Thermodynamic bounds on the magnetic fluctuation energy in unstable anisotropic plasmas

Abstract

Thermodynamic bounds on the magnetic fluctuation energy in unstable anisotropic plasmas are obtained. The spatial variation in equilibrium and perturbed quantities is assumed to be exclusively in the z direction, which coincides with the direction of a uniform external magnetic field B₀ê_z. In addition, the positive ions are assumed to form a fixed (m_i→ ∞) background providing overall charge neutrality, and only the electron dynamics are included in the analysis. For this configuration it has recently been shown that the non-linear Vlasov–Maxwell equations support two independent energy constants for arbitrary-amplitude electromagnetic disturbances propagating parallel to B₀ ê_z. In this paper, Fowler's method for calculating thermodynamic bounds on the field energy in unstable plasmas is generalized to incorporate two energy constraints (in addition to entropy and number conservation). An upper bound on the magnetic fluctuation energy is obtained for arbitrary initial distribution function f_e0 = f_e(z, v, 0), and the results are applied to unstable (i) bi-Maxwellian and (ii) loss-cone distributions. Depending on the parameter regime, it is found that the bounds obtained by enforcing two energy constraints can be an order-of-magnitude lower than the bounds obtained by enforcing a single (total) energy constraint. The theoretical bounds are compa red with the maximum change in field energy measured in computer simulation experiments for the case where f_e0 is bi-Maxwellian. For sufficiently large initial anisotropy, it is found that the bound obtained with two energy constraints is less than 1·5 times the measured change in field energy.