MHD Fundamentals

8/6/2019 MHD Fundamentals

1/26

History Equations Energy Dimensions Approximations Effects

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
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


2/26


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


3/26


A short history of MHD

The hydrodynamics chain

The electromagnetism chain

A short history of magnetohydrodynamics


4/26



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
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


5/26



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
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


6/26



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


7/26



Ohm (17891854) Die galvanische Kette,mathematisch bearbeitet, 1827

j = E

Faraday (17911867) the law of induction,1831

V = d

dt



8/26



Maxwell (18311879) synthesis of electromag-netismOn Faradays lines of force, 1855-1856



9/26



Faraday (17911867) the dynamo disk



10/26


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



11/26



Siemens (18161892) self-excited dynamo,1866



12/26

s or qu o s rg s o s ro o s c s


Siemens (18161892) self-excited dynamo,1866

Anyos Jedlik (18001895) first (?) self-exciteddynamo, 1861



13/26

q


Self-excited Faraday disk



14/26


Larmor (18571942), dynamo action for theSun and the Earth, 1919

Hartmann stabilizing effect of externally im-posed magnetic fields, 1937

http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


15/26


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



16/26


Shercliff (19271983) structure of flows underan imposed magnetic field



17/26


Kulikovskii (1933) characteristic surfaces,1971

dl

B= Cst



18/26


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



19/26

Governing equations



20/26

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



21/26

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



22/26

Lorentz force

qE + j B



23/26

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.



24/26

Magneto-static approximation /


25/26

Magneto-static approximation /


26/26

End of lecture 1
http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

MHD Fundamentals

Documents

Transcript of MHD Fundamentals