fusion reactor technology i
TRANSCRIPT
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Fusion Reactor Technology I(459.760, 3 Credits)
Prof. Dr. Yong-Su Na
(32-206, Tel. 880-7204)
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Week 1. Magnetic Confinement
Week 2. Fusion Reactor Energetics (Harms 2, 7.1-7.5)
Week 3. Tokamak Operation (I):
Basic Tokamak Plasma Parameters (Wood 1.2, 1.3)
Week 4. Tokamak Operation (II): Startup
Week 5. Tokamak Operation (III): Tokamak Operation Mode
Week 7-8. Tokamak Operation Limits (I):
Plasma Instabilities (Kadomtsev 6, 7, Wood 6)
Week 9-10. Tokamak Operation Limits (II):
Plasma Transport (Kadomtsev 8, 9, Wood 3, 4)
Week 11. Heating and Current Drive (Kadomtsev 10)
Week 12. Divertor and Plasma-Wall Interaction
Week 13-14. How to Build a Tokamak (Dendy 17 by T. N. Todd)2
Contents
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Week 1. Magnetic Confinement
Week 2. Fusion Reactor Energetics (Harms 2, 7.1-7.5)
Week 3. Tokamak Operation (I):
Basic Tokamak Plasma Parameters (Wood 1.2, 1.3)
Week 4. Tokamak Operation (II): Startup
Week 5. Tokamak Operation (III): Tokamak Operation Mode
Week 7-8. Tokamak Operation Limits (I):
Plasma Instabilities (Kadomtsev 6, 7, Wood 6)
Week 9-10. Tokamak Operation Limits (II):
Plasma Transport (Kadomtsev 8, 9, Wood 3, 4)
Week 11. Heating and Current Drive (Kadomtsev 10)
Week 12. Divertor and Plasma-Wall Interaction
Week 13-14. How to Build a Tokamak (Dendy 17 by T. N. Todd)3
Contents
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4
âą Neoclassical Transports
Tokamak Transport
- Rarefied plasma at high temperature:trapped particles are the main contributors to transport.Diffusion and thermal conductivity are dominated by the collisions which correspond to transferring the particles from being trapped to transit ones and vice versa.
enl / ,/ LtreffTeff qrxqRv »D»D>>=
( ) ennn // 2|| »» ^ vveff e^vv ~||
- Effective collision frequency:
- Transport coefficients:
( ) 222/32Lefftr rqx nene -=D Trapped particle fraction = Δ1/2
- The banana diffusion region is limited by the condition:
1/
//2/3*
2/3
<<=
>>==-
T
TeffTeff
vqR
qRvv
nen
nenele Trapped particle fraction = Δ1/2
e/Ldtr qrtvx »»D
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5
âą Neoclassical Transports
Tokamak Transport
- In the plateau region, 1 < Μ* < Δ-3/2
(Δ3/2 < ΜRq/vT < 1 or Δ3/2vT/Rq < Μ < vT /Rq )- The average collision frequency is less than the mean bounce
frequency â only slow-transit particles contribute to the transport- The relative number of slow-transit particles: Îœ/vT
qRvrqvv TLTeff /~/ 222nD Slow-transit particle fraction = v/vT
- Transport coefficients:
22 / vvTeff nn »- Effective collision frequency:
- Displacement: vvqr TL /»D
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6
âą Neoclassical Transports
Tokamak Transport
- In the Pfirsch-Schlueter region,
- Diffusion flux in a uniform field
drdp
Bnnv ^-= h2
1
drdpBjBBj / 1-^ -=ÂźĂ=ÂŽ
rr
evBjejBvE =Âź=ÂŽ+ ^^hh rrrr Friction force = Lorentz force (ExB drift
not contributing to diffusion)
- Modified diffusion flux by the additional flux due to longitudinalcurrent, so-called, the Pfirsch-Schlueter current owing to the toroidal effect
- Compared with a uniform magnetic field, the flux in toroidal plasma is enhanced by a factor (1+q2).
