1 Modelling of Configuration Optimization in the HL-2A Tokamak Gao Qingdi Zhang Jinhua Li Fangzhu Southwestern Institute of physics, P O Box 432, Chengdu

2 Introduction Main parameters of HL-2A

3 Introduction  Optimizing the tokamak configuration is a key point for improving the plasma performance  Reversed magnetic shear (RS) plasma configuration with internal transport barrier (ITB) is one of the most promising ways to achieve high performance regimes in tokmaks.  In HL-2A, the various schemes of auxiliary heating and current drive combined with the device flexibility offers the opportunity to optimize the plasma configuration.

4 Fig.3.1 Plasma current wave - form The simulated discharge is a deuterium discharge with the plasma current rising to ~0.3MA in 0.3s Neutral beam injection (NBI) begins with a lower power ( MW) at  0.15s, then the additional NBI heating power of 1.5 MW is injected.

5 Fig.3.2 The energy confinement time  E and  ITER89 vs. Time. RS begins at t=1.0s The central electron density increases from  2.7  m -3 to  1.0  m -3 with gradually peaking profile (= ), and the temporal evolution of line averaged density following the specific experimental wave - forms In the high NBI power phase the heat diffusivity models are assumed in terms of the TFTR experiment results.

6 Fig.3.3 q-profiles (a), and thermal pressure profiles (b) at t=1.08s, 1.30s, and 1.50s RS configuration is formed with q min evolving from q min >3.0 to q min <2.0 and q min located at x min (r min /a)  0.3

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8 Fig.3.4 Temporal evolution of ohmic current, bootstrap current, and NBI driven current, and their profiles

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13 Fig.3.6 Poloidal projection of the eigen displacement vector (n=1.5) for the equilibrium (a)at t=1.08s, and (b)at t=1.3s

14 Fig.3.7 Poloidal harmonics of the perpendicular displacement for the instability shown in Fig.3.6a Fig.3.8 Instability growth rate versus n

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17 Fig.3.11 Dependence of -D s on the plasma radius, showing the unstable window where -D s <0 The low-n MHD modes revealed by the ERATO(SWIP) analysis can not be interchange modes because they locate in the vicinity of shear reversal point (x ~ 0.3) where is not covered by the unstable window against interchange modes.

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19 Fig.3.9 Critical pressure gradient for the ideal ballooning stability (dotted dash line) and actual pressure gradient (full line) for the equilibrium (a) at t=1.08s, and (b) at t=1.30s

20 Fig.3.10 Fraction of the plasma radius on which ballooning modes are unstable versus time during the whole RS discharge

21 1.LH wave propagation and absorption 2.Quasi-stationary RS operation established by profile control 3.LH wave absorption in the quasi-stationary RS plasmas 4.Profile control for shaped plasmas 5.Development of electron transport barriers by LHH

22 1. LH wave propagation and absorption It is assumed that the wave electric field can be decomposed into a set of components (WKB approximation is valid), Suppressing the index j, each mode locally satisfies the wave matrix equation For a non-trivial solution,

23 To proceed, it is convenient to choose a local Cartesian system such that For LH waves, the electron cyclotron frequency is much higher than the wave frequency, which is much higher than the ion cyclotron frequency: which is assumed throughout. In the Hermitian part of the dielectric tensor we keep only cold plasma terms, except that the dominant warm plasma term is carried to guard against singularity near the lower hybrid resonance. The anti-Hermitian part of the tensor is retained as a perturbation. For the case at hand, the principal such term enters as an imaginary correction to K zz, and describes the interaction between the component of the wave electric field parallel to B and electrons whose speed along B matches that of the wave (Landau damping). Thus the plasma dielectric behavior is described by the following tensor elements:

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25 The thermal term designated by  would be important near lower hybrid resonance. The wave-particle interaction responsible for electron heating and current drive is in K zz,i. In the event, the lower hybrid resonance should become important, a thermal term representing the ion wave-particle interaction would have to be added to K xx and K yy, but here we assume such terms vanish. In the Cartesian co-ordinate system, we decompose the dispersion relation into its real and imaginary parts, D = D r + iD i = 0, where

26 The extension of the local solutions to spatially inhomogeneous plasma is accomplished by the eikonal method, with the useful result that an initial wave field at r with an initial propagation vector k evolves according to Hamiltonian equations which preserve the local dispersion relation D r = 0 along the ray trajectories As in Hamiltonian mechanics, the spatial coordinates denoted by r are canonically conjugate to the wave-number coordinates denoted by k. In the study of the axisymmetric tokamak, it is the cylindrical coordinate system that the most natural. In the cylindrical frame one has (R, Z,  ) and (k R,k Z, n), where R and Z have dimensions of length, k R and k Z have dimensions of inverse length and n is the dimensionless toroidal mode number. The canonical momentum n is constant along the ray path.

