Recently-developed theoretical insights and numerical methods will be used to generate and solve new mathematical models for interpreting measured spatial and temporal distributions of waves and associated particle velocity distributions in the magnetosphere and heliosphere. Emphasis will be on the evolution of beam-excited waves, wave-accelerated particles, and beam relaxation, using kinetic, fluid and hybrid models. Innovative higher dimensional two-time-scale fluid codes designed to mimic linear and nonlinear kinetic effects will be developed and solved on the NCAR supercomputers, with lower dimensional versions tested against Vlasov codes. In collaboration with experiments from the Berkeley Space Sciences Laboratory, theoretical models will be constructed and numerically analyzed, and the results compared with in-situ rocket and satellite measurements of electrons, ions, magnetized- electron-plasma waves and lower-hybrid wave structures in the magnetosphere. Relevant magnetospheric wave and particle data will come from the Alaska-88, Bidarca and Alaska-95 rockets, the imminent Fast Explorer and satellite, the Swedish satellite Freja, and the Cornell Topaz rocket campaigns. We will construct and numerically analyze closely related mathematical models, which describe similar fundamental underlying processes common to beam- driven Langmuir wave turbulence found in the electron foreshock of Earth's bow shock and in the solar wind, during solar radio emissions, and postulated as the source of the heliospheric emission measured by Voyager 1 and 2. Theoretical predictions will be compared with high-time resolution data on particle distributions and Langmuir waves in the electron foreshock and in the solar wind expected from the WIND satellite, to be launched in November of 1994.