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

Abstract

We simulate stellar convection at high Reynolds number (Re$\lesssim$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 $\ell_d \sim \langle u^3 \rangle /\langle \epsilon \rangle$. This "dissipative anomaly" (Onsager 1949) fixes the value of the "mixing length parameter", $\alpha = \ell_{\rm MLT}/H_P =\overline{\langle\Gamma_1\rangle}$, which is $\sim\, 5/3$ for complete ionization. The estimate is numerically robust, agrees to within 10% with estimates from stellar evolution with constant $\alpha$. For weak stratification $\ell_d$ 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).