tokamak fusion basics and the mhd equations -...
TRANSCRIPT
MHD Simulations for Fusion Applications
Lecture 1
Tokamak Fusion Basics and the MHD Equations
Stephen C. JardinPrinceton Plasma Physics Laboratory
CEMRACS ‘10
Marseille, France
July 19, 2010
1
Fusion Powers the Sun and Stars
Can we harness Fusion power on earth?
The Case for Fusion Energy• Worldwide demand for energy continues to increase
– Due to population increases and economic development
– Most population growth and energy demand is in urban areas• Implies need for large, centralized power generation
• Worldwide oil and gas production is near or past peak– Need for alternative source: coal, fission, fusion
• Increasing evidence that release of greenhouse gases is causing global climate change . . . “Global warming”– Historical data and 100+ year detailed climate projections
– This makes nuclear (fission or fusion) preferable to fossil (coal)
• Fusion has some advantages over fission that could become critical:– Inherent safety (no China syndrome)
– No weapons proliferation considerations (security)
– Greatly reduced waste disposal problems (no Yucca Mt.)
Controlled Fusion uses isotopes of Hydrogen in a High Temperature Ionized Gas (Plasma)
Deuterium
Tritium
Helium nuclei(α-particle) … sustains reaction
Neutron
Deuterium exists in nature (0.015% abundant in Hydrogen)
α
TTritium has a 12 year half life: produced via 6Li + n T + 4He
Lithium is naturally abundant
Lithium
proton
neutron
key
Need ~ 5 atmosphere @ 10 keV
Controlled Fusion Basics
Create a mixture of D and T (plasma), heat it to high temperature, and the D and T will fuse to produce energy.
PDT = nDnT <σv>(Uα+Un)
at 10 keV, <σv> ~ T2
PDT ~ (plasma pressure)2
Operating point ~ 10 keV
Note: 1 keV = 10,000,000 deg(K)
Toroidal Magnetic Confinement
Charged particles have helical orbits in a magnetic field; they describe circular orbits perpendicular to the field and free-stream in the direction of the field.
TOKAMAK creates toroidal magnetic fields to confine particles in the 3rd
dimension. Includes an induced toroidal plasma current to heat and confine the plasma
“TOKAMAK”: Russian abbreviation for “toroidal chamber”
• 500 MW fusion output• Cost: $ 5-10 B • Originally to begin operation in 2015 (now 2028 full power)
ITER is now under construction
scale
International Thermonuclear Experimental Reactor:
• European Union• Japan• United States• Russia• Korea• China• India
• World’s largest tokamak • all super-conducting coils
Tore Supra
ITER
ITER has a site…Cadarache, France
June 28, 2005Ministerial Level Meeting
Moscow, Russia
Progress in Magnetic Fusion Researchand Next Step to ITER
Years
Meg
awat
ts
10
1,000
100
10
1,000
100
10
100
1,000
Kilo
wat
tsW
atts
Mill
iwat
ts
1,000
100
10
Fusion Power
1975 1985 1995 2005
Data from Tokamak Experiments Worldwide
2015
TFTR(U.S.)
JET(EU)
2025
ITER(Multilateral)
Start of ITER
Operations
Operation with full power test
0123456789
10
Power Gain
TFTR/JET ITER
0
50
100
150
200
250
300
350
400
450
500
Power (MW) Plasma Duration(Seconds)Power
(MW)Duration(Seconds)
Power Gain(Output/Input)
A Big Next Step to ITERPlasma Parameters
Simulations are needed in 4 areas
• How to heat the plasma to thermonuclear temperatures ( ~ 100,000,000o C)
• How to reduce the background turbulence
• How to eliminate device-scale instabilities
• How to optimize the operation of the whole device
10-10 10-2 104100
SEC.
CURRENT DIFFUSION
10-8 10-6 10-4 102ωLH
-1Ωci
-1 τAΩce-1 ISLAND GROWTH
ENERGY CONFINEMENTSAWTOOTH CRASHTURBULENCEELECTRON TRANSIT
(a) RF codes(b) Micro-turbulence codes (c) Extended-
MHD codes(d) Transport Codes
These 4 areas address different timescales and are normally studied using different codes
Extended MHD Codes solve 3D fluid equations for device-scale stability
10-10 10-2 104100
SEC.
