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(1)

T opics in Fusion and Plasma Studies 2

(459.667, 3 Credits)

Prof. Dr. Yong-Su Na

(32-206, Tel. 880-7204)

(2)

-

J. Freidberg,

"Ideal Magnetohydrodynamics

(Modern Perspectives in Energy)", Springer, 1st edition (1987)

- T. Takeda and S. Tokuda,

"Computation of MHD Equilibrium of Tokamak Plasma", Journal of Computational Physics,

Vol. 93, p1~107 (1991)

Textbook

(3)

Week 1-2. The MHD Model, General Properties of Ideal MHD Week 3. Equilibrium: General Considerations

Week 4. Equilibrium: One-, Two-Dimensional Configurations Week 5. Equilibrium: Two-Dimensional Configurations

Week 6-7. Numerical Solution of the GS Equation Week 9. Stability: General Considerations

Week 10-11. Stability: One-Dimensional Configurations Week 12. Stability: Multidimensional Configurations

Week 14-15. Project Presentation

Contents

(4)

Week 1-2. The MHD Model, General Properties of Ideal MHD Week 3. Equilibrium: General Considerations

Week 4. Equilibrium: One-, Two-Dimensional Configurations Week 5. Equilibrium: Two-Dimensional Configurations

Week 6-7. Numerical Solution of the GS Equation Week 9. Stability: General Considerations

Week 10-11. Stability: One-Dimensional Configurations Week 12. Stability: Multidimensional Configurations

Week 14-15. Project Presentation

Contents

(5)

The Origin of the Star Energy

Nobel prize in physics 1967

“for his contribution to the theory of nuclear reactions, especially his

discoveries concerning the energy production in stars”

Hans Albrecht Bethe

(1906. 7. 2 – 2005. 3. 6)

(6)

To build a sun on earth

(7)

Magnetic confinement

plasma pressure gravitation

Equilibrium in the sun

plasma pressure

Plasma on earth

much, much smaller & tiny mass!

magnetic field

• Imitation of the Sun on Earth

(8)

JET (Joint European Torus): R

0

=3m, a=0.9m, 1983-today

Tokamak

(9)

JET (Joint European Torus): R

0

=3m, a=0.9m, 1983-today

Tokamak

How to describe fusion plasmas?

(10)

Single Particle Approach

• Single particle motion of the plasma

Magnetic

field

(11)

• Single particle motion of the plasma

Strong magnetic field:

r

L

<< a

Magnetic field

Electron Ion

- The single particle approach gets to be complicated.

- A more statistical approach can be used because we cannot follow each particle separately.

Single Particle Approach

(12)

Kinetic Approach

c

u

t

f f B

u m E

f q t u

f

 

 

 

 (    )

Ludwig Boltzmann (1844-1906)

• Kinetic model of the plasma

(13)

Fluid Approach

• Plasmas as fluids

- Introduce the concept of an electrically charged

current-carrying fluid.

(14)

What is Ideal MHD?

• Ideal MHD

- Single-fluid model

• Nonideal effects (e.g. electrical resistivity)

- allow the development of slower, weaker instabilities - Ideal:

Perfect conductor with zero resistivity - MHD:

Magnetohydrodynamic (magnetic fluid dynamic)

(15)

• Ideal MHD:  = 0 • Resistive MHD:   0

What is Ideal MHD?

(16)

• Ideal MHD:  = 0 • Resistive MHD:   0

What is Ideal MHD?

(17)

Applications

X

X

X X

X

Force PF coils

Plasma Eqauilibrium and Stability

(18)

How does a given magnetic geometry provide forces to hold a plasma in equilibrium?

Why are certain magnetic geometries much more stable against macroscopic disturbances than

others?

Why do fusion configurations have such

technologically undesirable shapes as a torus, a helix, or a baseball seam?

Applications

(19)

• Ideal MHD equations

 0

 

v

t

 

 Mass continuity equation

p B

dt J v

d       

 Single-fluid equation of motion

 0

 

 

p dt

d

 0

v B E   

Maxwell equations

0

0

 

B

J B

t E B

 

Ohm’s law: perfect conductor → “ideal” MHD Energy equation (equation of state):

adiabatic evolution

How to Derive the Ideal MHD Model?

