On a model for weak turbulence in synchrotrons

In this work we formulate a theory of weak turbulence for the longitudinal degree of freedom in a stored, coasting beam. We employ a perturbative approach to the fully nonlinear Vlasov equation that is well suited to a set of well separated eigenfrequencies and study the case where the beam is marginally stable to longitudinal oscillations. We derive an averaged equation that describes the nonlinear flow of the fluctuation power among the multiplicity of harmonic modes, characterized by the properties of the machine impedance. In addition, we introduce the Schottky noise of the beam that acts as a thermal reservoir with which the wake-driven oscillations must come into equilibrium. We find steady-state solutions of the nonlinear equation set that indicate significant enhancement of the Schottky spectrum is possible.