I ntroduction to Nuclear Fusion

I ntroduction to Nuclear Fusion

Prof. Dr. Yong-Su Na

2

How to describe a plasma?

3

• 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

- Ideal:

Perfect conductor with zero resistivity

= 0

⋅

∇

∂ +

∂ v

t

ρ r ρ

p B

dt J v

d r = r × r − ∇ ρ

= 0

 

 



 ρ

p dt

d

= 0

× + v B E

r r r

0

0

=

⋅

∇

=

×

∇

∂

− ∂

=

×

∇

B

J B

t E B

r

r r

r r

µ

4

What is plasma equilibrium?

5

Equilibrium? Yes! Forces are balanced

Equilibrium and Stability

http://www.dezinfo.net/foto/12220-kreativ-devida-nicshe-99-foto.html

6

We need a fusion device which confines the plasma particles to some region for a sufficient time period by making equilibrium.

Equilibrium and Stability

pressure gravitation

Equilibrium in the sun

7

• Basic Equations

- MHD equilibrium equations:

time-independent with v = 0 (static)

= 0

⋅

∇

∂ +

∂ v

t

ρ r ρ

p B

dt J v

d r = r × r − ∇ ρ

= 0

 

 



 ρ

p dt

d

= 0

× + v B E

r r r

0

0

=

⋅

∇

=

×

∇

∂

− ∂

=

×

∇

B

J B

t E B

r

r r

r r

µ

→ Force balance

→ Ampere‘s law

→ Closed magnetic field lines

B J p

r r

×

=

∇

J B

r r

µ

=

×

∇

= 0

⋅

∇ B r

Equilibrium

8

→ Force balance

→ Ampere‘s law

→ Closed magnetic field lines

B J p

r r

×

=

∇

J B

r r

µ

=

×

∇

= 0

⋅

∇ B r

• Plasma Equilibrium

kinetic pressure

balanced by JxB (Lorentz) force

, 2

nqB B v

p

r

r ∇ ×

−

∇

=

, 2

,

B

p v B

q n v

q n

J

× ∇

= +

=

r r

r r

Diamagnetic current

= 0

∇

⋅ p J

r

= 0

∇

⋅ p B

r

- If B is applied, plasma equilibrium can be built by itself due to induction of diamagnetic current.

Magnetic and Kinetic Pressure

induced by the pressure gradient:

causing a decrease in B → diamagnetism

B ion

←∇p x

B J p

r r

×

=

∇

9

/

)

( B B µ

p = ∇ × ×

∇

0 2

/ 2 )] / (

)

[( B ⋅ ∇ B − ∇ B µ

=

0 0

2

/ 2 ) ( ) / ( p + B µ = B ⋅ ∇ B µ

∇

Assuming the field lines are straight and parallel

constant 2

2

= + µ p B

Total sum of kinetic pressure

and magnetic field energy density

(magnetic pressure) will be a constant

0

* 2

2

2 µ

B BH

V E

=

=

→ Force balance

→ Ampere‘s law

→ Closed magnetic field lines

B J p

r r

×

=

∇

J B

r r

µ

=

×

∇

= 0

⋅

∇ B r

• Plasma Equilibrium

kinetic pressure

balanced by JxB (Lorentz) force

Magnetic and Kinetic Pressure

10

• Concept of Beta

0 2

0

2

/ 2

) 2

/ µ µ

β B

kT n

(n B

p

+

=

=

magnetic Pressure (balloon tension) plasma

pressure

- The ratio of the plasma pressure to the magnetic field pressure

- A measure of the degree to which the magnetic field is holding a

non-uniform plasma in equilibrium.

- In most magnetic configurations,

fusion plasma confinement requires an imposed magnetic pressure significantly exceeding the particle kinetic pressure.

Magnetic and Kinetic Pressure

11

What is plasma stability?

12

Equilibrium?

Stable?

Yes! Forces are balanced No!

Equilibrium and Stability

http://www.dezinfo.net/foto/12220-kreativ-devida-nicshe-99-foto.html

13

Equilibrium?

Stable? No! The system cannot recover.

Yes! Forces are balanced

We need a fusion device which confines the plasma particles to some region for a sufficient time period in a stable way.

Equilibrium and Stability

14

Stability

http://www.amazon.co.uk/11Inch-Latex-Orange-Wedding-Balloons/dp/B004JUQG4Q http://www.psdgraphics.com/backgrounds/blue-water-drop-background/

15

• Definition of Stability

metastable

- Generation of instability is the general way of redistributing energy which was accumulated in a non-equilibrium state.

- A small change (disturbance) of a physical system at some instant changes the behavior of the system only slightly at all future times t.

- The fact that one can find an equilibrium does not guarantee that it is stable. Ball on hill analogies:

linearly unstable non-linearly unstable stable

linear: with small perturbation non-linear: with large perturbation

Stability

16

• Definition of Stability

- Assuming all quantities of interest linearised about their equilibrium values.

( ⁱ )^{t i t t}

i _{r i r i}

e e

r Q e

r

Q

=

=

( r )

( r )

1

~ /

0 1

Q <<

Q )

,

~ ( ) ( )

,

( r t Q

r Q

r t

Q r r r

+

=

Im ω > 0 (ω_i > 0): exponential instability Im ω ≤ 0 (ω_i ≤ 0): exponential stability

Stability

metastable

linearly unstable non-linearly unstable stable

i r

i ω ω

ω = +

t

e

r Q t

r

Q ~ ( , ) = ( )

1 1

r

r

17

• Gravitational Instability

Stability

http://user.uni-frankfurt.de/~schmelin/presentations/Rayleigh-Taylornew.html

Rayleigh-Taylor instability

18

• Gravitational Instability

Rayleigh-Taylor instability

Stability

, 2 E

B

D

B v E ×

2

=

,

qB

m

g D

B v g ×

=

qB

B v F ×

=

x B

g ++

+

++ +

- - -

- - -

E

E

ExB ExB

19

What is plasma transport?

