J. Killeen

Multi-species Fokker-Planck calculations for D-T and D-3He mirror reactors

Values of the confinement parameter ntau and the figure of merit Q (the ratio of thermonuclear power to injected power) are obtained for postulated reactor conditions. The heart of the work is the numerical solution, under tested assumptions, of the coupled Fokker-Planck equations for as many as six charged species. A study is made of D-T and D-3He fuel cycles, including reaction products, in the following parameter ranges: mirror ratio, 2<or=Rm<10; injection energy, 50 keV<or=E0<or=1500 keV; equilibrium 3He fraction for the D-3He cycle, 0.05<or=fHe<or=0.35. The computed Qs fall in the ranges 0.5<QDT<1.5, 0.1<QDHe<0.3. The present results are compared with previous work, with analytical/empirical approximations, and with a separate code treating velocity space more rigorously for a single positive species.

Effect of equilibrium flow on the resistive tearing mode

The resistive tearing instability of an incompressible plasma is investigated for the plane sheet pinch in which the equilibrium magnetic field, xBx0+ẑBz0, depends only on y. The usual assumption is to take v0=0, but here the effect of a nonzero v0 is studied. A linear, time‐dependent model is used in which perturbations take the form f1(y,t)exp [i (kxx+kzz)]. A new initial‐value code has been developed to solve the resulting higher‐order system of equations. For a symmetric magnetic equilibrium and modes α<1, where α=a (kx2+ky2)1/2, an exponential growth develops. The growth rate, p=ωτr, is computed as a function of α and S=τr/τh, for several values of v0. The effect is to reduce p for all α, and to reduce the marginal α for instability for values of v0 of the order of the resistive diffusion velocity. Results for larger values of v0 are briefly discussed. For asymmetric tearing, the effect of the diffusion velocity depends on its sign. The velocity may have either a stabilizing or destabilizing influen...

Computation of E‐Layer and Plasma Equilibria in Astron

The equilibrium magnetic field for astron configurations is computed; a given vacuum magnetic field and the fields due to the E layer and plasma are included, assuming axial symmetry and Bθ≡0. The components Br(r, z) and Bz(r, z) are derived from the stream function ψ(r, z) = rAθ. The E‐layer is described by a distribution function that depends on the canonical angular momentum and the energy. Various models of the E layer have been used and are discussed. The function ψ(r, z) satisfies a nonlinear partial differential equation (Amperes law) that is solved by finite‐difference methods, using an iterative procedure. The object is to find equilibrium solutions with a reversed field in the central region that can contain a plasma. Examples of reversed field equilibria corresponding to various E layer distributions are presented. Simple models for the plasma are assumed, and equilibria of the coupled system of E layer plus plasma are computed.

Non-linear saturation of the tearing mode in a reversed field pinch

A non-linear, two-dimensional, resistive MHD computer code to study the non-linear behaviour of the m = 0, 1, and 2 tearing modes in a reversed field pinch is used. For the cases m ≠ 0, a co-ordinate transformation which assumes helical symmetry is employed to reduce the dimensionality of the problem from three to two. The force-free Bessel function model equilibrium is used. It is found that the most dangerous resistive instability in this case is the m = 1 tearing mode, because of its larger growth rate and extended period of exponential growth.

The non-linear evolution of resistive interchange modes in a reversed-field pinch

A two-dimensional resistive MHD computer code is used to study the non-linear behaviour of resistive interchange modes in finite-β Reversed-Field Pinches (RFP). This model is applied to a specific equilibrium that is known to be stable to both resistive and ideal current-driven modes as well as Suydam modes. It is found that the m = 1 resistive interchange saturates at extremely low amplitude when the singular surface lies outside the field reversal point and is more active non-linearly, but still fairly localized, if the singular surface is inside the field null. In contrast, it is found that, at high β, the m = 0 resistive interchange can lead to significant flux surface distortion and interchange vortices of large radial extent. The effect of temperature-dependent resistivity on this mode is presented. In addition, the possibility of Ohmic heating of the pinch in the presence of the m = 0 resistive interchange mode is discussed. It is found that, if the plasma is initially in a low-β state, significant Ohmic heating can occur.

NUMERICAL SOLUTION OF THE FOKKER--PLANCK EQUATIONS FOR A HYDROGEN PLASMA FORMED BY NEUTRAL INJECTION.

Abstract We describe a program for the solution of the time-dependent Fokker-Planck equations for electron and ion distribution functions in velocity space. We consider the formation of a plasma in a magnetic mirror configuration by the injection of energetic neutral atoms. In the equations we have a source of energetic protons and cold electrons, and we include losses due to scattering and charge exchange. The coupled nonlinear partial differential equations are solved by an implicit difference method which is described. Numerical results are presented for two cases of interest.

Computation of Hydromagnetic Equilibria with Finite Pressure in Minimum‐B Systems

The distortion of a magnetic well in a minimum‐B system by a plasma is calculated. The equations of hydromagnetic equilibria with a tensor pressure are used. The pressure tensor is assumed to be a function of B only. The axisymmetric case is considered in detail by using a stream function for the magnetic field. The resulting partial differential equation is solved by finite‐difference methods. Numerical solutions are presented for the stuffed‐cusp configuration. The effect of increasing the magnitude of the plasma pressure is discussed.

Computer simulation of pulse trapping and pulse stacking of relativistic electron layers in astron

During the past two years computer simulation has been used extensively to obtain a detailed description of pulse trapping and pulse stacking of relativistic E layers in astron. Resistors are essential to good trapping, in agreement with experiment. In the code, pulse trapping can easily be arranged to be 100% efficient—in marked contrast to the experiment. Details of pulse stacking are dependent on resistor configuration, degree of charge neutralization, and external well shape, but the field reversal increase invariably runs into a saturation due to axial expansion of the layer. This process can be described as a phase space exclusion or, alternatively, as nonadiabatic axial heating. The pulse‐stacking process involves a tearing and bunching, and it is also nonadiabatic in the radial motion; as a consequence, radial expansion always occurs, and this can also act as a limitation to field reversal unless the resistor locations allow considerable radial room. Very tightly focused (axially) pulses can resul...

Numerical simulation of relativistic electrons confined in an axisymmetric mirror field

Abstract A time-dependent numerical model of the astron, with which injection and trapping can be studied in detail, has been developed. The effects due to the resistors and neutralization have been included. The model is axially symmetric. The E -layer electrons are simulated by many thousands of finite-size superparticles, which move in the r−z domain and have velocity components ν r , ν θ , and ν z . The model is relativistic and the electromagnetic fields are obtained by solving four wave equations — three for the vector potential and one for the scalar potential. The E -layer current and the current induced in the resistor wires are included in the above field equations. The computed self-fields are added to the external field to give the field configuration as a function of time. Results of multiple pulse injection are presented.

A computation for studying the formation of the relativistic electron layer in astron

Abstract In the controlled fusion experiment, Astron, relativistic electrons are injected at the end of a long cylindrical tank across an applied axial magnetic field. A rotating cylindrical sheath of electrons is formed. The self-consistent field problem is solved by integrating the time-dependent Vlasov-Maxwell equations numerically using finite-difference methods. The problem described is four dimensional and requires a large scale computing system for the solution of the difference equations.