Radio Frequency Plasma Heating

Transcript

PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Class material- distribution forbidden
Radio Frequency Plasma
Heating
Giuseppe Vecchi
Credits/thanks:
Riccardo Maggiora & Daniele Milanesio
1
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Ohmic Heating
The plasma current is driven by a toroidal electric field induced by transformer
action, due to a flux change produced by current passed through the primary coil
Initial heating in all tokamaks comes from the ohmic heating caused by the
toroidal current (also necessary for plasma equilibrium)
PΩ = η j 2
: ohmic heating density
Limitations:
• on current density to avoid instabilities and disruptions
• by plasma resistivity
−3
η ∝T
2
Additional heating needed
2
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Method
Principle
Heated species
Neutral Beam
Injection
Injecting a beam of neutral atoms at
high energy across magnetic field
lines
Electrons, ions
Electromagnetic
Waves
α-Particles
•
•
•
•
Exciting of plasma waves that are
damped in plasma
Alfven waves
ion cyclotron waves
lower hybrid waves
electron cyclotron waves
Collisions
Auxiliary Heating and Current Drive (H&CD) Methods
Electrons
Electrons, ions
Electrons
Electrons
Electrons, ions
At ignition, only α-particles sustain the fusion reaction
3
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Electromagnetic Wave H&CD
•
•
•
•
Excitation of a plasma wave at the plasma edge
Wave transports energy into the plasma
At a resonance the wave is transformed into kinetic
energy of resonant particles
Collisions distribute the energy
Method
Advantages
How does this work?
Courtesy of D. Hartmann
Disadvantages
Internal solid antennas,
Ion Cyclotron
Direct ion heating, possible
Resonance Heating current drive, high efficiency, minority heating, low plasma
coupling
low cost
(ICRH&CD)
Lower Hybrid
Current Drive
(LHCD)
Localized current drive useful
in current profile control,
waveguide antenna
Low power capability, low
plasma coupling
The Ion Cyclotron, Lower Hybrid and Alfven Wave Heating Methods
R. Koch - Transactions of Fusion Science and Technology 53 (2008)
4
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
∂
−∇× E = B
∂t
∂
∇ × H = D + J + J src
∂t
D (E )
Accounts for bound charges (dielectric)
J (E )
Accounts for free charges (conduction)
In a (fully ionized) plasma: free charges dominate
J (E )
D = ε0 E
Couples kinetic effects (Coulomb+Lorentz) to EM fields
Maxwell Equations
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Linearity
F = q( E + v × B )
B = B 0 + B RF
E = E RF
BRF ≈ 10-3 T « B0 ≈ 3 T.
RF electric field ≈ 20 kV/m << Vti×B0≈1.5MV/m
Likewise one can show that also RF perturbation on // motion of
particles is << thermal velocity
(We can use the unperturbed trajectories)
(Koch 2008)
typical parameters of an ICRF system:
• frequency: f ≈ 10-100 MHz
• Power: 2 MW/antenna strap
• Voltage: 10-50 kV at the antenna
• Antenna current: IA ≈ 1 kA
• Central conductor: width ≈ 0.2m, length ≈ 1m,
distance to the plasma 5cm, to the wall 20cm
• Typical RF electric field: 20kV/m
• Typical RF magnetic induction: 10-3T
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Non-collisional (1)
ion an electron collision frequencies: νe≈10kHz, νi≈100Hz.
electron mean free path: 3km or 150 toroidal revolutions.
ion mean free path: 5km or 250 toroidal revolutions.
(Koch 2008)
J can be approximated as contribution from (average) charge motion of
all species (electrons, one or more ion species)
Motion can be considered “single particle” (collective effects neglected at
first order)
Typical machine size: JET-type machine
R0 = 3m, 2πR0 ≈ 20m; ap=1.5m, 2πap = 10m
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Non-collisional
In bulk of a hot plasma,
e.g. Te≈Ti≈5keV, n=5×10^19m-3 collision frequency ν ≈ 20kHz
RF frequency f above 30 MHz, v/f<<1
B-lines are guiding
νe≈10kHz, νi≈100Hz
electron cyclotron gyration: 10ps
ion cyclotron gyration: 40ns
During one gyration: electron travels 0.4mm in the toroidal direction
and the ion 2cm.
Electron: 1µs for one toroidal turn= 50,000 cyclotron gyrations,
ion: 40µs= 1,000 cyclotron gyrations
(Koch 2008, 2006)
Wave energy absorption is not by collision drag
POLITECNICO
DI TORINO
PFA
Plasma Facing
Antenna Group
Unperturbed: thermal motion (equilibrium)
Perturbed: RF fields (much smaller fields or effetcs)
Particle motion linearizaton
POLITECNICO
DI TORINO
PFA
Plasma Facing
Antenna Group
Time-harmonic Maxwell equations
E ( r, t ) = Re[ E ( r; ω ) exp( −iωt )]
− ∇ × E = −iωµ0 H
∇ × H = −iωε 0 E + J + J src
Important notes:
1) the RF field here is strictly sinusoidal (time-harmonic), it is so produced
by the RF generators
(in radio communications, it is nearly sinusoidal)
2) Since the problem is linear, the frequency is the same everywhere and
“no matter what”
For (small perturbation) linearized RF field
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Cold Plasma Approximation
For a static magnetic field (B0) along z axis
 σ s σ xy 0 


