T opics in Fusion and Plasma

T opics in Fusion and Plasma Studies 2

(459.667, 3 Credits)

Prof. Dr. Yong-Su Na

(32-206, Tel. 880-7204)

-

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

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

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

Introduction

• Basic physics areas in magnetic fusion

- Equilibrium and stability: plasma beta

- Heating and current drive: Lawson criterion

- Transport: Lawson criterion

Introduction

• Single particle motion of the plasma

Magnetic

field

Introduction

• Single particle motion of the plasma

•

Magnetic field lines trace out “magnetic surfaces”,

Strong magnetic field:

r

_L

<< a

Magnetic field

Electron Ion

Introduction

• Plasmas as fluids

- The single particle approach gets to be complicated.

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

- Now introduce the concept of an electrically charged current-carrying fluid.

→ Magnetohydrodynamic

(magnetic fluid dynamic: MHD) equations

Introduction

• Ideal MHD

- Single-fluid model

- how magnetic, inertial, and pressure forces interact within an ideal perfectly conducting plasma in an arbitrary magnetic geometry

- Any fusion reactor must satisfy the equilibrium and stability set by ideal MHD.

• Nonideal effects (e.g. electrical resistivity)

- allow the development of slower, weaker instabilities

Introduction

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

Introduction

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

The MHD Model

• Questions

1. The plasma is assumed to be collision dominated.

- Is it true in fusion plasmas?

- The ideal MHD provides a very accurate description of most macroscopic plasma behaviour.

cf) collisionless MHD

2. Ideal MHD must be viewed as an asymptotic model in that specific length and time scales must be assumed for

the derivation to be valid.

- Even when the collision-dominated assumption is not involved,

there are many situations where MHD is used to described

phenomena on time scales far beyond its strict range of

applicability.

The MHD Model

• 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

- Resistive instabilities

- alpha-particle behaviour

The MHD Model

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?

Description of the Ideal MHD Model

• 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   

Maxwell equations



0







 







J B

t E B







 



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

adiabatic evolution

Description of the Ideal MHD Model

• Validity of the ideal MHD model

- Characteristic length: a

- Characteristic speed: V

_Ti

= (2T

_i

/m

_i

)

^1/2

- Characteristic time: τ

_M

= a/V

_Ti

~ μs

Derivation of the Ideal MHD Model

c

u

t

f f B

u m E

f q t u

f 



 







 















 



_





 



 (    )

0 0 2

1 0















 















 







E

t E J c

B B

t E B



 





 

 

 





  



u d f u q J





 

• Starting Equations

current density

Homework



 



 









 



C

t f

c

 0





 ^C

^ee

^{d u   C}

ⁱⁱ

^{d u   C}

^ei

^{d u   C}

^ie

^{d u  }

 0

 ^m

^e

^{u  C}

^ee

^{d u   m}

ⁱ

^{u  C}

ⁱⁱ

^{d u  }  ¹ ₂ ^m

^e

^u

²

^C

^ee

^{d u  }  ₂ ^{1 m}

ⁱ

^u

²

^C

ⁱⁱ

^{d u   0}



 ^  ^{ }  ^{1 }

Derivation of the Ideal MHD Model

• Collision operator for elastic collisions

1. Conservation of particles between like and unlike particle collisions

2. Conservation of momentum and energy btw. like particle collisions

3. Conservation of total momentum and energy btw. unlike particle

collisions

• Two-Fluid Equations

Derivation of the Ideal MHD Model

 ^{m n v}   ^{m n u u}  ^{q n E v B m u C d u}

t



 

 



























^{      } 



 ( )

 0

 

 



 



 







 

 ^{Q df} _dt ^f _t ^{d u}

c i







 0





 







n



v

t

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 ^ _ ^{    } _ ^{ } _ ^{   } 

 









 



2 / 1

2 3

2 1

u m Q

u m Q

Q









 

^mass

momentum energy



 u f d u t

r

v   1   )

,

( ^{Q  1}  ^{Qf d u }

• Considering the random thermal motion of particles

Derivation of the Ideal MHD Model

2

3

1 n m w p

_



_{ }

I p P

w m n

P   























²

w d C w m R

w w m n h

n p T





 





















2

1 2 1

/













0

), ,

( 



 u v r t w w     



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.

