jmn2008.17 physics of rf heating j.-m. noterdaeme with the support of m. brambilla, r. bilato, d....

jmn2008.17

Physics of RF Heating

J.-M. Noterdaeme

with the support ofM. Brambilla, R. Bilato, D. Hartmann, H. Laqua, F. Leuterer, M. Mantsinen, F. Volpe, R. Wilhelm

Max Planck Institute for Plasmaphysics, Garching

Avanced Course for the European Fusion Doctorate

October 2008

jmn2008.17.2

Neutral Beam Injection

Ion source

Neutral

beam

Electricity -> other form(kinetic energy of particles)

Transport to plasma (outside part)

Neutraliser

Magnetic

filter

Beam duct

(inside part)

Accellerator

Ionisation Thermalisation

jmn2008.17.3

Wave heating

Electricity -> other form(electromagnetic oscillations)

Transport to plasma (outside part)transmission linesantenna

(inside part)waves

Thermalisation

Antenna

Wave to particles

Resonance zone

R

€

ω =Ωc

€

Ωc =Z eB

m

€

ω

jmn2008.17.4

Wave propagation and absorption sets the frequency range that can be used.

Wave heating: very tight combination of physics and technology

Physics Technology

Electromagnetic energy

Transmission lines

Antenna

Coupling

Waves

Waves -> Particles

Transfer to bulk

jmn2008.17.5

What will be addressed here?

• plasma– (coupling)– waves– absorption

how, why those frequencies, mechanisms, practical applications

• with side glances at– technology, to show that we can do it– experiments, to show that it works

• emphasize– concepts– physical understanding

• goal– working knowledge for experiments

jmn2008.17.6

Waves

• waves in neutral gas– pressure waves

– EM waves in vaccuum

• waves in medium with free charged particles, magnetised

successive approximations/simplifications

– electrons, several ion species, arbitrary distribution function

plasma kinetic theory

– electrons, ions (one or more types), fluid equations

cold plasma -> two fluid -> CMA diagram

– one electrically conducting fluid

MHD

jmn2008.17.7

In general: approximations in plasmas

Kinetic equation (full distribution function)

how importantare collisions?

not important important

induced fields wrtapplied fields?

how dominantare collisions?

not important important

partice orbit theory

cold plasma wave theory

not dominant ω > dominant ω <

warm plasma wave theory

MHD theory

jmn2008.17.8

Approximations

• coupling

– cold plasma

• propagation

– cold plasma

– (warm plasma)

• power absorption

– warm plasma

elctrons + (multiple) ionscold -> “fluid” equations, no temperature effectwarm -> finite larmor radius plays a role

EC

(IC)

EC IC

(EC)

IC

jmn2008.17.9

Power Absorption

• energy in the wave can be affected by energy transferbetween wave

and particles (wave-particle resonance)

or other wave (wave resonance)

• absorption mechanismalways particles

• note :– wave-particle resonance : only few particles fulfill resonance conditions

– wave resonance :collective effect : all particles fulfill conditions

jmn2008.17.10

Interaction between Absorption and Waves

• wave fields

• absorption

• change in

distribution function

Absorption affected by change in


Wave fields affected by change in


jmn2008.17.11

Transport of energy in plasma: waves



(inside part)waves Thermalisation

Antenna

Wave to particles

Resonance zone

R

€

ω =Ωc

€

Ωc =Z eB

m

€

ω

jmn2008.17.12

Wave Equation

generalized Ohm‘s law

€

j = j(E,B)

€

jω ,k = σ ω,k( ) ⋅Eω,k

tensortyconductivi : σ

€

∇×E = −∂B

∂t

€

∇×B = μ0ε0

∂E

∂t+ μ0 j

Maxwell‘s Equations

€

∇⋅E = ρ ε0

€

∇⋅B = 0

€

∇×

jmn2008.17.13

Dispersion relation

• set of homogenous, linear equations for Ex, Ey, and Ez,

• has non trivial (different from 0) solutions

provided the determinant vanishes

• det = 0 is known as the dispersion relation

jmn2008.17.14

Plane waves

velocityphase : k

v ph

ω=€

˜ E (r, t) = ˜ E ω ,k eik r−iωt

ω, k

∑

€

k : wave vector

ω : angular frequencyPlane waves

k

External magnetic field B0k||

k

fronts of constant phase

index refractive : ω⋅

==ck

c

vN ph

ω fixed by „generator“k response of plasma

equivalent k=k(ω) dispersion relation D(ω,k)=0

jmn2008.17.15

Wave Equation


€

j = j(E,B)

€

jω,k

= σ ω,k( ) ⋅Eω ,k


Plane waves

€

∇×E = −∂B

∂t

€

∇×B = μ0ε0

∂E

∂t+ μ0 j

Maxwell‘s Equations

€

∇⋅E = ρ ε0

€

∇⋅B = 0

€


ω, k

∑

€

∇× → ik ×

€

∂∂t

→ − iω

jmn2008.17.16

Wave Equation


€

j = j(E,B)


