G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 1

Source Regulation of Fast EnergeticParticle Driven Alfvén Modes

Dynamics

G. Vlad, S. Briguglio, G. Fogaccia, F. ZoncaAssociazione Euratom-ENEA sulla Fusione, C.R. Frascati - C.P. 65 - I-

00044 - Frascati, Rome, Italy

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 2

Introduction• Next generation Tokamaks (e.g., ITER), should approach the so called

ignition condition: heating due to fusion α particles able to sustain theburning plasma

• Good confinement of the α particles is crucial in getting such condition

• Interaction of such modes with energetic-particle can induce outward α-particle transport

Fusion α (“Hot”) particles aregenerated withvΗ ≈ vA=B/(4πnimi)1/2

and peaked density profile

free-energy source for theresonant destabilization ofshear Alfvén modes (TAE,EPM, …)

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 3

Introduction-cont.• Transport codes aimed to define burning-plasma scenarios do not include

the possibility of Alfvén mode - α-particle interactions• If α-particle pressure profile in presence of fully saturated modes is:

close to initial one scenario is consistent

very different from initialone

scenario could be inconsistent

• Particle simulations presented at the IAEA 2004 Conference (IT/P3-31)have shown that Reversed-Shear SC4 ITER scenario could beinconsistent with the α-particle collective dynamics

• In that paper the stability of a given scenario was analyzed assumingthe reference equilibrium: no build up of the α-particle profile and noselfconsistent modifications induced by Alfvén mode dynamicsconsidered

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 4

Introduction-cont-2.• A proper multiple-time-scale approach would describe the evolution of a

succession of quasi-equilibria characterized by increasing values of the αpressure gradient

• The fast Alfvén mode dynamics affecting each quasi-equilibrium coulddistort the α distribution function in such a way to modify the next quasi-equilibrium dynamics

• The final scenario configuration, reached in this way, could then differfrom that obtained previously, possibly with milder effects on the αredistribution and transport

• In this paper we try to address this point, by simulating the building up ofthe βH radial profile (with βH being the ratio between α and magneticpressures)

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 5

The Model• Hybrid MHD-Gyrokinetic Code (HMGC):

– thermal plasma described by zero pressure reduced O(ε3)Magnetohydrodynamic (MHD) equations (circular shifted magneticflux surfaces);

– energetic particles described by nonlinear guiding-center Vlasovequation (k⊥ρΗ<<1) solved by particle-in-cell (PIC) techniques;

– energetic particles are loaded according to an isotropic slowing-downdistribution function, with birth energy Efus=3.52 MeV and criticalenergy Ecrit≈33.0 Te(r) (Stix);

– nD=nT=ni/2 and ni=ne assumed;– energetic particles and thermal plasma are coupled through the α−

particle pressure tensor, which enters the MHD momentum equation;– mode-mode coupling neglected (only particle nonlinearities retained).

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 6

The Model-cont.• A source term is added to the Vlasov equation (dF/dt = S)

such to yield, in the absence of nonlinear effects, F(t,Z)=ν(t)Feq(Z), with Feq(Z) being the equilibrium distribution functionrelated to the initial conditions: S(t,Z)=(dν/dt) Feq(Z)

• For these source simulations, we assume:– ν(t) = 1 0 ≤ t ≤ ton

– ν(t) = 1+ (βH0fin-βH0

in )/βH0in (t-ton)/(toff- ton) ton < t < toff

– ν(t) = βH0fin/βH0

in toff ≤ t• βH0

in: initial value of on-axis βH0

• βH0fin: final (nominal) value of on-axis βH0

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 7

ITER scenario considered• Reversed shear scenario (“scenario 4”, SC4, from ITER Joint Work

Site): steady-state, 9 MA, weak-negative shear, with about 300 MWfusion power and Q=5; qmin ≈ 2.4 and rqmin/a ≈ 0.68.

• Simulations are performed retaining on-axis equilibrium magnetic field,major and minor radii, the safety-factor q, the plasma density ne, theelectron temperature Te and the α-particle density nH.

SC4

0.50rα−H,max/a

0.600αH,max=max{-R0q2β´H}

0.92βH0 (%)0.62nH0(1018m-3)23.9Te0(keV)0.73ne0(1020m-3)5.183BT(T)5.13q95

6.34R0(m)1.859a(m)

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 8

No-ramp simulations: linear ...• Toroidal mode number n=2 is found to

be the most unstable.• The mode is located close to rqmin/a

radius, and close to the tip of the lowerAlfvén continuum in frequency.

