Nonlinear saturation of Shear Alfvén Modes and energetic ion transports in Tokamak equilibria with hollow-q profiles G. Vlad, S. Briguglio, F. Zonca, G. Fogaccia Associazione Euratom-ENEA sulla Fusione, C.R. Frascati, Rome, Italy III Convegno Nazionale su La Fisica del Plasma in Italia L Aquila, 20-22 Maggio 2002 1
1 Introduction Shear-Alfvén waves in tokamaks Energetic (hot) ions in plasmas close to ignition conditions have vh va = B/ 4πnimi. They can resonate with shear-alfvén waves and drive them unstable. Alfvénic turbulence can, in turn, enhance energetic-ion losses. Discrete MHD modes (Toroidal Alfvén Eigenmodes, or TAE s) can exist in the gaps of the shear-alfvén frequency spectrum and can be easily driven unstable by the resonance with energetic particles. For large resonant drive (larger than the continuum damping), even continuum oscillations can be destabilized: a new unstable mode appears, which would not exist in the absence of the energetic particles: the Energetic-Particle (continuum) Mode (EPM). The nonperturbative character of EPM s and the interest in the nonlinear evolution of both TAE and EPM (effect on α-particle confinement) require particle simulation, able to fully retain kinetic effects in a self-consistent way. 2
2 Reversed Shear in Tokamaks Tokamak equilibria with hollow q profiles are relevant because they allow to obtain improved plasma confinement regimes. There are at least two consequences of such equilibria on the Shear Alfvén mode dynamics: 1. The formation of steep thermal-plasma temperature profiles causes, in plasmas close to ignition conditions, much steeper α-particle radial distribution. This means a high value of the gradient of energetic ( hot ) ion pressure, β H ; that is a large source of instability for the Shear Alfvén modes. 2. The shape of the q profile affects: both the toroidal gap and the resonant-excitation frequencies; the size of the energetic-ion orbits; the radial extension and the toroidal coupling of different poloidal harmonics; the magnitude of the energetic-ion drive (α = R0q 2 β H). Experimental evidence on JET, during the current ramp-up phase of Optimized Shear discharges characterized by hollow q profiles. Evidence of a variety of fast ion driven Alfvénic modes (Alfvén cascades (H. Berk)) with frequency increasing in time (chirping). 3
Frequency chirping interpreted, on the basis of a linear treatment (H. Berk, D. Borba), as the effect of equilibrium variations on Alfvén gap modes, localized, radially, near the minimum-q (qmin) magnetic surface and, in frequency, just above the local maximum in the shear Alfvén continuous spectrum. Here, we address mainly the second of these elements, discussing the results of particle simulations performed by the Hybrid MHD-Gyrokinetic Code (HMGC). We investigate the saturation mechanisms and energetic particle transports (nonlinear dynamics) of fast ion driven Alfvénic modes. The code solves reduced O(ɛ 3 ) MHD equations for a low-β core plasma and Vlasov equation for energetic ions. These equations are coupled together by an energetic-ion pressure term in the MHD eqs. HMGC uses several approximations: large aspect ratio expansion; only shifted circular magnetic surfaces; isotropic Maxwellian distribution for energetic ions. Much cautiousness should be adopted in comparing the simulation results with the experimental (e.g., JET) ones. 4
3 HMGC simulation In all the simulations presented here, we will consider the same shape for the βh profile. We will instead consider two different hollow q profiles: a deeply hollow profile (a) and a moderately hollow one (b). We will also consider a ordinary (monotonic) q profile for comparison (c). 1 6 β H /β H (0) q 0.8 5 0.6 b) 4 β a) q H 3 0.4 0.2 2 c) 1 0 0 0 0.2 0.4 0.6 0.8 1 r/a 5
Two simplifications we adopt are the following: radially constant energetic-ion temperature is assumed (the βh profile is only due to the energetic-ion density dependence); the nonlinear mode-mode coupling among different-n modes is neglected (single n dynamics; in this case n = 4) In the following simulations we assume: ɛ a/r0 = 0.1 (the inverse aspect ratio); ρh/a = 0.01 (normalized energetic ion Larmor radius); vh/va,r=0 = 1 (energetic ion thermal speed normalized to the Alfvén velocity, on-axis value); both energetic and thermal ion species are assumed to be deuterium. 6
3.1 Nonlinear saturation process Results presented in the form of movies. A typical frame: Energy rnh φm,n(r, t) vs r φ(x, t) ϕ=0 m φm,n(r, ω) 2 φm,n(r, t) vs r, m 7 Several quantities represented: the energy content of the different poloidal harmonics; the contour plot of the scalar potential in the poloidal plane, at a fixed toroidal angle; the energetic-ion line density, rnh(r), versus r; the contour plot of the power spectrum m φm,n(r, ω) 2 in the (r, ω) plane, with the toroidicity induced gap also shown; the amplitude of the different poloidal harmonics versus radius; the contour plot of the same quantity in the (r, m) space.
