An alternative approach to field-aligned coordinates for plasma turbulence simulations

Turbulence simulation codes can exploit the flute-like nature of plasma turbulence to reduce the effective number of degrees of freedom necessary to represent fluctuations. This can be achieved by employing magnetic coordinates of which one is aligned along the magnetic field. This work presents an approach in which the position along the field lines is identified by the toroidal angle, rather than the most commonly used poloidal angle. It will be shown that this approach has several advantages. Among these, periodicity in both angles is retained. This property allows moving to an equivalent representation in Fourier space with a reduced number of toroidal components. It will be shown how this duality can be exploited to transform conventional codes that use a spectral representation on the magnetic surface into codes with a field-aligned coordinate. It is also shown that the new approach can be generalised to get rid of magnetic coordinates in the poloidal plane altogether, for a large class of models. Tests are carried out by comparing the new approach with the conventional approach employing a uniform grid, for a basic ion temperature gradient (ITG) turbulence model implemented by the two corresponding versions of the ETAI3D code. These tests uncover an unexpected property of the model, that localized large parallel gradients can intermittently appear in the turbulent regime. This leaves open the question whether this is a general property of plasma turbulence, which may lead one to reconsider some of the usual assumptions on micro-turbulence dynamics.Comment: 19 pages (once in pdf format). 1 LaTeX file and 10 eps figures in the zip folde

Gyrokinetic Equations for Strong-Gradient Regions

A gyrokinetic theory is developed under a set of orderings applicable to the edge region of tokamaks and other magnetic confinement devices, as well as to internal transport barriers. The result is a practical set equations that is valid for large perturbation amplitudes [q{\delta}{\psi}/T = O(1), where {\delta}{\psi} = {\delta}{\phi} - v_par {\delta}A_par/c], which is straightforward to implement numerically, and which has straightforward expressions for its conservation properties. Here, q is the particle charge, {\delta}{\phi} and {\delta}A_par are the perturbed electrostatic and parallel magnetic potentials, v_par is the parallel velocity, c is the speed of light, and T is the temperature. The derivation is based on the quantity {\epsilon}:=({\rho}/{\lambda})q{\delta}{\psi}/T << 1 as the small expansion parameter, where {\rho} is the gyroradius and {\lambda} is the perpendicular wavelength. Physically, this ordering requires that the E\times B velocity and the component of the parallel velocity perpendicular to the equilibrium magnetic field are small compared to the thermal velocity. For nonlinear fluctuations saturated at "mixing-length" levels (i.e., at a level such that driving gradients in profile quantities are locally flattened), {\epsilon} is of order {\rho}/L, where L is the equilibrium profile scale length, for all scales {\lambda} ranging from {\rho} to L. This is true even though q{\delta}{\psi}/T = O(1) for {\lambda} ~ L. Significant additional simplifications result from ordering L/R =O({\epsilon}), where R is the spatial scale of variation of the magnetic field. We argue that these orderings are well satisfied in strong-gradient regions, such as edge and screapeoff layer regions and internal transport barriers in tokamaks, and anticipate that our equations will be useful as a basis for simulation models for these regions.Comment: Accepted for publication in the Physics of Plasmas, 12/30/201

An Asymptotic Preserving Scheme for the Euler equations in a strong magnetic field

This paper is concerned with the numerical approximation of the isothermal Euler equations for charged particles subject to the Lorentz force. When the magnetic field is large, the so-called drift-fluid approximation is obtained. In this limit, the parallel motion relative to the magnetic field direction splits from perpendicular motion and is given implicitly by the constraint of zero total force along the magnetic field lines. In this paper, we provide a well-posed elliptic equation for the parallel velocity which in turn allows us to construct an Asymptotic-Preserving (AP) scheme for the Euler-Lorentz system. This scheme gives rise to both a consistent approximation of the Euler-Lorentz model when epsilon is finite and a consistent approximation of the drift limit when epsilon tends to 0. Above all, it does not require any constraint on the space and time steps related to the small value of epsilon. Numerical results are presented, which confirm the AP character of the scheme and its Asymptotic Stability

Particle Simulation of Coulomb Collisions: Comparing the Methods of Takizuka & Abe and Nanbu

The interactions of charged particles in a plasma are in a plasma is governed by the long-range Coulomb collision. We compare two widely used Monte Carlo models for Coulomb collisions. One was developed by Takizuka and Abe in 1977, the other was developed by Nanbu in 1997. We perform deterministic and stochastic error analysis with respect to particle number and time step. The two models produce similar stochastic errors, but Nanbu's model gives smaller time step errors. Error comparisons between these two methods are presented

Suppression of turbulence and subcritical fluctuations in differentially rotating gyrokinetic plasmas

Differential rotation is known to suppress linear instabilities in fusion plasmas. However, even in the absence of growing eigenmodes, subcritical fluctuations that grow transiently can lead to sustained turbulence. Here transient growth of electrostatic fluctuations driven by the parallel velocity gradient (PVG) and the ion temperature gradient (ITG) in the presence of a perpendicular ExB velocity shear is considered. The maximally simplified case of zero magnetic shear is treated in the framework of a local shearing box. There are no linearly growing eigenmodes, so all excitations are transient. The maximal amplification factor of initial perturbations and the corresponding wavenumbers are calculated as functions of q/\epsilon (=safety factor/aspect ratio), temperature gradient and velocity shear. Analytical results are corroborated and supplemented by linear gyrokinetic numerical tests. For sufficiently low values of q/\epsilon (<7 in our model), regimes with fully suppressed ion-scale turbulence are possible. For cases when turbulence is not suppressed, an elementary heuristic theory of subcritical PVG turbulence leading to a scaling of the associated ion heat flux with q, \epsilon, velocity shear and temperature gradient is proposed; it is argued that the transport is much less stiff than in the ITG regime.Comment: 36 pages in IOP latex style; 12 figures; submitted to PPC