€

k × k × Eω ,k( ) +ω2

c 2K ⋅Eω ,k = 0

€

K : dielectric tensor

€

K =1+σ

iωε0

Plane waves

€

∇×E = −∂B

∂t

€

∇×B = μ0ε0

∂E

∂t+ μ0 jMaxwell‘s Equations

€

∇⋅E = ρ ε0

€

∇⋅B = 0

€


ω, k

∑

€

∇× → ik ×

€

∂∂t

→ − iω

€

∇×

€

jω,k

= σ ω,k( ) ⋅Eω ,k

jmn2008.17.17

Wave Equation

Dispersion relation

€

det N × N ×1( ) + K(ω,N)[ ] = 0€

k × k × Eω ,k( ) +ω2

c 2K ⋅Eω ,k = 0

€


€

K =1+σ

iωε0

€

N =c

vph

=c ⋅kω

€

N : refractive index

•the equation is exact

•will become approximate only in terms of the model used for

€

k × k × Eω ,k( ) +ω2

c 2K ⋅Eω ,k = 0

€


jmn2008.17.18

Propagation in cold unmagnetized Plasmasimply depends on ωp

(current carried only by electrons)

envj −=

€

−iω v =−e ⋅E

me

€

j =−i ⋅e2ne

meωE

2

2

02

2

0

111ωω

εωωεσ p

e

e

minie

iK −=

−+=+=

€

k ⋅E( ) k − k 2 E +ω2

c 21−

ωp2

ω2

⎛

⎝ ⎜

⎞

⎠ ⎟E = 0 €

ω p,e2 =

e2ne

ε0me

Plasma frequency:

Langmuir oscillations pωω =

:Ek ⊥ EM waves 2

2

2

222 1

ωω

ωpkc

N −==

ω

k

pω

:E||k

€

j = σ E

(equ. of motion of electrons)

€

k × k × Eω ,k( ) +ω2

c 2K ⋅Eω ,k = 0

€

dv

dt=

qs

ms

⋅(E + v × B)

jmn2008.17.19

Wave cutoff and resonance

1. N 0 „cutoff“reflexiontunnellingvph>c!

2. N „resonance“ vph -> 0wave „gets stuck“wave energy dissipation

€

N =c

vph

=c ⋅kω

€

N : refractive index

€

N → ∞ ; vph → 0 ; λ → 0 ; k → ∞

€

N → 0 ; vph → ∞ ; λ → ∞ ; k → 0

€

det N × N ×1( ) + K(ω,N)[ ] = 0

jmn2008.17.20

Wave resonance thermal effects become important

• wave resonance– Energy density ~ A2

– Energy flux vA2 =>

– when v -> 0 then A must increase -> damping mechanisms amplified

– v -> 0 also means -> 0 <

• simple example

x

k2perp

k2inf

jmn2008.17.21

Ohm‘s law

Goal: determine j=j(E) Small perturbation

€

v = v 0 + v1

€

E = E 0 + E1

€

B = B0 + B1

0

€

j1

= qsnsv1,s

s

∑€

−iωv1 =qs

ms

⋅ E1 + v1 × B0( )

€

dv

dt=

qs

ms

⋅(E + v × B)cold plasmaequation of motionof particles

€

v th <<ω

k

€


€

K =1+σ

iωε0

€

det N × N ×1( ) + K(ω,N)[ ] = 0

currentcoldsolve for v1 as function of E1

jmn2008.17.22

Generalisation

warm plasma

linearized Vlasov equation

€

∂∂t

f1 + v ⋅∂

∂rf1 +

q

mv × B0[ ]

∂

∂vf1 = −

q

mE1 + v × B0( )

∂

∂vf0

€

f1 = ...

j1 = qs

s

∑ v∫ ⋅ f v( ) d3v

€

∂∂t

f + v ⋅∂

∂rf +

q

mE + v × B[ ]

∂

∂vf =

∂

∂tf

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟coll

Fokker - Planck

Vlasov

€

∂∂t

f + v ⋅∂

∂rf +

q

mE + v × B[ ]

∂

∂vf = 0

6 waves:-2 cold-pressure driven-electron and ion Bernstein-accoustic branch

jmn2008.17.23

Continuing the cold plasma case

€

det N × N ×1( ) + K(ω,N)[ ] = 0

€

K =1+σ

iωε0

€

K =

S − iD 0

iD S 0

0 0 P

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟€

S =1

2(R + L ) ; D =

1

2(R − L )

€

R = 1 −(ωpe /ω )2

1 − ωce /ω−

(ωpi /ω )2

1 +ωci /ωi

∑

€

L = 1 −(ωpe /ω )2

1 +ωce /ω−

(ωpi /ω )2

1 − ωci /ωi

∑

€

P = 1 −ωpe

ω

⎛

⎝ ⎜

⎞

⎠ ⎟

2

−ωpi

ω

⎛

⎝ ⎜

⎞

⎠ ⎟

2

i

∑

€


jmn2008.17.24

Characteristic frequencies

€

ωpe2 =

e2ne

ε 0me

€

ωpi2 =

(Zie)2 ni

ε 0mi

€

ωce =eB

me

€

ωci =ZieB

mi

Plasmafrequencies

Cyclotron frequencies

€

R = 1 −(ωpe /ω )2

1 − ωce /ω−

(ωpi /ω )2

1 +ωci /ωi

∑

€

L = 1 −(ωpe /ω )2

1 +ωce /ω−

(ωpi /ω )2

1 − ωci /ωi

∑

€

P = 1 −ωpe

ω

⎛

⎝ ⎜

⎞

⎠ ⎟

2

−ωpi

ω

⎛

⎝ ⎜

⎞

⎠ ⎟

2

i

∑

€

ωuh2 = ωpe

2 + ωce2

€

ω lh2 =

1

ωpi2 + ωci

2+

1

ωciωce

⎡

⎣ ⎢

⎤

⎦ ⎥

−1

Upper Hybrid frequency

Lower Hybrid frequency

jmn2008.17.25

Ordering in tokamaks

€

ω p j =n j Z je( )