Derivation of the Ideal MHD Model













 



q n E v B P R

dt v n d

m      













 



 



 ( )

 0





 



 













n v

dt

dn 



 



     



 



m



v T v P h q n v E Q v R

dt

n d       



















 



 



 



 



 

2 3 2

2





 

 

 

 









t v dt

d 

• 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 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







 







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.

0

0 /

: 0

0 0 0















E t E 









• Low-Frequency, Long-Wavelength, Asymptotic Expansions

Derivation of the Ideal MHD Model

T T

T_{e i e}

m e

n

m T

V c V

V

c k















) /

(

,

) /

2 (

,

, /

2 / 1 2

2 / 1











• First asymptotic assumption: c → ∞

(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



Conditions for validity:

displacement current neglected

Derivation of the Ideal MHD Model

ce T

Le Le

ce

d pe

V

e

r a r

a













/

,

, ,











• Second asymptotic assumption: m_e → 0

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

Conditions for validity:

e e

e

e

E v B P R

en     















 ( )

0

• Subtle effect:

- Neglect of electron inertia along B can be tricky.

- For long wavelengths, electrons can still require a finite response time even though m_e is small. This is region of the drift wave.

- We shall see that MHD consistently treats || motion poorly, but for MHD behavior, remarkably this does not matter!!

- To treat such behavior more sophisticated models are required.

• The Single-Fluid Equations

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























/ ) ( 







 



  m

_e

 0 , n

_i

 n

_e

 n

Introduce single fluid variables mass density fluid velocity current density

total pressure total temperature

Derivation of the Ideal MHD Model

 0





 

 v

t

 



)

(

_{i e}

p B

dt J v

d               



 0



 J 

Navier-Stokes equation

This momentum density conservation equation for species resembles in parts the one of conventional hydrodynamics, the Navier-Stokes equation.

Yet, in a plasma for each species the Lorentz force appears in addition,

coupling the plasma motion (via current and charge densities) to Maxwell„s equation and also the various components (electrons and ions) among

u p

u t u

u

₂

] ) (

[       



 



charge conservation

P B

u E nq u

t u

u         



 ( ) ] ( )

 [

• The Single-Fluid Equations

Derivation of the Ideal MHD Model

 0





 

 v

t

 



)

(

_{i e}

p B

dt J v

d               



 0



 J 

charge conservation

 

 

 





 

 



 

 















 

 















 



 

























e e

i i

i i

J p v J

h p Q

d

v h

p Q dt

d

 

 



 



: 1 2

3 :

2 0

0















 







B

J B

t E B







 

 )

1 (

e e

e

R

p B

en J B

v

E       























Ohm‟s law

• The Single-Fluid Equations

• The Ideal MHD Limit

Derivation of the Ideal MHD Model

• Assumptions leading to ideal MHD

1. Philosophy: Ideal MHD is concerned with phenomena occurring on certain length and time scales.

2. Ordering: Using this, we can order all the terms in the one fluid equations.

After ignoring small terms, we obtain ideal MHD.

3. Status: At this point only the assumptions c → ∞, m_e → 0 have been used in the equation.

Ti

k a x

a V t

~ 1

~

~

~

~





 







• Characteristic length and time scales for ideal MHD

macroscopic MHD phenomena

1 /

~ V

_{T ii}

a 

ii _i





Derivation of the Ideal MHD Model

• Collision dominated limit

- Isotropic pressure is expected to arise in systems where many collisions take place on a time scale short compared to those of interest. → Maxwellian - Evolution of full pressure tensor plays only a minor role in equilibrium and stability problems.

