Simulation studies of plasma waves in the electron foreshock: The generation of Langmuir waves by a gentle bump‐on‐tail electron distribution

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Abstract
The generation of Langmuir waves by a gentle bump‐on‐tail electron distribution is analyzed. It is shown that with appropriately designed simulation experiments, quasi‐linear theory can be quantitatively verified for parameters corresponding to the electron foreshock. The distribution function develops a plateau by resonant diffusion, and changes outside this velocity range are negligible, except for the contribution of nonresonant diffusion to acceleration of bulk electrons. The dispersion relation is solved for the evolving distribution function and exhibits the dynamics of wave growth and changes in real frequency. The integral of the quasi‐linear equations is also used to relate the evolution of distribution function and wave spectrum and gives agreement with the simulations. Even in extremely long simulation runs there is practically no evolution in wave energy or the distribution function, once a plateau has been formed. The saturated field levels are much lower than the estimates that are generally used to assess the importance of additional weak or strong turbulence effects. These effects cannot prevent plateau formation and are only noticeable if ions are also included in the model. They then lead to a redistribution of the spectrum toward low wave number modes which propagate mainly opposite to the beam. This occurs long after plateau formation and plays no significant role in the overall system dynamics or energy balance. One will have to live with quasi‐linear theory as a key ingredient for a global model of foreshock wave phenomena.