Ă·Ă·Ăž
öççÚ
ĂŠ+-=
^^
2||2
211 q
drdp
Bnnv
hh
h in H and D plasmas||2hh »^
D. Pfirsch and A. Schlueter, Der Einflussder elecktrischen Leitfaehigkeit auf das Gleichgewichtsverhalten von Plasmenniedrigen Drucks in Stellaratoren, Max-Planck-Institut, Report MPI/PA/7/62 (1962)
2BkTn
D Ă„^^ =
h
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7
âą Pfirsch-SchlĂŒter Current
Tokamak Transport
â charge separation
No charge accumulation on a flux surface (quasineutrality)
A
B
BpJ Ă
=^
r
BA JJ ^^ >rr
0=ĂĂ Jr
J|| needed: Pfirsch-SchlĂŒter current
Diamagnetic current or current needed to establish an equilibrium,
0Ë22 ÂčĂ·Ă·
Ăž
öççÚ
ĂŠ-=
ĂÂŽ=^ y
jddpR
BBI
BpBJ
rrr
pBJ Ă=ÂŽrr
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8
âą Pfirsch-SchlĂŒter Current (J. Wesson, Tokamaks)
Tokamak Transport
pB
RBddp
BRB
ddp
Bp
Bj pp Âą-=-=Ă-=Ă=^ y
yy
11
^--= jBB
jBB
j pp
f||
ppp BFBddFj Âą==y
0 =Ÿ¶¶
-=ÂŽĂ ĂČ dsEtBE ps
ffh EBB
EBB
j psp +=||||
p
p
p
p
p
p
p
pps
p
BB
BBE
BB
BpF
BBBpF
BBBBE
EBBB
pFBj
F
/
/
/
/1
//
//
/1
2||
20
20
2||||
220||
hm
mhh
m
ff
ff
+Âą=
Âą++=
Âą+=Âą
ĂČĂČ= dsxdsx /
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9
Tokamak Transport
Ă·Ă·
Ăž
ö
çç
Ăš
ĂŠ-Âą= B
BB
BB
pFjp
pPS /
/1120m
BBB
BBEB
BB
BB
pFjp
p
p
p
/
/
/
/112
||20|| h
m ff+Ă·Ă·
Ăž
ö
çç
Ăš
ĂŠ-Âą=
- This flow is dominant in the SOL region (high collisionality regime).- Pfirsch-SchlĂŒter current removes the main part of the charge
separation caused by the curvature and gradient drifts (but residual charge separation still causes transport)
For the circular CX large aspect ratio toroidal configuration
qef cos10
+=
BBqq
cos12drdp
Rr
BjPS -=
âą Pfirsch-SchlĂŒter Current (J. Wesson, Tokamaks)
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10
âą Shafranov Shift
Tokamak Transport
- The Pfirsch-SchlĂŒter current produces vertical field BZ,0
â Plasma shifted outwardsâ Shafranov shift
0,0
ZBBRi
»D
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11
Tokamak Transport
Ă·Ă·
Ăž
ö
çç
Ăš
ĂŠ-+
Ă-=
-Ă
-=
Ă-
-=
^^
^^
^^^
fffff
ff
ff
hh
hh
h
EBBB
BBEBB
pjBB
B
BE
Bpj
BBB
Bp
BBEBE
v
p
p
pPS
p
pp
pPS
/
/1
22||
2||||
22
âą Pfirsch-SchlĂŒter Diffusion (J. Wesson, Tokamaks)
22 Bp
BBEv Ă-
ÂŽ= ^
^h
rrr
BBB
BBEjj
p
pPS /
/2
|||| h
ff+=
ffh EBB
EBB
j psp +=||||
Ă·Ă·
Ăž
ö
çç
Ăš
ĂŠ-Âą= B
BB
BB
pFjp
pPS /
/1120m
ĂČĂČ ^^ ==G dsRvnRdsvn pp 22
Ă·Ă·
Ăž
ö
çç
Ăš
ĂŠ-Âą-=^
p
p
pPS BB
BBB
RBpFRv
/
/11220||
fmh
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12
Tokamak Transport
For the circular CX large aspect ratio toroidal configuration
âą Pfirsch-SchlĂŒter Diffusion (J. Wesson, Tokamaks)
2||
2
0
/2q
hBdrdp
Rr
RRv
PS ÷Þö
çÚÊ-=^
( ) 2||2
20
2/BBE
qBdrdp
RRv qfhh -+-= ^
^
q
f
RBrB
q =
Ă·Ă·Ăž
öççÚ
ĂŠ+=
^
2||21 qDD C
hh
cos12drdp
Rr
BjPS -=
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13
âą Neoclassical Transports
Tokamak Transport
TnrTq
nrnD
E
ĂD
-»Ă-=
ĂD
-»Ă-=G
tk
t2
2
)(
)(: Fickâs law
: Fourierâs law
Thermal diffusivity
ct
tkc
22
)( aDrn E
E
»Ÿ»D
»Âș
J. Wesson, Tokamaks (2004)
2BkTn
D Ă„^^ =
h
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14
âą Ware Pinch
Tokamak Transport
- Inward particle transport due to the toroidal electric field
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15
âą Ware Pinch (J. Wesson, Tokamaks)
Tokamak Transport
- The inward flow occurs for trapped particles and their behaviourfollows directly from the toroidal equation of motion.