CURRENT DIFFUSION
10-8 10-6 10-4 102ωLH
-1Ωci
-1 τAΩce-1 ISLAND GROWTH
ENERGY CONFINEMENTSAWTOOTH CRASHTURBULENCEELECTRON TRANSIT
• Sawtooth cycle is one example of global phenomena that need to be understood
• Can cause degradation of confinement, or plasma termination if it couples with other modes
• There are several codes in the US and elsewhere that are being used to study this and related phenomena:
• NIMROD
• M3D
Quicktime Movie shows Poincare plot of magnetic field at one toroidal location
• Example of a recent 3D calculation using M3D code
• “Internal Kink” mode in a small tokamak (Sawtooth Oscillations)
• Good agreement between M3D, NIMROD, and experimental results
•500 wallclock hours and over 200,000 CPU-hours
Excellent Agreement between NIMROD and M3D
Kinetic energy vs time in lowest toroidal harmonics
M3D NIMROD M3D NIMROD
Flux Surfaces during crash at 2 times
15
2-Fluid MHD Equations:
( )
0
2
2
1
( ) 0 continuity
0 Maxwell
( ) momentum
Ohm's law
3 3 electron energy2 23 32 2
i
ee e
ii
e
e
i
GV
pne
n nt
t
nM p
J Q
t
p p pt
p p pt
μ
η
μ
η Δ
∂+ ∇ • =
∂∂
= −∇× ∇ = = ∇ ×∂
∂+ • ∇ + ∇ = × −
∂
+ × = +
∂ ⎛ ⎞+ ∇ =
∇
×
− ∇⎜ ⎟∂ ⎝ ⎠∂ ⎛ ⎞
+
+ ∇ =
∇
+ − ∇ +
− ∇
− ∇⎜ ⎟∂ ⎝ ⎠
Π
J
V
B E B J B
V V V J V
J
B
E BB
V
V
q
V
V
i
i i
i
i
i 2 ion energyiV Qμ Δ+ ∇ − ∇ −V qi i
Resistive MH2-
Idea
fluiD
d
l MHD
MHD
number densitymagnetic fieldcurrent densityelectric field
mass densityi
n
nM ρ≡
ΒJE
fluid velocityelectron pressureion pressure
electron charge
e
i
e i
ppp p pe
≡ +
V
0
viscosityresistivity
heat fluxesequipartitionpermeability
Qμ
μη
Δ
i eq ,q
16
Ideal MHD Equations:
0
( ) 0 continuity
0 Maxwell
( ) momentum
0 Ohm's law3 3 energy2 2
i
n nt
t
nM pt
p p pt
μ
∂+ ∇ • =
∂∂
= −∇× ∇ = = ∇×∂
∂+ • ∇ + ∇ = ×
∂+ × =
∂ ⎛ ⎞+ ∇ = − ∇⎜ ⎟∂ ⎝ ⎠
V
B E B J B
V V V J B
E V B
V V
i
i i
Ideal MHDnumber densitymagnetic fieldcurrent densityelectric field
mass densityi
n
nM ρ≡
ΒJE
fluid velocityelectron pressureion pressure
e
i
e i
ppp p p≡ +
V
0 permeabilityμ
17
Ideal MHD Equations:
0
5/3
( ) 0 continuity
0 Maxwell
( ) momentum
0 Ohm's law3 3 energy2 2
0 entropy
t
t
pt
p p pt
ss p st
ρ ρ
μ
ρ
ρ −
∂+ ∇ • =
∂∂
= −∇× ∇ = = ∇×∂
∂+ • ∇ + ∇ = ×
∂+ × =
∂ ⎛ ⎞+ ∇ = − ∇⎜ ⎟∂ ⎝ ⎠∂
≡ ⇒ + ∇ =∂
V
B E B J B
V V V J B
E V B
V V
V
i
i i
i
Ideal MHDnumber densitymagnetic fieldcurrent densityelectric field
mass densityi
n
nM ρ≡
ΒJE
fluid velocityelectron pressureion pressure
e
i
e i
ppp p p≡ +
V
0
can be eliminated/ is redundant
is redundantpermeability
tρ
μ
∂ ∂∇
E,J
Bi
18
Ideal MHD Equations:
( )
( )
( )
0
3/5
1( )
3 32 2
0
/
t
pt
p p pt
s st
p s
ρμ
ρ
∂= ∇ × ×
∂∂
+ • ∇ + ∇ = ∇ × ×∂
∂ ⎛ ⎞+ ∇ = − ∇⎜ ⎟∂ ⎝ ⎠∂
+ ∇ =∂
=
B V B
V V V B B
V V
V
i i
i
mass densitymagnetic fieldfluid velocity
entropy densityfluid pressure
sp
ρΒV
0
is redundantpermeabilityμ
∇ Bi
Quasi-linearSymmetric real characteristicsHyperbolic
Ideal MHD characteristics:The characteristic curves are the surfaces along which the solution is propagated. In 1D, the characteristic curves would be lines in (x,t)
0s sut x
∂ ∂+ =
∂ ∂
Boundary data (normally IC and BC) can be given on any curve that each characteristic curve intersects only once:
Cannot be tangent to characteristic curve
To calculate characteristics in 3D, we suppose that the boundary conditions are given on a 3D surface and ask under what conditions this is insufficient to determine the solution away from this surface. If so, is a characteristic surface.
