Rare Event Algorithm Links Transitions in Turbulent Flows with Activated Nucleations

Many turbulent flows undergo drastic and abrupt configuration changes with huge impacts. As a paradigmatic example we study the multistability of jet dynamics in a barotropic beta plane model of atmosphere dynamics. It is considered as the Ising model for Jupiter troposphere dynamics. Using the adaptive multilevel splitting, a rare event algorithm, we are able to get a very large statistics of transition paths, the extremely rare transitions from one state of the system to another. This new approach opens the way for addressing a set of questions that are out of reach through direct numerical simulations. We demonstrate for the first time the concentration of transition paths close to instantons, in a numerical simulation of genuine turbulent flows. We show that the transition is a noise-activated nucleation of vorticity bands. We address for the first time the existence of Arrhenius laws in turbulent flows. The methodology we developed shall prove useful to study many other transitions related to drastic changes for the turbulent dynamics of climate, geophysical, astrophysical, and engineering applications. This opens a new range of studies impossible so far, and bring turbulent phenomena in the realm of nonequilibrium statistical mechanics.

Figure 1

Right panels: typical snapshots of the vorticity fields (the colors show the vorticity values, with red for positive ones, blue for negative ones and black for intermediate ones; the range of vorticity is $[-1,1]$). Middle panels: Hovmöller diagrams of zonally averaged vorticity (the horizontal axis is $\alpha t$, the vertical one is $y$, and the color represents the $x$-averaged vorticity, $\alpha =1.20\times {10}^{-3}$). Left panels: time and zonally averaged vorticity (red) and velocity (green). The top plots show a two jet state for $\beta =5$, while the bottom ones show a three jet one for $\beta =10$.

Figure 2

Rare transitions between turbulent attractors with, respectively, 2 and 3 alternating jets. Upper panel: Hovmöller diagram of the zonal mean vorticity. Lower panel: time series of the modulus of $q_2$ (red) and $q_3$ (black), the zonal mean vorticity Fourier components, for wave number 2 and 3, respectively, versus the rescaled time $\alpha t$ ($\alpha =1.2\times {10}^{-3}$ and $\beta =5.26$).

Figure 3

(a) Sketch of the adaptive multilevel splitting (AMS) algorithm, one of the most versatile rare event algorithms. The aim of the algorithm is to compute the very small probability to go from a set $\mathcal{A}$ (for instance a two jet state) to a set $\mathcal{B}$ (for instance a three jet state). $N$ initial trajectories are computed from the model ($N$ is typically a few thousands). A score function $\mathcal{Q}$ measures how far each trajectory goes in the direction of $\mathcal{B}$. The worst trajectory is deleted (trajectory No. 1 on the sketch). It is replaced by a new trajectory (purple trajectory) whose initial condition is picked from one of the other previous trajectories (trajectory 2 on the sketch) at the time (red dot on the sketch) when it was crossing the $\mathcal{Q}$ level with the maximum value of $\mathcal{Q}$ for the deleted trajectory. This last step is called resampling or cloning. As the new set of $N$ trajectories has been obtained by selecting $N-1$ trajectories among $N$, and computing a new trajectory which has the same probability as the $N-1$ other ones, the new set has a probability $1-1/N$. The resampling step is iterated $K$ times, leading to a new trajectory set with probability ${(1-1/N)}^K$. This very efficiently produces extremely rare transitions from one attractor to another and gives an unbiased estimate of their probability (see the Supplemental Material [19], PDF file for more precise explanations).

Figure 4

Nucleation of a new jet. The panels display the same quantities as Fig. 1. A set of vortices is able to nucleate a new band of blue vorticity, a very unlikely process, leading to the birth of a new jet seen on the green velocity curve. As in nucleation processes in condensed matter, once the nucleated structure is large enough the new jet will be stable and persist for extremely long times, as seen on the Hovmöller diagram (see also the Supplemental Material [19], video).

Figure 5

(a) Zonally averaged velocity during the nucleation of a third jet in a $2\to 3$ jet transition: The blue, red and black curves show the velocity field at the start, at an intermediate stage, and at a more advanced stage of the nucleation, respectively. (b) Zonally averaged velocity during the coalescence of two jets in a $3\to 2$ transition: The blue, red and black curves show the velocity field before, just before, and after the merging, respectively. These 4 plots illustrate that while transitions appear at random time, their dynamic is predictable.

Figure 6

Instantons: The reactive tubes corresponding to the distribution of transition paths for the $2\to 3$ (red) and $3\to 2$ (blue) transitions. They illustrate the concentration of transition paths typical of an instanton phenomenology (see the main text) ($\beta =5.26$ and $\alpha =1.2\times {10}^{-3}$).

Figure 7

Arrhenius law. Logarithm of the mean first transition time $T$ from the two-jet to the three-jet attractors versus $1/\alpha$ ($\beta =5.5$). This result suggests that mean transition times might follow an Arrhenius law $T\propto e^{\mathrm{\Delta }U/\alpha }$.