( ) ( )[ ]fff BvEevmdtd
jj ÂŽ+=
Zero steady state time average of the left-hand side term for trapped particles (the integral between bounces is zero)
( ) ff EBv -=ÂŽ
q
f
BE
v -=^ ( ) qf BvBv ^=ÂŽ
q
feBE
n2/1~G Trapped particle fraction = Δ1/2
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16
âą Ware Pinch (J. Wesson, Tokamaks)
Tokamak Transport
- The modified equation of motion along the magnetic field line
j
jb m
Ees
dtsd fw +-= 22
2
mean angle:
2sinbj
jbb m
Eetss
ww f+=
( ) qq rBBs /=rBmEBe
tbj
jbb 2sin
wwqq fq+=
rBmEBe
bj
j2w
q fq=
The effect of the âB and curvature drift is not symmetric about the mid-plane
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Tokamak Transport
rBmEBe
bj
j2w
q fq=
fq
wwqq
q
ErBmBve
tvv
vv
bj
djjbbdjdj
djr
2sin~
sin
--=-
-=
Resulting radial velocity for deeply trapped particles for which Ξ is small
q
ff
q
ww
BE
ErBBv
vb
djcjr -=-= 2
( )( ) 2/1b
2 2// ,/21 RrqRvRBevmv jjdj ^^ == w
q
f
BE
vr -=
âą Ware Pinch (J. Wesson, Tokamaks)
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18
Tokamak Transport
- The drift velocity is controlled by the balance of two forces, the electrical field force and the Lorentz force.
- vW ~ 0.2 m/s for E = 0.1 V/m, BΞ = 0.5 T- The effect is much larger (1/Δ2) for trapped particles than that
experienced by passing particles. HW: Why?
âą Ware Pinch (J. Wesson, Tokamaks)
rBmEBe
bj
j2w
q fq=
q
f
BE
vr -=
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âą Bootstrap current
Tokamak Transport
- Named after the reported ability of Baron von Munchausen to lifthimself by his bootstraps (Raspe, 1785)
- Suggested with âAlice in Wonderlandâ in mind where the heroine managed to support herself in the air by her shoelaces.
http://en.wikipedia.org/wiki/Bootstrapping
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âą Bootstrap
Tokamak Transport
MEANING:verb tr.: To help oneself with one's own
initiative and no outside help.noun: Unaided efforts.adjective: Reliant on one's own efforts.
ETYMOLOGY:While pulling on bootstraps may help with putting on one's boots, it's impossible to lift oneself up like that. Nonetheless the fanciful idea is a great visual and it gave birth to the idiom "to pull oneself up by one's (own) bootstraps", meaning to better oneself with one's own efforts, with little outside help. It probably originated from the tall tales of Baron MĂŒnchausen who claimed to have lifted himself (and his horse) up from the swamp by pulling on his own hair.