Perform a coordinate transformation: and look for power series solution away from the boundary surface
0( , )tφ φ=r
( )( , ) , , ,t φ χ σ τ→r0φ φ=
( ) ( ) ( ) ( ) ( ) ( )0 00 0
0 0 0 0 0, , , , ,φ φφ φ
φ χ σ τ χ σ τ φ φ χ χ σ σ τ τφ χ σ τ
∂ ∂ ∂ ∂= + − + − + − + −
∂ ∂ ∂ ∂v v v vv v
These can all be calculated since they are surface derivatives within 0φ φ=
If this cannot be constructed, then is a characteristic surfaceφ
φ
20
Ideal MHD characteristics-2:
0 0 0 00 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0
0 0 0 0 0 0 0
z A x A x S
z A
z S
z A
z A
x A
x S z S
u n V n V n cu n V
u n cn V u
n V un V un c n c u
u
− −⎡ ⎤⎢ ⎥− −⎢ ⎥⎢ ⎥−⎢ ⎥− −⎢ ⎥= ⎢ ⎥− −
⎢ ⎥−⎢ ⎥
⎢ ⎥−⎢ ⎥−⎢ ⎥⎣ ⎦
A0
0
0
1
x
y
z
x
y
z
S
V
V
V
B
B
B
c p
s
ρ
ρ
ρ
ρ μ
ρ μ
ρ μ−
⎡ ⎤′⎢ ⎥
′⎢ ⎥⎢ ⎥
′⎢ ⎥⎢ ⎥′⎢ ⎥= ⎢ ⎥′⎢ ⎥⎢ ⎥′⎢ ⎥
′⎢ ⎥⎢ ⎥
′⎢ ⎥⎣ ⎦
X
= ⋅ ⋅ ⋅A Xi
( )
( ) ( )
0Introduce a characteristic surface ( , ) ˆspatial normal /
characteristic speed: /t
t
u
φ φ
φ φ
φ φ φ
φ
=
= ∇ ∇
≡ + ∇ ∇
∂′ =∂
rn
Vi
( )( )
0,0,ˆ
ˆis in z direction
propagation in (x,z),0,x z
B
n n
=
=
B
n
B
0
53
/
/A
S
V B
c p
μ ρ
ρ
≡
≡Ideal MHD
All known quantitiesAll terms containing derivatives involving φ
if det = 0 is characteristic surfaceφ→A
21
Ideal MHD wave speeds:
( ) ( )
( ) ( )
2 20
2 2
1/222 2 2 2 2 2 2 21 12 2
1/222 2 2 2 2 2 2 21 12 2
entropy disturbance
Alfven wave
slow wave
fas
0
4
t wave4
A An
s A S A S An S
f A S A S An S
u u
u u V
u u V c V c V c
u u V c V c V c
= =
= =
⎡ ⎤= = + − + −⎢ ⎥⎣ ⎦
⎡ ⎤= = + + + −⎢ ⎥⎣ ⎦
( ) ( )2 2 2 4 2 2 2 2 2
det 0
0An A S An SD u u V u V c u V c
=
⎡ ⎤= − − + + =⎣ ⎦
A ( )( )
0
53
2 2 2
0,0,ˆ ,0,
/
/
x z
A
S
An Z A
B
n n
V B
c p
V n V
μ ρ
ρ
=
=
≡
≡
≡
B
n
2 2
2 2
2 2
2 2
2 2 2
Alfven wave
slow wave
fast wave
A An
s z S
A x Sf
u u V
u u
u u
n c
V n c
= =
=
= +
In normal magnetically confined plasmas, we take the low-β limit 2 2
S Ac V
22
Ideal MHD surface diagrams
2 2
2 2
2 2
2 2
2 2 2
Alfven wave
slow wave
fast wave
A An
s z S
A x Sf
u u V
u u
u u
n c
V n c
= =
=
= +
( )( )
0
53
2 2 2
0,0,ˆ ,0,
/
/
x z
A
S
An Z A
B
n n
V B
c p
V n V
μ ρ
ρ
=
=
≡
≡
≡
B
n
Reciprocal normal surface diagram Ray surface diagram
0
0
0
1
x
y
z
x
y
z
S
V
V
V
B
B
B
c p
s
ρ
ρ
ρ
ρ μ
ρ μ
ρ μ−
⎡ ⎤′⎢ ⎥
′⎢ ⎥⎢ ⎥
′⎢ ⎥⎢ ⎥′⎢ ⎥= =⎢ ⎥′⎢ ⎥⎢ ⎥′⎢ ⎥
′⎢ ⎥⎢ ⎥
′⎢ ⎥⎣ ⎦
X
10 0 1 000 1 0 000 0 0 100 0 1 000 1 0 010 0 0 0/0 0 0 101 0 0 0
S Ac V
⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
±⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥±⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥±⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
± ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦
entropy Alfven
fastnx = 1nz = 0
fastnx = 0nz = 1
