Quality Metric for Spitzer–Braginskii and Grad 8 Moment Heat Flux Closures

Quality metrics for Spitzer–Härm and Grad closures are presented based on the percentage of the heat flux moment supported only by nonnegative, physical, phase space densities $F>0$ underlying the closure. The Spitzer and Grad qualities exceed 95% for the perturbative regimes where Spitzer's formulation is analytically known to be convergent. Beyond this regime both heat flux qualities fall about 30% per decade increase of ϵ > 0.01. In the solar corona the first decade's decrease in quality straddles the radius of the coronal temperature maximum and spans the initial acceleration of the solar wind. By the end of the second decade of increase of ϵ the observer is between 5 and 10R _⊙, already in conditions comparable to those at 1 au with ≃60% degradation of quality. These strong radial decays of closure quality show that integrating the fluid equations using such closures must represent a very poor assay of the role and effects of ∇ · q had the heat flux been described throughout with a uniformly high quality closure procedure. For small ϵ, $F<0$ occurs for cosine of pitch angle μ < 0 opposed to q at speeds above 2 thermal speeds and are omnipresent (but ignorable) for truly perturbative closures. Above a computed threshold in ϵ unphysical $F<0$ occurs for speeds below 2 thermal speeds with μ > 0. The present work graphically shows $F<0$ regimes becoming increasingly pervasive as ϵ increases, first crossing ≃4 thermal speeds at μ < 0 and then representing ever larger unphysical incursions within the needed velocity sphere required to accurately determine the heat flux.