Transcript EFTC10 Oral Presentation

Plasma Instabilities and Turbulence-II: Fusion
Plasmas, particularly tokamaks
A Thyagaraja
UKAEA/EURATOM Fusion Association
Culham Science Centre, Abingdon, OX14 3DB, UK
QUAMP Lecture, Durham, September 18, 2006
1
Plasmas: what are they and why study them?
• Fully or partially ionized collections of charges; dominated by collective, longrange Coulomb forces; ubiquitous in the Universe
• Debye’s theory of electrolytes; electron gas in metals are “cold plasmas”.
Technologically important plasmas are discussed by Professor Braithwaite.
“Classical”, high temperature, fully ionized plasmas common to Astrophysics
and Fusion studies.
• Fusion can only occur at high temperature: Coulomb repulsion of positively
charged nucleii vs. nuclear forces.
• Individual charged particle motions in E,B fields understood; “collective
behaviour” very complex many-body problem- “Final frontier” of classical
physics!
2
Plasma is the fourth state of matter
At high temperature, the atoms in
a gas dissociate into nuclei and
electrons: a “plasma”.
Plasmas are electrically
conducting, and can therefore be
controlled by magnetic fields.
Most of the Universe consists of
plasma.
Picture coutesy of Stephen Haigh,
Culham Electromagnetics and
Lightning
3
4
The DT fusion reaction cycle
The DT fusion reaction produces He and a neutron. The latter reacts
with Lithium producing Tritium, which is re-cycled.
5
6
Kinetic picture of plasma dynamics: FokkerPlanck/Maxwell system with sources
q
f e,i
f e,i
v
v
 f e,i 
[E   B].
 Si,e  Ci,e
t
v
m
e,i
e,i
 B0
B   j
0
1 E
c
2
t
B
   E
t
(q n i  q n e )
 E
i

e
0
 f dv dv dv  n
i,e
x
y
z
i,e
 f vdv dv dv  n u
i,e
x
y
z
i,e
i,e
q in i  q en e "quasi  neutrality"
j  q in iu i  q en eu e
7
Fokker-Planck/Maxwell system with sources
• Rather formidable system with extremely different scales!
• “Collisionless kinetics”-Vlasov-Maxwell system. Leads to plasma waves and
“Landau (collisionless) damping” by wave-particle interactions and phase-mixing.
• Quasi-neutrality applies to all length-scales much larger than the Debye shielding
length and frequencies smaller than plasma frequency.
2 1/ 2
1/ 2
V
th e,i
2T e,i
V
th,e
e
n
e
(
;
) ; ω pe (
=
) ; λ Debye
me,i
ω pe
 me
B
e,i

;

eB
;
m
ρ Larmor  V Ω
1/ 2 Ω
0
V
Alfven
( 0mine)
ce ,i
e ,i
th e ,i
ce ,i
8
Individual charged-particle behaviour in
specified E and B fields
• Neglecting inter particle interactions, individual charged particle motions in given E
and B fields well-described by Newton-Lorentz equations.
• Charged particles with charge e and mass m “gyrate” about field lines with Larmor
frequency (eB/m) and a Larmor (or gyro) radius (mv/eB). In the presence of “slowly
varying” fields, they possess an adiabatic invariant, the magnetic moment μ.
2
mv 
(


