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The effect of passing particles on the confinement of particles trapped in a magnetic mirror with confining electrostatic potential is examined. Exact relations between the particle and energy fluxes and the distribution functions for two auxiliary boundary value problems are derived. Approximate analytic expressions for these fluxes, asymptotically valid for large ratio of confining potential energy to trapped-particle temperature and either large or small mirror ratio, are obtained. The results are applied to the determination of the ambipolar potential and the temperature difference between trapped and untrapped electrons in a stream-stabilized conventional mirror machine such as 2XIIB and in the plug cells of a tandemmirror machine. An important result is that, under a wide range of conditions, the potential difference Φ between the plug and the barrier of a thermal-barrier tandem-mirror machine is given by eΦ = Tep ln [(np/nb) (Tec/Tep)1/2], a modification of the Maxwell-Boltzmann result.

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We report the results of a new, systematic study of nonlinear longitudinal wave propagation in a collisionless plasma. Based on the decomposition of the electric field E into a transient part T and a time-asymptotic part A, we show that A is given by a finite superposition of wave modes, whose frequencies obey a Vlasov dispersion relation, and whose amplitudes satisfy a set of nonlinear algebraic equations. These time-asymptotic mode amplitudes are calculated explicitly, based on approximate solutions for the particle distribution functions obtained by linearizing only the term that contains T in the Vlasov equation for each particle species, and then integrating the resulting equation along the nonlinear characteristics associated with A, which are obtained via Hamiltonian perturbation theory. For "linearly stable" initial Vlasov equilibria, we obtain a critical initial amplitude (or threshold), separating the initial conditions that produce Landau damping to zero (A [triple bond] 0) from those that lead to nonzero multiple-traveling-wave time-asymptotic states via nonlinear particle trapping (A not identical with 0). These theoretical results have important implications about the stability of spatially uniform plasma equilibria, and they also explain why large-scale numerical simulations in some cases lead to zero-field final states whereas in others they yield nonzero multiple-traveling-wave final states.