Kinetic Approach of Plasmas

I ntroduction to Nuclear Fusion

Prof. Dr. Yong-Su Na

2

How to describe a plasma?

3

• Thermonuclear fusion with high energy particles of 10-20 keV energy → plasma

• Three approaches to describe a plasma

- Single particle approach - Kinetic theory

- Fluid theory

Description of a Plasma

http://http://dz-dev.net/blog/tag/eruption-solaire

4

Kinetic Approach of Plasmas

3

1020

~ m^ n_e

18 3

~ m

Volume

1021

# of electrons: ~

3 GHz of personal computer:

3x10⁹ #/s ~ 10¹⁷ #/year

→ 10⁴ years

5

• Boltzmann equation

dt du u

f dt

du u

f dt

du u

f dt

dz z f dt

dy y f dt

dx x f t

f dt

df _z

z y

y x

x 

 



 



 



 



 



 



  ^{      }



Kinetic Approach of Plasmas



 u f a f

t f

u



  

 

 





 

 E u B f

m f q

t u f

u

 

     

 

  ( )

t c

f 

 







  ^



^{t x y z u}_x ^u_y ^u_z



f

f_  _ , , , , , , Distribution function: number density of particles found ‘near’

a point in the 6-D space (x, u)

6

• Boltzmann equation

Ludwig Boltzmann (1844-1906)

t E J c

B

t E B

B E



 









 



















 



 





0 2 0

1 0







James Clerk Maxwell (1831-1879)

c u

u t

f f B

u m E

f q t u

f f a

f t u

f dt

df 



 







 















 

 











 

  _ ^



 

 

 

   _     _

) (

Kinetic Approach of Plasmas

 

 









i e

i e

u d f q

u d f u q J

, ,

  

  

 ^



 

current density

charge density

7

• Boltzmann equation

Kinetic Approach of Plasmas



^{t x y z u}_x ^u_y ^u_z



f

f_  _ , , , , , , Distribution function: number density of particles found ‘near’

a point in the 6-D space (x, u)



 Qf du

Q n 





1



 f du t

r

n  



 ( , ) number density of particles in physical space



 uf du t n

r

v   







) 1 ,

( Mean (fluid) velocity of particles in physical space



 m u f du t

r

p   

 ² 

) 3 ,

( Scalar pressure of particles in physical space

8

• Boltzmann equation

Kinetic Approach of Plasmas

Maxwellian velocity distribution in thermal equilibrium

Scalar pressure of particles in physical space



 







 



 

T u m T

n m u

f_M

exp 2 ) 2

(

2 2 /

3 







 f du t

r

n  



 ( , ) number density of particles in physical space



 uf du t n

r

v   







) 1 ,

( Mean (fluid) velocity of particles in physical space



 m u f du t

r

p   

 ² 

) 3 ,

( Scalar pressure of particles in physical space

nT p 

m u T

d t u r f n u

v v

v_x²  _y²  _z²  ¹



_x² _M ^{(, , ) }  Mean square velocity

http://www.maisondelasimulation.fr/smilei/collisions.html

9

• 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.

- Introduce the concept of an electrically charged current-carrying fluid.

Fluid Approach of Plasmas

http://www.tower.com/music-label/gulan-music-studio Album art of music OST “Plasma Pong”

• Two fluid equations



 



 









 

 C

t f

c

0

, )

,

(  

 v r t w w u    



 0

 

 



 



 







 

 ^{Q df} _dt ^f _t ^{d u}

c i







2 / 1

2 3

2 1

u m Q

u m Q

Q









  ^mass

momentum energy

Plasmas as Fluids

25

c

u t

f f B

u m E

f q t u

f dt

df 



 







 















 

  _ ^



 



     _

) (



 f du t

r

n  



( , )



 uf du t n

r

v   







) 1 , (



 Qf du

Q n 





1

• Continuity equation

 

^{ 0}





 





 nv

t

n 



 



 









 

 C

t f

c

0

, )

,

(  

 v r t w w u    



0

1  



 



 



 







 

  ^df _dt ^f _t ^{d u}

c







Plasmas as Fluids

25

c

u t

f f B

u m E

f q t u

f dt

df 



 







 















 

  _ ^



 



     _

) (



 f du t

r

n  



( , )



 uf du t n

r

v   







) 1 , (



 Qf du

Q n 





1

 

^{  0}



 







 



 











 



  



^f_t ^{du u f du} _m^{q E u B f}_u ^{du f}_t ^du

c



 

 

 



 _{ }



 



t u n

d t f

u t d f



 



 









^{    }

 

_{  }

 

_

du uf du f udu uf du nv

f

u  



   



    



   



mass

1 1 Q

• Two fluid equations

 0





 _{ }

 n v

dt

dn 









 



 q n E v B P R

dt v m d

n      













 ( )







 

 P v h Q

dt

n dT        2 :

3

w2

m n P_  _{ }

w d C w m

Q 







 ²

2



1



w d C w m

R  





 ^



w w m n

h ₂ 

2 1





 



 



 









 

 C

t f

c

0

, )

