Lecture 6 - Magnetohydrodynamics I
o
Topics in today’s lecture:
o Introduction to MHD
o Flux Tubes
o Fundamental Equations
o Induction Equation
o Equation of Motion
Lecture 6 - MHD I
Magnetic Field Effects
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E.g., Simple sunspot.
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B exerts a force on plasma.
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What is equilibrium condition?
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What makes it unstable?
Lecture 6 - MHD I
Magnetic Field Effects
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E.g., prominence/filament.
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Magnetic tube with cool plasma.
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B provides stability and thermal
shielding.
o
What is global equilibrium?
o
What drives it to erupt?
Lecture 6 - MHD I
Magnetic Field Effects
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E.g., Coronal Mass Ejection (CME)
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Erupting magnetic tube with hot plasma.
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B determines shape and acceleration.
o
An example of an MHD instability?
Lecture 6 - MHD I
Magnetic Field Effects
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E.g., Solar flare.
o
B stores energy.
o
During a flare, energy of B is converted
to other forms.
o
Another example of an MHD instability?
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How is energy converted from B to
thermal energy or kinetic energy?
Lecture 6 - MHD I
Magnetic Field Lines and Flux Tubes
o
o
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Magnetic field line: Curve with tangent
in direction of B.
dy By
= , " dx = dy
dx Bx
Bx By
dx dy dz
or in 3D:
=
=
Bx By Bz
In 2D:
Magnetic!
flux tube: Surface generated by
set of field lines intersecting simple
closed curve.
!
o
Magnetic flux crossing a section: F =
o
But, !"B=0 => F = const. along tube.
# B " dS
!
Lecture 6 - MHD I
Magnetic Field Geometry for an “X-point”
o
Sketch field lines for Bx = y, By = x.
o
We know that field lines must satisfy
dy By
=
dx Bx
dy x
=>
=
dx y
" ydy = " xdx
!
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Field lines must therefore have equation:
x2 - y2 = const
!
o
What about directions? Use slopes.
Lecture 6 - MHD I
!
Magnetic Field Geometry
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What about field spacing?
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Magnitude of B is:
B = Bx2 + By2 = x 2 + y 2
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If x = 0 => |B| = y.
! o Bx increases with y and By increases
with x.
o
At origin, Bx = By = 0.
o Called a neutral or null point.
Lecture 6 - MHD I
Fundamental MHD Equations
o
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Maxwell’s Equations:
" # B / µ = j + $ D / $ t,
".B = 0,
" # E = % $ B / $ t,
".D = & c ,
where B = µ H, D = ' E, E = j / ( .
Fluid Equations:
dv
= # $p,
dt
d"
Continuity
+ "$.v = 0,
dt
Perfect gas
p = R " T,
Lecture 6 - MHD I
Energy eqn.
.............
where d / dt = % / %t + v.$
Motion
"
Basic Assumptions for MHD
o
Assume v<<c => Neglect ! D/! t . Therefore,
" # B/µ =
o
= j/!
" (2)
Add magnetic force:
!
o
$ (1)
Include extra E on moving plasma:
E + v!B
o
j
dv
dt
= " #p +
j! B
Taking the curl of Eqn. 2 and substituting for j from Eqn. 1:
"B
= #$ % E = $ % (v % B # j/& )
"t
= $ % (v % B) - '$ % ($ % B)
-(3)
where ! = 1/"µ is the magnetic diffusivity.
!
Lecture 6 - MHD I
Induction Equation
o
Simplifying Eqn. 3, we get:
"B
= # $ (v $ B) + % # 2B
"t
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Consequences:
1. Can calculate B if v is known.
2. In MHD, v and B are the primary variables: basic physics can be derived
from induction equation and equation of motion.
3.
j(= !#B/µ) and E (=v # B + j/$) are secondary variables.
4.
B changes due to advection and diffusion.
Lecture 6 - MHD I
Induction Equation
5.
Magnetic Reynolds number is
Rm ~
" # (v # B) L0v 0
~
$" 2B
$
eg, In corona, % = 1 m2/s, L0 = 105 m, v0 = 103 m/s => Rm ~ 108.
6.
!
Advective term >> diffusive term for most of Universe => Most plasmas have
high Rm.
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Fields are frozen into plasma.
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Except where large gradients in field (!B large).
Lecture 6 - MHD I
Induction Equation Limits
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If RM >> 1, then ! << 1 and
"Bˆ ˆ
= # $ ( vˆ $ Bˆ )
"t
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The field is frozen into the plasma.
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Field configuration is!
determined by the flow field.
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Typical of conditions in and below the photosphere.
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Particularly important in active regions, where strong flows are most evident.
Lecture 6 - MHD I
Induction Equation
o Magnetic flux conservation:
o Magnetic field line conservation:
Lecture 6 - MHD I
Induction Equation
o If RM << 1, then ! >> 1
"Bˆ
ˆ 2Bˆ
= #$
"t
o Field diffuses thought the plasma.
o Which implies that !
the field diffuses away on a time-scale of
L20
"D =
#
o With ! ~ 1 m2 s-1 and L0 = 1 Mm (e.g., a sunspot), #D ~ 30,000 years.
o For flares, #D ~ 100
!sec => L0 ~ 10 m!
Lecture 6 - MHD I