Turbulent Mixing and Layer Formation in Double-Diffusive Convection: Three-Dimensional Numerical Simulations and Theory

Abstract

Double-diffusive convection, often referred to as semi-convection in astrophysics, occurs in thermally and compositionally stratified systems which are stable according to the Ledoux-criterion but unstable according to the Schwarzchild criterion. This process has been given relatively little attention so far, and its properties remain poorly constrained. In this paper, we present and analyze a set of three-dimensional simulations of this phenomenon in a Cartesian domain under the Boussinesq approximation. We find that in some cases the double-diffusive convection saturates into a state of homogeneous turbulence, but with turbulent fluxes several orders of magnitude smaller than those expected from direct overturning convection. In other cases the system rapidly and spontaneously develops closely-packed thermo-compositional layers, which later successively merge until a single layer is left. We compare the output of our simulations with an existing theory of layer formation in the oceanographic context, and find very good agreement between the model and our results. The thermal and compositional mixing rates increase significantly during layer formation, and increase even further with each merger. We find that the heat flux through the staircase is a simple function of the layer height. We conclude by proposing a new approach to studying transport by double-diffusive convection in astrophysics.

A new vision of giant planet interiors: Impact of double diffusive convection

“…Rosenblum et al (2011) show that in this regime Nu T 2, meaning that most of the energy is transported by pure diffusion. We thus assume that the thermal gradient to be used in these regions is given by the gradient needed to transport the whole outgoing energy flux by diffusion:…”

“…However, as mentioned in Rosenblum et al (2011), turbulent transport due to the double-diffusive instability can yield compositional Nusselt numbers, Nu μ 5 , around 2 to 4, yielding even stronger constraints, as summarized in Table 1. These low values of α justify a posteriori the approximation of continuous thermal and heavy element profiles when considering the planet's entire internal structure.…”

“…4, soon after they form, layers are expected to merge, leading to thicker layers, until the layer height reaches either the planet size, yielding a standard adiabatic interior, or an equilibrium value, which is the semiconvective case considered here. Numerical simulations by Radko (2005) and Rosenblum et al (2011) tend to show that the equilibration timescale of the staircase is much shorter than the typical timescale for the evolution of the planet. Thus, if an equilibrium height is reached, as discussed in Sect.…”

“…Analytical and numerical arguments show that the size of the fastest growing layers is equal to 10-100 d > ∼ l crit (Radko 2003;Rosenblum et al 2011). The efficient convection regime is thus appropriate in the planetary domain, where Pr ∼ 10 −2 −10 −1 .…”

“…Its exact value, however, depends on the characteristics of the medium in a nontrivial way and is difficult to estimate (Rosenblum et al 2011;Mirouh et al 2012). For smaller ∇ μ gradients, the medium is unstable according to the Ledoux criterion, and the thermal forcing is strong enough to force large-scale overturning convection.…”