radio frequency plasma heating · 2011-03-20 · •electron cyclotron waves electrons electrons,...

Plasma FacingPFAAntenna Group

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Radio Frequency Plasma

Heating

Credits/thanks:

Riccardo Maggiora & Daniele Milanesio

Giuseppe Vecchi


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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)

: ohmic heating density

Limitations:• on current density to avoid instabilities and disruptions• by plasma resistivity

Additional heating needed

2jP η=Ω

23−

∝Tη


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Method Principle Heated species

Neutral Beam Injection

Injecting a beam of neutral atoms at high energy across magnetic field

linesElectrons, ions

Electromagnetic Waves

Exciting of plasma waves that are damped in plasma

• Alfven waves

• ion cyclotron waves

• lower hybrid waves

• electron cyclotron waves

Electrons

Electrons, ions

Electrons

Electrons

α-Particles Collisions Electrons, ions

Auxiliary Heating and Current Drive (H&CD) Methods

At ignition, only α-particles sustain the fusion reaction


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How does this work?• 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

Electromagnetic Wave H&CD

Courtesy of D. Hartmann

Method Advantages Disadvantages

Ion Cyclotron Resonance Heating

(ICRH&CD)

Direct ion heating, possible current drive, high efficiency,

low cost

Internal solid antennas, minority heating, low plasma

coupling

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 MethodsR. Koch - Transactions of Fusion Science and Technology 53 (2008)


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iddensrcJJD

tH

Bt

E

++∂∂

=×∇

∂∂

=×∇−

)(ED Accounts for bound charges (dielectric)

)(EJ Accounts for free charges (conduction)

In a (fully ionized) plasma: free charges dominate ED 0ε=

Maxwell Equations

)(EJ Couples kinetic effects (Coulomb+Lorentz) to EM fields


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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

BRF ≈ 10-3 T « B0 ≈ 3 T.

RF electric field ≈ 20 kV/m << Vti×B0≈1.5MV/m

)( BvEqF ×+=

RFEE =

RFBBB += 0

Likewise one can show that also RF perturbation on // motion of particles is << thermal velocity(We can use the unperturbed trajectories)

Linearity

(Koch 2008)


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Non-collisional (1)Typical machine size: JET-type machine R0 = 3m, 2πR0 ≈ 20m; ap=1.5m, 2πap = 10m

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)


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Non-collisional

(Koch 2008, 2006)

Wave energy absorption is not by collision drag

In bulk of a hot plasma,e.g. Te≈Ti≈5keV, n=5×10^19m-3 collision frequency ν ≈ 20kHzRF frequency f above 30 MHz, v/f<<1

B-lines are guidingνe≈10kHz, νi≈100Hzelectron cyclotron gyration: 10psion 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


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Unperturbed: thermal motion (equilibrium)Perturbed: RF fields (much smaller fields or effetcs)

Particle motion linearizaton


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src0

0

JJEiH

HiE

++−=×∇

−=×∇−

ωε

ωµ

Time-harmonic Maxwell equations

For (small perturbation) linearized RF field

)]exp();(Re[),( tirEtrE ωω −=

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”


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Cold Plasma Approximation

EEJ ⋅= )()( ωσ

=

||00

0

0

)(

σσσσσ

ωσ syx

xys

For a static magnetic field (B0) along z axis

src0 JJEiH ++−=×∇ ωε

srcJEiH +⋅−=×∇ εωσω

εεi−

+=1

0


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12

−

=

P

SiD

iDS

00

0

0

0εε

( )∑ +−≡

s cs

psR

ωωω

ω 2

1( )∑ −

−≡s cs

psL

ωωω

ω 2

1 ∑−≡s

psP

2

2

1ω

ω

( )LRS +≡2

1 ( )LRD −≡2

1

The dielectric tensor results as:

Stix parameters are defined as:

Cold Plasma Approximation

The cold-plasma approximation provides a good description of wave propagation even in quite hot plasmas, except for the reason where absorption takes place


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Plane wave solution

Look for a solution of the kind

])[exp()(Re[),( rktiEtrE ⋅−−= ωω

To be determined in such a way that the solution satisfies (source-free) Maxwell eqs.