)
• The perpendicular kinetic energy is μB :
This leads to “mirror force”
2B
and particle “trapping” in inhomogeneous B-fields. From these we can distinguish
“trapped particle” and “passing particle” regions in velocity space
• The particles “feel” various drifts: “E x B”, “grad B”, “curvature”, “inertial” etc.
• “Collective behaviour” involves “back reaction” by particles on fields via their
currents and charge densities (usually negligible); very complicated!
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Magnetically confined Fusion...
Picture courtesy of NASA/GSFC.
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Two-fluid model: evolution equations
n
.(nv)  Sp    "Continuity"
t
dv
m i n  (pi  pe)  j  B  Feff    "Ion  momentum"
dt
3 T i,e v
n[
 i,e.T i,e]  nT i,e .v i,e . q i,e P i,e    "Energy"
2
t
E  v e  B   p / en  R e    "Electron  momentum"
e
11
Characteristics of inhomogeneous turbulence
• Driven, dissipative systems with very large numbers of degrees of freedom
exhibit “turbulent evolution”.
• Energy from the driving sources is redistributed among the unstable and
stable degrees of freedom and dissipated/transported out of the system.
• Free energy redistribution occurs in both real space (“turbulence
spreading”) and wave number space (“cascading”). Enstrophy (vorticity)
cascades to high-k!
• Typically excitation occurs at macro/meso scales.
• Dissipation usually occurs at very short length scales.
12
Nature of plasma instabilities
• Plasma waves of importance in tokamak: Alfvén waves in which the “tension”
of magnetic field provides the restoring force. Sound waves due to
compressibility of plasma. Drift waves - sound waves modified by magnetic
field structure and gradients of density and temperature
• The plasma stores energy in the magnetic field, pressure and possibly flows.
These sources of free energy are tapped by a wide variety of “current” and
“pressure” and “entropy gradient/curvature” -driven linear and nonlinear
instabilities (tearing, ballooning, drift collisional and collisionless, trappedparticle, sawteeth, disruptions, ELM’s NTM’s, RWM’s,…).
• This zoology of unstable plasma modes are due both to velocity and position
space effects. The most dangerous ones are big MHD modes. After “taming”
them with careful manipulation of current and field (safety factor q) residual
losses are due to electromagnetic turbulence in the “mesoscale”.
13
Tokamaks: equilibrium, macro-stability, transport
• Equilibrium: “force balance” and field distribution; flows; electric field: “macrostructure” sustained by sources
• Macro-stability against large-scale “disruptive” instabilities (MHD, ELM’s,
NTM’s, RWM’s,…)
• Strictly “collisionless” systems do not exist! Collisional dissipation sets lower
limit to losses.
• “Micro-structure”: enormous range; fundamentally nonlinear though may be
“triggered” linearly.
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15
16
A MAST plasma
MAST
Spherical Tokamak
Temperature ~
15million oC
~ 3m
17
Produce significant amounts of
fusion power (at least ten times
the power required to heat the
plasma up).
Plasma duration ~30 minutes
Aim at demonstrating steadystate operation.
Develop fusion reactor relevant
technologies
Project involves Europe, Japan, USA, Russia, China, S Korea. India
18
Approved July, 2005, start-up 2015? Construction at Cadarache, France
Tokamaks: spectral transfer mechanisms
• Electromagnetic turbulence is due to linear/nonlinear instability and
spontaneous symmetry breaking-results in both direct and inverse spectral
cascades.
• Sheared flows and Alfvén waves cascade (particularly enstrophy) to high radial
k. Landau and other damping “kill” fine-scale structures smaller than
collisionless skin depth (if they exist, “where are they?”)
• Two high-k linearly growing modes can “beat” to populate the low-k and can
also decay strongly by modulational instability: a fundamental “inverse spectral
cascade” process [ cf. Lashmore-Davies et al PoP (2005).]
• Powerful means to “self-generate” equilibrium flows, currents and populate lowk spectral regions “forming condensates”.
19
Characteristics of tokamak “plasma climatology”
• Universal electromagnetic turbulence (dn/n and dj/j comparable!), <
system size and >ion gyro radius; <confinement (s) and >Alfvén (ns)
times.
• Strong interactions between large and small scales.
• Plasma “self-organizes”, like planetary atmospheres (Rossby
waves=Drift waves).
• Transport “barriers” connected with sheared flows, rational q’s,
inverse cascades/modulational instabilities (Hasegawa).
• Analogous to, “shear sheltering” (J.C.R Hunt et al):
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Spectral transfer mechanisms: schema
Nonlinearity; phase mixing by flows & Alfven waves
Direct cascade
ExB;jxB
Inverse cascade
Zonal
flows
Streamers
Dynamo
currents
Macroscale
Mesoscale
Microscale
21
Modulational Instabilities; Wave beating
A basic turbulence characteristic
• The free energy flux into the unstable modes is redistributed by the
nonlinearity (saturation).
• All turbulent transport occurs with “effective diffusivity” (turbulent
Prandtl number O(1)) of order:
Dturb  (Lturb) turb
2
22
Plasma turbulence
• Turbulence provides a typical “decorrelation-rate”, (effective collision
frequency) via the saturated RMS Ex B flow vorticity [1/T].
qR
L turb   s[ r ]
Poloidal gyro-radius (meso
lengthscale)
2
| 2
2
|
k



