The evolution of nonlinear Alfvén waves subject to growth and damping

<jats:p>Results of a numerical study of Alfvén waves are presented subject to nonlinearity, dispersion, growth, and damping. The model presented is the derivative nonlinear Schrödinger equation, modified to include linear growth and damping processes. The processes that are considered are wave amplification by streaming particle distributions, and damping resulting from ion-cyclotron resonance absorption. These growth and damping mechanisms are dominant in different portions of wavenumber space. The primary role of nonlinearity is the transfer of wave energy from growing or amplified wavenumbers to those which are damped. A nonlinear saturation mechanism thereby results, in which instability of low wavenumber modes may be quenched. A simple phenomenological model is developed, which accounts for many of the salient features of the numerical calculations. The application of these results to observations of Alfvén waves upstream of the Earth’s bow shock is briefly considered. It is suggested that the short wavelength ‘‘shocklet’’ structures resemble the soliton-like pulses that emerge from the driven derivative nonlinear Schrödinger equation. However, the nonlinear effects discussed in this paper do not seem responsible for limiting the amplitude of the ‘‘low-frequency’’ waves in the foreshock region.</jats:p>