)exp()( rkirf ⋅= )()( rfkirf =∇

EiH

HiE

⋅−=×∇

−=×∇−

εω

ωµ0

EHk

HEk

⋅−=×

=×

εω

ωµ0

nkk 0=normalize

xe

yeze

k

ϑzeBB ˆ

0=

static magnetic field


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0

sin0sincos

0

sincoscos

222

2

222

,=

⋅

−

−

−−

=⋅

z

y

x

k

E

E

E

nPn

nSiD

niDnS

EM

ϑϑϑ

ϑϑϑ

ω

0)()()(0),(det 24 =+−⇒= θθθω CnBnAkM

where: ( )

=

++=

+=

PRLC

PSRLB

PSA

ϑϑϑϑ

22

22

cos1sin

cossin

0)( =⋅+×× EEnn ε : wave in homogeneous plasma

Wave Equation and Dispersion Relation

xe

yeze

k

ϑ


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Note: setting k and ϴ means choosing the wavevector nkk 0=

Consider first vacuum (or air) 0)( =+×× EEnn 12 ==⋅⇒ nnn

Plane waves

Observe:There is ONE solutions for n^2There are two solutions for n and k, corresponding to counter-propagating wavesIf you fix frequency and angle, then n is “chosen” by the physicsand this gives the wavelength (spatial period of wave oscillations) n=1 means k=k0

Recall: frequency is a constant everywhere (enforced by generator, linear problem)

1±=n

)exp()( 0nxikxf =||

2

0 nk

πλ =


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Example: Consider simple medium with (slowly) varying material properties

0)()( =+×× ExpEnn

)(2 xpnnn ==⋅⇒

Plane waves

|)(|

2)(

0 xnkx

πλ =

))(exp()( 0 xxnikxf =

)()( 0 xpx εε =

0 2 4 6 8 101

1.2

1.4

1.6

1.8

2

x

n(x

)

0 2 4 6 8 10-1

-0.5

0

0.5

1

x

f(x)

f(x)=cos(2π n(x) x)


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0)()()(0),(det 24 =+−⇒= θθθω CnBnAkM

where: ( )

=

++=

+=

PRLC

PSRLB

PSA

ϑϑϑϑ

22

22

cos1sin

cossin

0)( =⋅+×× EEnn ε : wave in homogeneous plasma

Wave Equation and Dispersion Relation

xe

yeze

k

ϑ

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

Recall: frequency is a constant everywhere (enforced by generator, linear problem)


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0=ϑ

2

πϑ =

: parallel propagation

: perpendicular propagation

Langmuir wave

Ionic whistler

Electronic whistler

Slow (O) wave

Fast (X) wave

( )BE //

( )BE ⊥

xe

yeze

k

ϑ zeBB ˆ0= : static magnetic field

Dispersion Relation Solutions


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Note: setting k and ϴ means choosing the wavevector nkk 0=Who chooses k and ϴ?

Consider first vacuum (or air) 0)( =+×× EEnn 12 ==⋅ nnn- 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..)

Actually, we never launch a single plane wave, we launch a fieldwith some plane-wave “spectrum” e.g. we consider its Fourier transform

e.g. 1D case

θcos

)exp()()( 0

=

−= ∫u

dxuuxikuAxa

Plane waves and plane wave spectrum


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nkk 0=Who chooses k and ϴ?

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…

Plane waves and plane wave spectrum


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Wave PropagationDispersion relation for plane waves: ( )ωkk =

“Index of refraction”:

(wavenumber normalized to vacuum value)

ωckn =

Phase velocity:k

vphω= Group velocity:

At which

energy and information travel

kvg ∂∂= ω

Cutoff:

02 <nNote: when frequency or angle is such that αin =

)exp()( xrf α−= Evanescent wave

02 =n ∞→phv

“Resonance”: 0, →∞→ phvn Wave slows down enormously, filed can now interact with thermal velocity (intuitive), absorption mechanisms favored


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Wave Propagation

( )ωkk =

Cutoff: Resonance: ∞→→ phvn ,0 0, →∞→ phvn

2n

Space

propagation

evanescence

2n

propagation

propagation

Space

Dispersion at fixed frequency and non-homogeneous plasma (density and/or B field vary in space)


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Ion Cyclotron Resonance

Frequency range: 40÷80 MHz

pecepici ωωωωω ,, <<≈

Generators: tetrode tubes

Principle: absorption of the wave by ions (cyclotron resonances) or by electrons (ELD - TTMP)

Tore Supra ICRH antenna

s

scs

m

Bq 0=ω

Courtesy of CEA-Cadarache: http://www-cad.cea.fr

0

2

εω

s

ssps

m

qn=


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Improved resonance condition in IC range

0//// =+− vkn cihωω

≥

=

2

1

h

h

n

n : first harmonic heating

: second (or higher) harmonic heating

Adding effect of parallel motion due to RF field (v||)It is a Doppler effect


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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!!!

⇒+= ////vkn cihωω

≥

=

2

1

h

h

n

n : first harmonic heating

: second (or higher) harmonic heating

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!!!


