I ntroduction to Nuclear Fusion
(409.308A, 3 Credits)
Prof. Dr. Yong-Su Na
(32-206, Tel. 880-7204)
Week 1. Fundamentals of Nuclear Fusion I
- Present Status and Future Prospect Week 2. Fundamentals of Nuclear Fusion II
- Fusion Reactions
Week 3. Fundamentals of Nuclear Fusion III - Thermonuclear Fusion Conditions Week 4. Review of Plasma Physics
- Plasma Confinement, Transport, Equilibrium, and Stability
Week 5. Inertial Confinement Week 6. Magnetic Confinement
- Mirror, Pinches, and Stellarator
Contents
Week 9. Tokamaks I
- Plasma Equilibrium and Stability Week 10. Tokamaks II
- Plasma Transport
Week 11. Plasma Heating and Current Drive
- OH, NBI, RF, Adiabatic Compression, and Alpha Self-heating
Week 12. Plasma Wall Interaction
Week 13. Overview of Fusion Power Plants Week 14. Critical Issues in Fusion Researches
Contents
Plasmas as Fluids
) (
α βαβ α α
αβ
m n ν u u
R G G G
−
−
=
∑
+
⋅
∇
−
× +
⎟ =
⎠
⎜ ⎞
⎝
⎛ + ⋅ ∇
∂
∂
β αβ α
α α
α α
α α α
α
u u n q E u B P R
t n u
m G G G G G G I G
) (
) (
• Two-fluid equations
The continuity equation will apply separately to each of the different species. The momentum balance equation must consider the fact that particles of one species can collide with particles of another species, thereby transferring momentum between the different species.
αβ
βα
R
R G G
−
=
βα β β αβ
α
α
n ν m n ν
m =
The rate at which momentum per unit volume is gained by species α due to collisions with species β.
ναβ : collision frequency of α on β
The rate at which momentum per unit volume is gained by species β due to collisions with species α.
Plasmas as Fluids
) (
e iei e e
ei
m n u u
R G G G
−
−
= ν
||
0 = − n
eeE
||+ R
ei• Plasma resistivity
The acceleration of electrons by an electric field applied along (or in the absence of) a magnetic field is impeded by collisions with non- accelerated particles, in particular the ions, which, because of their much larger mass, are relatively unresponsive to the applied electric field. Collisions between electrons and ions, acting in this way to limit the current that can be driven by an electric field, give rise to an
important plasma quantity, namely its electrical resistivity, η.
∑
+
⋅
∇
−
× +
⎟ =
⎠
⎜ ⎞
⎝
⎛ + ⋅ ∇
∂
∂
β αβ α
α α
α α
α α α
α
u u n q E u B P R
t n u
m G G G G G G I G
) (
) (
Homogeneous (neglecting the electron pressure and velocity gradients along B)
Plasmas as Fluids
) (
|| ||||
n
ee u
eu
iJ = − −
||
2 ||
||
||
||
( ) J J
e n u m
e u E m
e ei e i
e ei
e
ν − = ν = η
−
=
e
2n m
e ei e
ν η =
J e n u
u e n u
u n
m
R G
ei e e eiG
eG
i eG
eG
i eG η
η
ν − = − − =
−
= ( )
2 2( )
Momentum gained by electrons due to collisions with ions Simplified
Ohm’s law
Ohm’s triangle
• Plasma resistivity
||
0 = − n
eeE
||+ R
eiR G
eim
en
e ei( u G
eu G
i)
−
−
= ν
Plasmas as Fluids
• Single-fluid magnetohydrodynamics (MHDs)
A single-fluid model of a fully ionised plasma, in which the plasma is treated as a single hydrogynamic fluid acted upon by electric and magnetic forces.
