AssociationEuratom-CEA

Variational descriptionof low frequency waves in

magnetically confined plasmas

R. Dumont

CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France.

September 9, 2008

AssociationEuratom-CEA Outline

I Low frequency waves in magnetic fusion devicesI Plasma heating by ICRF wavesI MHD (Alfven) wavesI Modelling aspects

I Global modelling of low frequency wavesI Wave-field calculationI Plasma kinetic response descriptionI Towards a self-consistent treatmentI Variational approach

I The EVE-3D codeI Setup and numerical implementationI BenchmarksI Applications

I Conclusions and prospects

AssociationEuratom-CEA RF heating of magnetic fusion plasmas

I Advanced scenarios require electromagnetic waves excited byexternal antennas

I Ion and/or electron heatingI Non inductive current driveI Rotation, flow drive (reduction/suppression of turbulence)I Alpha-channelling

I Three base ingredients of modellingI Calculation of electromagnetic field: wave codeI Description of plasma response to the RF power: kinetic codeI Antenna / edge plasma interaction: antenna code

I Room for improvement in each of these subtopics

I Needs to work towards self-consistent loop

AssociationEuratom-CEA Plasma heating by ICRF waves

I ICRH (= Ion Cyclotron Resonance Heating)

I General principle:

Antenna

Ion cyclotron resonance

I A (fast magnetosonic)wave is excited on the LowField Side of the tokamak

I Its frequency is in the rangeω ∼ Ωci (f ≈ 50MHz)

I Resonance relation:ω = pΩci + k‖v‖,p integer

I In most scenarios, it isused for ion heating

I Creation of superthermalpopulations (fast ions)

AssociationEuratom-CEA ICRF heating of fusion plasmas

ICRF: Ion Cyclotron Range of Frequency (f ∼ 30− 80MHz)

ICRF PowerICRF Power

• Fast Wave + Cycl. Res.• Fundamental absor.• Harmonic absorption

• Fast Wave + Cycl. Res.• Fundamental absor.• Harmonic absorption

• Fast Wave• ELD• TTMP

• Fast Wave• ELD• TTMP

• Ion Bernstein Wave• ELD

• Ion Bernstein Wave• ELD

Thermalions

Thermalions

Thermal electronsThermal electronsSuperthermal

ionsSuperthermal

ions

E < Ec E > Ec

Ionheating

Electronheating

AssociationEuratom-CEA MHD waves

I Magnetically confined plasmasfeature Alfven waves(compressional / shear)

I Toroidal effects result in acoupling of these waves

I Within the frequency gaps lieglobal, regular, modes:Alfven Eigenmodes (AE)

I These modes may bedestabilized by fast ions (fusionborn alphas / fast ICRF ions)

Alfven eigenmodes are crucial toITER operation and performance

AssociationEuratom-CEA MHD / Kinetic simulation of LF waves

AssociationEuratom-CEA Low frequency RF waves modelling: a summary

I Low frequency waves in fusion plasmasI Alfven eigenmodesI ICRF waves

I Three basic components of RF waves modelling

absorptionAntenna Wave propagation / Plasma

response

PSfrag replacements Jant Prf

fs,0

I Modelling of ICRF waves propagation and absorption in ongoing andfuture experiments should include

I 2D / 3D effectsI Non-thermal particle distributionsI Finite orbit effects

AssociationEuratom-CEA Global modelling of LF waves

I Wave excitationI Current flowing in the antenna

structure (frequency ω)I Excitation of a fast magnetosonic

wave (evanescent in vacuum)I Coupling is determined by the

radiative resistance(Picrf = Rrad I 2

ant/2)

I PropagationI Driven oscillation at frequency ωI Space / time dispersionI Global electromagnetic field

I Plasma responseI Modification of distribution

functionsI Collisional effects

AssociationEuratom-CEA Linear calculation of wave-field

I Wave equationI System driven at frequency ω:

∇×∇× E− ω2

c2

(E +

i

ωε0j

)= iωµ0jant

I Non-local response:

j(r, t) =∑

s

∫d3r′

∫ t

−∞dt ′

=σs(r, r′, t, t ′, fs,0)︸ ︷︷ ︸

Conductivity kernel

·E(r′, t ′)

I Linear conductivity kernel:=σs ∝ δfs

I AnisotropyI Time dispersionI Space dispersion

I Vlasov equation (collisionless)

dfs

dt≡ ∂fs

∂t+ v · ∂fs

∂r+

qs

ms(E + v × B) · ∂fs

∂v= 0.

