Turbulence comes in bursts in stably stratified flows

There is a clear distinction between simple laminar and complex turbulent fluids; however, in some cases, as for the nocturnal planetary boundary layer, a stable and well-ordered flow can develop intense and sporadic bursts of turbulent activity that disappear slowly in time. This phenomenon is ill understood and poorly modeled and yet it is central to our understanding of weather and climate dynamics. We present here data from direct numerical simulations of stratified turbulence on grids of ${2048}^3$ points that display the somewhat paradoxical result of measurably stronger events for more stable flows, not only in the temperature and vertical velocity derivatives as commonplace in turbulence, but also in the amplitude of the fields themselves, contrary to what happens for homogenous isotropic turbulent flows. A flow visualization suggests that the extreme values take place in Kelvin-Helmoltz overturning events and fronts that develop in the field variables. These results are confirmed by the analysis of a simple model that we present. The model takes into consideration only the vertical velocity and temperature fluctuations and their vertical derivatives. It indicates that in stably stratified turbulence, the stronger bursts can occur when the flow is expected to be more stable. The bursts are generated by a rapid nonlinear amplification of energy stored in waves and are associated with energetic interchanges between vertical velocity and temperature (or density) fluctuations in a range of parameters corresponding to the well-known saturation regime of stratified turbulence.

Figure 1

Evolution in time of vertical velocity variations $\delta w$ in the model of Eqs. (4) and (5) for $\ell =0.2$ and $N=0$ (no stratification, solid line), 2 (dotted line), 4 (dashed line), 12 (dash-dotted line), 20 (dash–triple-dotted line), and 30 (long-dashed line). Note the faster evolution towards negative and strong vertical gradients at intermediate values of $N$, as it increases from 2 to 12, before oscillatory behavior takes over for large enough $N$ (here, corresponding to $N=20$ and 30).

Figure 2

The top plot shows the parallel total energy spectrum for the run with $N=12$. Times correspond to $t=4.94$ (triangles, red), $t=7.54$ (squares, magenta), and $t>12$ (the remaining curves), when small scales have reached a turbulent steady state. A $\sim k_z^{-3}$ scaling is shown as a reference. The inset shows the time evolution of the potential energy $E_P$ and of the kinetic energy $E_K$ in runs with $N=4$ (solid line) and $N=12$ (dashed line). Note the oscillations associated with internal gravity waves for $N=12$. The middle plot shows the time evolution of the potential energy for $k=$10 (squares, black), 20 (triangles, blue), 30 (dash-dotted, green), 40 (dashed, red), and 80 (solid, magenta) in the run with $N=4$. The bottom plot is the same as the middle but for the run with $N=12$. Note the larger fluctuations and bursts for large $k$ in this run reflected by the values of the standard deviation: ${\sigma }_{N=4}(k=10)=1.0\times {10}^{-3}$ and ${\sigma }_{N=12}(k=10)=8.4\times {10}^{-4}$, whereas, for larger values of $k$, ${\sigma }_{N=4}(k=40)=4.2\times {10}^{-5}$ and ${\sigma }_{N=12}(k=40)=1.1\times {10}^{-4}$.

Figure 3

The top plot shows normalized histograms (in semilogarithmic coordinates) evaluated shortly after the peak of dissipation, for the temperature fluctuations $\theta$ and for the vertical component of the velocity $w$ for high-resolution simulations of a stratified flow with Froude number $\text{Fr}\approx 0.1$ ($N=4$) and $\text{Fr}\approx 0.03$ ($N=12$). A normal distribution is shown (inner black curve) as a reference. The bottom plot shows PDFs of vertical derivatives for the same quantities. In all cases, the more strongly stratified flow with $N=12$ has larger probability of developing extreme events, as illustrated by the wider wings in the PDFs. For the fields themselves, the velocity has stronger tails than the temperature and the converse is true for their vertical derivatives.

Figure 4

The top plot shows integrated in time PDFs (in semilogarithmic coordinates) of the temperature $\theta$ for three snapshots after the peak of enstrophy and for the simulations with Froude number $\text{Fr}\approx 0.1$ ($N=4$) and $\text{Fr}\approx 0.03$ ($N=12$). The bottom plot is the same but for the vertical velocity $w$. In both panels, the inner black solid line corresponds to a normal distribution.

Figure 5

Integrated in time PDFs (in semilogarithmic coordinates) of the perpendicular velocity $v_{\perp }$ in the simulations with Froude number $\text{Fr}\approx 0.1$ ($N=4$) and $\text{Fr}\approx 0.03$ ($N=12$). As a reference, the outer black curve corresponds to a normal distribution.

Figure 6

The top plot shows accumulated moments $C_p\left(w\right)$ for the vertical velocity in the simulation with $\text{Fr}\approx 0.1$ ($N=4$). The curves, from top to bottom, correspond to moments of order $p$ from 1 to 8. The bottom plot is the same but for the simulation with $\text{Fr}\approx 0.03$ ($N=12$).

Figure 7

Shown above the grayscale bar are two-dimensional cuts in a ${[0,2\pi ]}^3$ box in the $x$ and $z$ directions for the flow with $\text{Fr}\approx 0.1$ ($N=4$, top row) and $\text{Fr}\approx 0.03$ ($N=12$, bottom row). The white segment is of unit length. The left-hand column corresponds to the temperature and the right-hand column to the vertical velocity. Note the strata in the temperature (the more strata, the higher the value of $N$). For $\text{Fr}\approx 0.03$, sporadic overturning in the vertical cut is clearly visible. The flow with stronger stratification is less complex, but extreme values of the fields and their gradients are higher, leading to the development of turbulent bursts and localized mixing. Shown below the grayscale bar are details of some extreme events: Kelvin-Helmholtz (KH) instability in the velocity field for $\text{Fr}\approx 0.03$ ($N=12$, top), eddies in the temperature field for $\text{Fr}\approx 0.1$ ($N=4$, bottom left), and KH instability in the temperature field for $N=12$ (bottom right).

Figure 8

Two-dimensional cuts in a ${[0,2\pi ]}^3$ box in the $x$ and $y$ (horizontal) directions for the flow with $\text{Fr}\approx 0.1$ ($N=4$, top row) and $\text{Fr}\approx 0.03$ ($N=12$, bottom row). The white segment is of unit length. The left-hand column corresponds to the temperature and the right-hand column to the vertical velocity. Note the stripes in the fields in the run with $N=12$, associated with the overturning events in Fig. 7, this time shown from the top.