Turbulent Transport due to Kinetic Ballooning Modes in High-Beta Toroidal Plasmas

Transcription

1 1 TH/P-3 Turbulent Transport due to Kinetic allooning Modes in High-eta Toroidal Plasmas A. Ishizawa 1, S. Maeyama, T.-H. Watanabe 1, H. Sugama 1 and N. Nakajima 1 1 National Institute for Fusion Science, Toki, Gifu, 59-59, Japan Japan Atomic Energy Agency, Rokkasho, Aomori, 39-31, Japan Corresponding Author: ishizawa@nifs.ac.jp Abstract: Turbulent transport due to kinetic ballooning modes (KMs) in high-beta toroidal plasmas is investigated by means of an electromagnetic gyrokinetic model and a newly developed electromagnetic hybrid model consisting of the gyrokinetic equation for ions and fluid equations for electrons. Full gyrokinetic simulation results for Cyclone base case tokamak with β = % and η e = is quickly and accurately reproduced by the hybrid simulation. Agreement between the hybrid and full kinetic simulations are confirmed on detailed transport mechanisms: electrostatic heat and particle fluxes and electromagnetic heat and particle pinches for ions and electrons. The numerical solutions satisfy the entropy balance equation, and entropy variable is transferred from ions to electrons through electromagnetic perturbation. The hybrid model enables us to simulate KM turbulence in high-beta Large Helical Device (LHD) plasmas, for which full gyrokinetic simulation is difficult because of large computational cost. The critical beta of KM in the standard LHD plasma is about 1.5% (1%) for η e = (η e = 3). 1 Introduction In high-beta torus plasmas micro-turbulence due to kinetic ballooning modes (KMs) causes anomalous transport of heat and particles, while the growth rate of ion temperature gradient (ITG) instability is suppressed by magnetic field line bending as plasma beta increases [1], where plasma beta β is the normalized plasma pressure. allooning modes are related to the degradation of confinement in many torus plasma experiments such as the Large Helical Device (LHD) experiments []. The β scaling of turbulent transport is one of the central issues in fusion plasma research. In order to understand the scaling the impact of magnetic perturbation on heat transport through affecting the balance between zonal flow and micro-turbulence should be understood. In addition the validity of Rechester-Rosenbluth model describing direct effects of magnetic perturbation on heat transport is another important issue.

2 TH/P-3 Two of authors studied KM driven turbulent transport and found the profile stiffness of externally heated tokamak plasmas by means of global two-fluid simulations [3]. In order to investigate electromagnetic turbulent transport more quantitatively, we need to include kinetic effects. Gyrokinetic simulations including both kinetic ions and electrons are applied to the studies of turbulence in finite-beta plasmas [4, 5, 6]. In this work, we investigate electromagnetic turbulent transport in high-beta tokamak and LHD plasmas by means of newly developed simulation codes: GKV+/EM solving gyrokinetic ion and electron equations and GKV+/EMH solving gyrokinetic ion and fluid electron equations. The code is developed by extending GKV [7] that solves gyrokinetic equations with adiabatic electrons. Simulation model.1 Electromagnetic gyrokinetic equations Micro-turbulence in a radially localized flux tube plasma along magnetic field line is investigated. Temperature and density gradients are uniform and direct to x-axis, and temperature gradient is represented by the parameter η s = L n /L Ts in terms of density scale length L n = (d ln n/dx) 1 and temperature scale lengths L Ts = (d ln /dx) 1, where the subscript s denotes particle spices. The distribution functions are divided into the Maxwellian n part and a perturbed part, F s = F Ms + δf s, where F Ms = (π/m s) exp( msv 3/ µ ), and the perturbed part is represented by δf s = k δf sk exp(is k ), where S k = k. The model consists of the gyrokinetic equation of perturbed part of distribution functions, Dδf sk t + v b δf sk = iv ds k (δf sk + q s F Ms φ k J s ) v q s F Ms E k J s + µ m s b δf sk v the gyrokinetic Poisson and Ampere s equations, k φ k = 4π s + iv s k q s F Ms (φ k v c A k)j s + C s (δf sk ), (1) q s (δn sk q sn (1 Γ s )φ k ), A k k A k = 4π c q s n δu sk, () where E k = b φ k 1, δn c t sk = δf sk J s d 3 v, n δu sk = v δf sk J s d 3 v, q i = e, q e = e. In Eq. (1) Df k = f k + c [φj Dt t s,f] k and b f k = b f k 1 [A J s,f] k, where [f,g] k = k,k δ k,k +k b k k f k g k. The drift velocities are v ds = c b (µ + q s m s v b b) and v s = cts b ln F q s Ms, and C s (δf sk ) is the Lenard-ernstein collision operator. In Eq. () Γ s = e b sk I (b sk ) and b sk = ρ sk, where I are the zeroth order modified essel function. The entropy balance equation is obtained from Eqs. (1)-() [8] and is written as ( ) d δs s + W es + W em = Q es,s + Q em,s + (Γ es,s + Γ em,s ) + D s, (3) dt L Ts L ps s s

