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Fundamentals of MHD
Thierry Alboussiere
LGIT, University of Grenoble, France
Ecole dete sur la Dynamo
Les Houches
30 July 23 August 2007week 1 and week 4
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Plan
1. A short history of MHD
2. Governing equations
3. Energetics
4. Dimensionless parameters5. MHD approximations
6. MHD physical effects
7. Stability of MHD flows
8. MHD turbulence
9. Measurement techniques
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A short history of MHD
The hydrodynamics chain
The electromagnetism chain
A short history of magnetohydrodynamics
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The hydrodynamics chain
Newton (16421727) Principia mathematicaphilosophi naturalis, 1687
f = ma
Euler (17011783) Memoires de lacademie des
sciences de Berlin, 1757
u
t+ (u )u = p
t + (u) = 0
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The hydrodynamics chain
Navier (17851836) French Engineer. Navier-Stokes equation, 1821
u
t+ (u )u = p+ 2u
Stokes (18191903) On the theories of the in-ternal friction of fluids in motion, 1845
H E E D A E
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The electromagnetism chain
Volta (17451827) voltaic pile or battery,letter to the Royal Society, 1800
Ampere (17751836) unification of electricityand magnetism, 1820
H E E D A E
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The electromagnetism chain
Ohm (17891854) Die galvanische Kette,mathematisch bearbeitet, 1827
j = E
Faraday (17911867) the law of induction,1831
V = d
dt
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The electromagnetism chain
Maxwell (18311879) synthesis of electromag-netismOn Faradays lines of force, 1855-1856
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A short history of magnetohydrodynamics
Faraday (17911867) the dynamo disk
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Faradays six principles
Have a little pad and take notes at all times
Exchange letters with other scientists
Have collaborations Check everything
Avoid controversy
Never make general assumptions too quickly, speak and write
as precisely as possible
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A short history of magnetohydrodynamics
Siemens (18161892) self-excited dynamo,1866
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s or qu o s rg s o s ro o s c s
A short history of magnetohydrodynamics
Siemens (18161892) self-excited dynamo,1866
Anyos Jedlik (18001895) first (?) self-exciteddynamo, 1861
History Equations Energy Dimensions Approximations Effects
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q
A short history of magnetohydrodynamics
Self-excited Faraday disk
History Equations Energy Dimensions Approximations Effects
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A short history of magnetohydrodynamics
Larmor (18571942), dynamo action for theSun and the Earth, 1919
Hartmann stabilizing effect of externally im-posed magnetic fields, 1937
History Equations Energy Dimensions Approximations Effects
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A short history of magnetohydrodynamics
Alfven (19081995) the mechanism of Alfvenwaves, 1942
Alfven wave demonstration Lundquist
(1949), Lehnert (1953) and Jameson (1964) inliquid sodium. Bostik and Levine (1952), Allenet al. (1959), De Silva (1961) and Spillman(1963) in plasmas
Elsasser (19041991) father of Earths dynamomagnetism
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A short history of magnetohydrodynamics
Shercliff (19271983) structure of flows underan imposed magnetic field
History Equations Energy Dimensions Approximations Effects
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A short history of magnetohydrodynamics
Kulikovskii (1933) characteristic surfaces,1971
dl
B= Cst
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A short history of magnetohydrodynamics
Demonstration of dynamo action Lowes and Wilkinson (1963) in
an homogeneous solid with rotating cylinders. Gailitis (1999), Muhler(1999) and VKS team (2007) with liquid sodium
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Governing equations
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Governing equationsNavier-Stokes
ut
+ (u.)u = p+ g + Lorentz force+ [u]
t+ (u) = 0
Maxwell B = 0
E = q/
B = j + E
t
E = B
tq
t+ j = 0
Ohm
s law
History Equations Energy Dimensions Approximations Effects
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Ohms law
Correct Ohms law (in a material reference system)
j = E
Non-relativistic change of coordinates (G. Rousseaux, EuroPhys.
Lett. 71, 2005)
E = E + u B and B = B
General Ohms law (with Hall effect)
j = qu + (E + u B) +|B|
j B
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Lorentz force
qE + j B
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Electrical charge equation
q
t+ (u )q
+ q = (u B)
When / is much smaller than any timescale of the flow, then
q = (u B)
It can then be seen that qu is negligible in Ohms law. Similarly,qE is negligible in the Lorentz force and the displacement current
E/t is negligible in Maxwells equations.
This is the so-called magneto-static approximation.
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Magneto-static approximation /
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Magneto-static approximation /
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End of lecture 1
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