σ (ω ) = σ yx σ s 0 
 0
0 σ || 
∇ × H = −iωε 0 E + J + J src
∇ × H = −iωε ⋅ E + J src
1
ε = ε0 +
σ
− iω
J ( E ) = σ (ω ) ⋅ E
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Cold Plasma Approximation
The dielectric tensor results as:
The cold-plasma approximation
provides a good description of
wave propagation even in quite
hot plasmas, except for the
reason where absorption takes
place
0
0 
P 
 S − iD
ε = ε 0 iD S
 0
0
Stix parameters are defined as:
ω ps2
R ≡ 1− ∑
s ω (ω + ωcs )
ω ps2
L ≡ 1− ∑
s ω (ω − ωcs )
1
S ≡ (R + L )
2
ω ps2
P ≡ 1− ∑ 2
s ω
1
D ≡ (R − L )
2
12
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Plane wave solution
Look for a solution of the kind
To be determined in such a way that the solution satisfies (sourcefree) Maxwell eqs.
f ( r ) = exp(i k ⋅ r )
∇f ( r ) = i k f ( r )
− ∇ × E = −iωµ0 H
k × E = ωµ0 H
∇ × H = −iωε ⋅ E
k × H = −ωε ⋅ E
êx
normalize
k = k0 n
k
B = B0 eˆz
ϑ
ê y
êz
static magnetic field
E ( r, t ) = Re[ E (ω ) exp( −i[ωt − k ⋅ r ])
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Wave Equation and Dispersion Relation
: wave in homogeneous plasma
 S − n 2 cos 2 ϑ

M k ,ω ⋅ E = 
iD
 n 2 cos ϑ sin ϑ

det M ( k , ω ) = 0 ⇒
êx
where:
k
ϑ
ê y
êz
− iD
S − n2
0
n 2 cos ϑ sin ϑ   E x 
  
0
 ⋅  Ey  = 0
P − n 2 sin 2 ϑ   E z 
n × (n × E ) + ε ⋅ E = 0
A(θ )n 4 − B (θ )n 2 + C (θ ) = 0
 A = S sin 2 ϑ + P cos 2 ϑ

2
2
sin
ϑ
1
cos
ϑ
B
=
RL
+
PS
+

C = PRL

(
)
14
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Plane waves
Note: setting k and ϴ means choosing the wavevector
k = k0 n
Consider first vacuum (or air)
n × (n × E ) + E = 0 ⇒ n ⋅ n = n 2 = 1
n = ±1
Recall: frequency is a constant everywhere (enforced by generator,
linear problem)
Observe:
There is ONE solutions for n^2
There are two solutions for n and k, corresponding to counter-propagating
waves
If you fix frequency and angle, then n is “chosen” by the physics
and this gives the wavelength (spatial period of wave oscillations)
n=1 means k=k0
f ( x ) = exp(ik0nx )
2π
λ=
k0 | n |
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Plane waves
Example:
Consider simple medium with
(slowly) varying material properties
1.6
1.4
1.2
1
0
⇒ n ⋅ n = n = p( x )
2
2
4
6
8
10
8
10
x
f(x)=cos(2π n(x) x)
1
f ( x ) = exp(ik0n( x ) x )
0.5
f(x)
n × (n × E ) + p( x ) E = 0
2π
λ ( x) =
k0 | n ( x ) |
1.8
n(x)
ε ( x ) = ε 0 p( x )
2
0
-0.5
-1
0
2
4
6
x
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Wave Equation and Dispersion Relation
det M ( k , ω ) = 0 ⇒
: wave in homogeneous plasma
A(θ )n 4 − B (θ )n 2 + C (θ ) = 0
n × (n × E ) + ε ⋅ E = 0
Observe:
There are TWO solutions for n^2 (only one in vacuo)
If you fix frequency and angle, then n is “chosen” by the physics
where:
êx
k
ϑ
ê y
 A = S sin 2 ϑ + P cos 2 ϑ