AssociationEuratom-CEA Unperturbed orbits

I Linear response (formal solution of Vlasov equation)

δfs = − qs

ms

∫ t

−∞dt ′[δE(r′, t) + v′ × δB(r′, t)

]· ∂fs,0∂v′

.

Characteristics of Vlasov equation

dr′

dt ′= v′,

dv′

dt ′=

qs

ms(E0(r′, t) + v′ × B0(r′, t)).

I Unperturbed orbits

r = rg (µ,H,Pφ)+rc (µ,H,Pφ, φc )

AssociationEuratom-CEA Fast ion orbits

Trajectory (guiding center)of a trapped ion

→ banana orbit

I Banana width:

∆b ∼v‖0ωb

I Confinement is determined by

∆b/a0

I Orbit effects crucial when

∆b ∼ r

I Ex: α particle, E = 3.5MeVI JET: 2∆b/a0 ∼ 0.8I ITER: 2∆b/a0 ∼ 0.2

AssociationEuratom-CEA Wave-field calculation in a nutshell

Equilibrium distributionfunction

Equilibrium fields

Unperturbed orbits

Wave-field

Dielectric tensor

Perturbed current

Perturbed distributionfunction

AssociationEuratom-CEA Plasma kinetic response

I Kinetic equation for fs :

∂fs∂t

+ v · ∂fs∂r

+qs

ms(E + v × B) · ∂fs

∂v=

(∂fs∂t

)coll

I Linearization: fs(r, v, t) ≡ fs,0(r, v) + δfs(r, v, t)

I Linear treatment (wave-field calculation):I fs,0 stationaryI Collisions are neglected

I Quasilinear treatment (kinetic response calculation):I fs,0 varies slowly compred to 2π/ω (secular effects)

〈δfs(t)〉 = 0 → 〈fs(t)〉 = fs,0(t)

where 〈·〉 means averaging over fast time scaleI Collisions need to be included

AssociationEuratom-CEA Kinetic response

I Averaged kinetic equation⟨dfs

dt

⟩=

dfs,0

dt≡ ∂fs,0

∂t+v· ∂fs,0

∂r+

qs

ms(E0 +v×B0)· ∂fs,0

∂v= C (fs,0)+Q(fs,0)

I C (fs,0): Collision operator (Fokker-Planck)I Q(fs,0): Quasilinear operator

Q(fs,0) = − qs

ms

⟨(δE + v × δB) · ∂δfs

∂v

⟩.

Competitionwave / collisions

AssociationEuratom-CEA Antenna phase and orbit effects

−+

− +Antenna phasing

-20 0 20 40Nφ

0

5

10

15

p RF [a

.u.]

Toroidal number: Nφ = −15

I RF waves have a strongimpact on the fast ion orbits

I The unperturbed trajectoriesare modified

AssociationEuratom-CEA Kinetic response calculation in a nutshell

Wave-field

WaveQL diff. coefficient

Distribution function

F.P. Equation

Collisions

CollisionalQL diff. coefficient

Intrinsicchaos

Extrinsicchaos

HeatingCurrent DriveInstabilities

AssociationEuratom-CEA Waves in a plasma: the whole story (almost)

Unperturbedorbits

Eq. distributionfunction

Collisionaldiff. coefficient

HeatingCurrent DriveInstabilities

F.P. Equation

Wavediff. coefficient

Wave equation

Extrinsicchaos

Intrinsicchaos

PSfrag replacements

(A0, ϕ0)

(δA, δϕ)

fs,0

fs,0

=

Dcoll

=

Dwave

I Challenges:

1. Unperturbed orbits2. Consistency of wave and kinetic calculations3. Phase decorrelation processes

AssociationEuratom-CEA Waves in a plasma: the whole story (almost)

Unperturbedorbits

Eq. distributionfunction

Collisionaldiff. coefficient

HeatingCurrent DriveInstabilities

F.P. Equation

Wavediff. coefficient

Wave equation

Extrinsicchaos

Intrinsicchaos

PSfrag replacements

(A0, ϕ0)

