## Periodic Reporting for period 1 - TransTurb (Large Deviations and Rare Transitions in Turbulent flows)

Reporting period: 2017-12-01 to 2019-11-30

Turbulent flows, like planetary atmospheres or flow around an airfoil, undergo strong fluctuations. Sometimes, those fluctuations may flip the system to an entirely different flow configuration. Such transitions seem to occur at random times, in an unpredictable manner.

These extreme events are crucial for weather, climate, and many engineering applications. One may think for instance about the occurrence of heat waves or cold spells in the mid-latitudes, related to fluctuations of the strong turbulent jet going around the planet, the Jet Stream, which have a large impact on society and global economy. Similarly the most important factor for designing devices like wind turbines is not the average mechanical forces they will be subjected to, but rather the strongest ones. Finally, the existence of tipping points, leading to abrupt climate change, is a major question for climate projections in the 21st century. Because turbulent flows have in general several metastable attractors, it can be expected that such abrupt transitions exist in the ocean and atmosphere, due solely to their turbulent nature.

There are two overarching difficulties in these problems: one is essentially technical, the other more fundamental.

The first deadlock is that we are interested in rare events, for which, by definition, we have few observations. Direct numerical simulations of the system do not really alleviate the problem, because models for turbulent flows or the climate system are computationally expensive. Specific algorithms have been developed over the past few years to solve this sampling problem. The first main goal of the project was to show that they can be adapted to address relevant questions for rare events in turbulent flows.

The second major aspect of the problem is to understand which properties of rare events are predictable, and which properties are not. For noise-induced transitions, for instance, transition times are unpredictable, but the dynamics of the transition is: the path to the rare event is always the same. The second main goal of the project was to test whether such ideas, inspired from statistical physics, hold for complex systems such as turbulent flows and climate models. To start with, we intended to establish on a solid basis if noise-induced transitions between bistable states existed at all in the atmosphere.

These extreme events are crucial for weather, climate, and many engineering applications. One may think for instance about the occurrence of heat waves or cold spells in the mid-latitudes, related to fluctuations of the strong turbulent jet going around the planet, the Jet Stream, which have a large impact on society and global economy. Similarly the most important factor for designing devices like wind turbines is not the average mechanical forces they will be subjected to, but rather the strongest ones. Finally, the existence of tipping points, leading to abrupt climate change, is a major question for climate projections in the 21st century. Because turbulent flows have in general several metastable attractors, it can be expected that such abrupt transitions exist in the ocean and atmosphere, due solely to their turbulent nature.

There are two overarching difficulties in these problems: one is essentially technical, the other more fundamental.

The first deadlock is that we are interested in rare events, for which, by definition, we have few observations. Direct numerical simulations of the system do not really alleviate the problem, because models for turbulent flows or the climate system are computationally expensive. Specific algorithms have been developed over the past few years to solve this sampling problem. The first main goal of the project was to show that they can be adapted to address relevant questions for rare events in turbulent flows.

The second major aspect of the problem is to understand which properties of rare events are predictable, and which properties are not. For noise-induced transitions, for instance, transition times are unpredictable, but the dynamics of the transition is: the path to the rare event is always the same. The second main goal of the project was to test whether such ideas, inspired from statistical physics, hold for complex systems such as turbulent flows and climate models. To start with, we intended to establish on a solid basis if noise-induced transitions between bistable states existed at all in the atmosphere.

"A first goal of the project was to develop rare event algorithms to make them suitable for studying complex systems such as turbulent flows or the climate system. The idea of one of these algorithms, called ""adaptive multilevel splitting"" (AMS), is to bias the statistics of an ensemble of simulations in a controlled manner, by progressively replacing the least performing trajectories (with respect to the distance to the rare event of interest) with mutations of the better performing ones. In the standard formulation of this algorithm, each simulation must be integrated until it hits one of two sets in phase space, corresponding either to typical conditions or to a rare event of interest. This is cumbersome for high-dimensional complex systems. We have given a different formulation of the algorithm, where the trajectories are always integrated for a fixed duration. It amounts at evaluating the probability that a given event occurs within a given time. This is well suited to many problems in climate science. We have also shown that the algorithm allows for evaluating ""return times"" for rare event, i.e. the typical time between two occurrences. This metric is commonly used in applications, for heat waves, floods, etc. As an example of application to complex systems, we have computed return times for extreme drag force acting on an object immersed in a turbulent flow. These results led to a journal article in J. Stat. Mech, published in 2018, and have been presented in conferences and various seminars.

Another limitation is the need to construct the distance to the rare event, called ""score function"", which is the crucial object in the selection step of the AMS algorithm. We have outlined a possible strategy, using machine learning methods to iteratively improve the score function. As a first step, we have compared direct methods and machine learning to estimate ""committor functions"", the mathematical object computed by the AMS. These results have been presented at a conference and shall lead to a journal publication soon.

