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055902 2 Zhang et al Phys Plasmas 17 055902 2010, ergy and pitch angle can thus be accurately calculated using trapped particle transport in Ref 40 are erroneous We ob. the random walk model We find that the diffusivity de serve that the central claim in the letter of Ref 40 has been. creases drastically for high energy particles due to the aver published previously by the same authors as a regular article. aging effects of the large gyroradius and drift orbit width of Ref 41 and later in Refs 42 and 43 together with simu. and the fast decorrelation of the energetic particles with the lation results from gyrokinetic continuum flux tube GENE. ITG oscillations By performing the integration in phase code40 supporting the erroneous energy scaling Considering. space we can calculate the diffusivity for any distribution the fact that the claim of a spatial scale separation for the. function At high energy regime the energetic particle trans orbit averaging contradicts the canonical perturbation theory. port scales as E T 1 for purely passing particles due to the the gene simulation results in Refs 40 43 consistent with. orbit averaging and fast decorrelation of parallel resonance the erroneous energy scaling are thus questionable. E T 2 for trapped particles due to gyroaveraging banana. orbit averaging and wave particle decorrelation of drift II SIMULATION APPROACH. bounce resonance and E T 1 for particle with isotropic. velocity distribution since passing particles dominate trans A Fully self consistent ITG turbulence simulations. port in this energy regime This result not only reconciles the The energetic particle transport by the ITG turbulence in. differences between the older experiments4 with a higher burning plasma is investigated using GTC with multispecies. born energy and the newer experiment13 14 with a lower born capability A global field aligned unstructured mesh20 is uti. energy relative to the plasma temperature but also explains lized in GTC to provide the maximal computational effi. many features of recent DIII D experiments 15 16 ciency without any approximation in physics or geometry to. describe nonlocal geometric effects and the toroidal eigen. modes with anisotropic structures The tokamak with con. A Orbit averaging, centric flux surface is described by magnetic coordinates. Canonical perturbation theory for a periodic Hamiltonian r where r is the radial coordinate labeling the flux. system22 24 shows that orbit averaging follows strictly surfaces is the poloidal angle and is the toroidal angle. from the existence of a time scale separation between respectively Representative parameters of DIII D tokamak. equilibrium and perturbed motion The orbit averaged theory H mode core plasmas44 has been used in the simulation. has been well established and widely applied in plasma which have a peak ITG at a radial position r 0 5a with the. physics 7 10 25 39 However in a recent letter 40 the authors following local parameters R0 LT 6 9 R0 Ln 2 2 q 1 4. present a heuristic argument and simulation results contra s r q dq dr 0 78 Te Ti 1 a R0 0 36 and. dicting to the existing literatures 7 10 25 30 that the concept of a i 500 Here R0 is the major radius a is the minor radius. orbit averaging is invalid for the calculation of turbulent LT d ln T dr 1 and Ln d ln n dr 1 are the temperature. transport of energetic particles in a tokamak plasma if the and density gradient scale lengths Ti and Te are the ion and. particle orbit size r is larger than the turbulence eddy size electron temperatures q is the safety factor s is the magnetic. c This claim is of both fundamental and practical signifi shear is the inverse large aspect ratio i vi i is the. cance Fundamentally since the orbit averaging effect is a gyroradius of thermal ions vi is the thermal velocity and. necessary result of the orbit averaged theory the claim im i eB mic is the gyrofrequency These parameters give rise. plies that the orbit averaged theory is only valid if c r to a strong ITG instability Our global simulations uses. Eqs 3 and 4 of Ref 40 The requirement of the spatial simplified physics models including a parabolic profile of. scale separation contradicts the textbook notion22 24 that only q 0 854 2 184 r a 2 a temperature gradient profile of. the time scale separation is required for the validity of the exp r 0 5a 0 28a 6 a circular cross section electro. orbit averaged theory 7 10 25 39 Practically the requirement of static fluctuations with an adiabatic electron response fixed. the spatial scale separation leads to an energy scaling in Ref boundary conditions with electrostatic potential 0 en. 40 different from that of the orbit averaged theory10 30 for the forced at r 0 1a and r 0 9a no externally driven plasma. turbulent transport of energetic trapped particles in a burning flows or collisions 45 and an effective collision operator mod. plasma such as ITER eling a heat bath21 