(δA, δϕ)

fs,0

fs,0

=

Dcoll

=

Dwave

H0(J) δH(J,Φ)

I Challenges:

1. Unperturbed orbits2. Consistency of wave and kinetic calculations3. Phase decorrelation processes

Hamiltoniandescription

AssociationEuratom-CEA Wave-field calculation: variational approach

I Electrical current / charge conservation jant + jpart =1

µ0∇×∇× A + ε0∂t(∂tA +∇ϕ) ≡ jmaxw ,

ρant + ρpart = −ε0∇ · (∂tA +∇ϕ) ≡ ρmaxw

I Three gauge-invariant functionals

I Antenna functional

Lant ≡ µ0

∫d3r jant · A∗ − ρantϕ

∗

I Maxwellian functional

Lmaxw ≡ µ0

∫d3r jmaxw (A, ϕ) · A∗ − ρmaxw (A, ϕ)ϕ∗

I Plasma functional

Lpart ≡ µ0

∫d3r jpart(A, ϕ) · A∗ − ρpart(A, ϕ)ϕ∗

AssociationEuratom-CEA Wave-field calculation

I Variational statement ≡ extremalization of

Lpart(A, ϕ,A∗, ϕ∗) + Lmaxw (A, ϕ,A∗, ϕ∗) + Lant(A∗, ϕ∗)

I Decomposition of electromagnetic potential: (A, ϕ) ≡∑

i αi (ai , φi )I Decomposition of functionals:

L(A, ϕ,A∗, ϕ∗) ≡ Lpart + Lmaxw =∑

ij

Lijαiα∗j , Lant(A∗, ϕ∗) =

∑j

Kjα∗j ,

withLij ≡ L(ai , φi , a

∗j , φ∗j ), Kj ≡ Lant(a∗j , φ

∗j )

I Electromagnetic field calculation

δ[

(Lij ,part + Lij ,maxw )αi + Kj

]α∗j

δα∗j= 0

−→ (A, ϕ) is obtained by solving (Lij ,part + Lij ,maxw )αi = −Kj .

AssociationEuratom-CEA Energy balance and functionals

I Global energy balanceI Power coupled by the antenna

Want ≡⟨∫

d3r E · jant

⟩=

ω

2µ0=(Lant)

I Power transferred to plasma species

Wpart ≡⟨∫

d3r E · jpart

⟩=

ω

2µ0=(Lpart)

I Maxwellian functional

Lmaxw = −2µ0

∫d3r

(ε0|E|2

2− |B|

2

2µ0

)is a real quantity,

−→ Want + Wpart =ω

2µ0=(Lant + Lpart) = 0

I Local energy balance (Poynting theorem)

−iωWfield +Spoynting−want−wpart =iω

2µ0Lmaxw (s)+Lant(s)+Lpart(s)

AssociationEuratom-CEA Integration with kinetic calculation

I Wave calculationI Variational approach∑

s

Lpart,s + LMaxw . + Lant = 0,

I Resonant plasma functional

L(res)part,s = µ0

∑N1=p,N2,N3=N

(2π)3

∫d3J

ω

ω − Ni Ωi

∂fs,0

∂Ji|δhp,N2,N |2

I Kinetic response calculationI Fokker-Planck equation

∂fs,0

∂t= C fs,0 +

∂

∂JiD

(QL)ij

∂fs,0

∂Jj

I Wave quasilinear diffusion coefficient

D(QL)ij = 2π

∑N1=p,N2,N3=N

NiNj |δhp,N2,N |2δ(ω − Ni Ωi )

AssociationEuratom-CEA The eve code

I Physics featuresI Wave equation formulated in terms of potentialsI 2nd order Larmor radius code + Order Reduction AlgorithmI Resolution of 2 (E‖ = 0) or 4 variables (E‖ 6= 0)I Uses quasi-local plasma functional

I Numerical featuresI 3D version functional (no coupling of toroidal modes)I Radial direction: finite elements (cubic + quadratic)I Fourier expansion in the toroidal and poloidal directionsI Uses code generator for some partsI Core in Fortran 90 - Post-processing in Python

I ObjectivesI Main element of a wave + kinetic packageI ICRF module for integrated modellingI Detailed physics studies of wave-particle interactions

