Most stars host convection zones in which heat is transported directly by fluid motion, but the behavior of convective boundaries is not well-understood. Here, we present 3D numerical simulations that exhibit penetration zones: regions where the entire luminosity could be carried by radiation, but where the temperature gradient is approximately adiabatic and convection is present. To parameterize this effect, we define the “penetration parameter” , which compares how far the radiative gradient deviates from the adiabatic gradient on either side of the Schwarzschild convective boundary. Following Roxburgh and Zahn, we construct an energy-based theoretical model in which controls the extent of penetration. We test this theory using 3D numerical simulations that employ a simplified Boussinesq model of stellar convection. The convection is driven by internal heating, and we use a height-dependent radiative conductivity. This allows us to separately specify and the stiffness of the radiative–convective boundary. We find significant convective penetration in all simulations. Our simple theory describes the simulations well. Penetration zones can take thousands of overturn times to develop, so long simulations or accelerated evolutionary techniques are required. In stars, we expect ≈ 1 , and in this regime, our results suggest that convection zones may extend beyond the Schwarzschild boundary by up to ∼20%–30% of a mixing length. We present a MESA stellar model of the Sun that employs our parameterization of convective penetration as a proof of concept. Finally, we discuss prospects for extending these results to more realistic stellar contexts.
High Peclet number, turbulent convection is a classic system with a large timescale separation between flow speeds and the thermal relaxation time. In this paper, we present a method of fastforwarding through the long thermal relaxation of convective simulations, and we test the validity of this method. This Accelerated Evolution (AE) method involves measuring the dynamics of convection early in a simulation and using its characteristics to adjust the mean thermodynamic profile within the domain towards its evolved state. We study Rayleigh-Bénard convection as a test case for AE. Evolved flow properties of AE solutions are measured to be within a few percent of solutions which are reached through Standard Evolution (SE) over a full thermal diffusion timescale. At the highest values of the Rayleigh number at which we compare SE and AE, we find that AE solutions require roughly an order of magnitude fewer computing hours to evolve than SE solutions.
Large-scale convective flows called giant cells were once thought to transport the Sun's luminosity in the solar convection zone, but recent observations have called their existence into question. In place of large-scale flows, some authors have suggested the solar luminosity may instead be transported by small droplets of rapidly falling, low entropy fluid. This "entropy rain" could propagate as dense vortex rings, analogous to rising buoyant thermals in the Earth's atmosphere. In this work, we develop an analytical theory describing the evolution of dense, negatively buoyant thermals. We verify the theory with 2D cylindrical and 3D cartesian simulations of laminar, axisymmetric thermals in highly stratified atmospheres. Our results show that dense thermals fall in two categories: a stalling regime in which the droplets slow down and expand, and a falling regime in which the droplets accelerate and shrink as they propagate downwards. We estimate that solar downflows are in the falling regime and maintain their entropy perturbation against diffusion until they reach the base of the convection zone. This suggests that entropy rain may be an effective nonlocal mechanism for transporting the solar luminosity.
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