2

m jε0

€

ωc j =Z jeB

m j

for the electrons

€

ω pe2

ωce2

=e − motion //B

e − motion ⊥B

€

=1.029 ×10−19 ne

B2= O(1)

further

€

ωc i =me

mi

ωc e

€

ω p i =me

mi

ωp e

€

ωc i << ωp i << ωc e ≈ ωp e

€

ωc i

2π=

15MHz

AB T[ ]

€

ωc e

2π= 28GHz ⋅B T[ ]

€

ω lh2 =

1

ωpi2 + ωci

2+

1

ωciωce

⎡

⎣ ⎢

⎤

⎦ ⎥

−1

≈1

ωpi2

+1

ωciωce

⎡

⎣ ⎢

⎤

⎦ ⎥

−1

€

ωk//

≈ v th e →ω

2π=

1.3 GHz Te keV[ ]λ // cm[ ]

jmn2008.17.26

• ECRH– electron cyclotron resonance heating

• LH– lower hybrid frequency

• ICRF– Ion cyclotron range of frequencies

Transport from outside plasma to inside: wave propagation (wave cut-off and resonance)

Transfer of energy from wave to particles: resonance condition (wave-particle)

Wave propagation and absorption

-

E

~ B0

+

ion unmagnetized, oscillate with E1

electrons oscillate with E1 x B0 drift

€

ωc i <<ω << ωc e

jmn2008.17.27

Electron / Ion Cyclotron Range

• electron cyclotron– only electron dynamics (ions fixed background)

– v >> vthe

– relativistic effects

• ion cyclotron– ions (multiple) and electron dynamics

– characteristic propagation velocity is the Alfven speed

– comparison with the electron thermal velocity

€

vA =B

μ0 mi ni

= cωci

ωpi

€

vA2

v the2

=c 2

v the2

ωci2

ωpi2

=me

mi

β −1

€

vA > v the for β <me

mi

€

vA < v the for β >me

mi

jmn2008.17.28

Approximations made

• plane wave: ignored initial conditions• Te= Ti = 0• infinite plasma: ignored boundary conditions• homogeneous in space: equilibrium values are constant• B = B0, uniform, static• no free streaming• no dissipative effects, no collisions, no forces quadratic in v• small amplitude B0 >> B1

Consequences• no finite temperature effects• no streaming effects such as

– sound waves– particle bunching– Landau damping– shock waves

jmn2008.17.29

Dispersion relation, cold plasma case

€

A ⋅N 4 + B ⋅N 2 +C = 0

€

A = S ⋅sin2 Θ + P ⋅cos2 Θ

B = R ⋅L ⋅sin2 Θ + P ⋅S ⋅(1+ cos2 Θ)

C = P ⋅R ⋅L

€

tg2Θ = −(N 2 − R) ⋅(N 2 − L) ⋅P

(S ⋅N 2 − R ⋅L) ⋅(N 2 − P)

€

det N × N ×1( ) + K(ω,N)[ ] = 0

with

2 solutions for N2

form of solution depends on S, P, R, L,

€

N =c

vph

=c ⋅kω

€

S =1

2(R + L ) ; D =

1

2(R − L )

€

R = 1 −(ωpe /ω )2

1 − ωce /ω−

(ωpi /ω )2

1 +ωci /ωi

∑

€

L = 1 −(ωpe /ω )2

1 +ωce /ω−

(ωpi /ω )2

1 − ωci /ωi

∑

€

P = 1 −ωpe

ω

⎛

⎝ ⎜

⎞

⎠ ⎟

2

−ωpi

ω

⎛

⎝ ⎜

⎞

⎠ ⎟

2

i

∑

jmn2008.17.30

Resonance

• 2 solutions for (k/k0)2, function of – if > 0 -> propagating– if < 0 -> non-propagating

• propagating solution k/k0 = +/-– waves travelling in opposite directions

• if two propagating solutions– smaller v = ω /k -> slow wave– larger -> fast wave

• (k/k0)2 can change sign – by going through 0 -> cut-off

• reflection• evanescent wave

– by going through infinity -> resonance• absorption• reflection and transmission

• cut-off -> independent of angle• resonance -> depending on angle

x

k2perp

k2inf

Cut-off

€

N → 0 ; vph → ∞ ; λ → ∞ ; k → 0

€

N → ∞ ; vph → 0 ; λ → 0 ; k → ∞

jmn2008.17.31

Classification of waves

• phase velocity– fast

– slow

• direction of propagation– k parallel to B0: according to polarisation

(with respect to B0, in other wrt propagation direction)