AssociationEuratom-CEA Geometric setup

I Vacuum chamberI Perfect conductorI Arbitrary shapeI ψ = const. surface

I AntennaI Must coincide with a virtual flux

surfaceI Current flowing in 3 directionsI No feedback of the plasma

I PlasmaI Cold / hot (FLR 2nd )

/ hot (ORA) plasmaI Analytical / numerical (helena)

equilibriumI Analytical / numerical profiles

AssociationEuratom-CEA Numerical implementation

I Spectral decomposition of state vector for periodic directions

s

θ

φ

uk (s, θ, φ) ≡∑mn

ukmn(s)e i(mθ+nφ)

I Finite elements in radial direction

ukmn(s) ≡∑

jp

αkpjmnhp(s − sj )

2nd order FLR kinetic equations only involve ∂su and ∂2ssu

→ (hp) are quadratic / cubic Hermite finite elements

AssociationEuratom-CEA Numerical implementation (2)

I Variational principle yields directly Garlekin weak form

I Bilinear functionals

Lpart + LMaxw . = µ0

∫d3r j(A, ϕ) · A∗ − ρ(A, ϕ)ϕ∗

= µ0

∫dsdθdφJLkk

eq × uku∗k

= µ0δn,n

∫dsJLkk

eq m−mhpjhpj × αkpjmn

(αk p

jmn

)∗I Provide stiffness matrix elementsI Toroidal modes (n) are decoupled (in axisymmetric devices)I Poloidal modes (m) are coupled by equilibrium geometry

and wave/particle interaction

AssociationEuratom-CEA Numerical implementation (3)

I Antenna functional

Lant = µ0

∫d3r jant · A∗ − ρantϕ

∗

= µ0

∫dsdθdφJLkk

ant × u∗k

= µ0

∫dsJLkk

antm,nhpj ×(αk p

jmn

)∗I Provides source vector (RHS)I Antenna current is decomposed in toroidal harmonics:

jant =∑

n

jant,neinφ

I Separate computation for each nI 3D solution is constructed afterwards

AssociationEuratom-CEA Numerical implementation (4)

I Electromagnetic field calculationI Linear system solution (block matrix)

k,m,j

k,m,j

k,m,j

k,m,j

x = An

tenn

a

2

3

4

5

6

1

2

3

4

5

I Matrix elements calculation:I Fast Fourier Transforms (FFTW)I Radial integrals (Gauss quadrature)

I Code is partially generated by symbolic manipulation softwareI Boundary conditions and unicity directly applied to stiffness matrixI Parallel matrix inversion (ScaLAPACK)

AssociationEuratom-CEA Benchmarks in reference cases

I K. Appert and J. VaclavickPhys. Fluids 27(1984) 432

I Homogeneous plasma columnI Hydrogen plasmaI ne = 0.52× 1019m−3

I B0 = 1TI No eq. plasma current

ap

av

I Linearized MHD modelρ∂tδv = δj× B0

δE + δv × B0 = mi (δj× B0)/(eρ)∇× δB = µ0δj∇× δE = −∂tδB

I Perturbed quantities ∝ exp(i(kzz + mθ − ωt)→ Dispersion relation Dm,ap ,av (ω/ωci , kz ) = 0

AssociationEuratom-CEA EVE benchmarks: MHD waves

I Alfven frequencies: ω < ωcHI Divergence free helix antenna

I Single poloidal mode (m = −1)I k‖ = 15m−1

Dispersion relation

0.74 0.76 0.78 0.8 0.82 0.84ω/ωcH

-20

-10

0

10

20

Dis

pers

ion

m=-1

A-1,1 A-1,2 A-1,3

Frequency scan (EVE)

0.74 0.76 0.78 0.8 0.82 0.84ω/ωcH

103

104

105

106

107

Ant

enna

load

ing

[a.u

.]

A-1,1 A-1,2

A-1,3

m=-1

Peaks in the antenna loading are obtained atthe Alfven wave eigenfrequencies

AssociationEuratom-CEA EVE benchmarks: MHD waves

I Fast wave frequencies: ω > ωcHI Divergence free helix antenna

I Single poloidal mode (m = −1)I k‖ = 15m−1

Dispersion relation

4 5 6 7 8ω/ωcH

-20

-10

0

10

20

Dis

pers

ion

F-1,2 F-1,3

m=-1

Frequency scan (EVE)

3 4 5 6 7 8 9ω/ωcH

102

104

106

108

Ant

enna

load

ing

[a.u

.]