2
2
sin
ϑ
1
cos
ϑ
B
=
RL
+
PS
+

C = PRL

(
)
êz Recall: frequency is a constant everywhere (enforced by
generator, linear problem)
17
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Dispersion Relation Solutions
Langmuir wave
Ionic whistler
parallel propagation
ϑ =0:
Electronic whistler
ϑ=
π
2
Slow (O) wave
: perpendicular propagation
Fast (X) wave
êx
(E ⊥ B )
k
B = B0 eˆz : static magnetic field
ϑ
ê y
(E // B )
êz
18
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Plane waves and plane wave spectrum
Note: setting k and ϴ means choosing the wavevector
k = k0 n
Consider first vacuum (or air)
n × (n × E ) + E = 0
- The RF generator “chooses” (enforces) the frequency
- The “physics” chooses k (i.e. n), i.e. the wavelength
- The antenna chooses angle ϴ (if very directive..)
n ⋅ n = n2 = 1
Actually, we never launch a single plane wave, we launch a field
with some plane-wave “spectrum” e.g. we consider its Fourier
transform
e.g. 1D case
a ( x ) = ∫ A(u ) exp( −ik0ux )dxu
u = cos θ
Who chooses k and ϴ?
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Plane waves and plane wave spectrum
k = k0 n
Any source distribution corresponds (can be represented as) a
“collection” of plane waves with different wavenumber (PW spectrum)
Each component (each individual PW) will travel its own way
At a first approx, we consider only the peak of the plane wave
spectrum (like the “dominant” tone in a sound or color in light)
In fact, all ICRH antenna have a pretty broad spectrum…
Plasma propagation acts as a “filter”, some plane waves pass through
better than others, some get absorbed well etc.
We’d like to put all our power in those that get well absorbed…
Who chooses k and ϴ?
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Wave Propagation
Dispersion relation for plane waves: k = k (ω )
v ph = ω
k
“Index of refraction”: n = ck
ω
Group velocity: v g = ∂ω
∂k
At which
energy and information travel
Phase velocity:
(wavenumber normalized to
vacuum value)
Note: when frequency or angle is such that
f ( r ) = exp( −αx )
Cutoff:
n < 0 n = iα
2
Evanescent wave
n = 0 v ph → ∞
2
“Resonance”:
n → ∞, v ph → 0
Wave slows down enormously,
filed can now interact with
thermal velocity (intuitive),
absorption mechanisms favored
21
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Wave Propagation
Dispersion at fixed frequency and non-homogeneous plasma
(density and/or B field vary in space)
k = k (ω )
Cutoff:
n2
n → 0, v ph → ∞
Resonance:
n → ∞, v ph → 0
n2
propagation
propagation
evanescence
Space
propagation
Space
22
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Ion Cyclotron Resonance
Tore Supra ICRH antenna
Frequency range: 40÷80 MHz
qs B0
ωcs =
ms
ω ≈ ωci , ω pi << ωce , ω pe
ns qs2
ω ps =
msε 0
Generators: tetrode tubes
Principle:
absorption of the wave by ions (cyclotron
resonances) or by electrons (ELD - TTMP)
Courtesy of CEA-Cadarache:
http://www-cad.cea.fr
23
POLITECNICO
DI TORINO
PFA
Plasma Facing
Antenna Group
Improved resonance condition in IC range
ω − nhωci + k// v// = 0
Adding effect of parallel motion due to RF field (v||)
It is a Doppler effect
 nh = 1

