Alfvén Acoustic Channel for Ion Energy in High-Beta Tokamak Plasmas Revisited with a Linear Gyrokinetic Model (LIGKA)

1 I-10

Alfv´en Acoustic Channel for Ion Energy in High-Beta Tokamak Plasmas Revisited with a Linear Gyrokinetic Model (LIGKA)

A. Bierwage1, P. Lauber2, N. Aiba1, K. Shinohara3 and M. Yagi1

1Japan Atomic Energy Agency, Rokkasho, Aomori, Japan 2Max Planck Institute for Plasma Physics, EURATOM Association, Garching, Germany 3Japan Atomic Energy Agency, Naka, Ibaraki, Japan

Corresponding Author: [email protected]

Abstract: A recently proposed Alfvén acoustic self-heating channel for burning tokamak plasmas — where fast-ion-driven shear Alfvén waves transfer energy to sound waves which then heat the bulk ions through Landau damping — is reexamined using a linear gyrokinetic model (LIGKA code). An N-NB-driven JT-60U plasma with high toroidal beta (3.6% on axis) is used as a test case. The structure of the sound and Alfvén continua is studied with a local eigenvalue analysis. For the default case with Te 1.7 Ti, sound waves are strongly damped with γ/ω > 0.5, and no evidence for a coupling≈ to Alfvén× waves is found above the − compressibility-induced gap (ω>ωBAE). For Te 4.8 Ti, low-frequency MHD continua in the range of beta-induced Alfvén acoustic eigenmodes≈ × (BAAE) are recovered with damping as low as γ/ω 0.15. Changes in the Alfvén continua with ω>ωBAE are also seen. This suggests− that∼ Alfvén acoustic couplings and the associated self-heating channels in the frequency range ω>ωBAE may exist when Te Ti. The role of “hot sound” branches that are found in the presence of beam ions remains≫ to be clarified.

1 Introduction and review of MHD results

The heating of bulk plasma ions by fusion products, namely energetic alpha particles, is important for maintaining nuclear fusion conditions in burning plasmas. The existence of a new self-heating channel for burning tokamak plasmas as illustrated in Fig. 1(a) was recently proposed [1]. This idea was based on the discovery of couplings between MHD sound waves and shear Alfvén waves that occur in a high-beta tokamak plasma (with toroidal beta β0 & 1% on axis), where the continuous spectra of long-wavelength ion sound waves enter the frequency band occupied by shear Alfvén waves. Under such conditions, discrete MHD modes with mixed Alfvén acoustic polarization can form, which were called beta-induced Alfvén continuum modes (BACM). One such BACM is encircled with a dashed line in Fig. 1(b). In the MHD system, BACMs are weakly damped modes that arise when slowly decaying ion sound waves transfer energy to shear Alfvén waves. If I-10 2

(b) MHD continua and E×B spectrum for β = 3.6%, n = 3 (a) Alfven acoustic channel for bulk ion heating via BACM 0

m =0 Resonant S drive 0.5 Energetic ions Shear Alfven

(≤ 3.5 MeV) A0 m =1 waves S ω 0.4 BACM / m = 5, 6 ω A 0.3 m = 2 m =2 S Nuclear fusion Coupling (BACM) S 0.2 m =4 Frequency A m =5 Bulk ions Ion sound A 0.1 m =6 (≤ 10 keV) waves A Heating 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Minor radius r / a FIG. 1: Beta-induced Alfvén continuum modes (BACM) were proposed to provide a new energy channel between fast ions (such as 3.5 MeV alpha particles) and bulk ions via MHD waves as illustrated in panel (a). An example of a BACM obtained by simulating the MHD response of a high-beta JT-60U tokamak plasma is shown in panel (b) for toroidal mode number n =3. The spectrum of the radial E B velocity component δur (color contours), × full MHD continua (black), and shear Alfvén continua (magenta) are plotted. The BACM resulting from the coupling between the poloidal harmonics mA =5, 6 of the shear Alfvén branch with the mS =2 harmonic of the ion sound branch is encircled with a dashed line. See Ref. [1] for further details. The present paper deals with the structure of the n = 3 continuous spectra in a gyrokinetic description of the plasma for the same configuration.