20

Plasma Transport

http://www.amazon.co.uk/11Inch-Latex-Orange-Wedding-Balloons/dp/B004JUQG4Q

21

• Classical Transport

- Particle transport

random walk: no net flux (zero average)

with gradient: net flux down the gradient (diffusion)

http://functions.wolfram.com/Constants/Pi/visualizations/2/ShowAll.html

Plasma Transport

22

• Classical Transport

- Particle transport

random walk: no net flux (zero average)

with gradient: net flux down the gradient (diffusion)

Plasma Transport

23

The heat and momentum fluxes can be estimated in the similar fashion.

x x n x

n x

x

n ∂

∆ ∂ +

≈

∆

+ ) ( ) (

: Fick’s law

- Particle transport

• Classical Transport

Plasma Transport

n(x)

Γ+

Γ-

n(x+∆x)

x x+∆x

τ τ

x x

x Γ n

x x

Γ n + ∆ ∆

∆ =

=

+

2

) (

2 , ) (

: diffusion coefficient (m²/s)

( )

τ 2

x

D ∆

=

[ ^{( ) ( )} ]

2 x n x n x x Γ

Γ

Γ ∆ − + ∆

=

−

=

τ

s]

/m [#

v

Γ r n r

=

Particle flux:

( )

x D n x

n x

∂

− ∂

∂ =

∂

− ∆

= ² τ

2

Adolf Eugen Fick (1829-1901)

24

( ) x nD n

x q T

~

~

2

κ τ

κ

∆

∂

− ∂

=

- Heat transport

Heat flux

: thermal conductivity : Fourier’s law

• Classical Transport

Plasma Transport

Jean-Baptiste Joseph Fourier

(1768-1830)

[출처]: 과학자들이 들려주는 과학이야기 44

- Heat transport

• Classical Transport

Plasma Transport

26

• Classical Transport

( )

τ 2

x

D ∆

=

λ σ

n

m

n

x 1

=

=

∆

: particle approach

σ π

π

n

d vt n d n

n

vt 1 1

2

2

= =

Estimate transport coefficients: ∆x from mean free path

Plasma Transport

- Particle transport in weakly ionised plasmas

n(x)

Γ+

Γ-

n(x+∆x)

x x+∆x

d

m

n x x

n

Γ e

Γ e

Γ

−

−

≡

=

Γdx dΓ = − σ n

Neutral particles

27

• Classical Transport

( )

τ 2

x

D ∆

=

- Particle transport in weakly ionised plasmas

Estimate transport coefficients: τ from collision frequency with neutrals

Plasma Transport

Γ+

Γ-

n(x+∆x)

x x+∆x

Neutral particles

τ

τ λ ν

µ ν

α α

α α α

2

2

~

~

n n j

m v D kT

m q

=

≡

: Diffusion coefficient

• Classical Transport

Plasma Transport

- Particle transport in weakly ionised plasmas

( E v B ) p n m

( v v

)

nq v

t v m v

n r r r r r r r r

−

−

∇

−

× +

 =



 



 + ⋅ ∇

∂

∂

α α α α α α

α α

α α α

α

ν

α α α

α α

α

α

n v µ n E D n

Γ = = ± − ∇

r r r

α α α α α α

α

E kT n n m ν v

nq r

r

−

∇

−

= 0

α α

α α α

α α α

α

ν m ν ⁿ

kT m

E v nq

n

n n

∇

−

=

r

r

29 i

i e i e

i

i e e i a

a

T D D T D

D D

n Γ D

+ +

≡ +

∇

−

=

µ ~ µ

µ µ

r

• Classical Transport

Plasma Transport

- Particle transport in weakly ionised plasmas

Ambipolar Diffusion

e

i

Γ

Γ

r r

=

Faster electrons

slower ions → E-field induction

→ Electrons decelerated, ions accelerated

→ Charge separation

→ Electrons and ions diffuse together

30

( )

( )

τ ν ω

ν τ

ω τ ω µ µ

ω µ

ν

α α

2 2

2 2 2

2 2

2 2

2 2

~

~ 1 ~

1

/ 1

L th

L th c

j c

c

c D E

r v

v r m

kT D D

v v n n

D E n v

Γ n

= +

≡ +

+ + +

∇

−

±

=

=

⊥

⊥

⊥

⊥

⊥

⊥

r r r

r r

• Classical Transport

Plasma Transport

- Particle transport in weakly ionised plasmas with magnetic field

α α α

α α

α

α

n v µ n E D n

Γ = = ± − ∇

r r r

τ τ λ ν

µ ν

α α

α α

α

2

2

~

~

n j

n

m v D kT

m q

=

≡

B q r mv

m B q

L c

= ⊥

ω =

31

B

kT D n

n D v

Γ n

⊥

∑

⊥

⊥

⊥

⊥

=

∇

−

=

= η r r

τ from collision frequency

2 / 3 4

e e ei

ee m T

v ne v ≈ ∝

ee i e

ie v

m v m 



 



=

ee e

i e

ii

v

Ti T m

v m

2 / 2 3

/ 1

 

 



 



 



= 

• Classical Transport

Plasma Transport

- Particle transport in fully ionised plasmas with

magnetic field