Nonlinear evolution of firehose-unstable Afvén waves

The time evolution of firehose-unstable Alfvén waves is calculated within the usual weak-turbulence scheme. In § 2 the amplitude equations are established using a third-order solution of Vlasov's equation. From these a spectrum equation is obtained by discarding four-wave correlations; in addition, we derive an equation which characterizes the truncation error ( §3). Both these equations are integrated numerically, together with the velocity-moment equations (§ 4), forward in time. The results as presented in § 5 correspond to a rapid relaxation of the plasma to equilibrium. For large wave-vectors the relaxation time, as well as the equilibrium wave energy, are in good agreement with the quasilinear treatment, and the truncation error is small. But for low wave-numbers the relaxation is much faster, and the wave energy grows higher as predicted by quasilinear theory. This is because the nonlinear particle current leads to a high effective growth rate, especially at small wave-numbers. In this region the truncation error grows appreciably, and may sometimes reach the order of magnitude of the spectrum. But the overall picture as given by quasilinear theory has been confirmed. In § 6 comparison is made with a macroscopic model of Berezin & Sagdeev (1969), where the plasma noise was simulated by ‘computer noise’. For low amplitudes both methods agree qualitatively.