T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

Integrated Modelling of ICRH and AE Dynamics

T. Hellsten, T. Bergkvist, T. Johnson and M. Laxåback

Alfvén Laboratory, Royal Inst. of Technology,

SE-100 44 Stockholm, Sweden Association Euratom-VR.

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

ICRH is a versatile heating method that can provide:

Heating

Enhance fusion reactivity

Drive Currents

Induce rotation

Excite AEs

ICRH

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

ICRH requires self-consistent modelling of distribution functions and wave field; including effects of finite orbit width and RF-induced spatial transport of fast ions for waves with finite n.

Due to the different time scales this can be done by iterations.

ICRH Modelling

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

1 LION code L. Villard et al, Computer Physics Reports 4(1986)95 and Nucl. Fusion, 35(1995)1173

Define equilibrium, antenna spectrum, power.

Calculate the dielectric tensor and wave field for ICRH (LION code1) from the output of FIDO

Calculate changes in orbit invariants by collisions, and ICRH with the FIDO code. Remove lost ions, add NBI, -particles and edge source.

Create tables for the various interactions used in the Monte Carlo code with an orbit solver.

Output

The SELFO code calculates the ICRH wave field with the LION code and the distribution function in the invariant space (W, P, ) with the Monte Carlo code FIDO.

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

-90o phasing: trapped 3He ions displaced outwards. emission from turning points of trapped ions at

cyclotron resonance

+90o ICRH phasing:

trapped 3He orbits pinched,

then detrapped to co-current wide passing orbits at the low field side of the center

RF-induced pinch and detrapping of the orbits

T. Johnsson et al, IAEA Technical Meeting, Gothenburgh, 2001

Tomographic reconstruction of the -emission profiles from JET

Tomegraphic reconstruction by C. Ingesson

T. Hellsten et al Phys. Rev. Lett 1995

CoCounter

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

Comparison of the gamma emissitivity in the mid-plane z=0 between tomographic reconstructions (full line) dashed region (confidence interval) and the density of high-energy 3He ions calculated with the SELFO code (boxes)

+90-phasinglocation of the excited TAE modes indicated

-90-phasing

SELFO code modelling by T. Johnson

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

The excitation of Alfvén eigenmodes is sensitive to the details of the distribution function.

AEs excited in JET during ICRH with +90° and -90° phasing of the antennas

+90° -90° L.-G. Eriksson, et al Phys Rev. Lett 81 (1998) 1231M. Mantsinen et al Phys. Rev. Lett. 84(2002).

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

Splitting of the mode frequency

A. Fasoli et al Phys. Rev. Lett. 81(1998)5564

Fast damping when ICRH is switched off

The AEs are damped in a time period of about 0.1ms after the ICRH is switched off.

K. L. Wong,et al Phys. Plasmas 4 (1997) 393

Typical mode splitting of about 2kHz is seen during ICRH.The spitting is too wide to be due to restoration of the distribution function by Coulomb collisions.

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

Ion interaction with AEs

In the absence of Coulomb collisions and ICRH the interactions of a resonant ion with an AE lead to a superadiabatic oscillation in the phase space of the invariants of the equation of motion for the drift orbit along the AE characteristics

ΔPφ =nω

ΔW

Δμ=0

If the distribution function increases with energy around the resonance, energy will then be transferred from the ions to the mode and vice verse.

When the distribution function is flattened along all AE characteristics no net transfer of energy takes place. The mode will then be damped by different background damping mechanisms.

P

W

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

Decorrelation of the interactions leads to a diffusion of the orbits along the characteristics instead of a superadiabatic oscillation.

Ion cyclotron interactions and Coulomb collisions will partially restore the distribution function in the resonant regions and result in further transfer of energy from the resonant ions to AEs.

The decorrelation of the interactions and local renewal of the distribution function by ICRH increases with energy, whereas they decrease with energy for Coulomb collisions.