http://wordsmith.org/words/bootstrap.html
Baron MĂŒnchausen lifting himself up from the swamp by his own hair
Illustrator: Theodor Hosemann
In computing, booting or bootstrapping is to load a fixed sequence of instructions in a computer to initiate the operating system. Earliest documented use: 1891.1
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âą Bootstrap
Tokamak Transport
http://wordsmith.org/words/bootstrap.html
âI was still a couple of miles above the clouds when it broke, and with such violence I fell to the ground that I found myself stunned, and in a hole nine fathoms under the grass, when I recovered, hardly knowing how to get out again. Looking down, I observed that I had on a pair of boots with exceptionally sturdy straps. Grasping them firmly, I pulled with all my might. Soon I had hoist myself to the top and stepped out on terra firma without further ado.â- With acknowledgement to R. E. Raspe, Singular Travels, Campaigns and Adventures of Baron Munchausen, 1786. Edition edited by J. Carswell. London: The Cresset Press, 1948. Adapted from the story on p. 22(???).
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âą Bootstrap current
Tokamak Transport
Nature Physical Science 229 110 (1971)
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âą Bootstrap current
Tokamak Transport
IOH
INB+IOH
INB+IOH+IBS
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Toroidal direction
Ion gyro-motion
Fast ion trajectory
Poloidaldirection
Projection of poloidallytrapped ion trajectory
R
B
âą Neoclassical Bootstrap current
http://tfy.tkk.fi/fusion/research/
ASCOT
Tokamak Transport
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Currents due to neighbouring bananas largely cancel
orbits tighter where field
stronger
eb
e
q
pBS
BS
II
drdp
BJ
67.0/
2/1
=
-»Tokamak Transport⹠Neoclassical Bootstrap current
- More & faster particles on orbits nearer the core (green .vs. blue) lead to a net âbanana currentâ.
- This is transferred to a helical bootstrap current via collisions.
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âą Bootstrap Current
Tokamak Transport
M. Kikuch et al, PPCF 37 1215 (1995)
- Trapped-electron orbits and schematics of the velocity distribution function in a collisionless tokamak plasma
Small Coulomb collision smoothes the gap and causes particle diffusion in the velocity space.Collisional pitch angle scattering at the trapped-untrapped boundary produces unidirectional parallel flow/momentum input and is balanced by the collisional friction force between electrons and ions.
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âą Bootstrap Current
Tokamak Transport
( ) ( )[ ]( ) 0 ,,
0 ,,,
||||0
0
||||
>D¶
¶-»
>--=+ -+-+
vdrvrrfe
vdvrfrfeJJ
e
gege
vvvvv
( ) 2/12/12 /eLqrrr »D»D
rn
BTq
drn
BTq
vdvvrF
BqmJ
c
Met
¶¶
-»
¶¶
-=
>¶¶
-=
ĂČ
ĂČ-
^-
0
2/1
2/ 2
0
2/1
||||0
2/1
0
cossin23
0 ,
e
qqqe
e
p
q
v
eq 2cos =c
- The trapped electron magnetization current
Assumption:- Uniform temperature- Infinitely massive ions
not depending on collisions, generated solely because of the density (or temperature) gradient of the guiding centers
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âą Bootstrap Current
Tokamak Transport
( ) 0 ,,||||
0
0 >D¶
¶-» vdrv
rrfeJ e
p vv
Lqrr »D
rn
BTq
drn
BTq
vdvvrF
BqmJ
c
Mep
¶¶
-»
¶¶
-=
>¶¶
-=
ĂČ
ĂČ ^
0
2/ 2
0
||||00
cossin23
0 ,
p
qqqq
v
eq 2cos =c
- The passing electron magnetization current
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âą Bootstrap Current
Tokamak Transport
( )ce
eeeep
eptp
p
peptppep r
nqTJemn
enJ
mnumPwnnnn
¶¶
»-»÷÷Þ
öççÚ
ĂŠ-==D ||
( )ce
eeeet
etpt
t
tetpttet r
nqTJemn
enJmnumP
wne
ennn
¶¶
»-»÷÷Þ
öççÚ
ĂŠ-==D - 2/1
||
- The collision-driven bootstrap current
( ) ( )tp
PP ||2/1
|| D=D e( ) ( )tp
PP |||| DÂčD Collisional momentum balance violated
( ) ( ) ( ) ( )
( ) ( ) Ă·Ă·Ăž
öççÚ
ĂŠ +-
ĂșĂșĂ»
Ăč
ĂȘĂȘĂ«
Ă©+DĂ·
Þö
çÚÊ
¶¶
+»
ĂșĂșĂ»
Ăč
ĂȘĂȘĂ«
é -+-Ÿ÷
Ă·Ăž
öççÚ
ĂŠ +-=
^
^
^
^
^
^
2
2||
2
2||
||
||32/3
2
2||
2
32/32
2||
2
32/3
exp211
exp exp,
vvv
vuv
rrn
nvv
vrn
vuvv
vrn
vvv
vrn
rf
T
Bp
T
g
B
T
g
T
ggp
p
ppv Shifted
Maxwellian
- The shift must be in the passing particles since the trapped particles are âtrappedâ and thus are not allowed to drift toroidally.