3 3 TH/P-3 where δs s = k d 3 v Ts δf sk F sm, D s = k d 3 v Tsδf sk Cs F sm, ( ) k W es = k + n qs 4π [1 Γ (b sk )] δφk, W em = k [ ( Q es,s = Re 1 k δp s + δp s 5T ) ( ) ] ik yφ k c sδn s, [ ( Q em,s = Re 1 k δq ) ( ) ik ] s + δq ya k s, [ ( Γ es,s = Re k δn ikyφ k c s <> denotes the flux surface average. ) ], Γ em,s =. Electron fluid hybrid model Re k A k 4π, [ ( ) k n ikya ] k δu s, where In order to simulate KM turbulence efficiently we introduce a new hybrid model of gyrokinetic ions and fluid electrons and have developed GKV+/EMH code. Our new model enables us to simulate KM at high-beta more quickly than the conventional model with kinetic electrons. The model consists of the gyrokinetic equation of perturbed distribution functions of ions Eq. (1), the gyrokinetic Poisson and Ampere s equations Eq. () and a set of moment equations of the electron gyrokinetic equation obtained by assuming the electron Larmor radius is small, ρ e k 1, so that J e (ρ e ) = 1. The fluid equations obtained from the moment equations are density, parallel velocity, parallel pressure, δp s = m s v δf sj s d 3 v = n δt s + δn s, perpendicular pressure δp s = µδfs J s d 3 v = n δt s + δn s, parallel heat flux, n δq s = m s v 3 δf sj s d 3 v 3n δu s, and perpendicular heat flux n δq s = µv δf s J s d 3 v n δu s, equations [9, 1], Dδn ek Dt q en c + n b δu ek = iv δp e + δp ek + q e n φ k def k T e A k = q e n b φ k + b δp ek + (δp ek δp ek )b ln t iv ef k n q e φ k T e,(4) + q en c i(1 + η e)v ef k A k im e n T e v def k (δq e + δq ek + 4q e T e δu ek ), (5) DδT ek n + n T e b δu ek Dt + n b δq ek = iη ev ef k q e n φ k iv de k (4n δt ek + δp ek + q e n φ k ) n (q ek + T e δu ek )b ln, (6) DδT ek n + n b δq ek Dt = iq en η e v ef k φ k iv def k (3n δt e + δp ek + q e n φ k ) + n (q ek + T e δu e )b ln, (7) Dδq ek + (3 + β ) T e b δt ek = iv def k (3δq ek 3δq ek α n T e δu ek ) Dt m e Dδq ek Dt 3i q et e m e c η ev ef k A k v Te D q R δq ek + µ e δq ek, (8) + T e m e b δt ek = q et e m e c iη ev ef k A k + T e n m e (δp ek δp ek )b ln iv def k ( δq ek δq ek + n T e δu ek ) v Te D q R δq ek + µ e δq ek, (9)

