Closed-loop adaptive control of extreme events in a turbulent flow

Extreme events that arise spontaneously in chaotic dynamical systems often have an adverse impact on the system or the surrounding environment. As such, their mitigation is highly desirable. Here, we introduce a control strategy for mitigating extreme events in a turbulent shear flow. The controller combines a probabilistic prediction of the extreme events with a deterministic actuator. The predictions are used to actuate the controller only when an extreme event is imminent. When actuated, the controller only acts on the degrees of freedom that are involved in the formation of the extreme events, exerting minimal interference with the flow dynamics. As a result, the attractors of the controlled and uncontrolled systems share the same chaotic core (containing the nonextreme events) and only differ in the tail of their distributions. We propose that such adaptive low-dimensional controllers should be used to mitigate extreme events in general chaotic dynamical systems, beyond the shear flow considered here.

Figure 1

Schematic of the system attractor. The thick black curve shows a trajectory of the uncontrolled system undergoing an extreme event. The thin red curve marks the trajectory of the controlled system which evades the extreme event. The black dot marks the initial state of the system.

Figure 2

The time series of the energy dissipation rate $D$ at Reynolds number $\text{Re}=40$. (a) Uncontrolled system and (b) controlled system.

Figure 3

Prediction of extreme events at $\text{Re}=40$. (a) Conditional probability density function (PDF) of $\tilde{D}\left(t\right)={max}_{s\in [t+{\tau }_p,t+{\tau }_p+\mathrm{\Delta }{\tau }_p]}D\left(s\right)$ given as $\widehat{\mathbf{u}}(1,0,t)$. (b) Probability $P_{ee}$ of an extreme dissipation event occurring over the future time interval $[t+{\tau }_p,t+{\tau }_p+\mathrm{\Delta }{\tau }_p]$ given the current value of $|\widehat{\mathbf{u}}(1,0,t)|$.

Figure 4

The block diagram of the control strategy. The Fourier transforms of the external forcing, the velocity field, and the control term are denoted by $\widehat{\mathbf{f}},\widehat{\mathbf{u}}$, and $\widehat{\mathit{\xi }}$, respectively. The Fourier mode $\widehat{\mathbf{u}}(1,0)$ is used to measure the probability of upcoming extreme events $P_{ee}$. The control is proportional to the Fourier mode $\widehat{\mathbf{u}}(0,4)$ where coincidentally ${\mathbf{k}}_f=(0,4)$ is the wave number of the external forcing.

Figure 5

Controlled versus uncontrolled systems at $\text{Re}=40$. (a) Energy dissipation rate $D$ of the uncontrolled (black circles) and controlled (solid red) systems as a function of time. The horizontal dashed line marks the threshold $D_e=0.2$ for the extreme events. The top panel shows the probability $P_{ee}$ of upcoming extreme events. (b) The corresponding vorticity fields at times $t_1=27.0,t_2=35.2$, and $t_3=43.0$.

Figure 6

The PDF of the uncontrolled (black circles) and controlled (solid red) systems at Reynolds numbers $\text{Re}=40$, 60, 80 and 100. Each PDF is estimated from 50 000 data points.