3.2 Deeply hollow q profile Deeply hollow q profile (q(0) 5, qmin = 2.1, q(a) 5, profile (a)), βh(0) = 2.5%. ωgap/ωa,r=0 = 1/(2q(r) ρ/ρr=0): assume first a radially constant thermal-plasma density ρ radially constant Alfvén velocity (such an assumption will be removed later). see movie: http://fusfis.frascati.enea.it/ vlad/miscellanea/iaea-goteborg/n4 JET 7.mov After a transient initial phase, a mode localized around the maximum β H emerges at r 0.35a, with frequency well inside the continuum. We can identify this mode as an Energetic Particle continuum Mode (EPM). Its saturation takes places because of a strong (convective) outward radial displacement of the energetic ions. As such a displacement takes place, the local drive is reduced due to the flattening of the energetic-ion density profile. The drive is no longer able to overcome the strong continuum damping at the original frequency. The maximum of the power spectrum migrates towards the gap (in order to minimize the continuum damping), but it also moves outwards, following the displaced source, in order to maximize the drive. The mode reaches the gap and it localizes around the zero-shear, qmin surface (r 0.5a). 8
A strong (convective) outward radial displacement of the energetic ions is observed at saturation(see rγh(r), the radial energetic-ion flux times r plot). The qmin surface, for this case, encloses a good confinement region (see the space-time line density rnh plot). 9
3.3 Good confinement region for the energetic ions The good confinement region observed in the previous case could be ascribed: 1. to the stability of the outer region plasma with respect to Alfvénic perturbations, or 2. to the lack of efficiency of the gap mode in displacing a strong enough free energy source into that region. The former reason (1) is associated with the strength of continuum damping and the radial width of the frequency gap; the latter (2) depends on the effectiveness of wave-particle interactions, and it is related with the amplitude and the radial width of the mode and with the size of the particle orbits. Lets try to discriminate which is the most relevant among these two factors for the considered hollow q-profile equilibria. 10
3.4 Consider first the effect of continuum damping A more realistic thermal-plasma density profile (decreasing as r a) has been introduced in the present case, making the frequency gap wider in the outer region. see movie: http://fusfis.frascati.enea.it/ vlad/miscellanea/iaea-goteborg/n4 JET 18.mov The outer region now is affected by weaker continuum damping: modes with frequency close to the shear Alfvén accumulation point of the upper continuum are observed, corresponding to the upper branch of Kinetic Toroidal Alfvén modes, which is less damped than the lower one. In spite of this fact, the good confinement of the energetic ion population is not lost. We can conclude that, at least for the case examined here, the energetic-particle transport is mainly inhibited by the mode-particle interaction at the qmin surface. 11
3.5 Consider the effect of higher βh gradient The interaction between the gap mode and the energetic particles can be trivially enhanced by increasing the initial value of the βh gradient (i.e., the source of instability): βh = 5%. see movie: http://fusfis.frascati.enea.it/ vlad/miscellanea/iaea-goteborg/n4 JET 19.mov For such a high value of energetic-particle drive, the linear growth-rate of the EPM is even larger than the mode frequency itself, resulting in radially broader mode structures. The mode amplitude reached at saturation is larger than that of the lower βh case shown before. Both effects yield a larger energetic-ion displacement outer poloidal harmonics driven unstable, with frequency close to the upper continuum. Transport of energetic ions increases in the outer region, although their global confinement is not substantially degraded. 12
