MAGNETOHYDRODYNAMICS

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MAGNETOHYDRODYNAMICS
(Bozeman, July, 2005) Eric Priest
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Check Google for eric priest
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CONTENTS
1. Introduction 2. Flux Tubes Example 3.
Fundamental Equations 4. Induction
Equation Example 5. Equation of Motion
Example 6. Equilibria 7. Waves
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Traditionally
close link Mathematics - Astronomy
e.g., Gregory Chair of Maths James Gregory
(1638-1675)
Invented Reflecting Telescope
Co-founder of Calculus
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James Gregory

• Elected FRS (Academician) 1668 (age 30)

• Appointed 1st regius prof maths at St Andrews
  (1668)

• Given hall as lab -- see line for telescope
  along meridian

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James Gregory
- wrote 1st text book on Calculus -
discovered
-- General binomial theorem
-- Taylor expansions
-- Ratio test for convergence of a series
-- Series for sin x and tan x
-- Integral of log x and sec x
-- Differentiation is inverse of integration
-- How to use change of variable in integration
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1. INTRODUCTION
Role of Theory ?

• Not -- reproduce images
• Nor explain every observation

Understand Basic Processes
-- step-by-step -- simple -gt
sophisticated model

• Listen to Observers -- clues

• Diff. Types Theory -- complement
• -- analytical -- computational -- data interp.

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The Sun
Amazingly rich variety of MHD phenomena
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Magnetic Field Effects
E.g., A Sunspot
B exerts a force

• - creates intricate structure

What is equilibrium ? / nature of instabilities
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E.g., A Prominence
Magnetic tube w. cool plasma
B --gt Thermal Blanket Stability
What is global equilibrium? / fine structure
?
Why erupt ?
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E.g., A Coronal Mass Ejection

Magnetic instability ??
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E.g., A Solar Flare
(from TRACE)
B stores energy - converted to other forms

Why erupt ? How is energy converted?
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Magnetohydrodynamics

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

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(ii) But
--gt No flux is created/destroyed inside flux
tube So is constant along tube
(iii) If cross-section is small,

B lines closer --gt A smaller B increases
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EXAMPLE
Sketch the field lines for

• Eqn. of field lines

y2 - x2 constant

• Sketch a
• few field lines

? arrows, spacing
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(iii) Directions of arrows
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(iv) Spacing
At origin B 0.
A "neutral" or "null" point
Magnetic reconnection energy conversion
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? Studying Sun Entails
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? THOUGHT of MATHS FILL with CONFIDENCE
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Eqns of Magnetohydrodynamics
Model interaction of B and plasma (conts medium)
PDE's for v and B as functions of x,y,z,t
As beautiful as the Sun !! -- Nonlinear
Impossible to solve completely
So approximate in imaginative ways --
Construct models for many dynamic phenomena on Sun
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3. FUNDAMENTAL EQUATIONS of MHD

• Interaction of B and Plasma
• Unification of Eqns of
• (i) Maxwell

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

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

• (v) --

magnetic
Reynold number
eg, L0 105 m, v0 103 m/s --gt Rm
108
(vi) A gtgt B in most of Universe --gt
B frozen to plasma -- keeps its energy
Except SINGULARITIES -- j B large
Form at NULL POINTS, B 0
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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

• E.g. sunspot ( 1 m2/s, L0 106 m), td
  1012 sec
• for whole Sun (L0 7x108 m), td 5x1017
  sec

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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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Magnetic Flux Conservation Magnetic Field
Line Conservation
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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

Scale Height
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Example
MHS Eqm. along B
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On Earth H 9 km,
so on munro (1 km) p 0.9 p0
or on Everest (9 km) p 0.37 p0
T 5000 K, H T 2 x 106
K, H
150 km
50 Mm
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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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EXAMPLE
Find Magnetic Pressure force, Magnetic Tension
force and j x B force for
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Ex.
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Ex.
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Equation of Motion
(1) (2) (3) (4)

Plasma beta

Alfvén speed
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6. EQUILIBRIA
(1) (2) (3) (4)

• If v ltlt vA, then (1) ltlt (3) and so

( Equilibrium)
Magnetohydrostatic
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(2) (3) (4)

• If Lo ltlt H, then (4) ltlt (2) and

• If Lo ltlt 2H/ , then (4) ltlt (3) and

(Magnetostatic Equilibrium)
(Force-Free)

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Example
Shapes -
caused by magnetic field (force-free)
Fineness -
small scale of heat sources small kperp
Structure along loops -
hydrostatics/hydrodynamics (?H)
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7. WAVES
(i) Sound Waves (B0 0)

Dispersion Relation
Waves propagate with speed
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(ii) Magnetic Waves (p0 0)
Repeat, but uniform (B0) - include j x B
force - assume wave propagates at angle to B0
Either
Alfvén Waves
Incompressible - due to magnetic tension
Or Compressional Alfvén Waves
Compressible - due to magnetic pressure
- propagate at speed
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(iii) MHD Waves (p0 and B0 nonzero)

• Alfvén Wave is unaffected

• Compressional Alfvén Wave and Sound Wave are
  coupled

Slow Magnetoacoustic Wave (Slow-Mode) Fast
Magnetoacoustic Wave (Fast-Mode)
Propagate slower/faster than Alfvén Wave
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8. CONCLUSIONS

• Understanding how B interacts with plasma
• Key to many solar system phenomena

• Two main equations
• Induction equation -- advection diffusion
• Eqn. motion -- magnetic tension pressure
  forces

• --gt Theory for equilibria, waves,
  instabilities,
• and reconnection

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PS. SOLUTIONS
Ex. 1
--gt
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Ex. 2
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sunearth.gsfc.nasa.gov/sunearthday/2004/vt_edu2004
_venus_back_his.htm
In 1663, Rev. James Gregory (1638 - 1675) who
was considered one of the most important
mathematicians of the 17th century suggested that
a more accurate measurement of the Earth-Sun
distance could be made using the transit of
Venus. Sir Edmund Halley (1656-1742), the
namesake for Halley's Comet, made the same
suggestion 14 years later in 1677 and published
an important paper on the details of this
technique in 1716. In science, it is not always
'being first' or 'being correct' that gets you
into the history books.
But - James Gregory was not a minister.
Jeremiah Horrocks used transit in 1639 to obtain
1AU 60 million miles
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EXAMPLES 1 2
Sketch field lines for
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Lo lt H --- p const
Lo gt H --- p falls
- acts from high to low p - is normal to isobars
isobars
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EXAMPLE .
Diffusion of a 1D Field satisfies
Find B(x,t) if B0 for xgt0 B(x,0)
- B0 for xlt0
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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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Force-Free Fields

is constant along each field line
So
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Constant- or fields
linear force-free
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(iv) Shock Waves

• Nonlinear sound wave can steepen to a shock
  wave -- propagates at speed gt cs

• In MHD 3 modes

• (1) Slow-mode shock
• - propagates faster than slow-mode speed
• - turns B towards normal

(2) Fast-mode shock - propagates faster than
fast-mode speed - turns B away from normal
(3) Finite-amplitude Alfvén Wave - no change in p
- reverses tangential magnetic field
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Slow-mode Alfvén Fast-mode