A numerical simulation of supersonic turbulent convection relating to the formation of the Solar system

Abstract

A flux-corrected transport scheme due to Zalesak is used to carry out a numerical simulation of thermal convection in a two-dimensional layer of ideal, diatomic gas, which is heated from below and stratified gravitationally across many pressure scaleheights. The purpose of this calculation is to mimic the physical conditions in the outer layers of the protosolar cloud (PSC) from which the Solar system formed. The temperature T 0 at the top boundary (z = 0) and the dimensionless temperature gradient θ = (d/T 0)∂ T /∂z at the base of the layer of thickness d are kept fixed, with θ = 10. The initial atmosphere is uniformly superadiabatic, having polytropic index min = 1. Because the Reynolds number of the real atmosphere is so large, a subgrid-scale (SGS) turbulence approximation due to Smagorinsky is used to model the influence of motions with length-scale less than the computational grid size. The flow soon evolves to a network of giant convective cells, which span the whole layer. At cell boundaries the downflows are spatially concentrated and rapid while the upflows are broad and sluggish. The peak downflow Mach number is Mpeak = 1.1 at depth z = 0.55d. The descent of the cold gas eliminates much of the initial superadiabatic structure of the atmosphere for z 0.1d, thereby reducing the long-term mean temperature gradient d ¯ Tlt/dz and causing a net shift of mass towards the base. In the top 10 per cent of depth, SGS modelling causes d ¯ Tlt/dz to increase sharply. A steep density inversion occurs with the long-term mean density ¯ ρlt(0) at the top boundary rising to 3.5 times the initial density ρ0 there. This result gives new credibility to the modern Laplacian theory (MLT) of Solar system origin. Here a postulated 35-fold density increase at the surface of the PSC causes the shedding of discrete gas rings at the observed mean orbital spacings of the planets. Even so, further numerical simulations, corresponding to higher values of θ , which may yield values Mpeak � 3 and ¯ ρlt(0)/ρ0 � 35, are required before the MLT can be