(20)

The MHD Model - Limitations

• Major discoveries of modern physics

- Maxwell’s equations with wave propagation - Relativity

- Quantum mechanics

• Important processes in fusion plasmas

- Radiation - RF heating

- Resonant particle effect - Microinstabilities

- Classical transport

- Anomalous transport

(21)

Derivation of the Ideal MHD Model

c

u

t

f f B

u m E

f q t u

f

 

 

 

 (    )

• Starting Equations

Ludwig Boltzmann (1844-1906)

0 2

1 0

 

 

E

t E J c

B B

t E B

 

 

James Clark Maxwell (1831-1879)

(22)

• Two-Fluid Equations

Derivation of the Ideal MHD Model

 0

 

 

 

 

 

Q df dt f t d u

c i

m n v   m n u uq n E v B m u C d u

t

 

 



 ( )

 0

 

n

v

t

n

u d C u m E

v n q u

u n m u

n

m

2

1

2

   1

2

1

 

2 / 1

2 3

2 1

u m Q

u m Q

Q

 

mass

momentum energy

(23)

• Considering the random thermal motion of particles

Derivation of the Ideal MHD Model

0

), ,

( 

u v r t w w     

2

3

1 n m w p

I p P

w m n

P   

2

w d C w m Q

w d C w m R

w w m n h

n p T

 





2 2

2 1 2 1

/

Scalar pressure

Anisotropic part of the pressure tensor Total pressure tensor

Temperature

Heat flux due to random motion

Mean momentum transfer btw. unlike particles due to the friction of collisions Heat generated due to collisions btw.

unlike particles.

Brownian motion

(24)

Derivation of the Ideal MHD Model

q n E v B P R

dt v n d

m      

 

 

 ( )

 0

 

 

n v

dt

dn

m n v   m n u uq n E v B m u C d u

t

 

 



 ( )

 0

 

v t n

n

u d C u m E

v n q u

u n m u

n t m

 



2 2 2

2 1 2

1 2

1

 

(25)

• Final set of two-fluid equations

Derivation of the Ideal MHD Model

q n E v B P R

dt v m d

n      

 

 

 ( )

 0

 

 

n v

dt

dn

P v h Q

dt

n dT       

 

   

2 : 3

 

 

1

0 0 2

 

 

n e n

E

t E v c

n v n e B

t E B

e i

e e i

i

 

 

 

 0

 

v

t

 

p B

dt J v

d       

 0

 

 

p dt

d

 0

v B E   

0

0

 

B

J B

t E B

 

1. Certain asymptotic orderings are introduced which eliminate the very high-frequency, short-wavelength information in the model.

2. The equations are rewritten as a set of single-fluid equations by the introduction of

appropriate single-fluid variables.

3. The plasma is assumed to be collision dominated.

(26)

• Step 1. Remove high-frequency, short-wavelength information

Derivation of the Ideal MHD Model

• First asymptotic assumption: 0 → 0

(Full → low-frequency Maxwell’s equations)

Displacement current, net charge neglected

quasineutrality

 

0

1

0

2 0 0

 

 

e E n

n

t J E v c

n v n e B

e i

e e i

i

 

 

n n

n

i

e

c

 / k 

Conditions for validity:

displacement current

(27)

Derivation of the Ideal MHD Model

• Second asymptotic assumption: me → 0

(electron inertia neglected: electrons have an infinitely fast response time because of their small mass)

ce T

Le Le

ce

d pe

V

e

r a r

a

/

,

, ,









Conditions for validity:

0 )

(       

 

 

e e

e e

e e

e

en E v B P R

dt v m d

n      

(28)

Derivation of the Ideal MHD Model

i e

i e

e

e i

i i

T T

T

p p

nT p

en J

v v

v v en J

v v

n m

/

) (  

 

 

mass density

m

e

 0 , n

i

n

e

n

fluid velocity current density

total pressure total temperature

• Step 2. Introduce single fluid variables

(29)

Derivation of the Ideal MHD Model

 0

 

v

t

 

)

(

i e

p B

dt J v

d               

 0

J

 

 

 

 

 

 

 

 

 

 

 

 

 

 

e e

e e

e

i i

i i

J p en en

v J h

p Q dt

d

v h

p Q dt

d

 

 

 

: 1 3

2 3 :

2

) 1 (

e e

e

R

p B

en J B

v

E       

• The Single-Fluid Equations

 0

 

v

t

 

 0

 

 

p dt

d

 0

v B E   

p B

dt J v

d       

(30)

1 /

~ V

T ii

a 

ii i



Derivation of the Ideal MHD Model

1 /

) / (

~ m

e

m

i 1/2

V

T ii

a 

ee i

 

ee

~ ( m

e

/ m

i

)

1/2

ii

( T

e

~ T

i

)

ions

electrons

mean free path

• Step 3. Assume the plasma to be collision dominated

a V

T 



 ~

1 /

~

/ a V a 

V

T ii T ee

e

i

 

(31)