• right -> direction of rotation of electrons

• left -> direction of rotation of ions

– k perpendicular to B0

• ordinary: E1 // B0

• extraordinary: E1 perp to B0

jmn2008.17.32

Electron-Cyclotron-Wave

ECR

ICR

Ion-Cyclotron-Wave

Whistler-Wave

Alfvèn-Wave

L-Wave

R-Wave

k

ω

ωce

ωci

c

0

k || B

cut-off Resonance

Parallel and perpendicular propagation

ωUH

ω

LHR

UHR

k

ωLH

Alfvèn-Wave

0

O-Mode

X-Mode

LH-Wave

UH-Wave

c

ωpe

ωci

ωce

ECRH

LH

ICRH

k B

3 Frequency rangesfor Plasmaheating

€

k //B → Θ = 0

N 2 = R → R - wave

N 2 = L → L - wave

P = 0

€

k⊥B → Θ = π2

N 2 = P → O - mode

N 2 =RL

S→ X − mode

€

tg2Θ = −(N 2 − R) ⋅(N 2 − L) ⋅P

(S ⋅N 2 − R ⋅L) ⋅(N 2 − P)

jmn2008.17.33

€

A ⋅N 4 + B ⋅N 2 +C = 0

€

A = S ⋅sin2 Θ + P ⋅cos2 Θ

B = R ⋅L ⋅sin2 Θ + P ⋅S ⋅(1+ cos2 Θ)

C = P ⋅R ⋅L

Electron-Cyclotron-Wave

ECR

ICRIon-Cyclotron-Wave

Whistler-Wave

Alfvèn-Wave

L-Wave

R-Wave

k

ω

ωce

ωci

c

0

k || B

ωce

ωUH

ω

LHR

UHR

k

ωLH

Alfvèn-Wave

0

O-Mode

X-Mode

LH-Wave

UH-Wave

c

ωpe

ωci

k B

jmn2008.17.34

• 2 solutions for (k/k0)2, function of , n, B– if > 0 -> propagating– if < 0 -> non-propagating







R

L

B O X

L

R

B X

jmn2008.17.35

CMA diagram

ωp2

ω2=> Plasma density

ωce2

ω2=>Magnetic field ωci

2

ω2

1

1

cut-off

Resonanz

P=0

L=

S=0

R=

• 2 solutions for (k/k0)2, function of , n, B– if > 0 -> propagating– if < 0 -> non-propagating







jmn2008.17.36

Easier to show with mi / me = 2.5

jmn2008.17.37

CMA detail

jmn2008.17.38

Use of the CMA diagram

1 20

1

0

O-Modecut-off

„ECR“

O-/X-Mode with kB

ωceω Magnetic field

(=Density)

X- cut off

upper Hybrid -Resonanz

High field-

Low field-coupling

path through the Plasma!

B (R)

Wave

€

N 2 − P = 0

= "ordinary wave" (O-Mode) mit E || B

= “extra-ordinary wave”(X-Mode) mit E B

€

(S ⋅N 2 − RL) = 0

€

ω pe2 ω2


jmn2008.17.39

ECR

R

B(R)

ne

“HF-cut-off“

UH-Resonanz

EC-“Resonance“X1-ModeEC-“Resonance“

R

B(R)

ne

B

O-Mode

O-Mode cut-off ..if ne >ncrit

B (R)

Wave

1 20

1

0

O-Modecut-off

„ECR“

O-/X-Mode with kB


ωpe /ω22 (=Density)

X- cut off

upper Hybrid -Resonanz

High field-

Low field-coupling

path through the Plasma!

E || BE B

B

jmn2008.17.40

Wave heating




Antenna

Wave to particles

Resonance zone

R

€

ω =Ωc

€

Ωc =Z eB

m

€

ω

jmn2008.17.41

• Wave propagation– approximations– wave equations– dispersion relation– characteristic frequencies– classification of waves– parallel and perpendicular propagation– cut-off and resonances– CMA diagram– application to ECRH

• Absorption– interaction between wave and particles

jmn2008.17.42

Approximations

• coupling

– cold plasma

• propagation

– cold plasma

– (warm plasma)

• power absorption

– warm plasma

elctrons + (multiple) ionscold -> “fluid” equations, no temperature effectwarm -> finite larmor radius plays a role

EC

(IC)

EC IC

(EC)

IC

jmn2008.17.43

Force on an electron

€

md

r v

dt= e ⋅(

r E +

r v ×

r B ) ⋅exp(i(ωt −

r k

r x )) ≈ e ⋅

r E ⋅exp(i(ωt −

r k

r x ))

Integration along an unperturbed orbit gives for the momentum increase

€

mΔr v = e

r E dt exp iωt − ik⊥ρ sin(ωct + ϕ 0) − ik||v||0t[ ]

−∞

+∞

∫

= er E exp(inϕ 0)Jn (k⊥ρ)δ(ω − nωc − k||v||0)

−∞

+∞

∑

Energy transfer only if

€

ω −nωc = k||v|| is satisfied

Interaction between a wave and a charged particle

With relativistic effects we have

€

ωc = ωc0 1− v 2 /c 2

=> Interaction only with resonant particles in velocity space

The same is valid for ions.

jmn2008.17.44

Collisionless Damping

€

ω −nωc = k||v||

vf(v)

Resonance condition:

∂f (v)

∂v< 0

Condition for damping

Landau damping: Increase of parallel momentum

The deformation of the distribution function increases the energy of the electron system.


