II' MAGNETOHYDRODYNAMICS

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II. MAGNETOHYDRODYNAMICS
(Space Climate School, Lapland, March,
2009) Eric Priest (St Andrews)
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CONTENTS
1. Introduction 2. Flux Tubes 3. MHD
Equations 4. Induction Equation 5. Equation
of Motion 6. Solar MHD 7. 2D magnetic
reconnection 8. 3D reconnection Conclusions
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1. INTRODUCTION
Magnetic Field Effects
-- exerts a force (creates structure) --
provides insulation -- stores energy (released
in CME or flare)
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Magnetohydrodynamics

• MHD - the study of the interaction between a
  magnetic field and a plasma, treated as a
  continuous medium

• This assumption of a continuous medium is valid
  for
• length-scales

• Chromosphere
• Corona

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2. FLUX TUBES
Magnetic Field Line -- Curve w. tangent in
direction of B.
Equation
or in 3D
In 2D _ _ _ _ _ _
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Magnetic Flux Tube
Surface generated by set of field lines
intersecting simple closed curve.

• Strength (F) -- magnetic flux crossing a section
• i.e., _ _ _ _ __ _ _ _

(ii) But ---gt F is constant along tube
(iii) If cross-section is small, _ _ _ _ _ _ _
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Eqns of Magnetohydrodynamics
Model interaction of B and plasma (conts medium)
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3. FUNDAMENTAL EQUATIONS of MHD

• Unification of Eqns of
• (i) Maxwell

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(ii) Fluid Mechanics
or (D / Dt)
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In MHD

• 1. Assume v ltlt c --gt Neglect_ _ _

• 2. Extra E on plasma moving
• _ _ _ _

• 3. Add magnetic force
• _ _ _ _

• Eliminate E and j take curl (2), use (1) for j

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4. INDUCTION EQUATION

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Induction Equation
N.B. (i) --gt B if v is known
primary variables
(ii) In MHD, v and B are induction
eqn eqn of motion --gt basic physics
(iii) are secondary variables
(iv) B changes due to transport diffusion
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Induction Equation
A B
magnetic

• (v) --

Reynolds number
eg, L0 105 m, v0 103 m/s --gt Rm
108
(vi) A gtgt B in most of Universe --gt
B moves with plasma -- keeps its energy
Except SINGULARITIES -- j B large
Form at NULL POINTS, B 0 --gt reconnection
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(a) If Rm ltlt 1

• The induction equation reduces to

• B is governed by a diffusion equation
• --gt field variations on a scale L0
• diffuse away on time
  

with speed
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(b) If Rm gtgt 1
The induction equation reduces to
and Ohm's law --gt
Magnetic field is
frozen to the plasma
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5. EQUATION of MOTION
(1) (2) (3) (4)

• In most of corona, (3) dominates

• Along B, (3) 0, so (2) (4) important

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Magnetic force
Tension B2/ ----gt force when lines
curved
Magnetic field lines have a
Pressure B2/(2 )----gt force from high to low
B2
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Ex
Expect physically
(check mathematically)
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Ex
Expect physically
(check mathematically)
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Equation of Motion
(1) (2) (3) (4)

Plasma beta

Alfvén speed
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Typical Values on Sun
N (m-3) 106 N (cm-3), B (G) 104 B (tesla)
3.5 x 10 -21 N T/B2, vA 2 x 109 B/N1/2
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6. In Solar MHD
Waves,
Instabilities,
We study Equilibria,
Magnetic reconnection
in dynamos, magnetoconvection, sunspots,
prominences, coronal loops, solar wind,
coronal mass ejections, solar flares
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Example
Shapes -
caused by magnetic field (force-free)
Fineness -
small scale of heating process small
Structure along loops -
hydrostatics/hydrodynamics (--H)
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7. MAGNETIC RECONNECITON

• Reconnection is a fundamental process in a
  plasma
• Changes the topology
• Converts magnetic energy to heat/K.E
• Accelerates fast particles
• In Sun ---gt Solar flares, CMEs / heats Corona

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In 2D takes place only at an X-Point

• -- Current very large --gt ohmic heating
• -- Strong diffusion allows field-lines to break
• / change connectivity
• and diffuse through plasma

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Reconnection can occur when X-point collapses
Small current sheet width --gt magnetic field
diffuses outwards at speed


_ _ _
 
 
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If magnetic field is brought in by a flow
(vx - Ux/a vy Uy/a) then a steady balance
can be set up
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Sweet-Parker (1958)
Simple current sheet - uniform inflow
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Petschek (1964)

• Sheet bifurcates -
• Slow shocks - most of energy
• Reconnection speed ve --
• any rate up to maximum

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8. 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
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(ii) Global Topology of Complex Fields
In 2D -- Separatrix curves
In 3D -- Separatrix surfaces
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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.
In complex fields we form the SKELETON -- set of
nulls, separatrices -- from fans
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(iii) 3D Reconnection
Can occur at a null point or in absence of
null
At Null -- 3 Types of Reconnection
Spine reconnection
Fan reconnection
Separator reconnection
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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
Colour shows locations of strong Ep stronger Ep
Final twist
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9. CONCLUSIONS

• Reconnection fundamental process -
• - 2D theory well-developed
• - 3D new voyage of discovery
• topology
• reconnection regimes ( or - null)

• Coronal heating
• Solar flares