one considers the energy transfer in the opposite direction, from shear Alfvén waves to ion sound waves, BACMs provide a channel to transfer energy from fast ions that resonate with shear Alfvén waves to bulk ions that absorb sound wave energy through Landau damping. Reference [1] concluded with the conjecture that such a noncollisional energy channel may complement the usual collisional self-heating channel (fast ions to electrons, and electrons to bulk ions) and have an influence on the performance of burning plasmas with respect to fusion rates and fast ion confinement. In order to evaluate the efficiency of such Alfvén acoustic energy channels, kinetic effects of the bulk plasma and, ultimately, the energy flows at finite mode amplitudes (nonlinear regime) must be considered. In the present proceedings paper, results of recent analyses performed with a linear gyrokinetic model are presented and implications for the proposed Alfvén acoustic self- heating channel are discussed. The results of the previous MHD simulations were also reported at the conference but are not repeated here since they can be found in Ref. [1].

2 Simulation model and setup

The results presented in this paper were obtained with the code LIGKA [2], which de- scribes all species (bulk electrons, bulk ions and fast ions) using a linear gyrokinetic model. The following approximations and simpliﬁcations are made: 3 I-10

4 3 3 4

] n T (a) (b) e (c) e (d) Total −3 2.5 n 2.5 T N−NB ions i i m ) β

ρ 3 3 = 19 ) [%] 0 ρ 2 2 ( 3.6% β 2 1.5 1.5 2

1 Adjusted in 1 simulations

Safety factor q( 1 1 Hybrid such that Temperature [keV] 0.5 0.5 Toroidal beta simulation n = n − n Number density [10 i e h result 0 0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Minor radial coordinate ρ = ψ1/2 ρ ρ ρ

FIG. 2: JT-60U scenario studied. The radial proﬁles of the safety factor q(ρ), the number densities n(ρ) and temperatures T (ρ) of bulk electrons and bulk ions, and the toroidal beta 2 β(ρ)=2µ0P/B0 of the entire plasma and N-NB ions are shown. In order to reveal trends via parameter scans, some of these reference proﬁles will be scaled by constant factors.

Trapped particle effects are not yet included and may alter the low-ω response [3, 4]. • Geometric coupling between sound waves with different poloidal mode numbers m • is included up to first order (geodesic). Further gaps are expected for ∆m 2. | | ≥ Finite-orbit-width (FOW) effects are not yet included. With FOW, the effect of fast • ions is expected to change, since magnetic drifts are large in the scenario considered. Isotropic Maxwellian distributions are used for all species. The effect of fast ions is • expected to change when a more realistic slowing-down distribution is used. A pure Deuterium plasma is considered. Impurities (mostly Carbon) are ignored. • The simulation scenario is based on a high-beta JT-60U tokamak plasma that was driven by a pair of tangentially injected negative-ion-based neutral beams (N-NB) as described in Ref. [5]. The radial profiles of some relevant plasma parameters are shown in Fig. 2 as a function of ρ = √ψ, where ψ is the normalized poloidal flux. The bulk ion density profile in Fig. 2(b) is adjusted as required to ensure that the quasineutrality condition is satisfied: ni = ne nh. The density nh of the fast (“hot”) − ions is computed from the reference profile of the fast ion beta βh in Fig. 2(d) as nh = 2 B βh/(2Thµ0), assuming an isotropic Maxwellian distribution exp( E/T ) with temper- 0 − ature T = Th. The fast ion beta profile in Fig. 2(d) was originally obtained with a self-consistent long-time simulation using the hybrid code MEGA [6] with sources and collisions [7], and is formed by ions in the 150–400 keV range with a realistic slowing- −1 down distribution and narrow pitch angle range 0.2 . α/π . 0.4, where α = sin (vk/v).