E

V|| =Vph −δ V|| =Vph +δ

k

€

n = 0

€

ω −k||v|| = 0

jmn2008.17.45

Cyclotron Damping (Doppler shifted)

€

ω −nωc = k||v||

Resonance condition:


€

n =1

€

ω −k||v|| = ωc

B0E

Cyclotron Damping: increase of perpendicular momentum

jmn2008.17.46

Force on an electron

€

md

r v

dt= e ⋅(

r E +

r v ×

r B ) ⋅exp(i(ωt −

r k

r x )) ≈ e ⋅

r E ⋅exp(i(ωt −

r k

r x ))

Integration along an unperturbed orbit gives for the momentum increase

€

mΔr v = e


−∞

+∞

∫


−∞

+∞

∑


€

ω −nωc = k||v|| is satisfied

Interaction between a wave and a charged particle

With relativistic effects we have

€

ωc = ωc0 1− v 2 /c 2

=> Interaction only with resonant particles in velocity space

The same is valid for ions.

jmn2008.17.47

€

mΔr v = e


−∞

+∞

∫


−∞

+∞

∑

jmn2008.17.48


UH

R

O-wave

X1-wave

1r/a

ne

B0

EC-resonance

Upper hybrid resonance

X-mode Cutoff

X1-wave

X2-wave

ωωUH

ω

k

ωL

Alfvèn-Wave

0

O-Mode

X-Mode

Lower Hybrid Wave

Upper Hybrid Wave

ω=c

k

ωpe

ωR

ωωLHωci

ECRHLHICRH

3 frequency regionsfor plasma heating:

ωce

jmn2008.17.49

Mode Conversion OXB-Heating

O-mode

X-mode

Mode conversion processunder certain launch anglesand for minimum density.

O mode converts into X modeat O-mode cutoff.

X-mode converts into electrostatic electron (Bernstein) wave.

Bernstein wave absorbed by electron cyclotron damping.

No upper density limit.X-mode cutoff

O-mode cutoff

B-mode

UH-resonanceEC-resonance

jmn2008.17.50

ECRH: Operation Scenarios for W7-XPlasma density range

(m-3)

EkB

. . . . . .

k

Ev,j

B

ωc =eB

mCyclotronfrequency:

ωp2 =

e2ne

ε 0 me

Plasmafrequency:

Determines the microwave frequency:

(2.5 T, n=2, 140 GHz for W7-X)

Determines the density range

ω −nωc /γ − k||V|| = 0 γ =1/ 1 −v2

c2

1019 1020 1021

X2-Mode

O2-Mode

O-X-B-Mode

Plasma density

⊥BkkE | |

k ⊥ B , E ⊥ B

k ⊥ B , E || B

jmn2008.17.51

ECRH System

Collector

Resonator

Electron-beam

typ. 30A, 80kV

mm-Wave

quasi-optical

orwaveguide transmission

EC-Resonance

Gyrotronbis 1MW

80····170GHz

SuperconductingMagnet

Window

jmn2008.17.52

ECRH - Gyrotrons

superconductingcoils

diamondwindow

annularelectron beam

resonator

conversionto Gaussianbeam

collector

Presently development of 1 MW cw gyrotrons

jmn2008.17.53

Needs:-Local current drive-Synchronized injection-Fast detection of NTM by ECE-Extremely fast mirror control and power modulation

NTM Stabilisation

EC-resonance NTM-island

ECRH-Beam

jmn2008.17.54

Removal of the magnetic Island by ECCD

Current drive (PECRH / Ptotal = 10-20 %) results in removal

method has the potential for reactor applications

jmn2008.17.55

Lower Hybrid system

Klystron500-750kW

2,53,7 GHz

“Grill”

00 900 1800 ...

= “phased array”-Antenna

Stack of102-103

Waveguides

Wave

0,5 vacuum

N|| 2

Front view:

N||

NN

= conserved -quantity!

E~

0 90 1800 ....

evaneschent layer

ne increases

“Grill-Antenna“

jmn2008.17.56

2 solutions of dispersion relation: slow wave (exhibits lower hybrid res.)fast wave

ne > 1017m-3 at antenna, to enter plasmak|| > kc to reach center.

€

ωci << ωLH <<ωce

€

|| ≈ 2 −10cm, λ⊥ ≤1cm

Lower Hybrid Heating

k|| too low, power stays near plasma edge

sw

fw

k

radiusk|| sufficiently high, slow wave travels into plasma,absorption at LH or before

k

LowerHybridresonance

radius

jmn2008.17.57

LH - Wave Propagation

Depends onne and B.