4 TH/P-3 4 where the parameters are set to be β = 3/ and α = 3, so that the model is consistent with the entropy balance equation. The electron inertia terms in Eq. (5) are neglected by assuming small mass-ratio m e /m i 1. In the hybrid model δs e and D e in the entropy balance equation Eq. (3) are replaced to δs e = T e n k ( δn ek + δt ek + δt ek + 1 δq ek n Te Te 3 Te 3 /m e + δq ek T 3 e /m e ) and D e = k k n m e T e 3 Numerical results (( µe 3 + v Te q R D ) δq ek + (µ e + v Te q R D ) δq ek ). Growth rate γ, Frequency ω [v ti /L n ] Cyclone ase Case (Tokamak) γ Full kinetic ω/4 Full kinetic γ Hybrid ω/4 Hybrid eta β [%] Growth rate γ, Frequency ω [v ti /L n ] γ Full kinetic ω/4 Full kinetic γ Hybrid ω/4 Hybrid Standard LHD eta β [%] Figure 1: Growth rates and real frequencies as a function of beta for the CC tokamak with η e = and k y ρ i =. (left) and for the standard LHD with η e = and k y ρ i =.19 (right). Results by the full kinetic and the hybrid calculations are in good agreement for kinetic ballooning modes which appear at high-beta regime. y extending the gyrokinetic simulation code GKV+, which is electrostatic and adopts adiabatic electron response, we have developed GKV+/EM solving the full gyrokinetic equations [11] and GKV+/EMH solving the hybrid equations. The perpendicular wavenumber and drift frequencies are written in terms of the flux tube coordinate (x,y,z,v,µ), as k = (k x + ŝzk y ) + ky, v ds k /v ds = v dsf k /v dsf = (k x sin z + k y (cos z + ŝz sin z)), v s k /v s = v sf k /v sf = k y, where v ds = v dsf (m s v + µ)/, v s = v sf (1 + (m s v /() + µ/ 3/)η s ), v dsf = Tsc, v q sr sf = Tsc. The q sl n number of Fourier modes is In the z, v s and µ direction, 64, 64, 16 (56, 64, 16 for the LHD) grid points are uniformly distributed. The collision frequency and the Debye length are set to be ν i = 1 3 and λ Di /ρ i =, respectively. Calculations are carried out for the Cyclone base case (CC) tokamak parameters, q = 1.4, ŝ =.786, η i = 3.1, R/L n =., r /R =.18, T i = T e, except η e =, and a model configuration of standard LHD experiments q = 1.9, ŝ =.85, η i = 3, R/L n = 3.33, r /R =.11, T i = T e, except η e = by using the full kinetic code GKV+/EM and the hybrid code GKV+/EMH.

5 5 TH/P Linear analysis of finite-beta plasmas Growth rates and real frequencies as a function of β for k y ρ i =. mode are plotted in FIG. 1. The results of CC tokamak by the full kinetic calculation show the growth rate of ITG instability decreases with β up to β = 1 %, and then KM appears, when β is larger than 1.5 %. These KMs are reproduced by the hybrid calculation. The growth rate of KM increases with β, while its frequency decreases. The hybrid model also provide good approximation of growth rates and real frequencies for the LHD plasmas from the full kinetic calculation. The extrapolation of growth rates in FIG. implies that the critical beta of KM destabilization in the standard LHD is about 1.5% for η e = and is about 1% for η e = 3. Profiles of electrostatic potential φ and parallel component of vector potential A in the parallel direction z are also shown in FIG.. The oscillation in z direction is caused by the helical magnetic field. These results are qualitatively consistent with the linear analysis of KMs for LHD [1]. Full kinetic: Standard LHD Standard LHD Growth rate γ, Frequency ω [v ti /L n ] γ η e = ω/4 η e = γ η e =3 ω/4 η e = Re(A ) Im(A ) Re(φ) Im(φ) eta β [%] z Figure : Growth rates and real frequencies of KMs in the standard LHD from the full kinetic calculation with η e = and 3 (left), and profiles of φ and A of the KM (right). 3. Heat and particle transport in high-beta plasmas Nonlinear simulations of CC tokamak and of the standard configuration of LHD with β = % are carried out. In order to avoid electron temperature gradient instabilities the electron temperature gradient is set to be zero, η e = in the simulations. Figure 3 shows time evolution of square of electrostatic potential < φ k > for each k y =.5k/ρ i mode. Kinetic ballooning modes grow exponentially at the beginning, and then they get saturated around t = 5L n /v Ti. The amplitude of most unstable mode k = 5 (yellow) decreases after it gets saturated, and the k = 4 mode (light blue) dominates in the quasisteady state. The amplitude of zonal flow energy represented by k = mode (red line with crosses) is an order of magnitude smaller than the dominant k = 4 mode. The weak zonal flow in the KM turbulence is in contrast with strong zonal flow in ITG turbulence.