3 Results of local linear analysis

In this section, we present solutions of the local eigenvalue problem obtained by Nyquist contour integration around the zeros of the local dispersion relation. We focus on the I-10 4

β ≈ × (0)=1.7%,0.5 Te 1.7 Ti 0.5 0.5 (a) (b) Strongly Damped Unstable (c) damped 0.4 0.4 0.4

0.3 0.3 0.3 ω A0 A0 Compressional A0 MHD ω ω stabilization ω / / (e.g., m =5) / ω 0.2 ω 0.2 A ω 0.2

0.1 0.1 0.1 ω BAE 0 0 0 0.2 0.4 0.6 0.8 −6 −4 −2 0 2 0.2 0.4 0.6 0.8 ρ Γ = γ × τ ρ

ref ref FIG. 3: Results obtained with Ti = Ti and Te = Te in the absence of fast ions (βh =0), ref so that β0 =1.7% β /2. Panel (b) shows the eigenvalues in the complex (ω, Γ) plane. ≈ 0 The radial distribution of eigenfrequencies ω(ρ) is shown in (a) for Γ < 2 and in (c) for − Γ > 2. For comparison, the MHD continua ωMHD(ρ) are plotted as gray lines. − toroidal harmonic n = 3. For each radial location, the analysis yields a set of eigenvalues in the complex frequency plane ω+iγ for each radial location. When plotted as a function of the minor radial coordinate ρ, the sets of points ω(ρ) form the continuous spectrum of the kinetic plasma response. Negative values γ(ρ) < 0 are interpreted as the local damping which a global mode with frequency ωglob will experience at radii where ωglob = ω(ρ). In the following, the angular frequency ω and growth rate γ are normalized by the on-axis Alfv´en frequency ωA0 = vA0/R0 with vA0 = B0/√µ0nimi. Instead of γ, we plot Γ=2πγ/ω = γτ = ln Φ(t + τ)/Φ(t) , (1) | | where Φ is the electrostatic potential. For instance, when Γ . 2, the ﬂuctuation ampli- − tude is reduced to Φ(t+τ)/Φ(t) = exp(γτ) . 15% during one oscillation period τ =2π/ω, so we speak of strong damping. The Nyquist contour integration method used here is op- timized for speed, so it may miss some eigenvalues or produce artefacts, especially for Γ 0. The algorithm breaks down for Γ < 6 (i.e., γ/ω > 100%). Solutions with 6≪< Γ . 3 are only indicative, but those with− Γ & 3− are considered to be reliable. − − −

3.1 Increasing Te reduces damping of bulk ion sound branches In the default scenario based on JT-60U as shown in Fig. 2, the electron and ion temperatures are comparable: Te0 1.7 Ti0 on axis. Figure 3 shows the eigenvalues (ω, Γ) for ≈ × this case when the contribution of fast ions is excluded (βh = 0), so that the total plasma ref beta is only about half of the reference value: β0 =1.7% β0 /2 on axis. The shear Alfvén branches in Fig. 2(c) are marginally stable≈ (Γ 0) and coincide with ≈ the MHD continua; except near the low-frequency gap associated with plasma compressibility [8]. There, the damping increases as frequencies approach the accumulation points ωBAE that are known to give rise to beta-induced Alfvén eigenmodes (BAE) [9]. The gyrokinetic values of ωBAE lie below their MHD limit due to diamagnetic effects. Traces of sound branches can only be seen in the strongly damped domain shown in Fig. 3(a). 5 I-10

β ≈ × (0)=3.6%,0.5 Te 4.7 Ti 0.5 0.5 (b) Strongly Damped Unstable (c) (a) damped 0.4 0.4 0.4 Low−frequency 0.3 0.3 Alfven acoustic 0.3 A0 A0 A0

ω ω coupling (e.g., ω

/ / m =5 ↔ m =4, 6) / A S ω ω 0.2 ω 0.2 ω 0.2 BAE

0.1 0.1 0.1 ω BAAE

0 0 0 0.2 0.4 0.6 0.8 −6 −4 −2 0 2 0.2 0.4 0.6 0.8 ρ Γ = γ × τ ρ

ref FIG. 4: Results obtained with Ti = Ti and an increased electron temperature Te = ref ref 2.8 T , so that β = β even in the absence of fast ions (βh =0). Arranged as Fig. 3. × e

0.5 0.5 (a) (b) FIG. 5: Comparison be- 0.4 0.4 tween the distribution of (ω ,Γ ) 0.3 A A 0.3 complex local eigenvalues for A0 A0 ω ω

/ / Alfv´en branches (ωA, ΓA) in ω 0.2 ω 0.2 ref the cases with (a) Te = Te 0.1 0.1 ref ω and (b) Te =2.8 Te . Pan- BAE Noise? × 0 0 els (a) and (b) are enlarge- −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 Γ = γ × τ Γ = γ × τ ments of Figs. 3(b) and 4(b).