Antenna structure

jmn2008.17.58

Klystron and Grill

Beam dump

cathode

anode

-wave input

-waveoutput

3.7 GHz500 kW3 sec

klystron waveguide grill

jmn2008.17.59

LH - Wave Excitation

Fast wave

kB0

Ehf

Bhf

Slow wave

kB0

Ehf

BhfMultiplewaveguides

Ewg

ASDEX

jmn2008.17.60

Current drive on ASDEX

Example: Lower hybrid current drive, 1.3 GHz / 2.4 MW / 3 secLeuterer F., Eckhardt D., Soeldner F.X., et al., Phys. Rev. Lett. 55 (1985) 75

The plasma current can be ramped up or the OH-transformer can be recharged

with loop voltage = 0 with loop voltage < 0

jmn2008.17.61

Ion cyclotron system

Tube-amplifier

typ. 2 MW30-100 MHz

50 ΩKoax-Ltg

Matching-Tuner50 Ω 1··3 Ω

Plasma:Re(N) >>1

Dipole-antenne

Preamplifier

jmn2008.17.62


ωωUH

ω

k

ωL

Alfvèn-Wave

0

O-Mode

X-Mode

Lower Hybrid Wave

Upper Hybrid Waveω

=ck

ωpe

ωR

ωωLHωci

ECRHLHICRH

3 frequency regionsfor plasma heating:

ωce

jmn2008.17.63

Absorption

wave - particleions : cyclotron resonance

fundamental need correct polarisation E+ minority heating

harmonic need gradient in E+ preferentially fast particles wave other wave particle mode conversion

ω -k// v//i =(n+1)ωci

jmn2008.17.64



jmn2008.17.65

Larmor radius

= m vperp / Z e B

Cyclotron frequency

ω = Z e B / m = Z/A * e/mH * B

Cyclotron motion

vperp

B

15 MHz per Tesla * Z/A

jmn2008.17.66

Ion Cyclotron Resonance Heating

jmn2008.17.67

• particles are equally– accelerated and

– decelerated by the wave

• when – more particles at low velocity

• then– net transfer from wave to particles

Diffusion in velocity space

f(v)

v

jmn2008.17.68

0

0.2

0.4

0.6

0.8

1

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 100 %

p2

m2

z2

a

H to absorb but no correct polarisation

for pure H

at ω = ωcH

the left hand polarization is = 0

nH/ne = 100 %

jmn2008.17.69

0

0.2

0.4

0.6

0.8

1

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 0%

a

Correct polarisation but no H to absorb

nH/ne = 0 %

in pure D

at ω = ωcH

the left hand polarization exists but there is „no“ resonance

jmn2008.17.70

0

0.2

0.4

0.6

0.8

1

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 0%

a

What happens if we add a bit of H

nH/ne = 0 %

in plasmas with dominant D and a bit of H

the left hand polarization is set by the dominant D

but now we have the H to absorb!

0

0.2

0.4

0.6

0.8

1

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 0.1 %

p2

m2

z2

a

nH/ne = 0.1 %

jmn2008.17.71

How much is a bit before it starts to affect the polarisation

0

0.2

0.4

0.6

0.8

1

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 2 %

a

0

0.2

0.4

0.6

0.8

1

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 15 %

p2

m2

z2

a

dependence on concentration

at ω = Ωci the left hand polarization decreases

with increasing H concentration

nH/ne = 2 % nH/ne = 15 %

jmn2008.17.72

Need the correct polarization AND the species to absorb

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3

30 MHz, 2T, nphi

=6

p2

m2

z2

nh/n

e

amplitude with correct polarisation 0 !!nH/ne 10 %

jmn2008.17.73

0 100

2 103

4 103

6 103

8 103

1 104

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 0.1 %

Re2

Im2

n2perp

a

Need to check the propagation of the wave

jmn2008.17.74

Propagation of the wave

0 100

2 103

4 103

6 103

8 103

1 104

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 3 %

Re2

Im2

n2perp

a

jmn2008.17.75

At even higher nH concentration: cut-off

-1 104

-5 103

0 100

5 103

1 104

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 10 %

Re2

Im2

n2perp

a

jmn2008.17.76



jmn2008.17.77

Resonance/ Cut-off, Tunnelling, Mode conversion

x

k2perp

k2inf

-1 104

-5 103

0 100

5 103

1 104

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 10 %

Re2

Im2

n2perp

a

Resonance

Cut-off€

N → ∞ ; vph → 0 ; λ → 0 ; k → ∞

€

N → 0 ; vph → ∞ ; λ → ∞ ; k → 0

jmn2008.17.78

In the vicinity of the ion-ion hybrid layer, mode conversion to shorter wavelength waves occurs.

IBW : Ion Berstein WavePropagates towards the high field side

ICW : Ion Cyclotron WavePropagates towards the low field side

F.W. Perkins, Nucl. Fusion 17, 1197 (1977)