to prevent the relaxation of the tempera. From the point of view of the canonical perturbation ture profile The computational mesh consists of 32 parallel. theory 22 24 the guiding center drift orbit averaged theory is grids and a set of 384 3072 unstructured radial and poloi. identical conceptually and mathematically to the gyroaver dal grids with a perpendicular grid size of i These simula. aged theory Requiring c r for the orbit averaged theory tions evolved 4 108 thermal particles gyrocenters for. is equivalent to requiring c for the gyrokinetic theory 5000 time steps along with their interactions with the self. is the gyroradius which clearly contradicts the gyroki consistent electrostatic potential represented on the 4 107. netic theory 35 39 spatial grid points The default time step is 0 1LT vi. We clarify that the orbit averaged theory is valid for the The marker temperature and density are set up uniformly. turbulent transport of energetic particles in tokamak and in these simulations where very small random fluctuations. demonstrate that the energy scaling of the turbulent transport are launched upon start and grows exponentially due to the. predicted by the orbit averaged theory is consistent with re ITG instability as evidenced in the early time history of the. sults from large scale gyrokinetic particle simulations There ion heat conductivity shown in the lower panel of Fig 2. fore the heuristic claim on the spatial scale separation for Zonal flows are then generated through modulational. the orbit averaging and the associated energy scaling of instability 46 47 saturate the ITG instabilities after the time of. Author complimentary copy Redistribution subject to AIP license or copyright see http php aip org php copyright jsp. 055902 3 Scalings of energetic particle transport Phys Plasmas 17 055902 2010. FIG 1 The time history of heat flux between GTC Ref 19 solid and. XGC Refs 51 and 52 dotted for a i 250 and without heat source. t 250LT vi through random shearing of the zonal flows 48. Finally the nonlinear coupling of ITG zonal flows leads to a. fully developed turbulence after t 400LT vi with a steady. state transport level which is insensitive to the initial condi. tions and is characterized by an averaged ion heat conduc. tivity of i 3 1 GB over t 400 1000 LT vi The gyro, Bohm unit for the heat conductivity of the thermal ions are. defined as GB B where i a and B cTe eB, Here c and e are the speed of light and electron charge. respectively and B is the on axis magnitude of the magnetic FIG 2 Color Lower panel Time history of thermal ion heat conductivity. field The fluctuations in the steady state are nearly isotropic i driven by ITG turbulence solid or by particle noise dashed measured. in radial and poloidal directions21 with the perpendicular in different runs with time step t 0 2LT vi black t 0 1LT vi blue. and t 0 05LT vi red Upper panel Time history of radial excursion r2. spectrum peaks around k i 0 2 GTC simulations of these. for particle energy E Te 1 black 2 blue 4 green and 16 orange. ITG turbulence with similar parameters studying the when ITG turbulence is present solid or absent dashed where the time. energetic particle transport by microturbulence 10 the trans step is t 0 1LT vi. port scaling with respect to the device size20 and the turbu. lence spreading21 that underlies the transition of the scaling. have previously been carried out with extensive numerical support object composition very well and polymorphism can. convergences and cross code benchmarks 49 In particular be implemented manually 53. convergence with respect to the number of particles has been In object oriented design user defined types are created. carefully studied to ensure that the particle noise does not which provide a kind of basis set for structuring the code. affect the physics being studied We found that the noise data Each type should represent the important concepts ab. driven flux50 is consistently smaller than the ITG driven flux stractions used in the code The data inside the type is nor. by at least an order of magnitude when using 20 particles per mally kept secret encapsulated and changeable while the. cell as shown in the lower panel of Fig 2 Furthermore Fig public usage interface of the procedures using the type is. 1 shows the benchmark between GTC Ref 19 and X point kept simple and stable A class in FORTRAN90 consists of a. included Guiding Center Code XGC 51 52 where a similar module that contains a derived type definition and the pro. linear growth rate and steady state turbulence level is dis cedures which use that type plus any static data shared by. played for the case of a i 250 and without heat source the class An object is a variable of the derived type and a. constructor is a procedure which assigns an initial value to. an object Generally data cannot be modified outside this. B Multispecies simulation capability, class except by calling procedures provided by the author of.