The original value of the plasma beta can be recovered if we increase the bulk ion ref ref temperature by a factor 4.1: Ti = 4.1 T β0 = β =3.6% on axis. If this is done × i ⇒ 0 (not shown here), the values of Γ for the bulk ion sound branches increase but still lie below 2. At the same time, shear Alfvén continua in the BAE frequency range ω & ωBAE are destabilized− by the increased ion temperature gradient (ITG). These modes are known as Alfvénic ITG (AITG) instabilities [11]. Another way to recover the original value of the plasma beta without including fast ref ions is to increase the bulk electron temperature by a factor 2.8: Te =2.8 Te β0 = ref × ⇒ β0 =3.6% on axis. The electron temperature is then nearly 5 times greater than the ion ref temperature: Te0 6.0 keV 4.8 T . Results for this case are shown in Fig. 4. ≈ ≈ × i0 One can see in Fig. 4(b) that some of the eigenvalues that we interpret as bulk ion sound waves rise out of the “sea” of strongly damped solutions that are plotted in Fig. 4(a). They now appear in Fig. 4(c) around Γ 1, where they form structures that closely ∼ − resemble low-frequency MHD continua. In particular, one can see the gap induced by geodesic coupling between sound and Alfvén waves in the low-frequency band ω . 0.1. This may facilitate the formation of beta-induced Alfvén acoustic eigenmodes (BAAE) [10] near the accumulation points ω ωBAAE that are indicated in Fig. 4(c). ∼ The Alfvén continua near and above the BAE accumulation point, ω & ωBAE, are somewhat stabilized and their structure becomes more complicated as can be seen in the comparison shown in Fig. 5. The physical meaning of this result is still unclear. It is I-10 6

β (0)=3.6%,0.5 Th=200keV 0.5 0.5 "Hot" (b) (c) (a) sound 0.4 0.4 0.4 ω TAE,u

0.3 0.3 0.3 A0 A0 A0 ω ω ω / / /

ω 0.2 ω 0.2 ω 0.2 ω BAE ω 0.1 hS 0.1 0.1 Stable Unstable 0 0 0 0.2 0.4 0.6 0.8 −4 −2 0 2 4 0.2 0.4 0.6 0.8 ρ ix = y < −0.53/4*x; Γ = γ × τ ρ β (0)=3.6%,0.5 Th=400keV 0.5 0.5 "Hot" (e) (f) (d) sound ω 0.4 0.4 0.4 TAE,u

0.3 0.3 0.3 A0 A0 A0 ω ω ω

/ / / ω ω 0.2 ω 0.2 ω 0.2 BAE

0.1 ω 0.1 0.1 hS Stable Unstable 0 0 0 0.2 0.4 0.6 0.8 −4 −2 0 2 4 0.2 0.4 0.6 0.8 ρ ix = (y−0.1) < −0.57/4*x; Γ γ × τ ρ ix = (y < 0.1) & (x > −0.05); =

ref ref FIG. 6: Results obtained with Ti = Ti , Te = Te and a Maxwellian population of fast ions with Th = 200 keV (top) and 400 keV (bottom). The fast ion density nh is chosen such that β = βref. Arranged as Fig. 3. The shaded triangles in panels (a), (c), (d), (f) indicate the regions encompassed by what we interpret as “hot sound” branches ωhS.

possible that the changes seen in Fig. 5 are due to Alfv´en acoustic coupling in the band ω>ωBAE that we are looking for and which gave rise to BACMs in the MHD picture [1]. If so, this result indicates that such coupling may become signiﬁcant when Te Ti. ≫

3.2 Effect of fast ions Finally, Fig. 6 shows the results obtained in the presence of Maxwellian-distributed fast ions with Th = 200 keV (top) and 400 keV (bottom). One can see that the entire Alfvén continuum is locally destabilized. The largest growth rates are found near the BAE accumulation points, ω ωBAE. The values of these local growth rates merely constitute an upper limit. They are≈ expected to be much lower for any global mode and will also be reduced by FOW effects of fast ions. Nearly none of the bulk ion sound branches is visible in Fig. 6 because they are strongly damped, and we have not plotted solutions with Γ < 4. However, with the fast ions, − an additional branch of damped solutions appears along the boundaries of the shaded triangular regions shown in Figs. 6(a) and 6(d). Their dispersion relation appears to have the form ωhS = k vhS = nq m vhS and their slope dωhS/dρ increases with increasing | k| | − | fast ion temperature Th. Therefore, we interpret these branches as fast ion (“hot”) sound waves, as indicated by the subscript “hS”. 7 I-10