6 TH/P-3 6 < φ k > Full kinetic: CC tokamak 1e+ k= k=1 1e+1 k= k=3 k=4 k=5 1e+ k=6 k=7 k=8 1e-1 k=9 k=1 k=11 1e χ and D [ρ i vti /L n ] χ es ion χ em ion D es ion D es ele D em ion D em ele Full kinetic: CC tokamak χ and D [ρ i vti /L n ] χ es ion χ em ion D es ion D es ele D em ion D em ele Hybrid: CC tokamak χ and D [ρ i vti /L n ] χ es ion χ em ion D es ion D es ele D em ion D em ele Hybrid: standard LHD Figure 3: Time evolution of square of electrostatic potential for each k mode < φ k > (top, left) and of heat and particle transport coefficients, χ and D, of the CC tokamak by the full kinetic (top, right) and by the hybrid (bottom, left) simulations, and of the standard LHD by the hybrid simulation (bottom, right), with η e = and β = %. Figure 3 also shows time evolution of heat transport coefficients due to electrostatic (magnetic) perturbation, χ es,s = L Ts Q es,s / (χ em,s = L Ts Q em,s / ) and particle transport coefficients due to electrostatic (magnetic) perturbation, D es,s = L ps Γ es,s /n (D em,s = L ps Γ em,s /n ), where L ps = L n /(1 + η s ) and s denotes particle spices. The transport coefficients becomes large as KMs grow and get saturated around t = 5L n /v Ti for both of the CC tokamak and the standard LHD. The ion heat (particle) transport coefficient χ es,ion (D es,ion ) is about.6 (.5) ρ iv Ti /L n for the tokamak. The ion and electron particle transport caused by electrostatic (magnetic) perturbation are the same, D es,ion = D es,ele and D em,ion = D em,ele, because of the quasi-neutrality condition (Ampere s law). The transports due to magnetic perturbation causing magnetic-flutter is much smaller than those by electrostatic perturbation causing radial Ex flow convection. In addition, both of heat and particle transports by magnetic perturbation are negative, and thus the magnetic perturbation of KM turbulence has pinch effects.