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Minority H&CD (Multiple Ions)

cicicih vkvkn ωωωωωω ≈⇒+=⇒=−− //////// 0

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


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Main Collisionless Wave Damping Mechanisms

Landau damping

Transit time magnetic pumping (TTMP)

Force on magnetic moment: BF ∇−= µ

similar to Landau damping with substitution:EB

q

→∇

→µ

kv ω≅Strong interaction if

Slower particles are accelerated and faster

particles are decelerated


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ICRF Power Scheme

ICRF power

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


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Lower Hybrid H&CD

Frequency range: 1÷8 GHz

ceLHci ωωω <<<<

Generators: Klystrons

Principle: Landau absorption of the wave by fast electrons

Tore Supra LH antenna

2

2

2

2

1ce

pe

pi

LH

ωω

ωω

+≈

with


Courtesy of PSFC (MIT): http://www.psfc.mit.edu/

Alcator C-Mod LH antenna


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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

Group velocity: kv g ⊥ Polarization:

kE //

//nn >⊥

Accessibility criterion :

ceci

accnn

ωωω 2

2

//,

2

//

1

1

−=>>


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ICRF Overall Scheme

~Tuning and matching systems

Launcher

Generator

DC breaker

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

ITER IC antenna T&M scheme

Feed through


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Issues with Plasma Facing Antennas


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

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

Courtesy of JET: http://www.fusion.org.ukLH antennas


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• 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

Example : the Tore Supra ICRH Antenna


Some features:

Analysis of Tore Supra ICRF Antenna with TOPICAD.Milanesio, V.Lancellotti, L. Colas, R.Maggiora, G.Vecchi, V.KyrytsyaPlasma Physics and Controlled Fusion 49 (2007)

Loading capacitors


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Example : the JET ITER-Like Antenna

Courtesy of JET Task Force H

Some features:• Single cavity

• 8 straps with coax cable excitation, grouped in 4 resonant double loops

• Main parameters:– Major radius: 2.96 m– Minor radius: 1.25 m– Toroidal magnetic field: 1.9 T – Generator frequency: 42 MHz– Scenario: D(H) with 3% H minority

• Measured density/temperature profiles

Jet ITER-like Antenna Analysis using TOPICA codeD. Milanesio, R. Maggiora, F. Durodié, P. Jacquet, M. Vrancken and JET-EFDA contributors51st APS-DPP meeting, Atlanta (2009)


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Example : the ITER IC Antenna

Some features:• 24 straps grouped in poloidal triplets

• Complex antenna structure and matching scheme (never experienced before)

• Main parameters:– Major radius: 6.2 m– Minor radius: 2.1 m– Toroidal magnetic field: 5.3 T – Generator frequency: 40÷55 GHz– Main scenario: 50%D-50%T

• Expected density/temperature profiles

Proposed reference launcher

Side views


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Large Plasma-Antenna Distance Dependence

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

TOTAL power to plasma (MW)

Max. voltage in coax: 45kV


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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• disruptionThe 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.

Plasma-Surface Interactions


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-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

z (m)

y (

m)

Re(E//) (V/m for 1V @ feeder), x=5mm

-1 -0.5 0 0.5 1

-1.5

-1

-0.5

0

0.5

1

1.5 Upper box corner zone

Lower box corner zone

Electric Field Map and Rectified Potential

Electric field maps can be evaluated at every radial position in front of the

antenna mouth

1000

2000

3000

4000

5000

6000

7000

8000

x (m)y

(m

)

|VRF

| (V for 20MW coupled)

0 0.01 0.02 0.03 0.04 0.05

-1.5

-1

-0.5

0

0.5

1

1.5

Rectified potentials are influenced by plasma scenarios, by input

phasing and by the geometry of the front part of the launcher


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A significant increase in the antenna performances has been reached by optimizing some geometrical details

TOTAL power to plasma (MW)Max. voltage in coax: 45kV

TOPICA as an Optimization Tool

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

Reference antenna

Optimized antenna


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Proposed Design II: the ITER LH Launcher

Some features:• 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:– Major radius: 6.2 m– Minor radius: 2.1 m– Toroidal magnetic field: 5.3 T – Generator frequency: 5 GHz– Main scenario: 50%D-50%T

• Expected density/temperature profiles

Proposed reference launcher

Detailed view of a single module

Courtesy of ITER-LH working group


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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

R. Koch, “The Ion Cyclotron, Lower Hybrid and Alfven Wave Heating Methods”, Transactions of Fusion Science and Technology 53 (2008)

All-time classics

Tutorial, with nice application to RFH

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)

To fill in the gaps/to probe further

radio frequency plasma heating · 2011-03-20 · •electron cyclotron waves electrons electrons,...

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