Decorrelation of AE interactions and renewal of the distribution function

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

Renewal of the distribution function by ICRH

Distribution function f(w) along a characteristic

w

AE resonance

initial distribution function

distribution function flattened by an AE

The width of the resonance and the renewal rate increase with ICRH power

ICRH creates an inverted distribution function along the AE characteristics

High energy ions created by ICRH

Low energy ions removed by ICRH

The dynamics of the AE excitation depend not only on the growth rate of the AE and background damping, but also of the renewal rate of the distribution function and the decorrelation of the wave particle interactions.

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

1 LION code L. Villard et al, Computer Physics Reports 4(1986)95 and Nucl. Fusion, 35(1995)1173

Define equilibrium, antenna spectrum, power, type of AE mode etc.

Calculate the dielectric tensor and wave field for ICRH (LION code1) and amplitude of AEs

Calculate changes in orbit invariants by collisions, ICRH and AE with the FIDO code. Remove lost ions, add NBI, -particles and edge source.

Create tables for the various interactions used in the Monte Carlo code with an orbit solver.

Output

The SELFO code calculates the distribution function in the invariant space (W, P, ) with the Monte Carlo code FIDO and the ICRH wave field with the LION code. The AE field can either be calculated with the LION code or from a simplified model.

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

Monte Carlo code FIDO for calculating the distribution function

W(t+Δt)

Λ(t+Δt)

Pφ(t+Δt)

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

=

W(t)

Λ(t)

Pφ(t)

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ +

μWC

μΛC

μPφC

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ Δt+

AWWC 0 0

0 AΛΛC 0

APφWC APφΛ

C APφPφC

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

ζWC

ζΛC

ζPφC

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Δt+ωi

∑μWIC

μΛIC

μPφIC

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ Δt+ζW

IC

AWWIC

AΛWIC

APφWIC

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Δt

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

nφ

∑ +

Coulombcollisions ioncyclotroninteractions

+

μWMHD

μΛMHD

μPφMHD

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ Δt+ζW

MHD

AWWMHD

AΛWMHD

APφWMHD

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Δt

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

+.................+.............nφ ,m∑

J. Carlson et al, “Theory of Fusion Plasmas” Varenna 1996, L.-G. Eriksson and P. Helander Phys. Plasmas (1994), T. Bergkvist et al “Theory of Fusion Plasmas” Varenna 2004.

MHD interactions -particle lower ripple sawteething channelling hybrid diffusion

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

TAE Resonance regions

Amplitude variations of the variance of the energy for interactions with an TAE mode. Note that internal zeros of the variance appear.

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

The dynamics of TAE and frequency splitting

Fourier decomposition of the time evolution of the mode amplitude gives a characteristic frequency corresponding to the frequency separation of the side bands seen during ICRH.

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

Mode damping after ICRH switch off

The fast damping of the TAE of about 0.1ms as ICRH is switched off is consistent with experiments.

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

Conclusions

Self-consistent computations of wave field and distribution function are important for ICRH, in particular for power partition.

The effects of finite orbit width and RF-induced spatial transport are important for many phenomena.

The dynamics of the AEs are strongly affected by ICRH, which have to be taken into account when simulating AE excitation by thermonuclear alpha particles using ICRH ions.

The decorrelation by ICRH increases the width of the resonances and the renewal rate, making the interactions with AE much stronger.

T. Hellsten IEA Burning Plasma Workshop, July 2005 Tarragona Spain

Code for self-consistent modelling of heating

AE

NBI ICRH LH ECRH

Wave field

Power depositionFast ion Fusion reactionsCurrent profileMomentum

Power depositionCurrent profile

Distribution function for electrons f(,W,)3D-Finite element

Source

Ray tracing Ray tracing

Wave spectrum

Wave spectrum

Wave field

Distribution function for ions f(W,P)Monte Carlo method3D-Finite elementMHD

SawteethFishbones

Equi-librium,Loop voltage