7 7 TH/P Entropy balance in finite-beta plasmas Figure 4 shows time evolution of several groups of terms in the entropy balance equation Eq. (3): the time derivative group d(s i + S e + W es + W em )/dt, the transport group Q s /L T + Γ s /L ps, and the dissipation group D s, for the nonlinear simulation of KM in CC tokamak. The results by the full kinetic (left) and the hybrid (right) simulations are in good agreement. The numerical error representing difference between the lefthand-side and right-hand-side of Eq. (3) is much smaller than the ion transport group, Q i /L T + Γ i T i /L pi, in both simulations. Hence, our numerical results satisfy the entropy balance equation well. The ion transport group is much larger than electron transport group because KM is unstable in ion-scale. We remark that electron heat transport is zero because η e =. The small oscillation of the ion transport group is due to the shear Alfven wave and balances with that of the time derivative group. The time derivative group is dominated by ds i /dt and becomes small after t = 6, and then the system reaches a quasi-steady state. In the quasi-steady state the sum of ion heat and particle transport terms, Q i /L T + Γ i T i /L pi, almost balances with the sum of ion and electron dissipation terms D i +D e. On the other hand, the electron transport group, Q e /L T +Γ e T e /L pe, does not balance with the electron dissipation term D e. This implies that the entropy variable is transferred from ions to electrons through electromagnetic potential perturbation and the transferred variable is diffused by the electron dissipation as shown for slab ITG turbulence with β = in Ref. [13] Full kinetic: CC tokamak d(s i +S e +W es +W em )/dt Q i /L T +Γ i T i /L pi D i Q e /L T +Γ e T e /L pe D e Hybrid: CC tokamak d(s i +S e +W es +W em )/dt Q i /L T +Γ i T i /L pi D i Q e /L T +Γ e T e /L pi D e Figure 4: Time evolution of several groups of terms in the entropy balance equation Eq. 3 for the CC tokamak with η e = and β = % by the full kinetic (left) and the hybrid (right) simulations. The error representing the difference between the left-hand-side and right-hand-side in the equation is small. 4 Summary Electromagnetic simulation codes: solving the gyrokinetic ion and electron equations and solving the hybrid model of gyrokinetic ion and fluid electron equations are newly de-

8 TH/P-3 8 veloped. These codes are applied to the analysis of turbulent transport due to KM in high-beta CC tokamak and in a model configuration of standard LHD plasmas. The accuracy of the hybrid model is confirmed by comparing with linear results for the tokamak and LHD plasmas from the full kinetic code. Critical values of KM onset in the standard LHD configuration are about 1.5% for η e = and about 1% for η e = 3. In nonlinear calculations the hybrid code is about four times faster than the full kinetic code, and reproduces detailed transport mechanisms. The electrostatic heat (particle) transport coefficient of ion is about.6 (.5) ρ iv Ti /L n. The magnetic perturbation of KM turbulence has pinch effects on the heat and particle transport. Turbulent fluctuation of KM satisfies the entropy balance equation, and the entropy variable is transferred from ions to electrons. The hybrid code enables us to carry out nonlinear simulations of KM turbulence in high-beta LHD plasmas, for which full gyrokinetic simulation is difficult because of large computational cost, and shows the early stage of nonlinear saturation of KM at β = %. The work is supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant Nos and and by NIFS Collaborative Research Program (NIFS1KNTT15). References [1] Kim, J.Y., Horton, W., and Dong, J. Q., Phys. Fluids 5, (1993) 43. [] Ohdachi, S., Tanaka, K., Watanabe, K.Y., et.al., Cont. Plasma Phys. 5, (1) 55. [3] Ishizawa, A. and Nakajima, N., Nuclear Fusion 49, (9) [4] Candy, J., Phys. Plasmas 1 (5) 737. [5] Pueschel, M.J., Kammerer, M., Jenko, F., Phys. Plasmas 15 (8) 131. [6] Pueschel, M.J., and Jenko, F., Phys. Plasmas 17 (1) 637. [7] Watanabe, T.-H., Sugama, H., Margalet, S.F., Phys. Rev. Lett. 1, (8) 195. [8] Sugama, H., Watanabe, T.-H., Nunami, M., Phys. Plasmas 16 (9). [9] eer M.A., and Hammett, G.W., Phys. Plasmas 3,(1996) 446. [1] Scott,., Phys. Plasmas 7, 1845 (). [11] Maeyama, S., Ishizawa, A., Watanabe, T.-H., et.al., preparing. [1] Sugama H. and Watanabe, T.-H., Phys. Plasmas 11, (4) 368. [13] Ishizawa, A., Watanabe, T.-H., Nakajima, N., Plasma Fusion Res., 6 (11) 4387.