3D simulations and MLT: II. Onsager’s Ideal Turbulence

We simulate convective turbulence in stars, extending Arnett, et al (2019). Our implicit large eddy simulations (ILES) use the 3D Euler equations with shock capturing (Colella & Woodward 1984); we simulate an astrophysically-appropriate inviscid limit (Re.7000) with causal time stepping but no explicit viscosity. Anomalous dissipation of turbuent kinetic energy occurs as an emergent feature of advection (“Onsager damping”), a physical process caused by the moderate shocks, which terminate the turbulent kinetic energy spectrum. This differs from incompressible turbulence in which nonlinear fluid effects involving vorticity are supposed to control dissipation (Taylor 1937; P. Johnson 2021a). In strongly stratified stellar convection the asymptotic limit for the global damping length of turbulent kinetic energy is `d ∼ 〈u〉/〈 〉. This “dissipative anomaly” (Onsager 1949) fixes the value of the “mixing length parameter”, α = `MLT/HP = 〈Γ1〉, which is ∼ 5/3 for complete ionization. The estimate is numerically robust, agrees to within ∼10% with estimates from stellar evolution with constant α. For weak stratification `d shrinks to the depth of a thin convective region. For stellar atmospheres computed with “hyperviscosity stabilization” and variable Γ1 in regions of partial ionization, the agreement is ∼20% (Magic, Weiss, & Asplund 2016). Our ILES are filamentary, produce surfaces of separation at boundary layers, resolve the energy-containing eddies, and develop a turbulent cascade down to the grid scale (which agrees with the 4096 direct numerical simulations (DNS) of Kaneda, etal 2003). The cascade converges quickly (Onsager 1949), and satisfies a power-law velocity spectrum similar to (Kolmogorov 1941). Our ILES exhibit intermittency, anisotropy, and interactions between coherent structures, features missing from K41 theory. We derive a dissipation rate from Reynolds stresses which agrees with (i) our ILES, (ii) experiments (Warhaft 2002), and (iii) high Reynolds number DNS of the Navier-Stokes equations (Iyer, et al. 2018; Sreenivasan 2019).