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

a V

_{T }







 ~

_

1 /

~

/ a V a 

V

_{T ii T ee}

e

i

 

mean free path

Derivation of the Ideal MHD Model

• MHD Limit

1. Use the collision dominated assumption to obtain ideal MHD.

2. Several additional assumptions will also be required.

3. Various moments in the equations are approximated by classical transport theory of Braginskii.

4. Transport coefficients can also be derived in the homework problems.

a V v

v

Ti

jj

~ /

3 2 2

~ 

_{| | | |}

 



 



     

   

ii

nT

i



 ~



viscosity

Problems 2.2

)

(

_{i e}

p B

dt J v

d               



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

2. Mass conservation – OK 3. Momentum equation

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

ei e

e

ne

J m

en 1 R  ~   ,  ~

₂



B p

J  

/

~  ) 1

/

~ (

~

/

^{1/2 2}

 



 







 a

r m

m B

v J B

v

en

R

Li

ii i e e



 





 



1 / ~

) / (

:

^{1/2 2}

  

 



 



 



 



 













a V a

r m

m t

p

en

J

Li T ii

i e e

e

^

i



 1

/ ~ / )

( 









a r t

p

en p

J

_Li

e e



:   

^1/2

 

 ^ v ^ m V   ^ :  v ^  V  

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







 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

 

p

d    1    





i e

i e

ee e

e

nT m m m

^1/2 _{| |}

| |

~  / ( / ) 

 

)

(   

^1/2



  T m V 

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   

Maxwell equations



0







 







J B

t E B







 



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

adiabatic evolution

• Ideal MHD model

Derivation of the Ideal MHD Model

• Summary of assumptions

∞

a v m

x m a

y r

^{T ii}

e i

Li ^{1/2 i}



, 



 



 



Region of Validity

• Overall Criteria

1. High collisionality: x << 1 2. Small gyro radius: y << 1 3. Small resistivity: y²/x << 1

1 )

/ ( 10 0

.

3 

^{3 2}



 T an

x

1 )

/ ( 10 3

.

2 

^{2 2 1/2}





^

na

y 

1 /

10 8

. 1

/

^{7 2}

2

x  

^

aT 

y 

keV 50 keV

5 . 0

m 10 m

10^{18 3 22 3}







 ^



T n

 fixed(=0.05), a=1m, D, L=15

Region of Validity

• Ideal MHD model is not valid for plasmas of fusion interest.

- Reason: collision dominated assumption breaks down - But, large empirical evidence that MHD works very well

in describing macroscopic plasma behavior.

- Question: is this the result of some subtle and perhaps

unexpected physics?

Region of Validity

1. Momentum equation

a.  << p because of collision dominated assumption b. _⊥<< p from collisionless theory

c. _|| ~ p parallel to the field, the motion of ions is kinetic.

d. ∴⊥ momentum equation OK

|| momentum equation not accurate 2. Energy Equation

a. collision dominated assumption

b. _|| → ∞ rather than zero in collisionless plasma c. More accurate equation of state → B⋅∇T=0 d. ∴ energy equation not accurate

1 / ~

)

(

_{| | | |} ^1/2

| |

 



 















a V m

m t

p

T

_{T ii}

e

e i _i





• Where specifically does ideal MHD breakdown?

Region of Validity

1. Momentum equation

Ohm‟s law and Faraday‟s law Note that v_|| does not appear.

2. Errors appear in || momentum equation and energy equation.

3. However, it turns out that for MHD equilibrium and most MHD instabilities, the parallel motion plays a small or negligible role.

This is not obvious a priori.

4. Assuming this to be true, an incorrect treatment of parallel motion is unimportant, since no parallel motions are exerted:

the motions are incompressible.

a. B⋅∇ρ=0 no density compression along B b. B⋅∇T=0. _|| → ∞

p B

dt J v d



     







B t v

B   







 





Valid for

collisionless and collisional theory

• MHD errors in the momentum and energy equation do not matter, why?

What is

imcompressibilty?

d/dt=0

Region of Validity

• Conclusion

- Once incompressibility is accepted as the dominant motion of unstable MHD modes, then errors in ideal MHD do not enter the calculation.

- Ideal MHD gives the “same” answer as “collisionless MHD”.