Three-Dimensional MHD Analysis of Heliotron Plasma with RMP
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This study investigates the effects of Resonant Magnetic Perturbations (RMPs) on pressure-driven modes in the Large Helical Device (LHD) through three-dimensional equilibrium and dynamics analysis. The research explores coil configurations, equilibrium with RMP, and the impact of RMPs on pressure profiles in heliotron plasma. Detailed numerical simulations and comparisons with experimental results are discussed.
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Three-Dimensional MHD Analysis of Heliotron Plasma with RMP K. Ichiguchi1,2), Y. Suzuki1,2), M. Sato1), Y. Todo1,2), T. Nicolas1), S. Sakakibara1,2), S. Ohdachi1,2), Y. Narushima1,2), and B.A.Carreras3) National Institute for Fusion Science, Japan1) The graduate University of Advance Study, SOKENDAI, Japan2) Universidad Carlos III, Spain3) 25thIAEA Fusion Energy Conference (FEC2014), St Petersburg, Russian Federation, 13-18 October 2014 Acknowledgements This work is supported by the budget NIFS14KNST063 of National Institute for Fusion Science, and Grant-in-Aid for Scientific Research (C) 22560822 of Japan Society for Promotion Science. Plasma simulator (NIFS) and Helios(IFERC-CSC) were utilized for the calculations. 1
Outline 1. Introduction Motivation of this study Coil Configuration of Large Helical Device (LHD) 2. Three-dimensional equilibrium including resonant magnetic perturbation (RMP) Numerical Scheme of HINT2 code Difference of magnetic surfaces and pressure profiles between with and without RMP 3. Effects of RMP on pressure-driven mode dynamics Simulation procedure with MIPS code RMP dependence of linear modes Comparison of nonlinear dynamics with and without RMP Relation with LHD experiments 4. Summary 2
Motivation Effects of RMPs have been extensively studied in toroidal confinement devices. Tokamaks : Relaxation of the pressure gradient at pedestal region is studied for the mitigation of ELMs. (e.g. Evans et al. Nature Phys. 2006) Stellarator (LHD) : Penetration into the plasma and the effects on the global stability are studied. (e.g. Sakakibara et al. NF 2013) Numerical analyses of the effects of RMPs have also progressed. However, in most of previous numerical analyses for RMPs, equilibria with nested flux surfaces are employed, and then, RMPs are applied on the equilibria. (e.g. Garcia et al. NF2003, Strauss et al. NF2009, Saito et al. PoP2010, Becoulet et al. NF2012) The initial pressure profile corresponding to the nested surfaces is inconsistent with the magnetic field including the RMPs. In order to incorporate the pressure profile consistent with the magnetic field including RMP, three-dimensional (3D) analyses are indispensable. Here, we analyze the effects of RMPs on pressure driven modes in the Large Helical Device (LHD) by utilizing 3D equilibrium and dynamics codes. 3
LHD Configuration Large Helical Device (LHD) is a typical heliotron device composed of helical coils and poloidal field coils. Coil configurations Helical coils Poloidal coils Pole number : 2 3 pairs Field period : 10 RMP coils No net toroidal current is needed. Plasma is stable for current driven modes. Pressure driven modes are the most dangerous. RMPs are controlled by the currents in 10 pairs of RMP coils. Typical LHD Plasma 4
3D Equilibrium Calculation HINT2 code (Y. Suzuki, et al., Nuclear Fusion (2006) L19) Initial Condition including RMP pressure profile : The HINT2 code solves the 3D equilibrium equations without any assumptions of the existence of the nested flux surfaces. (suitable for the equilibrium analysis including RMPs) Step A : Relaxation of P (B fixed, field line tracing) ) Step B : Relaxation of B (P fixed, following MHD eqs.) ) An LHD configuration with an inwardly shifted vacuum magnetic axis and a high aspect ration is employed. (Rax=3.6m, =1.13) Calculation starts with the parabolic pressure profile with = = %. Convergence Equilibrium consistent with islands 5
3D LHD Equilibrium (1) Equilibrium without RMP Puncture plot of field lines 0.8 Bird s eye view and contour map of pressure 0.04 0.4 P 0.03 Z [m] Nested surfaces exist in the whole plasma region. 0.02 0.0 0.01 0 -0.4 -0.8 2.8 3.2 3.6 R [m] 4.0 4.4 0.8 0.4 Rotational Transform & Mercier Stability 0.0 4.4 2.0 4.0 Z [m] 4.0 -0.4 /2 DI 3.6 R [m] 3.2 -0.8 1.5 2.0 2.8 /2 DI 1.0 0.0 Pressure profile is smooth and the contours are also nested corresponding to the magnetic surfaces. 0.5 -2.0 0 0.2 0.4 0.6 0.8 1.0 Rational surface of /2 = = exists in the plasma. The /2 = = surface is Mercier unstable. 6
3D LHD Equilibrium (2) Equilibrium with a horizontally uniform RMP Puncture plot of field lines 0.8 Bird s eye view and contour map of pressure 0.04 0.4 An m=1/n=1 magnetic island is generated by the resonance with the /2 =1 field lines. P 0.03 Z [m] 0.02 0.0 0.01 0 -0.4 -0.8 2.8 3.2 3.6 R [m] 4.0 4.4 0.8 0.4 Comparison of pressure profile along Z=0 line 0.0 4.4 Z [m] 0.05 4.0 -0.4 w/o RMP w/ RMP 3.6 R [m] 3.2 0.04 -0.8 2.8 0.03 Shoulder like structure in the bird s eye view and island structure in the contour map are seen. P 0.02 Pressure profile is locally flat at the O-point, while the profile is steep at the X-point. 0.01 0.00 3.0 3.5 R (m) 4.0 7
3D MHD Dynamics Calculation MIPS code (Todo et al., Plasma Fus. Res. (2010) S2062) Solves the full MHD equations by following the time evolution. 4th order central difference method for (R, , Z) directions. 4th order Runge Kutta scheme for the time evolution. The most unstable mode is detected. Typical time evolution of kinetic energy Basic equations -5 log Ek -10 w/o RMP w/ RMP -15 0 500 1000 t/ A In either case, a pressure driven mode grows. After the linear phase, the modes are saturated nonlinearly. 8