0

0.05

0.1

0.15

0.2

0 0.5 1 1.5 2 2.5 3

!/"A0

nH0

/nH0, SC4

n=2

n=4

n=6

... and nonlinear dynamics

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 9

Ramp simulations• n=2 Reversed-shear SC4 ITER scenario• Neglecting fluid-fluid nonlinearities causes unstable tearing

modes to be unsaturated: limitation of the time scale we areable to simulate

• We need to extrapolate results related on real (very long)transport time scale from much shorter time scale numericalexperiments

• First numerical experiment:short βH radial profile ramp (toff- ton=100τA), fromβH0=0.0093 (reference SC4 scenario) to βH0=0.0109, withdifferent values of ton

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 10

Ramp from βH0=0.0093 → 0.0109toff- ton=100τA, ton=120τA, 200τA...

βH0(t)/[ν(t)βH0eq] vs t/τA

βH0(t) vs t/τA no-ramp,βH0=0.0109

no-ramp,βH0=0.0093(nominal SC4 scenario)

βΗ0 normalized to thenominal value of βΗ0 inabsence of Alfvén dynamics

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 11

Regulation of the pressure broadening isobtained if the system has time enough tofully develop the saturation dynamics(ton>1000 τA).

Ramp from βH0=0.0093 → 0.0109toff- ton=100τA, ton=120τA, 200τA...

!

d" H(t) = 1

a2

"H (r,t)

#(t)"H

eq(0)

$"H

eq(r)

"H

eq(0)

0

a

% rdr :

!

d "H

= limt#$

d"H

(t)}{ :

no-ramp,βH0=0.0109

no-ramp,βH0=0.0093(nominal SC4scenario)

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 12

Fast after-saturation ramps• Hypothesis: the suppression of the α-pressure profile broadening has to

be ascribed to the capability of the original - lower-βH0 - nonlineardynamics in modifying the α distribution function in such a way toreduce the reactivity of the system to the α-pressure increase

• Consequence: we expect this suppression is less effective if the startingconfiguration is characterized by a lower energy content (and, hence, aweaker nonlinear dynamics)

• Second numerical experiment: fixed increment βH0fin-βH0

in≈0.0011, fixedtoff- ton=100τA, various βH0

in values, large ton (after saturation)• Evaluate the suppression efficiency by measuring βH0

equiv, defined as thevalue of βH0|t=0 that would yield the same distortion of the α pressureprofile in a no-ramp simulation

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 13

βH0(t)/[ν(t)βH0eq] vs t/τA

βH0fin-βH0

in≈0.0011, fixed toff- ton=100τA,various βH0

in, large ton (after saturation)

βH0=0.00544

no-ramp cases(black, ■):

βH0=0.00653

βH0=0.00761

βH0=0.00870

A regulating effect of the nonlinear Alfvén dynamics yields, for a givensimulation, βH0

equiv < βH0fin: regulation is stronger for larger βH0

in

ramp cases (colours,●):

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 14

Long ramps, starting at t=0• Transition from non-suppressing to fully-suppressing Alfvén mode

dynamics with respect to further α redistributions suggests that, in arealistic (very slow) approach to the scenario α pressure profile, anadiabatic regulation, able to prevent ab initio any profile broadening,should not be observed

• We indeed expect that a certain broadening level will be reached, whilethe pressure content increases

• The broadening process will then slow down, as the Alfvén wavesuppressing dynamics becomes more effective; eventually, such aprocess will completely stop, yielding an upper limit in the overallnonlinear profile deformation

• Further increasing the α pressure should not cause a worsening of itsradial profile stability, unless additional nonlinear dynamics sets in

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 15

Long ramps, ton=0, fixed toff-ton=2000τA,different βH0

in, βH0fin

no-ramp cases(black, ■):

βH0=0.00761

βH0=0.00870

• A βH0equiv saturation clearly emerges at βH0≈0.0085, consistently with the

results of the previous numerical experiments

ramp cases (colours,●):βH0(t) vs t/τA

βH0=0.00979

βH0=0.01088

βH0in=0.00761

G. Vlad et al., P2.107 32nd EPS Plasma Physycs Conference. 27 June - 1 July 2005 Tarragona (Spain) 16

Conclusions• PIC simulations of energetic particle driven Alfvén modes, assuming the ITER

Reversed Shear scenario (SC4), has been found:– to be unstable;– to generate appreciable changes to the α-particle radial profile (possibly inconsistent

scenario).• PIC simulations of energetic particle driven Alfvén modes including a model building

up of the βH radial profile has been presented• A transition from non-suppressing to fully-suppressing Alfvén mode dynamics with

respect to further α redistributions has been observed• Alfvén wave suppressing dynamics becomes more and more effective as the pressure

content increases; the broadening process slows down, as the Alfvén wave suppressingdynamics becomes more effective yielding an upper limit in the overall nonlinearprofile deformation

• Further increase of the α pressure should not cause a worsening of its radial profilestability, unless additional nonlinear dynamics sets in