F-1,2F-1,1

m=-1

Peaks in the antenna loading are obtained atthe fast magnetosonic wave eigenfrequencies

AssociationEuratom-CEA Example: minority heating in JET

I D(H) minority heatingscenario

I R0 = 3m, a0 = 0.8m,B0 = 3.1T

I ne0 = 5× 1019m−3

I nH,b/ne = 4.75%,nH,f /ne = 0.25%

I Te0 = TD0 = TH,b0 = 5keV,TH,f 0 = 10keV

I Strap antennaI fFW = 46MHzI Ntor = 15, k‖,ant ≈ 4m−1

I Current perp. to B0

I Computation gridI Ns = 250 + 50, Nθ = 128I 49 poloidal modesI 1 toroidal mode

2.0 2.5 3.0 3.5 4.0

R [m]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Z [

m]

-14.4

-11.4

-8.4

-5.4

-2.4

0.6

3.6

6.6

9.6

12.6

<(E−) contours

AssociationEuratom-CEA ICRF waves features

Ion Bernstein wave:• excited by mode conversion• small wavelength• damps on thermal electrons• backward wave

Ion Bernstein wave:• excited by mode conversion• small wavelength• damps on thermal electrons• backward wave

AntennaAntenna

Fast Magnetosonic wave:• direct excitation by the antenna• large wavelength• damps on ions / electrons• forward wave

Fast Magnetosonic wave:• direct excitation by the antenna• large wavelength• damps on ions / electrons• forward wave

Ion cyclotron layer(Fundamental

Hydrogen)

Ion cyclotron layer(Fundamental

Hydrogen)

Mode conversion layer

(from cold plasma theory)

AssociationEuratom-CEA Artificial IBW damping

2.0 2.5 3.0 3.5 4.0

R [m]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Z [

m]

-3.75

-2.5

-1.25

0

1.25

2.5

3.75

5

6.25

7.5

<(E+) contours

I Electric field (equatorial plane)

-5

0

5

10

E+ [a

.u.]

2.4 2.8 3.2 3.6R [m]

-6-4-20246

E// [x

1000

a.u

.]

Real

Imag.

Real

Imag.

I Artificial damping of the IBWI Eases the convergenceI Does not affect the power

split (for these parameters)

AssociationEuratom-CEA He-3 mode conversion heating

I He-3 minority heatingI R0 = 2.96m, a0 = 0.9m,

B0 = 3.6TI ne0 = 3.5× 1019m−3

I Helium-3 in DeuteriumI nHe−3/ne between 2% to 30%I Te0 = TD0 = TH,b0 = 5keV

I Strap antennaI fFW = 37MHzI Ntor = 26 (dipole phasing)

2.0 2.5 3.0 3.5 4.0

R [m]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Y [

m]

-9.5

-7.5

-5.5

-3.5

-1.5

0.5

2.5

4.5

6.5

8.5

<(E−) contours

AssociationEuratom-CEA He-3 mode conversion heating

4% Helium-3

2.0 2.5 3.0 3.5 4.0

R [m]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Y [

m]

-9

-6.5

-4

-1.5

1

3.5

6

8.5

11

20% Helium-3

2.0 2.5 3.0 3.5 4.0

R [m]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Y [

m]

-21

-15

-9

-3

3

9

15

21

27

33

IBW on the HFS, damped by ELD

AssociationEuratom-CEA He-3 mode conversion heating

Antenna loading

0 0.05 0.1 0.15 0.2 0.25 0.3nHe-3/ne

0

0.2

0.4

0.6

0.8

1

Load

ing

resi

stan

ce [Ω

]

Electron power deposition

0 0.5 1

Normalized radius

0

20

40

60

80

p abs [k

W/m

3 ]

nHe-3/ne=2%

nHe-3/ne=10%

nHe-3/ne=20%

Power / species

0 0.05 0.1 0.15 0.2 0.25 0.3nHe-3/ne [%]

