3D Simulations and MLT: II. Onsager’s Ideal Turbulence. (arXiv:1810.04659v4 [astro-ph.SR] UPDATED)
<a href="http://arxiv.org/find/astro-ph/1/au:+Arnett_W/0/1/0/all/0/1">W. David Arnett</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Hirschi_R/0/1/0/all/0/1">Raphael Hirschi</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Campbell_S/0/1/0/all/0/1">Simon W. Campbell</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Mocak_M/0/1/0/all/0/1">Miroslav Mocák</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Georgy_C/0/1/0/all/0/1">Cyril Georgy</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Meakin_C/0/1/0/all/0/1">Casey Meakin</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Cristini_A/0/1/0/all/0/1">Andrea Cristini</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Scott_L/0/1/0/all/0/1">Laura J. A. Scott</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Kaiser_E/0/1/0/all/0/1">Etienne A. Kaiser</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Viallet_M/0/1/0/all/0/1">Maxime Viallet</a>
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”, $alpha = ell_{rm MLT}/H_P
=overline{langleGamma_1rangle}$, 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).
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”, $alpha = ell_{rm MLT}/H_P
=overline{langleGamma_1rangle}$, 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).
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