nh ≥ 2
: first harmonic heating
: second (or higher) harmonic heating
24
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Single Ion H&CD
First harmonic heating
Slow wave:
• sensitive to the fundamental
resonance
• not excitable in toroidal geometry
(evanescent)
Fast wave:
• excitable in toroidal geometry
• not sensitive to the fundamental
resonance
NOT WORKING!!!
: first harmonic heating
: second (or higher) harmonic heating
ω = nhωci + k // v//
 nh = 1
⇒ 
nh ≥ 2
Second harmonic heating
FW is sensitive to the harmonics of
the cyclotron frequency, but damping
strength strongly decreases with
harmonic number
High density and high
temperature needed
NOT EFFICIENT!!!
25
POLITECNICO
DI TORINO
PFA
Plasma Facing
Antenna Group
Minority H&CD (Multiple Ions)
ω − nhωci − k // v// = 0 ⇒ ω = ωci + k // v// ⇒ ω ≈ ωci
Propagation and polarization are determined by the majority ions
Good cyclotronic absorption on the minority ions (< 10%)
Possible mode conversion to Ion Bernstein Waves (IBW)
Ion Bernstein Waves:
• Perpendicularly propagating warm plasma waves with solutions
near each harmonic of the cyclotron frequency of each species
• Higher percentage of minority species (~ 15-20%)
• Landau damping on electrons
26
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Main Collisionless Wave Damping Mechanisms
v≅ω
k
Slower particles are
accelerated and faster
particles are decelerated
Transit time magnetic pumping (TTMP)
Force on magnetic moment:
F = − µ∇B
similar to Landau damping with substitution:
µ →q
∇B → E
27
Strong interaction if
Landau damping
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
ICRF Power Scheme
FW + cycl. res.
• Abs. fund. cycl.
• Abs. harm. cycl.
Fast Wave
• Abs. Landau
• TTMP
Ion Bernstein Wave
• Abs. Landau
Ions
Fast ions
Fast electrons
Ionic heating
Electronic heating
28
ICRF power
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Lower Hybrid H&CD
Tore Supra LH antenna
Alcator C-Mod LH antenna
Frequency range: 1÷8 GHz
ωci << ω LH << ωce
with
2
ω LH
≈
1+
ω pi2
2
ω pe
ωce2
Generators: Klystrons
Courtesy of PSFC (MIT):
http://www.psfc.mit.edu/
Principle:
Landau absorption of the wave by fast electrons
29
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Lower Hybrid H&CD
Original use: ionic heating by conversion of LH wave to a compressional wave
the best, experimentally proven,
current drive method
“Modern” use: electronic
heating by Landau
damping on fast electrons
In ITER: controlling current profile (in
addition to EC)
Propagation on a narrow cone of resonance
almost parallel to magnetic field when n⊥ > n //
Group velocity:
vg ⊥ k
Accessibility criterion :
Polarization
:
E // k
n//2 >> n//,2 acc =
1
2
ω
1−
ωciωce
30
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
ICRF Overall Scheme
Tuning and
matching
systems
~
DC breaker
Launcher
Generator
ITER IC antenna
T&M scheme
Feed through
T&M solutions (two elements):
• Resonant loop: the two feeding arms are set to the proper length
to achieve the desired phasing
• Hybrid: the two feeding arms are connected to the two output
ports of an hybrid device
• Conjugate T: the two feeding arms of equal length are connected
in order to minimize the imaginary part of the active input
impedance of the elements
31
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Plasma facing antennas are used in experiments
towards controlled nuclear fusion with
magnetically confined plasmas to transfer power
to the plasma and to control plasma current
ICRF antennas
LH antennas
Courtesy of JET: http://www.fusion.org.uk
These antennas are very complex geometries in a very
complex environment and they can not be tested before
being put in operation
A numerical predictive tool is necessary to determine the
system performances in a reasonable computing time and
to properly optimize the antenna
32
Issues with Plasma Facing Antennas
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Example : the Tore Supra ICRH Antenna
Some features:
•
2 adjacent cavities
2 center-fed straps
4 loading capacitors to resonate the
straps (resonant double loops)
Main parameters:
– Major radius: 2.355 m
– Minor radius: 0.725 m
– Toroidal magnetic field: 3.13 T
– Generator frequency: 48 MHz
– Scenario: D(H) with 10% H
minority
Analysis of Tore Supra ICRF Antenna with TOPICA