slownx = 0nz = 1
Only the fast wave can propagate perpendicular to the background field, and does so by compressing and expanding the field
The Alfven wave only propagates parallel to the magnetic field, and does so by bending the field. It is purely transverse (incompressible)
( )( )
0
53
2 2 2
0,0,ˆ ,0,
/
/
x z
A
S
An Z A
B
n n
V B
c p
V n V
μ ρ
ρ
=
=
≡
≡
≡
B
n
Ideal MHD eigenvectors
The slow wave does not perturb the magnetic field, only the pressure
Slow Wave Alfven Wave Fast Wave
Background magnetic field direction
propagation propagation prop
agat
ion
• only propagates parallel to B• only compresses fluid in parallel direction• does not perturb magnetic field
• only propagates parallel to B• incompressible• only bends the field, does not compress it
• can propagate perpendicular to B• only compresses fluid in ⊥ direction• compresses the magnetic field• This is the troublesome wave!
Tokamaks have Magnetic Surfaces, or Flux Surfaces
φ
Magnetic field is primarily into the screen, however it has a twist to it. After many transits, it forms 2D surfaces in 3D space.
Because the particles are free to stream along the field, the temperatures and densities are nearly uniform on these surfaces.
Only the Fast Wave can propagate across these surfaces, but it will have a very small amplitude compared to the other waves.
Tokamak schematic Tokamak cross section
• The field lines in a tokamak are dominantly in the toroidal direction.
• The magnetic field forms “flux surfaces”.
• Only the fast wave can propagate across these surfaces.• Since the gradients across surfaces are large (requiring high resolution), the time-scales associated with the fast wave are very short• However, the amplitude will always be small because it compresses the field.
The presence of the fast wave makes explicit time integration not practical
Must deal with Fast Wave
Summary• Nuclear fusion is a promising energy source that will be demonstrated in the coming decades by way of the tokamak (ITER)
• Global dynamics of the plasma in the tokamak are described by a set of fluid like equations called the MHD equations
• A subset of the full-MHD equations with the dissipative terms removed are called the ideal-MHD equations
• These have wave solutions that illustrate that there are 3 fundamentally different types of waves.
• Unstable plasma motions are always associated with the slow wave and Alfven wave.
• The fast wave is a major source of trouble computationally because it is the fastest and the only one that propagates across the surfaces
• Largely because of the fast wave, implicit methods are essential
28