ICRF Heating in DIII-D: Mode conversion

M. Brambilla, Plasma. Phys. Cont. Fusion 41, 1 (1999)

Mode conversion to ion Bernstein wave, electrostatic ion cyclotron wave

jmn2008.17.79

Ion-ion cut-off and resonances

• from low field side, sequence is always cut-off, then ion-ion resonance

• lies between cyclotron resonances of both ions

• lies closest to the cyclotron resonance with the lower concentration

• location of the cyclotron resonance wrt to pair thus varies

€

ω =Z eB

M

HFS

LFS-1 104

-5 103

0 100

5 103

1 104

-40 -20 0 20 40

30 MHz, 2T, nphi

= 6, nH/ n

e = 10 %

Re2

Im2

n2perp

a R

€

ω =ΩcH

€

ω =ΩcD

B

B1

B2

low H concentration

low D concentration

jmn2008.17.80

Multiple ions

• cut-off /resonance pair between each of ion cyclotron resonances

• location depends on the relative concentrations

€

ω =Z eB

M

HFS

LFS

R

€

ω =ΩcH

€

ω =ΩcD

B

B1

B2

€

ω =ΩcHe3

jmn2008.17.81

fundamental need correct polorisation E+ minority heating


jmn2008.17.82

“Second” harmonic

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

60 MHz, 2T, nphi

=12

p2

m2

z2

nh/n

e

2 ωcH, also in H plasma -> amplitude with correct polarisation 0

jmn2008.17.83

Cyclotron motion

B

vperp

B

vperp

B

vperp

B

jmn2008.17.84

Second Harmonic Heating

m d (vperp2 ) / 2 dt = Z E vperpcos

Ev. 7Ev 0 -.7Ev - Ev - .7Ev 0 .7 Ev

netto Ev- Ev = (E - E) v -> gradient in electric field

jmn2008.17.85

Second Harmonic Heating

jmn2008.17.86

• particles are equally– accelerated and

– decelerated by the wave

• when – more particles at low velocity

• then– net transfer from wave to particles

Diffusion in velocity space

f(v)

v

jmn2008.17.87

2nd and higher Harmonics damped if wave-field non-uniform on length-scale of Larmor radius

t=0 t=T/2 t=T

B B B

v

v

v

€

ω =2 ⋅ωcs

E

gradient of electric field

€

∂ l( )E

∂r l( )≠ 0

T = period of the wave

jmn2008.17.88

€

E+

How do we get this variation of E along the orbit ?

v ⊥

€

E+

€

E+

€

⊥

€

⊥

€

E+

€

E+

€

⊥ €

⊥

€

E+

€

⊥

€

⊥

E constant1

2

13

maximum effect 2= /2

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2

€

E+

€

E+

€

⊥

€

⊥

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2

4

zero effect again

jmn2008.17.89

€

mΔr v = e


−∞

+∞

∫


−∞

+∞

∑

maximum effect 2= /2 k= /2

jmn2008.17.90

At the second harmonic, the net acceleration depends on L/

Acceleration and deceleration by the wave along the ion orbit.

Net effect only if E+ is not constant, ie if L/ Lk is finite.

Effect becomes weak for L/ Lk near 1

jmn2008.17.91

H

sHStix n

tpT ∝

FLR effects play a key role in determining the fast ion distribution function, in agreement with theory

no more acceleration

Wave-particle interaction becomes weak for L/ Lk near 1 with k = F(n)

jmn2008.17.92

ICRF-acceleration of 4He beam ions for simulating fusion ’s

ITER: fusion-born 3.5-MeV alpha particles (4He ions)

JET simulation: ω3ω(4He) in 4He plasma.

For n = 3, damping on thermal ions

is small boost with high-energy 4He beams with finite L.

Strongest 4He tails with ICRF-

acceleration of highest energy (120

keV) 4He beams.

(Mantsinen, PRL 2002)

JET

jmn2008.17.93

Possible absorption scenarios

fundamental need correct polorisation E+ minority heating


in the ion cyclotron range of frequencies

jmn2008.17.94

Wide range of possibilities, beyond heating

Power • to ions• to electrons

• to thermal particles • to fast particles (tails)

• to particles going one way• or another

• on axis• off-axis

jmn2008.17.95

One of the first fast-ion effects: Sawteeth stabilisation

Sawteeth = periodic relaxations of plasma core temperature.

Central ICRH Strong sawtooth stabilisation due to ICRH-heated ions with ωDrift >ωmode.

Relevance for ITER: sawtooth stabilisation by fusion ’s.

Later: sawtooth destabilisation by ICRF.

(Campbell, PRL 1988; Phillips, Phys. Fluids 1992; Porcelli, PPCF 1991)

JET

jmn2008.17.96

Power deposition with ω = nωc depends on the finite orbit widths of fast ICRF-accelerated ions.

Fast ions orbits, and thus the power deposition, can be modified using toroidally directed waves.

Absorption of wave toroidal angular momentum ICRF-induced pinch of fast ions.

Turning points of trapped fast ion orbits are driven either outwards or inwards, depending on the direction of wave propagation.

Wave Ip Wave Ip Inward pinch Outward pinch

(Hellsten, PRL, 1995)

Eventually co-passing orbits residing on the low-field side of Rres

Control : ICRF-induced pinch modifies fast ion orbits

jmn2008.17.97

Experimental evidence for ICRF-induced pinch

(Kiptily, Nucl. Fusion, 2002; Mantsinen, PRL, 2002)

3He cyclotron resonance

jmn2008.17.98

Basics of ICRF power deposition localisation

The most narrow ICRF power deposition profiles can be obtained with ICRF mode conversion (MC).

Most common: MC in the vicinity of the ion-ion hybrid resonance of two ion species with comparable concentrations.

Real-time control of ion species mixture in routine use on JET to keep MC power deposition fixed in time.

