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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)