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.

Figure 1

DNS results. (a), (b) Snapshots of the vertical velocity field $w$ at times $t_1=2.21$ and $t_2=2.24$, along with the mean flow $\overline{u}$ (solid line) for $z>z_{\mathrm{NB}}=0.68$ (with $\overline{u}=0$ corresponding to $x=1$). Vertical velocity patterns show convective motions in the lower part of the domain ($z\le z_{\mathrm{NB}}$) and internal-wave motions in the upper part ($z\ge z_{\mathrm{NB}}$). Note that energy propagates upward along wave crests, so crests toward the upper left (right) correspond to retrograde (prograde) waves. (c) The mean flow $\overline{u}(t,z)$. The mean flow in the convective zone corresponds to the average of stochastic plumes emitted from the bottom boundary and hence reverses on a relatively rapid, convective timescale. In the stably stratified layer, $\overline{u}$ results from the nonlinear interaction of internal waves and oscillates on timescales $\sim 0.1$, much longer than the buoyancy period $\sim \pi {10}^{-4}$. Simulation details and videos are available in Supplemental Material [33].

Figure 2

Mean-flow rms ${\overline{u}}_{\mathrm{rms}}$ as a function of Pr. The mean flow becomes stronger as Pr decreases and is also more regular: the symbols’ area is inversely proportional to the frequency bandwidth of $\overline{u}$, defined as the difference $\mathrm{\Delta }f=f_{0.9}-f_{0.1}$ of the two frequencies $f_{0.1}$ and $f_{0.9}$ below and above which lies 10% of the mean-flow energy.

Figure 3

(a) Schematics of the DNS model ${\mathbf{M}}_1$ and the two reduced models ${\mathbf{M}}_2$ and ${\mathbf{M}}_3$. (1) We calculate the kinetic energy spectrum $\mathcal{K}$ of the fluctuations at height $z_{\mathrm{NB}}$ obtained in ${\mathbf{M}}_1$. (2) The forcing ($u^{\prime }$, $w^{\prime }$, $T^{\prime }$) is derived from $\mathcal{K}$, assuming that the fluctuations correspond to linear propagating internal waves. Propagation of the waves is solved (3) via DNS of the Boussinesq equations in ${\mathbf{M}}_2$, but is derived analytically (4) in ${\mathbf{M}}_3$ under WKB approximation. Thus, in ${\mathbf{M}}_3$ (5), we only need to solve for the mean-flow equation. As in full DNS, we use a damping layer for $1.35<z<1.5$, and boundary conditions for the mean flow are no slip. (b) $\overline{u}$ over one thermal time obtained for each model shown in (a). Physical parameters are as in Fig. 1. Note that the colormap has been changed in Fig. 3 compared to Fig. 1 to highlight differences between ${\mathbf{M}}_1$ and ${\mathbf{M}}_{2,3}$.