The damping of the hot sound branches decreases with decreasing frequency, so that they approach marginal stability (Γ = 0) in the low-frequency band ωhS . 0.2 ωBAE. ≈ This trend is enhanced with increasing fast ion temperature Th, as can be seen by com- paring Figs. 6(b) and 6(e). We do not see any evidence for Alfvén acoustic couplings in the form that we are familiar with from the MHD limit studied in Ref. [1]. However, an interesting effect that seems to be connected with the hot sound branches can be seen in Figs. 6(c) and 6(f): the shear Alfvén continua ωA(ρ) between neighboring ωhS(ρ) branches are strongly modified. In particular, the upper accumulation points ωTAE,u of the toroidicity-induced gap [12, 13] are flattened in the radial intervals where neighboring Alfvén continua lie below a hot sound continuum as ωA(m 1) <ωhS(m). In these regions, there appears to ± be a trend for the original minima at ωA = ωTAE,u to be turned into maxima. This effect may put kinetic toroidicity-induced Alfvén eigenmodes (KTAE [14]) into a new light.

4 Summary and conclusions

In this study, we reexamined the existence of Alfvén acoustic couplings in the frequency range ω>ωBAE, which were suggested as a channel for bulk ion heating in high beta tokamak plasmas [1]. While an actual transfer of energy from energetic fusion alpha particles to thermal bulk ions via waves (“alpha energy channeling”) would be a nonlinear process, the question motivating this work is whether the preconditions for such a channel are satisfied; namely, that both wave branches coexist and are able to interact. For this purpose, a local analysis was performed with the linear gyrokinetic code LIGKA in order to reveal the structure of the continuous spectra in a gyrokinetic description of the plasma and to compare it with the MHD results. Our present interpretation of the results presented in this paper is that the looked-for Alfvén acoustic coupling in the frequency range of interest, ω>ωBAE, may not be present when Te Ti, as is typical for present-day tokamaks. However, when Te is increased, the structure≈ of the continua in the complex frequency plane ω+iγ was found to change across the entire range of frequencies examined. At low frequencies, ω<ωBAE, geodesically coupled Alfvén acoustic branches appear in the moderately damped domain and coincide well with their MHD counterpart. Thus, we conjecture that changes at higher frequencies ω>ωBAE may also be due to couplings with sound waves whose damping is reduced with increasing Te. This motivates further study of the regime Te Ti. ≫ When a population of Maxwellian fast ions with temperatures up to 400 keV was included, new branches of solutions were found, which we interpreted as fast ion sound waves. Their role and relevance, as well as some interesting effects that were found in association with these “hot sound” branches remain to be understood. All results presented here were obtained with a local analysis, carried out point-by- point for each radial location. An analysis of global modes remains to be performed, and work is also underway to eliminate the approximations that were summarized in Section 2; in particular, trapped particle effects, higher-order geometric couplings between sound waves, effects of large magnetic drifts and realistic velocity distributions for the fast ions. I-10 8

Once it is advanced to the nonlinear regime, this work is expected to contribute to a better understanding of energy ﬂows in high-beta fusion experiments (such as JT-60SA) and burning plasmas (DEMO power plants).

Acknowledgments

This work has been partly carried out within the framework of the EUROfusion Con- sortium and has received funding from the European Unions Horizon 2020 research and innovation programme under grant agreement number 633053. The views and opinions expressed herein do not necessarily reﬂect those of the European Commission. One of the authors (A. B.) thanks Yasushi Todo (NIFS, Japan) for providing the code MEGA. The MEGA simulations reported in Ref. [1] (summarized in Fig. 1) were carried out using the supercomputer HELIOS at IFERC, Aomori, Japan, under the Broader Approach collaboration between Euratom and Japan, implemented by Fusion for Energy and JAEA. This work has been supported by Japan Society for Promotion of Science (JSPS) Grand-in-Aid for Scientiﬁc Research number 633053.

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