Georg Knorr

The integration of the vlasov equation in configuration space

Abstract A convenient, fast, and accurate method of solving the one-dimensional Vlasov equation numerically in configuration space is described. It treats the convective terms in the x and v directions separately and produces a scheme of second order in Δt . The resulting free-streaming and accelerating equations are computed with Fourier interpolation and spline interpolation methods respectively. The numerical method is tested witth linear and nonlinear problems. The method is very accurate and efficient. A new method of smoothing the distribution function is given. It reduces the computational effort by artificially increasing the entropy of the system. As a result, the distribution function is smooth enough to be well represented on a given mesh. The methods can be generalized in a straightforward way to deal with more complicated cases such as problems with nonperiodic spatial boundary conditions, two- and three-dimensional problems with and without external magnetic and/or electric fields.

Existence and stability of strong potential double layers

An elementary integral equation technique is used to construct strong and weak stationary shock solutions from the one-dimensional Vlasov equation. It is shown that the plasma is Penrose stable in all points in space under certain conditions.

Two‐dimensional turbulence in inviscid fluids or guiding center plasmas

Analytic theory for two‐dimensional turbulent equilibria for the inviscid Navier–Stokes equation (or the electrostatic guiding center plasma) is tested numerically. A good fit is demonstrated for the approach to a predicted energy per Fourier mode obtained from the two‐temperature canonical ensemble of Kraichnan: 〈‖u(k) ‖2〉 = (α+βk2)−1, where k is the wavenumber and α and β are reciprocal energy and enstrophy temperatures. Negative as well as positive temperature regimes are explored. Fluctuations about the mean energy per mode also compare well with theory. In the regime α<0, β≳0, with the minimum value of α+βk2 near zero, contour plots of the stream function reveal macroscopic vortex structures similar to those seen previously in discrete vortex simulations by Joyce and Montgomery. Kraichnan’s assertion that thermodynamic limits exist for the negative temperature states is questioned. Eulerian direct interaction equations, which can be used to follow the approach to inviscid equilibrium, are derived.

NUMERICAL INTEGRATION METHODS OF THE VLASOV EQUATION.

Abstract The methods of integrating the nonlinear Vlasov equation are reviewed, compared and interrelations are investigated. Another method is given which allows a truncation of the resulting infinite matrix without causing numerical instabilities. Its application to the linear and nonlinear Vlasov equation is discussed. It is shown that the cause for numerical instability is based on approximating a continuous eigenvalue spectrum by a discrete spectrum.

Numerical integration of the Vlasov equation

Abstract This work describes a numerical method for the solution of the nonlinear Vlasov equation. The distribution function is expanded in a series of orthogonal polynomials, namely the Chebyshev polynomials. The conditions under which this expansion is valid are discussed. Recurrence effects are eliminated by formally adding a damping term to the eigenvalues of the truncated system. Nonlinear effects have been simulated by an amount of information corresponding to less than 500 “particles.”

Nonlinear Behavior of the One‐Dimensional Weak Beam Plasma System

The effect of a weak beam of electrons on a Maxwellian plasma is studied by numerically integrating the Vlasov equation in one dimension. The solution is carried past the point of maximum electrostatic energy. After reaching its maximum value, the electrostatic energy fluctuates about its maximum. Some modes which are linearly stable grow due to mode coupling with the unstable modes. The relative amount of energy in the fastest growing mode is shown to depend on the periodicity length of the system.

Finite Larmor radius effects to arbitrary order

A representation of a finite Larmor radius plasma is proposed, which permits the transition rL → ∞ without becoming mathematically ill-posed. It is being used in a two-dimensional guiding center plasma spectral code and may have useful analytical applications. The ions are represented as guiding centers and the Larmor radius is averaged analytically for every Fourier-mode. Finite Larmor radius densities and velocities are thus obtained from guiding center quantities by application of a filter in wave vector space.

Plasma simulation with few particles.

Abstract A numerical method of plasma simulation is described which allows one to simulate Vlasov plasmas with very few degrees of freedom. Linear effects can be simulated by an amount of information corresponding to less than 100 particles, nonlinear effects require less than 500 “particles” in one spatial and one velocity dimension. The method is based on replacing the infinite real eigenvalue spectrum of the free streaming terms of the Vlasov equation by a finite discrete spectrum. An imaginary damping term is artificially added to the eigenvalues, and, thus, recurrence effects are minimized.

Use of the spectral method for two- and three-dimensional guiding center plasmas

Abstract The spectral method is shortly reviewed and then applied to equations of two- and three-dimensional guiding center plasmas. Some linear and nonlinear phenomena have been studied. It is shown that the spectral method can be used on a wide range of problems.

The role of fast electrons for the confinement of plasma by magnetic cusps

The influence of fast primary electrons on the leak width of a plasma in a picket fence or cusp geometry in the presence of a neutral gas background is discussed theoretically. Without primary electrons a leak width of the order of an ion Larmor radius (ai) is found, whereas with fast primary electrons a leak width of the order of a hybrid width (2 square root aiae) can be obtained. This is in agreement with experimental data in the literature.