e i
e i
e e i
i
M u n m u M u m u M m u m M u
n
v G G G G G G G
) / ( )
/(
) (
/ )
( + ≈ + + ≈ +
= ρ
nM m
M n m n M
n
i+
e≈ + ≈
= ( )
ρ
e n n
i e) ( − σ =
) (
)
( n
iu
in
eu
ene u
iu
ee
J G G G G G
−
≈
−
=
ne v J
ne u J M v m
u
i eG G G G
G
G ≈ + , ≈ −
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
ie i
i i
i
i
en E u B p R
dt u
Mn d G G G G G
+
∇
−
× +
= ( )
• The magnetohydrodynamic (MHD) equation
0 ) (
, ,,
+ ∇ ⋅ =
∂
∂
e i e i e
i
n u
t
n G
0 )
( =
⋅
∇
∂ +
∂ v
t
ρ G
ρ
Mass continuity equation= 0
⋅
∇
∂ +
∂ J
t σ G
Charge continuity equation
ei e
e e
e
e
en E u B p R
dt u
mn d G G G G G
+
∇
−
× +
−
= ( )
Equations of motion:
isotropic pressure assumed
p B
J E v
t v v dt
v
d ⎟ = + × − ∇
⎠
⎜ ⎞
⎝
⎛ + ⋅ ∇
∂
= ∂ G G G G G G
G
σ ρ
ρ
Single-fluid equation of motionmultiplied by M and m, respectively and added together
In the case, where a plasma is nearly Maxwellian (or at least nearly isotropic), the pressure
tensor term can be replaced by the gradient of a scalar pressure, ∇p
Plasmas as Fluids
Maxwell equation
J ne u
u e n u
u mn
R G
ei eiG
iG
eG
iG
eG η η
ν − = − =
= ( )
2 2( )
ne J p
B u
E G + G
e× G = G − ∇
eη
ne p B
J J B
v
E × − ∇
e+
=
× +
G G G
G G
G η
σ ε
μ
=
⋅
∇
=
⋅
∇
∂
− ∂
=
×
∇
∂ + ∂
=
×
∇
) (
0
1
0
0 2
E B
t E B
t E J c
B
G G
G G
G G G
Neglect electron inertia entirely:
valid for phenomena that are sufficiently slow that electrons have time to reach dynamical equilibrium in regard to their motion along the magnetic field Generalized Ohm’s law
ne v J
u
eG G
G ≈ −
Mass continuity equation
Single-fluid equation of motion
Maxwell equations
Ohm’s law: perfect conductor → “ideal” MHD Energy equation (equation of state):
adiabatic evolution
• Ideal MHD model
Plasmas as Fluids
ε0 → 0 assmued
(Full → low-frequency Maxwell’s equations) Displacement current, net charge neglected
= 0
⋅
∇
∂ +
∂ v
t
ρ G ρ
p B
dt J v
d G = G × G − ∇ ρ
= 0
⎟⎟ ⎠
⎜⎜ ⎞
⎝
⎛ ρ
γp dt
d
= 0
× + v B E G G G
0
0
=
⋅
∇
=
×
∇
∂
− ∂
=
×
∇
B
J B
t E B
G
G G
G G
μ
• 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
Plasmas as Fluids
• Ideal MHD: η = 0 • Resistive MHD: η ≠ 0
What is Ideal MHD?
• Ideal MHD: η = 0 • Resistive MHD: η ≠ 0
What is Ideal MHD?
Applications
X
●
●
X
X X
X
Force PF coils
Plasma Eqauilibrium and Stability
Equilibrium and Stability
Equilibrium?
Stable?
Yes! Forces are balanced No!
Equilibrium and Stability
Equilibrium?
Stable? No! The system cannot recover.
Yes! Forces are balanced
• Basic Equations
• MHD equilibrium equations:
time-independent with v = 0 (static)
Equilibrium
= 0
⋅
∇
∂ +
∂ v
t
ρ G ρ
p B
dt J v
d G = G × G − ∇ ρ
= 0
⎟⎟ ⎠
⎜⎜ ⎞
⎝
⎛ ρ
γp dt
d
= 0
× + v B E G G G
0
0
=
⋅
∇
=
×
∇
∂
− ∂
=
×
∇
B
J B
t E B
G
G G
G G
μ
→ Force balance
→ Ampere‘s law
→ Closed magnetic field lines
B J p G G
×
=
∇
J B G G
μ
0=
×
∇
= 0
⋅
∇ B G
• The Z Pinch
Equilibrium: 1-D Configurations
- 1-D toroidal configuration with purely poloidal field
longitudinal plasma current self field induced
by JZ
- Sequence of solution of the MHD equilibrium equations 1. The ▽·B = 0
2. Ampere’s law: μ0J = ▽xB
3. The momentum equation: JxB = ▽p
• The Z Pinch
Equilibrium: 1-D Configurations
1 0
∂ =
∂ θ
θB r
) 1 (
0
μ dr rB
θd J
z= r
dr B dp
J
z θ= − 0
) (
0
=
+
θ θμ dr rB d
r B dr
dp 0
2
02
0
2
⎟⎟ + =
⎠
⎜⎜ ⎞
⎝
⎛ +
r B p B
dr d
μ μ
θ θtension force by the curvature of the magnetic field lines
particle pressure + magnetic pressure force - Sequence of solution of the MHD equilibrium equations
1. The ▽·B = 0
2. Ampere’s law: μ0J = ▽xB
3. The momentum equation: JxB = ▽p
• The Z Pinch
Equilibrium: 1-D Configurations
- It is the tension force and not the magnetic pressure gradient that provides radial confinement of the plasma.
