The effects of velocity correlation times on the turbulent amplification of magnetic energy

This paper extends the quasilinear theory of Kulsrud & Anderson to assess the effects of realistically long velocity correlation times on the turbulent amplification of a very weak magnetic field. A computer simulation is presented that tracks the growth of the magnetic energy in a turbulent plasma at a single point moving with the turbulent flow. The velocities are assumed to conform to the ideas of Kraichnan concerning Lagrangian correlation times, and are modeled as a set of randomly generated pulses chosen to reproduce the correct two-time Lagrangian correlation tenser. The model is simple computationally and can be used to calculate the growth rate of the magnetic energy for arbitrarily high magnetic Reynolds numbers. The simulations show that the magnetic energy grows roughly half as fast as predicted in the short correlation time approximation of Kulsrud & Anderson's quasilinear theory. In a separate analysis, the effects of nonzero correlation times are considered using an analytic method developed by van Kampen. The growth rate is expanded, roughly speaking, in powers of the correlation time divided by the time required for the energy to exponentiate once. The first two terms in the series are calculated. In themselves, these two terms do not exactly determine the growth rate, but they are consistent with the numerical results. The analytic treatment is included mostly for completeness and because it offers some physical understanding of the problem. The main conclusion of the paper is that velocity correlation times do not play an important role in the growth of the magnetic energy. As a result, Kulsrud & Anderson's short correlation time analysis of the spectrum of amplified small-scale fields should be approximately correct.