All our codes have been published in a Python package, and a software metapaper describing it shall be submitted soon.

The second main goal was to study the possibility of abrupt transitions in atmospheric jets. We have identified a particularly interesting candidate for such phenomena: ""equatorial superrotation"", which corresponds to the appearance of a strong eastward jet at the equator. While tropical surface winds are easterly on current Earth, superrotation is observed on many planetary atmospheres, and might also be relevant for climates of the past. It might also be speculated that a transition to superrotation might occur due to anthropogenic climate change, providing another example of tipping point for global climate. While dynamical mechanisms leading to superrotation had been studied by dynamicists, the nature of the transition, abrupt or continuous, remained virtually unexplored. We have shown unambiguously, based on idealized models (an analytical model and a 2D numerical model) that a wave-jet feedback mechanism, sketched a few years ago by other researchers, indeed led to bistability and abrupt transitions. These results have been presented in several conferences and published in J. Atmos. Sci.

We have carried out numerical simulations with a full 3D General Circulation Model of the Atmosphere; while more work is still needed to understand the role of different parameters, this preliminary work seems to indicate that our findings still hold in a more realistic framework."

Another limitation is the need to construct the distance to the rare event, called ""score function"", which is the crucial object in the selection step of the AMS algorithm. We have outlined a possible strategy, using machine learning methods to iteratively improve the score function. As a first step, we have compared direct methods and machine learning to estimate ""committor functions"", the mathematical object computed by the AMS. These results have been presented at a conference and shall lead to a journal publication soon.

All our codes have been published in a Python package, and a software metapaper describing it shall be submitted soon.

The second main goal was to study the possibility of abrupt transitions in atmospheric jets. We have identified a particularly interesting candidate for such phenomena: ""equatorial superrotation"", which corresponds to the appearance of a strong eastward jet at the equator. While tropical surface winds are easterly on current Earth, superrotation is observed on many planetary atmospheres, and might also be relevant for climates of the past. It might also be speculated that a transition to superrotation might occur due to anthropogenic climate change, providing another example of tipping point for global climate. While dynamical mechanisms leading to superrotation had been studied by dynamicists, the nature of the transition, abrupt or continuous, remained virtually unexplored. We have shown unambiguously, based on idealized models (an analytical model and a 2D numerical model) that a wave-jet feedback mechanism, sketched a few years ago by other researchers, indeed led to bistability and abrupt transitions. These results have been presented in several conferences and published in J. Atmos. Sci.

We have carried out numerical simulations with a full 3D General Circulation Model of the Atmosphere; while more work is still needed to understand the role of different parameters, this preliminary work seems to indicate that our findings still hold in a more realistic framework."

All the developments of rare event methods mentioned above are new. We have showcased their interest for studying rare events in complex systems on examples from turbulence and climate, but we believe that they should be applicable much more broadly, in these fields and beyond. Our published software should make such endeavors easier.

Potential impacts beyond the academic community include the possibility to compute return times for heat waves, floods, etc (a crucial quantity for insurance and policy makers) much more efficiently using rare event algorithms (several orders of magnitude faster than direct methods), helping engineers use rare event algorithms to dimension industrial systems, and finally improving medium-range forecasts with committor functions.

On the problem of atmospheric bistability, we have shown explicitly that some feedback mechanisms leading to superrotation indeed resulted in bistability in idealized models.This gives credit to the idea that abrupt climate transitions might occur due to turbulent fluctuations. It opens an avenue of research on the physical mechanism leading to bistability (resonant interaction of equatorial waves with the mean-flow), underlying several problems of current interest. The question of whether an abrupt transition to a different atmospheric circulation (e.g. a superrotating one) might spontaneously occur, and with which probability, has clear socio-economic impacts. Although we are not able to provide an answer yet, our results are a step in this direction.

Potential impacts beyond the academic community include the possibility to compute return times for heat waves, floods, etc (a crucial quantity for insurance and policy makers) much more efficiently using rare event algorithms (several orders of magnitude faster than direct methods), helping engineers use rare event algorithms to dimension industrial systems, and finally improving medium-range forecasts with committor functions.

On the problem of atmospheric bistability, we have shown explicitly that some feedback mechanisms leading to superrotation indeed resulted in bistability in idealized models.This gives credit to the idea that abrupt climate transitions might occur due to turbulent fluctuations. It opens an avenue of research on the physical mechanism leading to bistability (resonant interaction of equatorial waves with the mean-flow), underlying several problems of current interest. The question of whether an abrupt transition to a different atmospheric circulation (e.g. a superrotating one) might spontaneously occur, and with which probability, has clear socio-economic impacts. Although we are not able to provide an answer yet, our results are a step in this direction.