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
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)
To build a sun on earth
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
JET (Joint European Torus): R
0=3m, a=0.9m, 1983-today
Tokamak
JET (Joint European Torus): R
0=3m, a=0.9m, 1983-today
Tokamak
How to describe fusion plasmas?
Single Particle Approach
• Single particle motion of the plasma
Magnetic
field
• 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
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
Fluid Approach
• Plasmas as fluids
- Introduce the concept of an electrically charged
current-carrying fluid.
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)
• Ideal MHD: = 0 • Resistive MHD: 0
What is Ideal MHD?
• Ideal MHD: = 0 • Resistive MHD: 0
What is Ideal MHD?
Applications
X
●
●
X
X X
X
Force PF coils
Plasma Eqauilibrium and Stability
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
• 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?
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
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)
• 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 u q 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
21
2 1
2
1
2 / 1
2 3
2 1
u m Q
u m Q
Q
massmomentum energy
• 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
2w 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
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 u q 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
• 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.
• 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
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
er 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
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 densitym
e 0 , n
i n
e n
fluid velocity current density
total pressure total temperature
• Step 2. Introduce single fluid variables
Derivation of the Ideal MHD Model
0
v
t
)
(
i ep 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
1 /
~ V
T iia
ii i
Derivation of the Ideal MHD Model
1 /
) / (
~ m
em
i 1/2V
T iia
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 eee
i
Derivation of the Ideal MHD Model
a V
v
v
Tijj
~ /
3 2 2
~
| | | |
ii
nT
i
~ 1
/
~
/
ip V
T iia
i
viscosity)
(
i ep 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
ia 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
Liii i e e
1 / ~
) / (
:
1/2 2
a V a
r m
m t
p
en
J
e Li T iie
i
1 / ~
/ )
(
a r t
p
en p
J
e Li1 / ~
:
1/2
a V m
m t
p
v
T iii e e
e
i
1
/ ~
:
a V t
p
v
T ii ii
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
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
| |
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 iie 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 iie i
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
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
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