Invariant manifolds in stratified turbulence

We present a reduced system of seven ordinary differential equations that captures the time evolution of spatial gradients of the velocity and the temperature in fluid elements of stratified turbulent flows. We show the existence of invariant manifolds (further reducing the system dimensionality), and compare the results with data stemming from direct numerical simulations of the full incompressible Boussinesq equations in the stably stratified case. Numerical results accumulate over the invariant manifolds of the reduced system, indicating the system lives at the brink of an instability. Finally, we study the stability of the reduced system, and show that it is compatible with recent observations in stratified turbulence of nonmonotonic dependence of intermittency with stratification.

Figure 1

(a) Rendering of the buoyancy $\theta$, in a small subvolume with local convection, in stably stratified turbulence with $N=8$ using $2048\times 2048\times 256$ grid points ($\approx 1/10$ of the domain length is shown). Note the entrainment and mixing between denser and lighter fluid in the region with rolls. (b) Rendering of $S={\partial }_z\theta$. Unstable regions have $S>N$ (light green, near the rolls), but many points have $S\approx N$ (blue). In both panels, a few particle trajectories are indicated in red.

Figure 2

(a) Isocontours of logarithmic levels of the joint PDF of $R$ and $Q$, for the DNS with $N=8$ and TG forcing. The gray curve indicates $Q=-{(27R^2/4)}^{1/3}$. (b) Same for $Q$ and $R_{\theta }$. The gray curve indicates $R_{\theta }=(2N^{2}Q-6Q^2)/\left(3N\right)$.

Figure 3

(a) Joint PDF of $T$ and $NA$, restricted to particles with $S\approx N$ in the DNS. The gray line is $T=NA$. (b) Same for $R_{\theta }$ and $NB$, restricted to the same particles. The gray line is $R_{\theta }=NB$. Insets show the corresponding joint PDF without any restriction.

Figure 4

(a) Joint PDF of $R_{\theta }/N-B$ and $Q$, restricted to particles with $S\approx 0$. The inset shows the same PDF without restriction. The gray straight lines indicate the manifold $Q=R_{\theta }/N-B$. (b) Time $t_{S=N}$ for an ensemble of Eqs. (5) to reach $S=N$, from initial values $\mathcal{O}\left({10}^{-3}\text{–}{10}^{-2}\right)$. Colors indicate whether the solution oscillates at $t_{S=N}$, or is already growing toward a blowup. The inset shows the time $t_{Q=-2}$ for the same ensemble to reach the value $Q=-2$.