RECONNECTION

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)

In 2D takes place only at an X-Point-- Current very large-- Strong dissipation allows field-lines to break

/ change connectivity

Sweet-Parker (1958)

Simple current sheet

- uniform inflow

€

Mass conservation: L v i = l vo

Advection / diffusion: v i = η / l

Accelerate along sheet: vo = vA

€

Rmi =L vA

η,

€

Recon. Rate M i =v i

vAi

=1

Rmi1/ 2

Petschek (1964)

SP sheet small - bifurcates

Slow shocks- most of energy

€

M e =ve

vA

=π

8 log Rm e

≈ 0.1

Reconnection speed ve --

any rate up to maximum

?? Effect of Boundary Conditions on Steady Reconnection

NB - lessons:

3. Global ideal environment depends on bc’s

5. Maximum rate depends on bc’s

1. Bc’s are crucial

2. Local behaviour is universal - Sweet-Parker layer

4. Reconnection rate - the rate at which you drive it

Newer Generation of Fast Regimes Depend on b.c.’s

Almost uniform Nonuniform

Petschek is one particular case -

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ηcan occur if enhanced in diff. region

Theory agrees w numerical expts if bc’s same

Nature of inflow affects regime

Converging Diverging

€

Me =f

Rme1/ 2

-> Flux Pileup regime

Same scale as SP, but different f,

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

Simplest B = (x, y, -2z)

Spine Field LineFan Surface

(i) Structure of Null Point

Many New Features

2 families of field lines through null point:

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

transfers flux from one 2D region to another.

In 3D, reconnection at separator

transfers flux from one 3D region to another.

In 2D, reconnection at X

? 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

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Separate --

Touching --

Enclosed

Higher-Order Behaviour

Multiple separators

Coronal null points

[? more realistic models corona: Longcope, Maclean]

(iii) 3D Reconnection

At Null -- 3 Types of Reconnection:

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Can occur at a null point (antiparallel merging ??)

or in absence of null (component merging ??)

Spine reconnection Fan reconnection

[Pontin, Hornig]

Separator reconnection[Longcope, Galsgaard]

Spine ReconnectionAssume kinematic, steady,

ideal. Impose B = (x, y, -2z)Solve E + v x B = 0 and curl E = 0 for v and E.

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--> E = grad FB.grad F = 0, v = ExB/B2

-> Singularity at Spine

Impose continuous flow on lateral boundary across fan

Fan Reconnection

(kinematic)

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Impose continuous flow on top/bottom boundary

across spine

[? Resolve singularities,

? Properties:

Pontin, Hornig, Galsgaard]

Separator Reconnection

(Longcope)

Numerical: Galsgaard & Parnell

In Absence of NullQualitative model - generalise Sweet Parker.

2 Tubes inclined at :

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ϑ

Reconnection Rate (local):

Varies with - max when antiparl

€

ϑ

Numerical expts: (i) Sheet can fragment

(ii) Role of magnetic helicity€

v i =vA

Rmi1/ 2 [2 sin 1

2ϑ ]1/ 2

Numerical Expt (Linton & Priest)

3D pseudo-spectral 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

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locations of strong Ep

stronger Ep

Final twist

€

π

Features

Reconnection fragments (cf Parnell & Galsgaard)

€

F 2 = 2 ×Φ

2πF 2

€

∴Φ=π

Complex twisting/ braiding created

Initial mutual helicity = final self helicity

Higher Rm -> more reconnection locations & braiding

Approx conservation of magnetic helicity:

? keep as tubes / add twist: Linton

(iv) Nature of B-line velocities (w)

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Inside D, w exists everywhere except at X-point.

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(E + w × B = 0)

Flux tubes rejoin perfectly

B-lines change connections at X

Outside diffusion region (D), v = w

[Hornig, Pontin]

In 3D : w does not exist for an isolated diffusion region (D)

∃ i.e., no solution for w to

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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

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Tubes

split

&

flip

Fully 3D Example

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Tubes split & flip -- but

don’t rejoin perfectly

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 -

€

Qmagnetic catastrophe/instabilitydrives reconnection

Reconnection heats loops/ribbons

[Forbes]

- rise / separate

5. HOW is CORONA HEATED ?

Bright Pts,

Loops,

Holes

Recon-nection 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 unconnectedInitial 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

Heating localised in space --

Impulsive in time

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(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

Time for all field lines to reconnect

only 1.5 hours

(Close, Parnell, Priest):

- each source connected to 8 others

Coronal Tectonics Model(Priest, Heyvaerts & Title)

Each "Loop" --> surface in many sources Flux from each source topology distinct -- Separated by separatrix surfaces

Corona filled w. myriads of separatrix/ separator J sheets, heating impulsively

As sources move, coronal fields slip ("Tectonics") --> J sheets on separatrices & separators

--> Reconnect --> Heat

Fundamental Flux Units

Intense tubes (B -- 1200 G, 100 km, 3 x 1017 Mx)

100 sources

10 finer loops

not Network Elements

Each network element -- 10 intense tubes Single ephemeral region (XBP) --

Each TRACE Loop --

80 seprs, 160 sepces

800 seprs, 1600 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 separatrixElementary (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

Reconnection on Sun crucial role - * Solar flares * Coronal heating

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

?? Extra Questions ??

? Threshold / conditions for onset of reconnection

? Occur equally easily at nulls or without

? Rate and partition of energy

? How does reconnection accelerate particles -

cf DC electric fields, stochastic accn, shocks

? Determines where non-null recon. occurs

? Role of microscopic processes

PS-Example from SOHO (EIT - 1.5 MK)

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Eruption

Inflow to reconnection site

Rising loops that have cooled

(Yokoyama)

Example from TRACE

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Eruption

Rising loops

Overlying current sheet (30 MK) with downflowing plasma

(Priest and Schrijver 1999)

Reconnection proceeds

New loopsform

Old loops cool

PS-B-Lines for 1 Tube

PS-Cause of Eruption

?Magnetic

Catastrophe

2.5 D Model

Numerical 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 LoopsEach flux element --> many neighbours

But in practice each source has 8 connections