0

20

40

60

80

100

Sin

gle-

pass

abs

orpt

ion

[%]

ElectronsDeuteriumHelium-3

I Increasing He3 concentration:I Minority → electron heatingI Poorer absorption for

nHe−3/ne & 15%

AssociationEuratom-CEA Kinetic effects: FWEH in JET

I Fast waveI “Cold”-like propagationI Same branch as the compressional

Alfven wave

I Fast Wave Electron HeatingI Electron absorption by

ELD + TTMPI Damping is highly sensitive to βe

I FWEH scenario in JETI 50% Hydrogen - 50% DeuteriumI B0 = 1.34T, ne0 = 2.5× 1019m−3

I Wave frequency: fFW = 48MHzI Parallel wavenumber:

k‖,ant ≈ 8m−1

I H-2 cyclotron layer on the HFS

2 2.5 3 3.5 4R [m]

-1.5

-1

-0.5

0

0.5

1

1.5

Z [

m]

H-2 Re(B//)

-2.00

-1.00

0.00

1.00

2.00

AssociationEuratom-CEA FWEH in JET: parasitic damping

Power deposition profiles

0 0.2 0.4 0.6 0.8 1r/a0

0

1

2

3

4

Pow

er d

ensi

ty [a

.u.]

0 0.2 0.4 0.6 0.8 10

1

2

3TH(0) = 4keVTH(0) = 30keV

Electrons

Hydrogen

Power split / species

0 10 20 30 40 50 60 70TH(0)

20

30

40

50

60

70

80

Abso

rbed

frac

tion

[%] Hydrogen

Electrons

I Electron vs ion dampingI Increase of ion damping with

T⊥,HI Bootstrapping processI Self-consistent wave + kinetic

simulation required

AssociationEuratom-CEA 3D effects: antenna

∆z

∆w

Toroidal direction

1I

I0

I ICRF antennas are comprised ofmetallic straps

I Tore Supra: 2 straps / antennaITER: 4 straps / antenna

I Oscillating current flowing in strapswith relative phase shifts

I I0 ∝ exp(i(kz0 − ωt)

)I I1 ∝ exp

(i(k(z0 + ∆z)− ωt + ϕ1)

)I . . .

I Antenna current is decomposed intoroidal harmonics

jant(φ) =∑

n

jant,neinφ

AssociationEuratom-CEA 3D effects: antenna phasing

+−

Tore Supra

2−straps ICRF antenna-200 -100 0 100 200

n (Toroidal number)

0.0

2.0

4.0

6.0

8.0

|σn|2 (

Sp

ectr

al in

t.)

Dipolephasing

-200 -100 0 100 200n (Toroidal number)

0.0

2.0

4.0

6.0

8.0

10.0

|σn|2 (

Sp

ectr

al in

t.)

Monopolephasing

-200 -100 0 100 200n (Toroidal number)

0.0

2.0

4.0

6.0

8.0

10.0

|σn|2 (

Sp

ectr

al in

t.)

"π/2"phasing

Adjustable phase shifts yield flexible antenna phasing

AssociationEuratom-CEA 3D reconstruction: dipole phasing

AssociationEuratom-CEA Antenna spectrum effects

-200 -100 0 100 200n (Toroidal number)

0.0

2.0

4.0

6.0

8.0

|σn|2 (

Sp

ectr

al in

t.)

Dipolephasing

-200 -100 0 100 200n (Toroidal number)

0.0

2.0

4.0

6.0

8.0

10.0

|σn|2 (

Sp

ectr

al in

t.)

"π/2"phasing

Antenna phasing determines the 3D field structure

AssociationEuratom-CEA Conclusions - Prospects

I Magnetic fusion plasmas feature low frequencyelectromagnetic waves

I ICRF waves are used for plasma heating / current driveI Alfven eigenmodes develop and impact plasma performance

I A variational approach is adapted to simulate thewave-field and plasma kinetic response self-consistently

I EVE-3D is a full-wave code built on a Hamiltonianformalism

I Describes ICRF scenarios and linear MHD wavesI Extensively benchmarked vs analytical models and other codes

I Ongoing and foreseen developements includeI Full integration to ITM frameworkI Kinetic response module (based on the same formalism)I Coupling of toroidal modes (non-axisymmetric plasmas)