,

(  

 v r t w w u    



 0

 

 



 



 







 

 ^{Q df} _dt ^f _t ^{d u}

c i







2 / 1

2 3

2 1

u m Q

u m Q

Q









  ^mass

momentum energy

Plasmas as Fluids

25

c

u t

f f B

u m E

f q t u

f dt

df 



 







 















 

  _ ^



 



     _

) (



 f du t

r

n  



( , )



 uf du t n

r

v   







) 1 , (



 Qf du

Q n 





1

• Two fluid equations

 0





 _{ }

 n v

dt

dn 









 



 q n E v B P R

dt v m d

n      













 ( )







 

 P v h Q

dt

n dT        2 :

3

w2

m n P_  _{ }

w d C w m

Q 







 ²

2



1



w d C w m

R  





 ^



w w m n

h ₂ 

2 1





 

Plasmas as Fluids

25

...

1 3

2

2 4 2



















 z y x

z y x

y x

Closure?

14

•

Single-fluid magnetohydrodynamics (MHDs)

A single-fluid model of a fully ionised plasma, in which the

plasma is treated as a single hydrodynamic fluid acted upon by electric and magnetic forces.

e i

e i

e e i

iMv n mv n Mv mv nM v m M v

n

v      

) / ( /

) (

/ )

(     

 

nM m

M n m n M

n_i  _e   

 ( )



e n n_{i e}) ( 

 

) (

)

(n_iv_i n_ev_e ne v_i v_e e

J    









ne v J

ne v J M v m

v_{i e}

 

 



   ,  

Hydrogen plasma, charge neutrality assumed

mass density charge density

mass velocity electron inertia neglected:

electrons have an infinitely fast response time because of their small mass

• The magnetohydrodynamic (MHD) equation

Plasmas as Fluids

15

Plasmas as Fluids

 

^{ 0}





 



i i

i n v

t

n 

 

^{ 0}





 



e e

e n v

t

n 

 

^{ 0}





 



i i

i Mnv

t

M n 

 

^{ 0}





 



e e

e mn v

t

m n 

 

^



^



^{ 0}







e e i

i e

i Mnv mn v

t mn

Mn  

 / ) (n_iMv_i n_emv_e v    

m n M

n_i  _e

 

 0





 

 v

t

 



16

Plasmas as Fluids



i



i ie

i i

i en E v B p R

dt v

Mn d    











 v  (n_iMv_i n_emv_e)/ 

m n M

n_i  _e

 



e



e ei

e e

e en E v B p R

dt v

mn d    















ne v J

ne v J M v m

v_{i e}

 

 



   ,  

ie i

i i i

i

i en E env B p R

ne J dt mn d dt

v

Mn d     











 



 



 

ei e

e e e

e

e en E en v B p R

ne J dt mn d dt

v

mn d     













 



 



 

p B

dt J v

d       



) (

)

(n_iv_i n_ev_e ne v_i v_e e

J    









17

Plasmas as Fluids

 / ) (n_iMv_i n_emv_e v    

m n M

n_i  _e

 



e



e ei

e e

e en E v B p R

dt v

mn d    















ne v J

ne v J M v m

v_{i e}

 

 



   ,  

) (

)

(n_iv_i n_ev_e ne v_i v_e e

J    









Neglect electron inertia entirely



e



e ei

e E v B p R

en    













 0

e e

e en

J p B

v

E        





^{v v}



^{n e}



^{v v}



^neJ

mn

R_{ei ei} _i _e _i _e 





    

 ^{2 2}

en

p B

J J B

v

E   

^e











 

 

 

18

Mass continuity equation (continuously flowing fluid)

Single-fluid equation of motion

Maxwell equations

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

adiabatic evolution

• Ideal MHD model

₀ → 0 assumed

(Full → low-frequency Maxwell’s equations) Displacement current, net charge neglected

 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







 



F. F. Chen, Ch. 5.7

Plasmas as Fluids

en p B

J J B

v

E   ^e











 

 

 

19









 



 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

 

 

0

1

0 0 2

















 











 







B

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







 



Plasmas as Fluids

20

• Magnetohydrodynamics (MHD): 1946 by H. Alfvén

Hannes Alfvén (1908-1995)

“Nobel prize in Physics (1970)”

Plasmas as Fluids

21

Plasmas as Fluids

• Magnetohydrodynamics (MHD): 1946 by H. Alfvén

22

• Ideal MHD

- MHD:

Magnetohydrodynamic (magnetic fluid dynamic) - Assumptions:

Low-frequency, long-wavelength collision-dominated plasma

- Single-fluid model

Plasmas as Fluids

Equilibrium and stability in fusion plasmas - Applications:

- Ideal:

Perfect conductor with zero resistivity

23

• Ideal MHD:



= 0 • Resistive MHD:



 0

What is Ideal MHD?

24

• Ideal MHD:



= 0 • Resistive MHD:



 0

What is Ideal MHD?

25

Applications

Plasma Eqauilibrium and Stability

https://fusion.gat.com/theory/File:Theory1.gif