Magdi Shoucri

Eulerian Vlasov codes

Abstract Four different Eulerian grid-based Vlasov solvers are discussed, namely a second order method and a fourth order method (symplectic integrator) using cubic splines for interpolation, the CIP (cubic interpolated propagation) method, and an Euler–Lagrange method applying two-dimensional cubic interpolants. The four methods will be presented by outlining their algorithm. The performance of the numerical methods will be compared by numerically solving the Vlasov–Poisson system for the distribution function on a fixed Eulerian grid, for the problem of a two-stream instability in a two-dimensional phase-space.

Study of the generation of a charge separation and electric field at a plasma edge using Eulerian Vlasov codes in cylindrical geometry

A one-dimensional (1D) Eulerian Vlasov code in cylindrical geometry is used to study the problem of the formation of a charge separation at a plasma edge, together with the self-consistent electric field, in the presence of a density gradient. The code applies a method of fractional steps for the solution of the Vlasov equation. The electric field calculated along the gradient is compared with the macroscopic values calculated from the same kinetic code for the gradient of the ion pressure and the Lorentz force term. We find that these quantities balance the electric field fairly well along the gradient. The extension of this method to a two-dimensional (2D) problem will be presented, together with the corresponding results.

High-Q Thermally Stable Operation of a Tokamak Reactor

Thermally stable subignited operation of a tokamak reactor, sustained in operation by a feedback-controlled supplementary heating source, is discussed. One-dimensional (radial) thermal stability analyses of model transport equations, together with numerical results from a one-dimensional (1-D) transport code, are used in studying the heating of deuterium-tritium (D-T) plasmas in the thermonuclear regime. The establishment of stability depends on a number of radially nonuniform nonlinear processes whose effect is analyzed. Nonuniform heat deposition resulting from plasma core supplementary heating is found to be a thermally more stable process than bulk heating. In the presence of impurity line radiation, however, core-heated temperature profiles may collapse, contracting inward from the limiter, the result of a radiation-induced instability. The effect of nonuniform transport coefficients is also discussed. Conditions are established for the realization of a subignited high-Q (Q ¿ 50) toroidal reactor plasma with appreciable output power (¿2000 MW thermal).

Transient response of a free‐burning discharge: Quantitative model

The work presented here deals with the application of the theoretical model of the nonequilibrium discharge developed by one of us to a free‐burning discharge in argon subjected to a current impulse. The paper includes a description of the system of transport equations and a discussion of the boundary conditions and numerical solutions. The response to a 0.5‐μs current ramp from 0.4 to 1.2 A is studied in detail. The variation of the basic parameters, such as the electron density, temperature, heat flux and diffusion velocity, the heavy‐particle temperature, and heat flux and the average mass velocity is shown as a function of time and space. Radial profiles of the initial and final states are shown to agree well with the results of the earlier steady‐state model.

The application of a fractional steps method for the numerical solution of the shallow water equations

A fractional steps technique is applied for the numerical solution of the shallow water equations. The method has the great advantage of solving the shallow water equations without the iterative steps involved in the multi-dimensional interpolation problems, and without the iteration associated with the intermediate step of solving the Helmholtz equation. The absence of iterative steps in the present technique makes it very suitable for problems in which small time steps and grid sizes are required (as in regional climate modeling), and the simplicity of the method makes it suitable to parallel computer.

Numerical solution of the Fokker-Planck equation with a dc electric field

Abstract Using the IMSL package TWODEPEP, the full non-relativistic Fokker-Planck equation is solved numerically for the runaway electron distribution function in the presence of a dc electric field. A set of partial integro-differential equations derived from a Legendre expansion of the Fokker-Planck equation is solved simultaneously for the electron distribution parts up to f 3 with f 0 non-Maxwellian and including the full electron-electron collision operator. The runaway distribution function and its moments yielding the non-linear conductivity, runaway rate, temperature and drift velocity are presented.

Different Fokker-Planck approaches to simulate electron transport in plasmas

Abstract An evaluation of different approximations of the Rosenbluth potentials is performed. A comparison is made between the Legendre expansion used in the FPI code with spherical Rosenbluth potentials, and an improved version (FPI+) with semi-anisotropic Rosenbluth potentials, and the new ( v , μ ) Fokker–Planck code “FPTrans” with fully anisotropic Rosenbluth potentials, for the problem of the relaxation of a bi-Maxwellian ( T ⊥ ≠ T ∥ ) electron distribution function, which confirms the validity of the semi-anisotropic approximation, but not of the isotropic one. The improved code FPI+ was also used to compute new propagators for nonlocal electron heat transport.

Numerical Solution of the Two-Dimensional Vlasov Equation

The two-dimensional Vlasov equation is solved by direct integration in phase space. Two problems, namely the nonlinear evolution of the two-dimensional electrostatic two-stream instability, and the nonlinear evolution of a monochromatic wave in a two-dimensional Vlasov plasma, are studied. Comparison with previously available results is given.

Relativistic Eulerian Vlasov simulations of the amplification of seed pulses by Brillouin backscattering in plasmas

We apply an Eulerian Vlasov code to study the amplification by Brillouin scattering of a short seed laser pulse by a long pump laser pulse in an underdense plasma. The stimulated Brillouin backscattering interaction is the coupling of the pump and seed electromagnetic waves propagating in opposite directions, and the ion plasma wave. The code solves the one-dimensional relativistic Vlasov-Maxwell set of equations. Large amplitude ion waves are generated. In the simulations we present, the density plateau of the plasma is ne=0.3 nc (nc is the critical density), which excludes spurious stimulated Raman scattering amplification (which can occur only if ne<nc/4). We also varied the duration and/or amplitude of the short input seed pulse to study how these influence its subsequent behaviour. An initially broad pulse grows more rapidly than an initially narrow pulse. Furthermore, for an initially broader seed pulse, towards the end of the simulation, it is seen to become narrower and to gradually detach from th...

Study of laser plasma interactions using an Eulerian Vlasov code

We present a one-dimensional Eulerian Vlasov code for performing kinetic simulations of laser-plasma interactions. We use the code to study parametric instabilities, in particular stimulated Raman scattering. For conditions similar to those of singlehot-spot experiments, we flnd that kinetic efiects are important in the saturation of this instability. Work is underway to make the code to 1 1 and 2D (resolving y and py) and parallelize it.