D.Milanesio, V.Lancellotti, L. Colas, R.Maggiora, G.Vecchi, V.Kyrytsya
Plasma Physics and Controlled Fusion 49 (2007)
•
•
•
Loading capacitors
33
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Example : the JET ITER-Like Antenna
Some features:
•
•
Single cavity
8 straps with coax cable excitation,
grouped in 4 resonant double loops
Main parameters:
– Generator frequency: 42 MHz
– Scenario: D(H) with 3% H minority
Measured density/temperature profiles
•
•
Courtesy of JET Task Force H
Jet ITER-like Antenna Analysis using TOPICA code
D. Milanesio, R. Maggiora, F. Durodié, P. Jacquet, M. Vrancken and JET-EFDA contributors
51st APS-DPP meeting, Atlanta (2009)
34
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Example : the ITER IC Antenna
•
•
•
Side views
•
24 straps grouped in poloidal triplets
Complex antenna structure and matching
scheme (never experienced before)
Main parameters:
– Generator frequency: 40÷55 GHz
– Main scenario: 50%D-50%T
Expected density/temperature profiles
Proposed reference
launcher
35
Some features:
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Large Plasma-Antenna Distance Dependence
TOTAL power to plasma (MW)
Max. voltage in coax: 45kV
Several plasma
profiles have been
loaded to predict
the antenna
performances in a
wide range of
input conditions
By increasing the
distance between
the antenna mouth
and the plasma,
results converge to
the vacuum case
36
POLITECNICO
DI TORINO
PFA
Plasma Facing
Antenna Group
Plasma-Surface Interactions
Why rectified potentials are so important?
RF-induced drifts accelerate ions that can hit the tokamak first wall, causing:
• hot spots
• sputtering (impurities)
• fuel dilution
• disruption
The heat flux attributed to accelerated ions is directly proportional to the DC
sheath (rectified) potential.
Solutions?
By accurately knowing the DC potential map resulting from the rectification
process due to RF fields in front of the antennas, one can try to mitigate this
effect modifying the antenna geometry itself.
37
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Electric Field Map and Rectified Potential
|VRF| (V for 20MW coupled)
Electric field maps can be
evaluated at every radial
position in front of the
antenna mouth
y (m)
6000
0.5
0.8
0.6
0
4000
-0.5
3000
2000
1
0.4
-1
0.5
0.2
-1.5
0
0
5000
1000
0
0.01
0.02
0.03
x (m)
0.04
0.05
-0.2
-0.5
-0.4
-1
-0.6
Lower box corner zone
-1.5
-1
-0.5
0
0.5
z (m)
1
-0.8
Rectified potentials are influenced
by plasma scenarios, by input
phasing and by the geometry of
the front part of the launcher
38
Upper box corner zone
1.5
7000
1
y (m)
Re(E//) (V/m for 1V @ feeder), x=5mm
8000
1.5
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
TOPICA as an Optimization Tool
Max. voltage in coax: 45kV
Reference
antenna
Optimized
antenna
The optimization process
has been focused on the
shape of the horizontal
septa and their position, on
the dimension of the feeder
and its transition with the
coaxial cable and on the
wideness of the straps
A significant increase in the antenna performances has
been reached by optimizing some geometrical details
39
TOTAL power to plasma (MW)
PFA
Plasma Facing
POLITECNICO
DI TORINO
Antenna Group
Proposed Design II: the ITER LH Launcher
•
•
•
Proposed reference
launcher
Detailed view of a
single module
•
Courtesy of ITER-LH working group
2352 waveguides, grouped in 4 blocks of
12 rows
Based on the PAM concept, i.e. on the
alternation between active and passive
waveguides
Main parameters:
– Generator frequency: 5 GHz
– Main scenario: 50%D-50%T
Expected density/temperature profiles
40
Some features:
POLITECNICO
DI TORINO
PFA
Plasma Facing
Antenna Group
To fill in the gaps/to probe further
R. Koch, “The Ion Cyclotron, Lower Hybrid and Alfven Wave Heating
Methods”, Transactions of Fusion Science and Technology 53 (2008)
Tutorial, tries to explain wave penetration in a Tokamak-like geometry
R. Koch, “The Coupling of Electromagnetic Power to Plasmas”,
Transactions of Fusion Science and Technology 49 (2006)
All-time classics
T.H. Stix, The Theory of Plasma Waves, McGraw-Hill, New York, 1962
T.H. Stix, Waves in plasmas, American Institute of Physics, New York, 1992
Tutorial, with nice application to RFH

Radio Frequency Plasma Heating

Transcript

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