27 A spectral component of power W experiences a change  W over time interval  : Given the velocity distribution and the profiles of macroscopic plasma parameters, the absorption of a lower hybrid spectrum can be computed. An actual incident wave spectrum is a continuous function of the parallel wave number. This continuous spectrum is approximated by assigning the input power to a number of discrete rays, each ray having a definite initial k // and launched power.

28 Fokker-Planck analysis Kinetic equation An electron kinetic equation can be written as The wave diffusion operator is the 1-D divergence of the RF induced flux: where D ql is the quasi-linear diffusion coefficient, and here it signifies a sum over all waves in existence on a flux surface, with the appropriate powers and velocities. A simple sum is used, which means that we assume there are no interference effects.

29 We employ a 1-D collision operator as given by Valeo and Eder, with the collisional diffusion and drag coefficient given by In solving for f e we set, because the time for equilibration between RF power and the electron distribution is short compared with the time for plasma to evolve. Then the solution for f e is an integral in velocity space,

30 Quasi-linear diffusion coefficient An incremental contribution to the quasi-linear diffusion coefficient D ql at velocity v // from a wave field of wave-number k // is given by where E // represents the amplitude of the wave field parallel to the static magnetic field. One instructive way to find the relationship between field E // and wave power W is to equate P ql, the energy per unit time per unit volume going into electrons and out of the wave from the quasi- linear point of view,

31 with the similar quantity from the ray point of view, obtaining the incremental D ql from a wave of power W traversing a flux shell of volume  V in time .

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34  The LHW power spectrum radiated by a multi-junction launcher is calculated with Brambilla’s coupling theory. Fig.2.1 Relative LH power versus toroidal index for the cases in which the relative wave-guide phase = (a) 90º, (b) 170º, (c) 180º

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39 Fig.2.2 Waveforms of the plasma current I p, loop voltage V p, the NBI power P NB, and the LH wave power P LH Fig.2.3Magnetic geometry of the discharge

40 Fig.2.3 The temporal evolution of LH wave driven current profile (a), and q profiles at different times (b) for the sustained RS discharge with I p = 265KA

41 Fig.2.4 Waveforms of the plasma currents I p, I LH, I BS, I NB, and I OH (a) and their profiles at t=1.0s (b) for the sustained RS discharge

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43 Fig.2.5 NBI driven current profiles 3) The approximately balanced beam injection produces only small beam driven current (I NB ~10kA compared with I NB ~45kA in the standard target plasma);

44 Fig.2.6 Electron density profiles 4) The density profile peaking factor decreases by ~25% by changing the density profile;

45 Fig.2.7 The temporal evolution of LH wave driven current profile (a), and q profiles at different times (b) for the target plasma i ) In all these cases nearly the same LH wave driven current profiles as in the standard case are produced, and sustained RS discharges are achieved.

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47 Fig.2.9 The ion temperature T i (full line), and magnetic shear s (dotted line) versus x

48 Fig.2.10 Time traces of a quasi-stationary RS discharge: (a) LHCD efficiency,  CD and non-inductive current fraction, (b) the H- factor, H 98(y,2) and normalized beta,  N, (c) the locations of the minimum q (full line) and the minimum  i (dotted line), (d) the central plasma temperatures (T i, T e ).

49 Fig.2.8 The temporal evolution of LH wave driven current profile (a), and q profiles at different times (b) for the discharge with I p =300KA When the plasma current increases to I p = 300kA, the RS discharge would not be sustained.

