2. MHD Equations

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2. MHD Equations
2.1 Introduction
Many processes caused by magnetic field (B)
Sun is NOT a normal gas
Sun is in 4th state of matter ("PLASMA")
behaves differently from normal gas
B and plasma -- coupled (intimate, subtle)
B exerts force on plasma -- stores energy
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MOST of UNIVERSE is PLASMA
Ionosphere --gt Sun (8 light mins)
Learn basic behaviour of plasma from Sun
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Magnetic Field Effects
E.g., A Sunspot
B exerts a force

• - creates intricate structure
• _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

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E.g., A Prominence
Magnetic tube w. cool plasma
B --gt Thermal Blanket Stability
What is global equilibrium? / fine structure
? _ _ _ _ _ _ _
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E.g., a Coronal Mass Ejection
_ _ _ _ _ _ _ _ _ _ _ _ _ _
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E.g., A Solar Flare
(from TRACE)
B stores energy - converted to other forms

• _ _ _ _ _ _ _
• _ _ _ _ _ _ _
• _ _ _ _ _ _ _

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2.2 Flux Tubes Field Lines
Magnetic Field Line -- Curve with 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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But
--gt No flux is created/destroyed inside flux
tube So is constant along tube
Ex 2.1 Prove the above result that, if ,
then is constant along a flux tube.
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If cross-section is small,
_ _ _ _ _
B lines closer --gt A smaller B increases
Thus, when sketching field lines, ensure they
are closer when B is stronger
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To sketch magnetic field lines

• Solve
• Sketch one field line
• Sketch other field lines, remembering that
  B increases as the field lines
• become closer
• (iv) Put arrows on the field lines

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EXAMPLE
Sketch the field lines for

1. Eqn. of field lines

_ _ _ _ _ _ _ _ _ _

• 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. _ _ _ _ _ _ _ _ _ _ _ _ _ _
Magnetic reconnection energy conversion
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Examples
Ex 2.2 Sketch the field lines for (a) Byx (b)
Bx1, Byx
Ex 2.3 Sketch the field lines for (a) Bxy,
Bya2x (b) Bxy, By-a2x
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2.3 Plasma Theory

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

• But there are different ways of modelling a
  plasma

(i) MHD -- fluid eqns Maxwell
(ii) 2-fluid-- electron/ion fluid eqns Maxwell
(iii) Kinetic -- distribution function for
each species of particle
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Eqns of Magnetohydrodynamics

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Magnetohydrodynamics (MHD)

• Unification of Eqns of
• (a) Maxwell

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(b) 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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2.4 Induction Equation
_ _ _ _ _ _
Describes how B moves with plasma / diffuses
through it
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N.B.

• In MHD, v and B are _ _ _ _ _ _ _ _ _ _

• Induction eqn
• eqn of motion
• --gt basic processes

• are secondary variables

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INDUCTION EQUATION
I II

• B changes due to transport diffusion

• -- _ _ _ _ _ _ _
• _ _ _ _ _ _

eg, L0 105 m, v0 103 m/s --gt Rm
108

• I gtgt II in most of Solar System --gt

B frozen to plasma -- keeps its energy
Except Reconnection -- j B large
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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 _ _ _ _ _ _ _ _ _ _ _ _ _
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Magnetic Flux Conservation Magnetic Field
Line Conservation
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2.5 EQUATION of MOTION
(1) (2) (3) (4)
_ _ _ _ _ _ _
_ _ _ _ _ _
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Typical Values on Sun
Photosphere Corona
N (m-3) 1023 1015
T (K) 6000 106
B (G) 5 - 103 10
106 - 1 10-3
vA (km/s) 0.05 - 10 103
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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Magnetic force
Tension B2/ ----gt _ _ _ _ _ _ _ _ _
_
Magnetic field lines have a
Pressure B2/(2 )----gt _ _ _ _ _ _ _ _ _ _
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EXAMPLE
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Examples
Find Magnetic Pressure force, Magnetic Tension
force and j x B force for Ex 2.4 (a) (b)
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Hydrostatic Equilibrium
(1) (2) (3) (4)

• In most of corona, (3) dominates

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

_ _ _ _ _ _ _ _ _
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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 _ _ _ _ _
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When is MHD valid ?
We assumed in deriving MHD eqns -- vltltc,
constant, and plasma continuous

• Can treat plasma as a continuous medium when

Chromosphere Corona

• When MHD can still be valid when
  particles collide with B

ri 1 m(corona)

• MHD equations can be derived by taking integrals
  of a
• kinetic equation for particles (but tricky)