Mode Structure in Linear Phase (1) Mode structure without RMP Mode pattern of perturbed pressure 0.8 0.4 Z [m] 0.0 -0.4 -0.8 2.8 3.2 3.6 4.0 4.4 R [m] The mode pattern is distributed around the /2 =1 surface almost uniformly. This pattern indicates that this mode is a typical interchange mode. The mode number is m=2/n=2 for the present parameters. 9
Mode Structure in Linear Phase (2) Mode structure with RMP Mode pattern of perturbed pressure 0.8 0.4 Z [m] 0.0 -0.4 -0.8 2.8 3.2 3.6 4.0 4.4 R [m] The mode is localized around the X-point showing a ballooning-like structure. This localization due to the deformation of the equilibrium pressure profile. The mode can utilize the driving force the most effectively by being localized around the steepest pressure gradient position. 10
Viscosity Dependence of Linear Mode Components with the higher mode number is stabilized the more effectively. =10-6 =6x10-5 0.8 0.8 0.4 0.4 Z [m] Growth Rate Z [m] 0.0 0.0 w/o RMP -0.4 -0.4 0.08 w/o RMP growth rate w/ RMP -0.8 -0.8 0.06 2.8 3.2 3.6 4.0 4.4 R [m] 2.8 3.2 3.6 4.0 4.4 R [m] Mode number of the most unstable interchange mode is decreased by the increase of the viscosity. 0.04 0.02 0.8 0.8 0.4 0.4 0.00 -6 -5 -4 Z [m] Z [m] log10 0.0 0.0 w/ RMP -0.4 -0.4 As the viscosity increases, the growth rate decreases largely In the case with the RMP than in the case without the RMP. -0.8 -0.8 2.8 3.2 3.6 4.0 4.4 R [m] 2.8 3.2 3.6 4.0 4.4 R [m] The mode structure is extended in the poloidal direction. This is due to the fact that the stabilization of the high mode components makes the localization weak. Since the mode structure extends to the small pressure gradient region. the reduction of the growth rate is enhanced. 11
Nonlinear Dynamics without RMP Time evolution of pressure and magnetic surfaces in the case without RMP Puncture plot of field lines Total Pressure According to the linear mode structure, the m=2/n=2 interchange mode starts to evolve from the equilibrium with the smooth pressure profile and the nested surfaces, and leads to the collapse. 12
Snapshots without RMP Snapshots of pressure and field lines in thee case without RMP t=1000 A t=1060 A t=960 A t=900 A Pressure profile Puncture plot of field lines The convection of the interchange mode generates a mushroom- like structure, which leads to the collapse in the core region. The interchange mode can be destabilized with an arbitrary phase in the poloidal and the toroidal directions. Spatial phase of the collapse can be changed. 13
Nonlinear Dynamics with RMP Time evolution of pressure and magnetic surfaces in the case with RMP Total Pressure Puncture plot of field lines According to the linear mode structure, the ballooning-like mode starts to evolve from the equilibrium with the locally flattened pressure profile and the m=1/n=1 magnetic island, and leads to the collapse. 14
Snapshots with RMP Snapshots in nonlinear phase in thee case with RMP t=600 A t=800 A t=700 A t=900 A Pressure profile Puncture plot of field lines Pressure collapse occurs at the X-point initially, and expands toward the core region. Field line structure becomes stochastic also from the X-point, however, the O-point structure survives even in the large collapse of the pressure. Spatial phase of the collapse should be fixed to the island. 15
Confirmation of Difference in Phase Property Phase of the initial condition is changed for the confirmation. Initial perturbations are given as : random function of toroidal angle : initial toroidal phase Constant pressure surface of =3.0% and pressure contour at enlarged cross section (x1.5) w/o RMP w/ RMP =0 t=900 A t=1000 A = /2 t=1000 A t=900 A Collapse phase is fixed independent of the initial phase. Collapse phase can change depending on the initial phase. 16
Relation with LHD Experiment Similar fixed phase is observed in the LHD experiments. In LHD, a natural error field exists, which works as an RMP. The m=1/n=1 component is dominant. This error field can be reduced by controlling the RMP coil currents. In either case of the reduction or the existence of the error field, pressure collapses are observed in the configurations similar to the present case. In the reduction of the error field case, the mode locations differ for every discharge. Mode location in collapses (toroidal angle where the O-point is located at the mid plane of the low-field side) In the existence of the error field case, the mode locations are concentrated at the phase of the error field. This phase property agrees with the present numerical results. To investigate the detailed mechanism, analyses including RMP penetration and plasma rotation, and precise comparison will be necessary. (S.Sakakibara et al. NF (2013) 0430101 ) 17
Summary For MHD analyses including resonant magnetic perturbations (RMPs), 3D equilibrium calculation is crucial, because RMPs can change the structure of pressure driven modes through the change of the equilibrium pressure profile. In the case of an LHD plasma, a horizontally uniform RMP changes the mode structure from an interchange type to a ballooning type localized around the X-point. The spatial phase of the nonlinear collapse is fixed corresponding to the geometry of the magnetic island. Similar fixed phase is observed in the LHD experiments with the error field. To investigate the detailed mechanism in the experiments, we need further analyses including RMP penetration and plasma rotation, and precise comparison. 18
