Role of whistler instabilities in regulating the heat conductivity in laboratory astrophysics experiments at NIF and OMEGA

A recent Discovery Science campaign at the NIF using the TDYNO platform has demonstrated a significantly reduced level of heat transport in plasmas resembling the intracluster medium - at least 100x less than the classical Spitzer value. Similar experiments performed at the Omega Laser Facility did not exhibit such reduction. The leading hypothesis to explain this difference is that, unlike on OMEGA, the NIF experiments were able to generate a magnetized, weakly collisional plasma likely susceptible to the whistler heat flux instability. The fluctuations generated by this instability readily scatter heat-carrying electrons to pin the heat flux to a marginal value that is less than Spitzer by a factor of the plasma beta, similar to the reduction observed in the experiments. Although plausible, more theoretical work is required to make this conclusion definitive. Here we investigate whistler-type instabilities in plasmas of various magnetization levels. The Wigner-Moyal equation that governs (exactly) the instability dynamics is derived, allowing us to obtain the exact dispersion relation/growth rate as the wave Hamiltonian function. These quantities contain additional terms that previous studies were unaware of, whose importance shall be highlighted. The back-action of this instability on the equilibrium temperature profile is also derived to understand the saturated heat flux. This equation is manifestly energy conserving and its constitutive terms have clear physical interpretations.