2 2 0 2
2 0 2
2 0 0
2 2 0 2
2 0 0
2 0 2 0 0
) (
8
) (
2
r r
r p I
r r
r J I
r r
I r B
z
= +
= +
= +
π μ
π π μ
θ
Bennett profiles (Bennett, 1934)
- The plasma is confined by magnetic tension.
- The magnetic pressure and plasma pressure each produce positive (outward forces) near the outside of the plasma.
Stability: General Considerations
• Definition of Stability
metastable
- Generation of instability is the general way of redistributing energy which was accumulated in a non-equilibrium state.
- Suggested by physics, where stability means, roughly speaking,
that 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:
linear unstable non-linear unstable
stable
linear: with small perturbation non-linear: with large perturbation
Stability
• Definition of Stability
Marginally stable
- assuming all quantities of interest linearised about their equilibrium values.
) ,
~ ( ) ( )
,
( r t Q
0r Q
1r t
Q G G G
+
=
t
r
iQ t
r
Q ~ ( , ) = ( ) ε
− ω1 1
G 1 G
~ /
0 1
Q <<
Q
small 1st order perturbation
Im ω > 0: exponential instability Im ω ≤ 0: exponential stability
Stability
• Various Approaches for Stability Analyses
1. Initial value problem using the general linearised equations of motion 2. Normal-mode eigenvalue problem
3. Variational principle 4. Energy Principle
Stability
1
~ /
0 1
Q <<
Q
• Initial Value Formulation
0 0
0 0
0 0
0
0 0
0
=
=
⋅
∇
×
∇
=
∇
=
×
v B
B J
p B
J
G G
G G
G G
μ ( , )
) ~ ( )
,
( r t Q
0r Q
1r t
Q G G G
+
=
v t
∂
= ∂ ξ
1
~
linearized
ξ: displacement of the plasma away from its equilibrium position
Aim: to express all perturbed quantities in terms of ξ and then obtain a single equation describing the time evolution of ξ
Stability
1
~ /
0 1
Q <<
Q
• Initial Value Formulation
0 0
0 0
0 0
0
0 0
0
=
=
⋅
∇
×
∇
=
∇
=
×
v B
B J
p B
J
G G
G G
G G
μ ( , )
) ~ ( )
,
( r t Q
0r Q
1r t
Q G G G
+
=
v t
∂
= ∂ ξ
1
~
linearized
ξ: displacement of the plasma away from its equilibrium position
) (
~ ) 1 (
) ~ 1 (
~ ~ ) ~
(
) (
0 0
1 1
1 2
2
ξ γ
μ ξ μ
ξ ξ ξ ρ
⋅
∇ +
∇
⋅
∇ +
×
×
∇ +
×
×
∇
=
∇
−
× +
×
=
∂ =
∂
p p
B Q
Q B
p B
J B
J F
t F
G G
G G
G
G
) 0 ,
~ ( ) 0 , (
, 0 ) 0 ,
( v
1r
t
r r G G
G =
∂
= ∂ ξ ξ
momentum equation
force operator
+ Boundary conditions
Formulation of the generalized stability equations as an initial value problem
Stability
• Normal-Mode Formulation
) (
) (
1 1
1
B B
Q
p p
p G G G
×
×
∇
=
≡
⋅
∇
−
∇
⋅
−
=
⋅
−∇
=
ξ
ξ γ
ξ
ρξ ρ
) exp(
) ( )
,
~ (
1
1
r t Q r i t
Q G = G − ω
) (
~ ) 1 (
) ~ 1 (
) (
) (
0 0
2
ξ γ
μ ξ ξ μ
ξ ρξ
ω
⋅
∇ +
∇
⋅
∇ +
×
×
∇ +
×
×
∇
=
=
−
p p
B Q
Q B
F
F
G G
G
G
normal-mode formulation- An eigenvalue problem for the eigenvalue ω2 conservation of mass conservation of energy Faraday‘s law
Stability
• Variational Principle
0 ) 1 ( )
0 (
0 )
(
=
=
=
−
⎟ +
⎠
⎜ ⎞
⎝
⎛
∂
∂ y y
y x g
f y dx
d λ
Classic eigenvalue problem
λ: eigenvalue
∫ ∫ ′ +
= y dx
dx gy
y f
2 2
2
)
λ (
Multiplied by yand integrated over the region 0 ≤ x ≤ 1 Why is this variational?