Derivation of the Ideal MHD Model

a V

v

v

Ti

jj

~ /

3 2 2

~ 

| | | |

 

 

     

   

ii

nT

i

 ~ 1

/

~

/  

i

p V

T ii

a

i

viscosity

)

(

i e

p B

dt J v

d               

• Reduction of single fluid equation 1. Maxwell equations – OK

2. Mass conservation – OK 3. Momentum equation

(32)

Derivation of the Ideal MHD Model

• Reduction of single fluid equation 1. Maxwell equations – OK

2. Mass conservation – OK 3. Momentum equation – OK 4. Ohm’s law

) 1 (

e e

e

R

p B

en J B

v

E       

resistivity momentum transfer due to collisions

ci T

Li e Li

V

i

a r r B

v

en

p / ~ 1 , / 

 

    / 

ci

~ r

Li

/ a  1

  

) 1 /

~ (

~

/

1/2 2

 

 

a

r m

m B

v J B

v

en

R

Li

ii i e e



 

 

(33)

1 / ~

) / (

:

1/2 2

  

 

 

 

 

 

a V a

r m

m t

p

en

J

e Li T ii

e

i

1 / ~

/ )

( 

a r t

p

en p

J

e Li

1 / ~

:

1/2

 

 

 

 

a V m

m t

p

v

T ii

i e e

e

i

 1

/ ~

:  

 

a V t

p

v

T ii i

i

i

Derivation of the Ideal MHD Model

• Reduction of single fluid equation 1. Maxwell equations – OK

2. Mass conservation – OK 3. Momentum equation – OK 4. Ohm’s law – OK

5. Energy equation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

e e

e e

e

i i

i i

J p en en

v J h

p Q dt

d

v h

p Q dt

d

 

 

 

: 1 3

2 3 :

2

(34)

T

h   

| |

| |

 

 

     

 

 

eq i e i

i

i

n T T

p T dt

d

 

) ) (

3 ( 2

| |

| |

 

 

     

 

 

eq i e e

e

e

n T T

p T dt

d

 

) ) (

3 ( 2

| |

| |

Derivation of the Ideal MHD Model

Dominated by parallel thermal conduction

 

| |

(35)

T

h   

| |

| |

 

 

     

 

 

eq i e i

i

i

n T T

p T dt

d

 

) ) (

3 ( 2

| |

| |

 

 

     

 

 

eq i e e

e

e

n T T

p T dt

d

 

) ) (

3 ( 2

| |

| |

Derivation of the Ideal MHD Model

Dominated by parallel thermal conduction

 

| |

en R J T

T Q n

T T Q n

e eq

i e e

eq e i

i

 

 

 

) (

) (

Equilibration and Ohmic heating

1

~

2 / 1 2

/ 1

 

 

 

 

 

a V m

m m

m

T ii

e i ii

e i eq

t



To derive a single fluid model

assuming small equilibration time

2 /

, 2

/ p p p

T T

T

i

e

i

e

If this is true, then

T

p dt

d

i

e | | | |

| |

| |

( )

3

1    

 

 

  

i e

i e

ee e

e

nT m m m

1/2 | |

| |

~  / ( / ) 

 

1 / ~

)

(

| | | | 1/2

| |

 

 

a V m

m t

p

T

i T ii

e i

(36)

Derivation of the Ideal MHD Model

 0

 

v

t

 

 Mass continuity equation

p B

dt J v

d       

 Single-fluid equation of motion

 0

 

 

p dt

d

 0

v B E   

 

B

E

 

Ohm’s law: perfect conductor → “ideal” MHD Energy equation (equation of state):

adiabatic evolution

• Ideal MHD model

(37)

Conclusion

• Ideal MHD

- Ideal:

Perfect conductor with zero resistivity - MHD:

Magnetohydrodynamic (magnetic fluid dynamic) - Assumptions:

Low-frequency, long-wavelength collision-dominated plasma

Equilibrium and stability in fusion plasmas - Applications:

- Single-fluid model

(38)

References

- http://www.dreamstime.com/stock-images-blue-3d-plasma-fluid- image698604

- http://www.runme.org/project/+plasmapong/

- http://www.what-is-this.com/blogs/archives/Science-blog/507424311- Dec-16-2008.html

- http://www.lhc-closer.es/pages/phy_1.html

- http://www.phy.cuhk.edu.hk/contextual/heat/tep/trans01_e.html

- http://twistedphysics.typepad.com/cocktail_party_physics/biophysics/

- http://www.antonine-

education.co.uk/Physics_GCSE/Unit_1/Topic_1/topic_1_how_is_heat_tran sferred.htm

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