3D Simulations and MLT: II. Onsager's Ideal Turbulence

We simulate stellar convection at high Reynolds number (Re≲7000) with causal time stepping but no explicit viscosity. We use the 3D Euler equations with shock capturing (Colella Woodward 1984). Anomalous dissipation of turbulent kinetic energy occurs as an emergent feature of advection ("Onsager damping"), caused by the moderate shocks which terminate the turbulent kinetic energy spectrum; see also (Perry 2021). In strongly stratified stellar convection the asymptotic limit for the global damping length of turbulent kinetic energy is ℓ_d ∼⟨ u^3 ⟩ /⟨ϵ⟩. This "dissipative anomaly" (Onsager 1949) fixes the value of the "mixing length parameter", α = ℓ_ MLT/H_P =⟨Γ_1⟩, which is ∼ 5/3 for complete ionization. The estimate is numerically robust, agrees to within 10 estimates from stellar evolution with constant α. For weak stratification ℓ_d shrinks to the depth of a thin convective region. Our flows 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 4096^3 direct numerical simulation of Kaneda (2003). The cascade converges quickly, and satisfies a power-law velocity spectrum similar to Kolmogorov (1941). Our flows 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 flows, (ii) experiment (Warhaft 2002), and (iii) high Re simulations of the Navier-Stokes equations (Iyer, et al. 2018).