3.6 Consider the effect of the safety factor profile: moderately hollow q profile A much more dramatic effect can be obtained by acting on the q profile: consider the moderately hollow q-profile (q(0) 5, qmin = 3.6, q(a) 5, profile (b)), βh(0) = 2.5%. see movie: http://fusfis.frascati.enea.it/ vlad/miscellanea/iaea-goteborg/n4 JET 11.mov The mode radial width scales, near the qmin surface, as 1/ nq. The typical orbit size is proportional to qmin. Decreasing the hollowness of the q profile, while taking q(0) and q(a) fixed, yields lower q and larger qmin values both the mode and the orbit widths becomes larger than in the deeply-hollow q-profile case. Moreover, the energetic-ion drive intensity, α, scales as q min. 2 The mode is then a more efficient scattering source for energetic-ion orbits even in the relatively low βh case. 13
Previous considerations and fair alignment of the frequency gap at different radial positions make the displaced energetic ion source effective in destabilizing an avalanche of outer poloidal harmonics. The confinement of the energetic particle population is significantly degraded. Space-time contour plot of the energetic ion line density is plotted in the (r/a, τ) plane for the three low-βh cases discussed above (βh = 2.5%). The degradation of energetic ion transport at the qmin surface is apparent in the moderately hollow q-profile case (right). deeply hollow q profile, flat thermal-plasma density deeply hollow q profile, decreasing thermal-plasma density moderately hollow q profile, flat thermal-plasma density 14
3.7 Comparison with monotonic-q equilibrium Consider a monotonic-q profile (q(0) 1.1, q(a) 4.5, profile (c)), with βh(0) = 5%. see movie: http://fusfis.frascati.enea.it/ vlad/miscellanea/iaea-goteborg/n4 JET 16.mov A strong redistribution of the energetic ions is observed, which, nevertheless, does not reach the plasma periphery. By comparing the previous simulations (reversed shear profiles (a) and (b)) with the monotonic-q equilibrium (profile (c)) we observe that even at the higher value of βh considered the monotonic-q equilibrium prevents the convective flux to reach the outermost magnetic flux surfaces. 15
3.8 Conclusions The saturation of EPMs presents a rich phenomenology. After the initial destabilization of an EPM within the qmin surface, a gap mode generally survives to the strong, EPM-induced, radial redistribution of the energetic ions. If the particle orbits are poorly affected by the interaction with the qmin gap mode and the gap is quite narrow in the outer region, both the Alfvénic coherent eddies and the energetic-ion gradient cannot propagate beyond a certain magnetic surface, within which the energetic ions are well confined, giving a barrier in the sense of transport. On the contrary, in the presence of an effective mode-particle interaction at the qmin surface, a significant fraction of the energetic ion population is displaced outside this surface. If the gap structure is sufficiently open to preserve outer modes from strong continuum damping, the displaced instability source can induce a strong excitation of outer poloidal harmonics and degradation of the energetic-ion confinement. Control parameters: β H (the obvious one); q, which reflects on the local mode width ( 1/ nq ); qmin itself, as the typical orbit size is proportional to q and the energetic-ion drive intensity scales as q 2. 16
4 Nonlinear excitation of zonal flow by EPM dynamics n = 8 EPM simulations: monotonic-q profile with q0 = 1.1, qa = 1.9 Npart 16.7 10 6 see movie: http://fusfis.frascati.enea.it/ vlad/miscellanea/epm MOVIES/n8 9 imirr1 13 zonal 3x4.mov Radial fragmentation of the EPM coherent eddies (kθ = k = 0, kr 0) is present (see, e.g., φ(x, t) ϕ=0 and φm,n(r, t) ). The fragmentation is associated with a diffusive transport of fast ions. Strong modification of the fast ion line density profile (particle radial convection) (see frame at t = 75R0/vA). Radial fragmentation of the EPM coherent eddies (kθ = k = 0, kr 0) induced by nonlinear EPM evolution (see frame at t = 75R0/vA). Evidence of spontaneous excitation of zonal flows. 17
Compute the non-self-consistent zonal flow; let x r/a and τ vat/r: eh τ x TH m δφzonal = i 1.6q 2 R 0 a 1/2 v H va ( ρ H a ) x 1/2 m e H 2 δφm 2,x x TH δφ m v2 A c δa,m 2 2,x δa,m c.c. Computing the decorrelation time (Hahm-Burrel) γe (r/q) r[(q/r)(c/b) rδφzonal] or, using current normalizations, γeτa R 0 a ρh vh rφzonal 2 a va See: dφzonal/dr vs. r and its power spectrum, and nonlinear E B shearing rate γe normalized to Alfvén time τa and its power spectrum in the movie. 18
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