- Substitute all allowable trial function y(x) into the equation above.
- When resulting λ exhibits an extremum (maximum, minimum, saddle point) then λ and y are actual eigenvalue and eigenfunction.
∫ ′ ∫ ′ + −
−
≈ y dx
dx y
g y
f y
2
0 0
0
) ( ) ]
[(
2 δ λ
δλ
Stability
• Variational Principle
)
*, (
)
*,
2
(
ξ ξ
ξ ξ ω δ
K
= W
)
2
ρξ ( ξ
ω F G
=
−
normal-mode formulationdot product with ξ* then integrated over the plasma volume
Any allowable function ξ for which ω2 becomes an extremum is an
eigenfunction of the ideal MHD normal mode equations with eigenvalue ω2.
∫
∫
∫
=
∇
⋅ +
⋅
∇
∇ +
×
×
∇ +
×
×
∇
⋅
−
=
⋅
−
=
r d K
r d p p
Q B
B Q
r d F
W
G
G G G
G G
G G
2
0 0
2 ) 1
*, (
)]
( )
1 ( )
1 ( [ 2 *
1
) ( 2 *
) 1
*, (
ξ ρ ξ
ξ
ξ ξ μ γ
ξ μ
ξ ξ
ξ ξ
δ = ∫ ′ ∫ +
dx y
dx gy
y f
2 2
2
)
λ (
Stability
• Variational Principle
)
*, (
)
*,
2
(
ξ ξ
ξ ξ ω δ
K
= W
)
2
ρξ ( ξ
ω F G
=
−
normal-mode formulationdot product with ξ* then integrated over the plasma volume
∫ ′ ∫ +
= y dx
dx gy
y f
2 2
2
)
λ (
2 + = 0
− ω K δ W
Kinetic energy
- Change in potential energy associated with the perturbation - Equal to the work done against the force F(ξ)
in displacing the plasma by an amount ξ.
Conservation of energy
Stability
• Energy Principle
2
= ≥ 0
K δ W
ω
stable≥ 0
δ W
stableV S
F
W W
W
W δ δ δ
δ = + +
( )
∫ ⋅ ⋅ ∇ +
=
⊥S S
d S n n p B
W [[ / 2 ]]
2 1
0 2 2
μ ξ
δ G G G
∫
=
VV
B r d W
0 2
ˆ
12 1
δ G μ
ˆ 0
1
=
⋅ B
rwn G
p
p r
r
B n n n n B
B
n ˆ ˆ ( ) ( )[ ( ) ˆ ]
1 1
1
= ⋅ ∇ ⋅ − ⋅ ⋅ ⋅ ∇
⋅ G
⊥G
⊥G G
G ξ ξ
Boundary conditions on trial functions
∫ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢
⎢
⎣
⎡
⋅
∇
∇
⋅ +
⋅
∇ +
×
⋅
−
=
⊥ ⊥ ⊥F P
J Q p p
Q r d
W
* 2 *0 2
) (
) 2 (
1 ξ γ ξ ξ ξ
δ μ G G
G
G
Energy required to compress the plasma: main source of potential energy for the sound wave
Energy necessary to compress the magnetic field: major potential energy contribution to the compressional Alfvén wave
Energy required to bend magnetic field lines: dominant potential energy contribution to the shear Alfvén wave
destabilising
current-driven (kinks) modes (+ or -) Pressure-driven modes (+ or -)
stabilising
∫ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢
⎢
⎣
⎡
⋅
×
−
⋅
∇
⋅
−
⋅
∇ +
⋅ +
⋅
∇ +
=
⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥F P
Q B p p J b Q
r d
W G G G G G
G 2 2 ( )( ) ( )
2
1
*||
2 * 2
0 2
0 2
ξ ξ
κ ξ
ξ γ
κ ξ μ ξ
δ μ
Stability
References
- R.J. Goldston and R.H. Rutherford, “Introduction to Plasma Physics”, Taylor&Francis (1995)
- J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, Plenum Publishing Corporation (1987)
- 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
- http://www.free-online-private-pilot-ground- school.com/Aeronautics.html
- http://serc.carleton.edu/introgeo/models/EqStBOT.html