The GTC code has grown over the years from a single this class As a result an author of a class can make changes. developer code designed for a specific problem to a promi to its inner structure without interfering with other classes. nent code with many users and contributors in the magnetic and authors. fusion energy community It became imperative to reengi In order that the code could be used for production while. neer refactor the code to allow multiple authors to contrib the changes were made we proceeded in stages Initially an. ute independently and to extend its capabilities to support upper layer of classes was implemented which called the. new physics such as the simulation discussed in this paper original GTC code under the hood The derived types. This was done by applying object oriented design principles pointed at existing data in GTC and the procedures called. to the FORTRAN90 code Although FORTRAN90 is an object the original GTC subroutines The original GTC code was. based language that does not support inheritance it does modified only slightly This allowed us to define and refine a. Author complimentary copy Redistribution subject to AIP license or copyright see http php aip org php copyright jsp. 055902 4 Zhang et al Phys Plasmas 17 055902 2010, set of classes which described the important features of the. GTC code in a new more abstract way with a minimum of. implementation The classes could be added one at a time. and at the end of each day the code continued to work cor. rectly Ten classes were ultimately defined and implemented. gyrokinetic and drift kinetic particles scalar and vector. fields a mesh Jacobian finite Lamor radius FLR equilib. rium interpolation and particle descriptors These classes. provided a stable interface to future modifications. In the second stage of this development some of the. scaffolding was removed The constructors in the class ob. jects now allocate the data and the array data in the original. GTC code was removed This isolates the components and. allows multiple instantiations of the objects to be created. such as the multiple ion species used in the simulation dis FIG 3 Radial profiles of thermal ion heat conductivity i and ITG intensity. cussed here Again the work was done incrementally of fluctuating potential I e T 2 where i a. one class at a time so that the code was always working. The third stage of the work still ongoing was to inte same figure The equilibrium spreads approach quickly to. grate the capabilities of the various versions of the GTC code fixed amplitudes that are proportional to the energy as. into one flexible and extensible version To do this we de expected This relaxation takes several bounce times. veloped a methodology to implement design patterns in 2 qR v which is inversely proportional to the. FORTRAN90 Design patterns are abstract solutions to ge square root of energy as clearly shown in Fig 2 In the linear. neric programming problems which allow one to handle in regime t 200LT vi the turbulence is so weak that the. creased complexity As an example the strategy pattern was equilibrium excursions dashed lines are identical to the per. used in GTC to support multiple solvers turbed excursions solid lines for all energy groups After. the turbulence reaches the steady state t 400LT vi in the. nonlinear regime the difference between the equilibrium and. III SCALINGS OF ENERGETIC PARTICLE, perturbed excursions is relatively small for high energy ions. e g E T 16 indicating that the turbulence exerts little. A Statistics of radial diffusion and transport influence on the high energy orbits On the other hand for a. Scalings of energetic particle transport by ion temperature gradient At high energy regime the energetic particle trans port scales as E T 1 for purely passing particles due to the orbit averaging and fast decorrelation of parallel resonance E T 2 for trapped particles due to gyroaveraging banana orbit averaging and wave particle decorrelation of drift bounce resonance and E T

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