50 Fig.3.1 LH wave propagation in the quasi-stationary RS plasma

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61 Fig.3.2 Region of LH power absorption by strong electron Landau damping(at t=1.0s): electron Landau damping limit (full line); n // -upshift boundary (dotted line); and boundary of the wave propagation domain ( dash line). (a) central deposition (I p = 300KA), (b) off-axis deposition (I p = 265KA)

62 Fig.3.3 Profiles of (a) LH wave driven current, (b) q, (c) geometry- factor for the case of central deposition (dotted line), and off-axis deposition (full line) (a) (b) (c)  The geometry-factor plays an important role in determining the location of LH wave deposition

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64 Fig.3.4 Temporal evolution of the LH wave driven current profile, and the regions of LH power absorption at two different times: t=0.8s, t=1.5s  A quasi-stationary RS configuration transits to another quasi-stationary RS configuration spontaneously

65 Fig.3.5 Two RS configurations at t=0.8s (full line), and t=1.05s (dotted line). (a) q - profile, (b) Te – profile, (c) Ti – profile.

66 Fig.3.6 Temporal evolution of (a) the location of shear reversal point, (b) geometry-factor, and (c) electron temperature

67 By tuning the phasing ( 75º ), a sustainable RS discharge with I p =320kA can be achieved, which, although not so stationary as the discharges with I p =265kA, has higher normalized parameters Fig.3.7 Temporal evolution of (a) normalized beta, and H-factor, and (b) the location of the shear reversal point (full line) and the minimum ion heat diffusivity (dashed line)

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69 Fig.4.1 Magnetic geometry of (a) a plasma with a nearly circular cross – section, (b) a plasma with the X point moving inward (D-shape, k 95 =1.08,  95 =0.44), (c) a plasma with modest elongation (elongated D-shape, k 95 = 1.21,  95 = 0.41). (a) (b) (c)

70 The triangularity variation with respect to the flux coordinate is dependent on the plasma current profile, but for both hollow and peaked current profiles it decreases rapidly at the plasma boundary region while moving towards the plasma center Fig.4.2 Triangularity of the D-shaped plasma,  versus the flux surface for the cases of hollow current profile (full line) and peaked current profile (dotted line).

71 Fig. 4.1 Boundary of a single x-point plasma (shot 1766#) determined by a filamentary model. The interior flux surfaces produced by solving Grad- Shafranov equation using the the boundary (99% flux surface) shown left

72  The H-mode transport barrier is localized at the plasma edge;  The pressure of the H-mode pedestal increases strongly with triangularity due to the increase in the margin by which the edge pressure gradient exceeds the ideal ballooning mode limit;  Therefore, the rather high triangularity located at the plasma edge is favorable to enhancing the confinement.

73  RS discharge with double transport barrier —The elongated D-shape plasma (  98 =0.43, k 98 =1.23) is used to model the RS discharge. The geometry of the boundary (98% flux surface of the diverted plasma) is specified as a general function of time. It evolves from circular to elongated D-shape during the current ramping-up phase and then keeping the same shaped boundary in the current flattop phase. The interior flux surfaces, which are computed by solving the Grad-Shafranov equation. —The standard target plasma described above is used, but the electron density profile has a modest change with a more obvious edge pedestal. — The current profile is still controlled by LHCD.

74 The double transport barrier is indicated by two abrupt decreases of the ion heat diffusivity, of which the two minima are located near the shear reversal point,  min  0.55, and near the plasma edge,   0.95, respectively. The elevated heat diffusivity between the two minima separates the two barriers. Fig.4.10 Profiles of q and ion heat diffusivity,  i (at t=1.0s) for the elongated D-shape plasma.

75 Fig.4.11 Profiles of the ion temperature and the gradient of the ion temperature,  T i  (at t=1.0s). The transport barriers are also shown on the ion temperature profile

76 Fig.4.12 Time traces of an RS discharge with double transport barrier: (a) normalized beta,  N, (b) H-factor, H 98(y,2), (c) locations of the double transport barrier (two dotted lines), and location of the shear reversal point (full line). The fainter lines indicate the results of the RS with L-mode edge.

77 Conclusion  The theoretical transport model, the NBI heating, and the LH current drive are all correlated with the plasma parameters forming a strong non-linear system, but the plasma temperature profiles and current profile evolve consistently, showing that the system is self-consistent and the formation of ITB is related to the magnetic shear reversal.  Quasi-stationary magnetic shear reversal can be established with LH waves even in moderate temperature plasmas in which the wave are weakly damped. Therefore, with the aid of plasma shaping and current profile control, stable steady state high performance modes are produced, implying that the underlying physics of enhanced confinement in the so-called ‘advanced tokamak’ scenarios can be explored in the future operation in HL-2A.