Order Out of Chaos: Slowly Reversing Mean Flows Emerge from Turbulently Generated Internal Waves

We demonstrate via direct numerical simulations that a periodic, oscillating mean flow spontaneously develops from turbulently generated internal waves. We consider a minimal physical model where the fluid self-organizes in a convective layer adjacent to a stably stratified one. Internal waves are excited by turbulent convective motions, then nonlinearly interact to produce a mean flow reversing on timescales much longer than the waves' period. Our results demonstrate for the first time that the three-scale dynamics due to convection, waves, and mean flow is generic and hence can occur in many astrophysical and geophysical fluids. We discuss efforts to reproduce the mean flow in reduced models, where the turbulence is bypassed. We demonstrate that wave intermittency, resulting from the chaotic nature of convection, plays a key role in the mean-flow dynamics, which thus cannot be captured using only second-order statistics of the turbulent motions

The energy flux spectrum of internal waves generated by turbulent convection

We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of convective--stably-stratified fluids. We demonstrate that in the limit of large Rayleigh number ($Ra\in [4\times 10^7,10^9]$) and large stratification (Brunt-V\"{a}is\"{a}l\"{a} frequencies $f_N \gg f_c$, where $f_c$ is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region (Lecoanet \& Quataert 2013 MNRAS 430 (3) 2363-2376). Specifically, we demonstrate that the wave energy flux spectrum scales like $k_{\perp}^4f^{-13/2}$ for weakly-damped waves (with $k_{\perp}$ and $f$ the waves' horizontal wavenumbers and frequencies), and that the total wave energy flux decays with $z$, the distance from the convective region, like $z^{-13/8}$.Comment: 13 pages, 6 figure

Shape and size of large-scale vortices: A generic fluid pattern in geophysical fluid dynamics

Planetary rotation organizes fluid motions into coherent, long-lived swirls, known as large-scale vortices (LSVs), which play an important role in the dynamics and long-term evolution of geophysical and astrophysical fluids. Here, using direct numerical simulations, we show that LSVs in rapidly rotating mixed convective and stably stratified fluids, which approximates the two-layer, turbulent-stratified dynamics of many geophysical and astrophysical fluids, have a generic shape and that their size can be predicted. We show that LSVs emerge in the convection zone from upscale energy transfers and can penetrate into the stratified layer. At the convective-stratified interface, the LSV cores have a positive buoyancy anomaly. Due to the thermal wind constraint, this buoyancy anomaly leads to winds in the stratified layer that decay over a characteristic vertical length scale. Thus LSVs take the shape of a depth-invariant cylinder with a finite-size radius in the turbulent layer and of a penetrating half dome in the stratified layer. Importantly, we demonstrate that when LSVs penetrate all the way through the stratified layer and reach a boundary that is no-slip, they saturate by boundary friction. We provide a prediction for the penetration depth and maximum radius of LSVs as a function of the LSV vorticity, the stratified layer depth, and the stratification. Our results, which apply for cyclonic LSVs, suggest that LSVs in slowly rotating stars and Earth's liquid core are confined to the convective layer, while in Earth's atmosphere and oceans they can penetrate far into the stratified layer

Turbulent convection in subglacial lakes

Subglacial lakes are isolated, low-temperature and high-pressure water environments hidden under ice sheets. Here, we use two-dimensional direct numerical simulations in order to investigate the characteristic temperature fluctuations and velocities in freshwater subglacial lakes as functions of the ice overburden pressure, pi, the water depth, h, and the geothermal flux, F. Geothermal heating is the unique forcing mechanism as we consider a flat ice–water interface. Subglacial lakes are fully convective when pi is larger than the critical pressure p∗≈2848 dbar, but self-organize into a lower convective bulk and an upper stably stratified layer when pi<p∗, because of the existence at low pressure of a density maximum at temperature Td greater than the freezing temperature Tf. For both high and low pi, we demonstrate that the Nusselt number, Nu, and Reynolds number, Re, satisfy classical scaling laws provided that an effective Rayleigh number Raeff is considered. We show that the convective and stably stratified layers at low pressure are dynamically decoupled at leading order because plume penetration is weak and induces limited entrainment of the stable fluid. From the empirical equation for Nu with Raeff, we derive two sets of closed-form expressions for several variables of interest, including the unknown bottom temperature, in terms of the problem parameters pi, h and F. The two predictions correspond to two limiting regimes obtained when the effective thermal expansion coefficient is either approximately constant or linearly proportional to the temperature difference driving the convection