(
)
turb
B

Turbulent RMS vorticity
( meso frequency scale)
23
Tokamaks: profile-turbulence interactions
• All instability, linear or nonlinear caused by thermal disequilibrium in a drivendissipative system-sources drive transport!
• Profiles and turbulence cross-talk: turbulence corrugates profiles; latter saturate
turbulence. Both electrostatic and magnetic components interact strongly and
play a role (cf. NSTX, MAST,..)
• Macroscale phenomena (pellets, sawteeth, ELM’s, ITB’s,..) influence and are
influenced by mesoscale turbulence (possibly also micro scale): nonlinear selforganization
• Momentum/angular momentum exchanges between turbulence and “mean
profiles” result in dynamo currents (electrons) and zonal flows (ions).
• No real “scale separation”-a continuum of scales in time and space
24
Zonal flows
• Poloidal E x B flows, driven by Reynolds/Maxwell
stresses against collisional damping.
• Modulational instability, “inverse cascade” , eg.
Generalized Charney Hasegawa Mima Equation (cf.
Lashmore-Davies et al, PoP, 2005).
• Highly sheared transverse flows “phase mix” and lead to
a “direct cascade” in the turbulent fluctuations.
• Enhances diffusive damping and stabilizes turbulence
linearly and nonlinearly.
• Confines turbulence to low shear zones. (“turbulence
sequestration”)
25
Typical advection-diffusion equation
f

t
f

f
v y (x) y  x ( D x )
Sheared velocity in combination with diffusion changes
spectrum
2
“Reynolds number”
Damping rate:
R  (v' y( x) L x) / D
 v'  D
2/ 3
y
1/ 3

 v'  R
1/ 3
y
Spectrum discrete, “direct cascade due to phase mixing”
“Jets” in velocity “ghettoized” to low shear regions
26
Zonal flow dynamics
u 
1  (r  )

(
)
  nc u nc u  

t
r r


“Poloidal momentum”


 e .(jxB)  / mi n
“Lorentz force”


   u u r 
“Reynolds stress”
1 pi
E r  (u Bzu zB)  en r
“Radial Ohm’s Law”
27
Neoclassical expressions
(cf. Rosenbluth and Hinton, 1999)
u
[
nc

1.17  0.35
1  0.7

*
][ 1  1.46
*
(r / R) ](
1
eB
)
dT i
dr
 nc  0.67[  i r / R ]
1
 (
*
2qR /  i V thi)(r / R )
3/ 2
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Zonal flow experimental analysis
u
(
u
1  (r  )
)  1 (
)[

 ncu nc r r
nc

1
]
LHS can be measured! (in
principle)

Higher the Z, easier it is to get
Er.
 n ImT
1
E r  (u B u zB )  Zen

Im
uz
u
z

Im
r
i
Estimation of non-neoclassical
poloidal torque needs
background flow measurements.
ion
z
T i  T Im
29
Mean poloidal field evolution
 B
t