(Mantsinen, Nucl. Fusion 2004)

Plasma Centre

jmn2008.17.99

Toroidally directed waves couple

asymmetrically to ions and electrons in the

v||-space due to their average finite k||.

This gives rise to

–ion cyclotron current drive (ICCD)

–fast wave electron current drive

(FWCD)

–ICRF mode conversion current drive

Current drive with ICRF waves

jmn2008.17.100

Ion cyclotron current drive (ICCD)

Ion cyclotron current with passing ions is dipolar, i.e. its sign reverses when crossing ω = nωci (Fisch, Nucl. Fusion 1981).

Flattening or peaking of the current profile depending on

- location of Rres versus R0

- toroidal direction of the wave.

While net current is small, local effect can be rather strong.

jmn2008.17.101

Sawtooth control with ICCD

ICCD at q = 1 on JET is used modify the current profile and thereby to stabilise or destabilise sawtooth activity.

Recent JET experiments:

ω = ωcH (Mayoral, Phys. Plasmas 2004;

Eriksson, Nucl. Fusion 2006)

ω = 2ωcH (Mantsinen, PPCF 2002)

Expanded capabilities and heavy use.

JET

(Start, EPS1992; Bhatnagar, Nucl. Fusion 1994)

jmn2008.17.102

Coupling with antennas




Resonance zoneAntenna

Wave to particles

jmn2008.17.103

Waves

wave excitation, propagation, absorption

– absorption in plasma

– wave excited in plasma and propagates

– in density layer in front of antennafields decay (evanescence)

– antenna creates E and B fields

jmn2008.17.104

Fast wave evanescent at the edge

0 100

2 103

4 103

6 103

8 103

1 104

-60 -40 -20 0 20 40 60

30 MHz, 2T, nphi

= 6, nH/ n

e = 100 %

Re2

Im2

n2perp

a -5 101

0 100

5 101

1 102

50 52 54 56 58 60a

jmn2008.17.105

Absorption... Coupling

z

Absorption

Wave

Coupling

Cut-off

excited by Antenna

boundary condition for Antenna

Evanescent WaveEco = f(k//) Eplasma/vacuum

Eplasma/vacuum ~ Bz (relation in plasma)

Ey/ Bz

x

z

y

jmn2008.17.106

Depth of the evanescent layer depends on k//

in the tenuous plasma, below the cut-off densitythe fast wave (k vector mostly perpendicular) is evanescent

€

B = B1 e i k r−iω t[ ]

with

€

k02 = k//

2 + k⊥2

we obtain

€

k02 small and k⊥

2 < 0

€

k⊥2 = k0

2 − k//2

€

ik⊥ ≈ k//

€

e i k r[ ] =e i k⊥x[ ] =e −k//x[ ]

thus relation between coupling and antenna spectrum due to the cut-off

jmn2008.17.107

-40 -20 0 20 400.0

0.2

0.4

0.6

0.8

1.0

nφ

2 , 90straps

2 , 180straps

4 , 90straps

4 , 180straps

nφ =Rk// =2πRλ

jmn2008.17.108

Antennas

strap 1strap 2

strap

1

stra

p 2

strap 1 strap 2

stra

p 1

stra

p 2

Bt

Ip

jmn2008.17.109

ICRF Antennas on JET

ICRF antennalaunches wave ω, k||

jmn2008.17.110

Dynamic matching or load isolation necessary

50 Ohm

matching

1 ms

D

R

0

08

8

2-10 +R Ohm + j X + X

Antenna

• timescales of variations

– particle/energy confinement time ms to s

– MHD events 100s

jmn2008.17.111

H

Power to antenna

Reflectedpower

Reflectedpower at generator

5 kW

100 kW

1 MW

3 dB couplers for ELM resilience3 dB couplers for ELM resilience

jmn2008.17.112

Overview

• steps: generation, transport, absorption, thermalization

• ICRF: minority, mode conversion, harmonic

• not just heating

• 3 dB couplers




Wave to particles

jmn2008.17.113

Summary

• for power to be absorbed into the plasma it must first get there

• wave propagation: range of frequencies– Electron cyclotron

– Lower Hybrid

– Ion cyclotron

• absorption in plasma: wave - particle interaction– cyclotron damping, also at harmonics

– Landau damping

• different types of approximations– cold plasma: typically two waves

– CMA diagram, parallel, perpendicular propagation

• very large number of possibilities, not just heating– current drive

– control of instabilities

– …

jmn2008.17.114

Literature

M. Brambilla

Kinetic Theory of Plasma Waves,

R. Koch

Physics and Implementation of ICRF of Fusion Reactors, Lab. Rep. 108, 1997, Brussels

Stix, Thomas H.

Waves in Plasmas, New york, American Institute of Physics, 1992

Wesson, John

Tokamaks, Oxford, Clarendon Press 1997, 2nd Ed.

The Oxford Engineering Science Series, Vol. 48

Swanson, D.G.

Plasma Waves, Boston,MA, Academic Press 1989

Cairns, R.A.

Radiofrequency Heating of Plasmas, Bristol, Hilger 1991

jmn2008.17 physics of rf heating j.-m. noterdaeme with the support of m. brambilla, r. bilato, d....

Documents