Factorizing a String into Squares in Linear Time

A square factorization of a string w is a factorization of w in which each factor is a square. Dumitran et al. [SPIRE 2015, pp. 54-66] showed how to find a square factorization of a given string of length n in O(n log n) time, and they posed a question whether it can be done in O(n) time. In this paper, we answer their question positively, showing an O(n)-time algorithm for square factorization in the standard word RAM model with machine word size omega = Omega(log n). We also show an O(n + (n log^2 n) / omega)-time (respectively, O(n log n)-time) algorithm to find a square factorization which contains the maximum (respectively, minimum) number of squares

Kinetic Simulations of Neoclassical and Anomalous Transport Processes in Helical Systems

Drift kinetic and gyrokinetic theories and simulations are powerful means for quantitative predictions of neoclassical and anomalous transport fluxes in helical systems such as the Large Helical Device (LHD). The δf Monte Carlo particle simulation code, FORTEC-3D, is used to predict radial profiles of the neoclassical particle and heat transport fluxes and the radial electric field in helical systems. The radial electric field profiles in the LHD plasmas are calculated from the ambipolarity condition for the neoclassical particle fluxes obtained by the global simulations using the FORTEC-3D code, in which effects of ion or electron finite orbit widths are included. Gyrokinetic Vlasov simulations using the GKV code verify the theoretical prediction that the neoclassical optimization of helical magnetic configuration enhances the zonal flow generation which leads to the reduction of the turbulent heat diffusivity χi due to the ion temperature gradient (ITG) turbulence. Comparisons between results for the high ion temperature LHD experiment and the gyrokinetic simulations using the GKV-X code show that the χi profile and the poloidal wave number spectrum of the density fluctuation obtained from the simulations are in reasonable agreements with the experimental results. It is predicted theoretically and confirmed by the linear GKV simulations that the E × B rotation due to the background radial electric field Er can enhance the zonal-flow response to a given source. Thus, in helical systems, the turbulent transport is linked to the neoclassical transport through Er which is determined from the ambipolar condition for neoclassical particle fluxes and influences the zonal flow generation leading to reduction of the turbulent transport. In order to investigate the Er effect on the regulation of the turbulent transport by the zonal flow generation, the flux-tube bundle model is proposed as a new method for multiscale gyrokinetic simulations

Polarization and magnetization in collisional and turbulent transport processes

Expressions of polarization and magnetization in magnetically confined plasmas are derived, which include full expansions in the gyroradius to treat effects of both equilibrium and microscopic electromagnetic turbulence. Using the obtained expressions, densities and flows of particles are related to those of gyrocenters. To the first order in the normalized gyroradius expansion, the mean part of the particle flow is given by the sum of the gyrocenter flow and the magnetization flow, which corresponds to the so-called magnetization law in drift kinetics, while the turbulent part contains the polarization flow as well. Collisions make an additional contribution to the second-order particle flow. The mean particle flux across the magnetic surface is of the second-order, and it contains classical, neoclassical, and turbulent transport processes. The Lagrangian variational principle is used to derive the gyrokinetic Poisson and Ampère equations, which properly include mean and turbulent parts so as to be useful for full-f global electromagnetic gyrokinetic simulations. It is found that the second-order Lagrangian term given by the inner product of the turbulent vector potential and the drift velocity consisting of the curvature drift and the ∇B drift should be retained in order for the derived Ampère equation to correctly include the diamagnetic current, which is necessary especially for the full-f high-beta plasma simulations. The turbulent parts of these gyrokinetic Poisson and Ampère equations are confirmed to agree with the results derived from the WKB representation in earlier works

The Eulerian variational formulation of the gyrokinetic system in general spatial coordinates

The Eulerian variational formulation of the gyrokinetic system with electrostatic turbulence is presented in general spatial coordinates by extending our previous work [H. Sugama et al., Phys. Plasmas 25, 102506 (2018)]. The invariance of the Lagrangian of the system under an arbitrary spatial coordinate transformation is used to derive the local momentum balance equation satisfied by the gyrocenter distribution functions and the turbulent potential, which are given as solutions of the governing equations. In the symmetric background magnetic field, the derived local momentum balance equation gives rise to the local momentum conservation law in the direction of symmetry. This derivation is in contrast to the conventional method using the spatial translation in which the asymmetric canonical pressure tensor generally enters the momentum balance equation. In the present study, the variation of the Lagrangian density with respect to the metric tensor is taken to directly obtain the symmetric pressure tensor, which includes the effect of turbulence on the momentum transport. In addition, it is shown in this work how the momentum balance is modified when the collision and/or external source terms are added to the gyrokinetic equation. The results obtained here are considered useful for global gyrokinetic simulations investigating both neoclassical and turbulent transport processes even in general non-axisymmetric toroidal systems

Radially local approximation of the drift kinetic equation

A novel radially local approximation of the drift kinetic equation is presented. The new drift kinetic equation that includes both E×B and tangential magnetic drift terms is written in the conservative form and it has favorable properties for numerical simulation that any additional terms for particle and energy sources are unnecessary for obtaining stationary solutions under the radially local approximation. These solutions satisfy the intrinsic ambipolarity condition for neoclassical particle fluxes in the presence of quasisymmetry of the magnetic field strength. Also, another radially local drift kinetic equation is presented, from which the positive definiteness of entropy production due to neoclassical transport and Onsager symmetry of neoclassical transport coefficients are derived while it sacrifices the ambipolarity condition for neoclassical particle fluxes in axisymmetric and quasi-symmetric systems

Improved linearized model collision operator for the highly collisional regime

The linearized model collision operator for multiple species plasmas given by Sugama et al. [Phys. Plasmas 16, 112503 (2009)] is improved to be properly applicable up to the highly collisional regime. The improved linearized model operator retains the conservation laws of particles, momentum, and energy, and it reproduces the same friction-flow relations as derived by the linearized Landau operator so that this model can be used to correctly evaluate neoclassical transport fluxes in all collisionality regimes. The adjointness relations and Boltzmann’s H-theorem are exactly satisfied by the improved operator except in the case of collisions between unlike particle species with unequal temperatureswhere these relations and H-theorem still hold approximately because there is a large difference between the masses of the two species with significantly different temperatures. Even in the unequal-temperature case, the improved operator can also be modified so as to exactly satisfy the adjointness relations, while it causes the values of the friction coefficients to deviate from those given by the Landau operator. In addition, for application to gyrokinetic simulations of turbulent transport, the improved operator is transformed into the gyrophaseaveraged form by keeping the finite gyroradius effect