E z
“Faraday’s Law”
r
Ez   ( j  j  j
nc
z
bs
dynamo
)
1 r B
 0 jz  r r
 j
nc
dynamo
 e z.(vxB) 
“Generalised Ohm’s Law”
“Ampère’s Law”
“Dynamo current”
30
Time-averaged Zonal Flow (-Er/B) and
Current density components (RTP)
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RTP tokamak: well-diagnosed, revealing subtle features of
transport, excellent testing ground
Step-like
changes
in Te(0)
“plateaux”
whenever
deposition
radius
crosses
“rational”
surfaces!
Sawtooth like oscillations
A
A’
Te(0)
A”
B
C
D
E
Hollow Te
ECH power deposition radius (Rho/a)
0.5
32
RTP Experimental Te profiles for different ECH
deposition radii
33
“Arithmetizing” two-fluid plasma turbulence:CUTIE
• Global, electromagnetic, two-fluid code.Co-evolves turbulence
and equilibrium-”self-consistent” transport.
• “Minimalist plasma climatology” : evolve Conservation Laws and
Maxwell’s equations for 7-fields, 3-d, pseudo spectral+radial
finite-differencing, semi-implicit predictor-corrector, fully nonlinear.
• Periodic cylinder model, but field-line curvature treated; describes
mesoscale, fluid-like instabilities, but no kinetics or trapped
particles (but includes neoclassics).
• Very simple sources/boundary conditions.
34
Equations solved: reduced forms
Continuity
Energy
Parallel
momentum
Potential vorticity
Quasi-neutrality
Ohm+Faraday
35
Off-axis ECH in RTP
[Phys Rev Letts.- de Baar et al, 94, 035002, (2005)]
• Ip=80 kA, B=2.24 T, qa=5.0, Hydrogen plasma
• neav ~ 3.0 E+19 m-3 PECH~350 kW, P ~80 kW
• PECH deposited at r/a = 0.55
• Resolution: 100x32x16; dt=25 ns ; simulated for >50 ms
   (%)  0.16;  *  0.007; *  1.0
36
Zonal flow (-Er/B) evolution (RTP): corrugations
37
Barriers and q
Off-axis Sawteeth simulated by CUTIE:
Te, q at r/a=0, 0.55
38
Sawtooth details and Magnetic and Electrostatic turbulence evolution
in CUTIE
(will be clearly shown in movies!)
Volume averaged magnetic and electrostatic turbulence measures
Note "precursors", compound crash, amplitude levels and phasing
39
No dynamo, no sawteeth!
With dynamo
No dynamo
Volume averaged magnetic turbulence measure and loop voltage
No "precursors" but "postcursors" in magnetic turbulence
40
Plasma turbulence: density fluctuations in R-Z plane and
poloidal mode spectrum
(Thyagaraja et al, Phys. Plasmas,12, 090907, 2005)
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“Ear choppers”: CUTIE vs. Expt.
42
Discussion of CUTIE's strengths and weaknesses
• Have shown only a small selection of results for many machines
• Some limitations: missing are, full toroidal geometry, trapped particle physics,
kinetic (finer-scale) dynamics, atomic physics effects, proper source terms, "real
time" (ie fast!) calculations.
•
Higher resolution in space (with correct physics!) would also be helpful to
answer worries about missing out on the "microscale".
•
CUTIE's "minimalist" model can be used globally to get a synoptic description
of a range of dynamic phenomena involving turbulence and transport.
• Provides broad, qualitative account of macro/mesoscale experimental
phenomena.
43
Conclusions
• CUTIE results bear a qualitative resemblance to experiments
(RTP, MAST, COMPASS, JET)
• Is there any quantitative agreement? Yes, partially!
• What have we learned from CUTIE simulations? Profileturbulence, electromagnetic effects are not negligible
• What are the limitations of minimalism and how can one proceed
further? Many defects, some addressed in CENTORI
• What are the lessons (if any) for the future? Mesoscale
evolution is crucial; trapped particles, electron-inertia
relevant
44
Terrestrial, man-made fusion offers the promise of limitless
energy supply with no environmental impact and no safety
issues.
JET has achieved everything it set out to do, to the point where we can
build ITER with confidence and design power-plant scenarios.
JET is a truly European experiment that operates very effectively. It is
run along the lines of a user facility - operated by UKAEA for teams of
visiting European scientists. It is part of a wide-ranging international
fusion research programme.
The world is ready to take the next step towards commercial fusion
power by building ITER.
45
This work was funded by EURATOM and the UK OST.