The damping of plasma oscillations in a weakly collisional plasma is revisited using a Fokker-Planck collision operator. It is shown that the Case-Van Kampen continuous spectrum is eliminated in the limit of zero collision frequency and replaced by a discrete spectrum. The Landau-damped solutions are recovered in this limit, but as true eigenmodes of the weakly collisional system. For small but nonzero collision frequency, the spectra and eigenmodes are qualitatively different from their counterparts in the collisionless theory. These results are consistent with recent experimental findings.

Turbulence is a ubiquitous phenomenon in space and astrophysical plasmas, driving a cascade of energy from large to small scales and strongly influencing the plasma heating resulting from the dissipation of the turbulence. Modern theories of plasma turbulence are based on the fundamental concept that the turbulent cascade of energy is caused by the nonlinear interaction between counterpropagating Alfvén waves, yet this interaction has never been observationally or experimentally verified. We present here the first experimental measurement in a laboratory plasma of the nonlinear interaction between counterpropagating Alfvén waves, the fundamental building block of astrophysical plasma turbulence. This measurement establishes a firm basis for the application of theoretical ideas developed in idealized models to turbulence in realistic space and astrophysical plasma systems.Introduction.-Turbulence profoundly affects many space and astrophysical plasma environments, playing a crucial role in the heating of the solar corona and acceleration of the solar wind [1], the dynamics of the interstellar medium [2][3][4], the regulation of star formation [5], the transport of heat in galaxy clusters [6], and the transport of mass and energy into the Earth's magnetosphere [7]. At the large length scales and low frequencies characteristic of the turbulence in these systems, the turbulent motions are governed by the physics of Alfvén waves [8], traveling disturbances of the plasma and magnetic field. Theories of Alfvénic turbulence based on idealized models, such as incompressible magnetohydrodynamics (MHD), suggest that the turbulent cascade of energy from large to small scales is driven by the nonlinear interaction between counterpropagating Alfvén waves [9][10][11][12]. However, the applicability of this key concept in the moderately to weakly collisional conditions relevant to astrophysical plasmas has not previously been observationally or experimentally demonstrated. Verification is important because the distinction between the two leading theories for strong MHD turbulence [11,12] arises from the detailed nature of this nonlinear interaction. Furthermore, verification is required to establish the applicability of turbulence theories, utilizing simplified fluid models such as incompressible MHD, to the weakly collisional conditions of diffuse astrophysical plasmas.

Landau damping and Bernstein-Greene-Kruskal ͑BGK͒ modes are among the most fundamental concepts in plasma physics. While the former describes the surprising damping of linear plasma waves in a collisionless plasma, the latter describes exact undamped nonlinear solutions of the Vlasov equation. There does exist a relationship between the two: Landau damping can be described as the phase mixing of undamped eigenmodes, the so-called Case-Van Kampen modes, which can be viewed as BGK modes in the linear limit. While these concepts have been around for a long time, unexpected new results are still being discovered. For Landau damping, we show that the textbook picture of phase mixing is altered profoundly in the presence of collision. In particular, the continuous spectrum of Case-Van Kampen modes is eliminated and replaced by a discrete spectrum, even in the limit of zero collision. Furthermore, we show that these discrete eigenmodes form a complete set of solutions. Landau-damped solutions are then recovered as true eigenmodes ͑which they are not in the collisionless theory͒. For BGK modes, our interest is motivated by recent discoveries of electrostatic solitary waves in magnetospheric plasmas. While one-dimensional BGK theory is quite mature, there appear to be no exact three-dimensional solutions in the literature ͑except for the limiting case when the magnetic field is sufficiently strong so that one can apply the guiding-center approximation͒. We show, in fact, that two-and three-dimensional solutions that depend only on energy do not exist. However, if solutions depend on both energy and angular momentum, we can construct exact three-dimensional solutions for the unmagnetized case, and two-dimensional solutions for the case with a finite magnetic field. The latter are shown to be exact, fully electromagnetic solutions of the steady-state Vlasov-Poisson-Ampère system.

Kinetic eigenmodes of plasma oscillations in a weakly collisional plasma, described by a collision operator of the Fokker-Planck type, are obtained in closed form for initial-value as well as for boundary-value problems. These eigenmodes, which are smooth and compose a complete discrete spectrum, play the same role for weakly collisional plasmas as the Case-Van Kampen modes do for collisionless plasmas.

Turbulence is a phenomenon found throughout space and astrophysical plasmas. It plays an important role in solar coronal heating, acceleration of the solar wind, and heating of the interstellar medium. Turbulence in these regimes is dominated by Alfvén waves. Most turbulence theories have been established using ideal plasma models, such as incompressible MHD. However, there has been no experimental evidence to support the use of such models for weakly to moderately collisional plasmas which are relevant to various space and astrophysical plasma environments. We present the first experiment to measure the nonlinear interaction between two counterpropagating Alfvén waves, which is the building block for astrophysical turbulence theories. We present here four distinct tests that demonstrate conclusively that we have indeed measured the daughter Alfvén wave generated nonlinearly by a collision between counterpropagating Alfvén waves.