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âą Bootstrap Current
Tokamak Transport
( ) eeBpp
epun
eJ
mP nĂ·Ă·Ăž
öççÚ
Ê+-»D ||
( ) ( )tp
PP |||| D=Dce
eeeepBe
ce
ee
rnqTnum
rnqT
wnen
wn
¶¶
=+¶¶ - 2/1
ĂșĂ»Ăč
ĂȘĂ«Ă©
¶¶
+¶¶
-= -
rT
Tn
rn
BTqJB 04.071.4
0
2/1eLarge aspect ratioCircular CXNon-massive ionsNon-uniform temperature
- A transport driven toroidal plasma current carried by the passingelectrons generated by collisional friction with the trapped electron magnetization current
rn
BTquenJ BpB ¶
¶-»-= -
0
2/1e1/Δ and 1/Δ1/2 larger than the trapped and passing particle magnetization current, respectively
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âą Bootstrap Current
Tokamak Transport
PPB
B GJJrf bebef
2/12/1 ~18.1)( -»Âș
( ) ( ) ( )¹¹+= qrBTnrG ln/ln04.0ln
- Bootstrap current fraction
- In high-ÎČ tokamak, ÎČp ~ 1/Δ, implying that fB ~ 1/Δ1/2 >>1:The bootstrap current can theoretically overdrive the total current
- No obvious âanomalousâ degradation of JB due to micro-turbulence- The bootstrap current is capable of being maintained in steady
state without the need of an Ohmic transformer or external current drive. This is indeed a favourable result as it opens up the possibility of steady state operation without the need for excessive amounts of external current drive power.
- This is critical since bootstrap current fractions on the order of fB > 0.7are probably required for economic viability of fusion reactors.
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Tokamak Transportâą 100% bootstrap discharges
Y. Takase, IAEA FEC 1996, S. Coda, IAEA FEC 2008
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âą Neoclassical Transports
Tokamak Transport
J. Wesson, Tokamaks (2004)
- May increase D, c up to two orders of magnitude:- ci 'only' wrong by factor 3-5- D, ce still wrong by up to two orders of magnitude!
2BkTn
D Ă„^^ =
h
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Tokamak Transportâą Transport in fusion plasmas is 'anomalousâ.
- Normal (water) flow: Hydrodynamic equations can developnonlinear turbulent solutions (Reynolds, 1883)
- Transport mainly governed by MHD turbulence: radial extent of turbulent eddy: 1 - 2 cmtypical lifetime of turbulent eddy: 0.5 - 1 ms
- Anomalous transport coefficients are of the order 1 m2/s
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References
- Francis F. Chen, âIntroduction to Plasma Physics and Controlled Fusionâ, 2nd Edition, Plenum Press, New York (1984)- Acad. M. A. Leontovich et al, âReviews of Plasma Physics, Volume 1â, Consultants Bureau, New York (1965)- Jeffrey P. Freidberg, âPlasma Physics and Fusion Energyâ, Cambridge University Press (2007)- Hartmut Zohm, âTokamaks: Equilibrium, Stability and Transportâ, IPP Summer University on Plasma Physics, Garching, 18 September, 2001- Tim Hender, âNeoclassical Tearing Modes in Tokamaksâ, 2009 Korean Physical Society/ Division of Plasma Physics (KPS/DPP) in Daejun, Korea, 24 April 2009- Mitsuru Kikuchi, âFrontiers in Fusion Researchâ, Springer (2011)