## CliMathNet Conference 2017, University of Reading

**Gualtiero Badin, University of Hamburg**

**Title: A study of surface semi-geostrophic turbulence: freely decaying dynamics**

**Abstract:**

In this study we give a characterization of semi-geostrophic turbulence by performing freely decaying simulations for the case of constant uniform potential vorticity, a set of equations known as the surface semi-geostrophic approximation. The equations are formulated as conservation laws for potential temperature and potential vorticity, with a nonlinear Monge?Ampère type inversion equation for the streamfunction, expressed in a transformed coordinate system that follows the geostrophic flow. We perform model studies of turbulent surface semi-geostrophic flows in a domain doubly periodic in the horizontal and limited in the vertical by two rigid lids, allowing for variations of potential temperature at one of the boundaries, and we compare the results with those obtained in the corresponding surface quasi-geostrophic case. The results show that, while the surface quasi-geostrophic dynamics is dominated by a symmetric population of cyclones and anticyclones, the surface semi-geostrophic dynamics features a more prominent role of fronts and filaments. The resulting distribution of potential temperature is strongly skewed and peaked at non-zero values at and close to the active boundary, while symmetry is restored in the interior of the domain, where small-scale frontal structures do not penetrate. In surface semi-geostrophic turbulence, energy spectra are less steep than in the surface quasi-geostrophic case, with more energy concentrated at small scales for increasing Rossby number. The energy related to frontal structures, the lateral strain rate and the vertical velocities are largest close to the active boundary. These results show that the semi-geostrophic model could be of interest for studying the lateral mixing of properties in geophysical flows. Work in collaboration with F. Ragone.

**Freddy Bouchet, ENS de Lyon and CNRS, France**

**Title: Large deviation theory applied to climate dynamics**

**Abstract:**

We will review some of the recent developments in the theoretical and mathematical aspects of the non-equilibrium statistical mechanics of climate dynamics. At the intersection between statistical mechanics, turbulence, and geophysical fluid dynamics, this field is a wonderful new playground for large deviation theory. It involves large deviation theory for stochastic partial differential equations, homogenization, and rare event algorithms.

We will discuss two paradigmatic examples. First, we will study extreme heat waves, as an example of rare events with a huge impact. We will explain how algorithms based on large deviation theory allow to sample rare heat wave at a numerical cost that is several orders of magnitude smaller than a direct numerical simulation. This will improve drastically the study of those events dynamics, in climate models, in relation with climate change.

A a second example, we will study rare trajectories that suddenly drive turbulent flows from one attractor to a completely different one, in the stochastic barotropic quasigeostrophic equation. This equation, a generalization of the stochastic two dimensional Navier–Stokes equations, models Jupiter's atmosphere jets. We discuss preliminary steps in the mathematical justification of the use of averaging, derive an effective action for large deviations, compute transition rates through Freidlin–Wentzell theory, and instantons (most probable transition paths).

This talk is based on works with Francesco Ragone and Jereon Wouters (heat waves) and Brad Marston, Eric Simonnet, Tomas Tangarife, and Eric Woillez (large deviations for quasi geomorphic turbulence).

**Mickael D. Chekroun, UCLA**

**Title: Closures for Stochastic Partial Differential Equations Driven by Degenerate Noise**

**Abstract:**

Stochastic partial differential equations (SPDEs), with bilinear drift and driven by a degenerate additive noise, will be considered. For such equations, we will present new analytic formulas for the parameterizations of the scales lying beyond a cutoff wavenumber. The derivation of these formulas takes place within a variational approach relying on the theory of stochastic parameterizing manifolds whose main tools and concepts will be introduced. The relationships with the ergodic theory of SPDEs will be discussed and applications to closure in the context of “Burgulence” will be presented. The role of path-dependent, non-Markovian coefficients arising in the related closure systems will be also discussed.

(Joint work with Honghu Liu, James C. McWilliams, and Shouhong Wang.)

**Davide Faranda, French National Centre for Scientific Research**

**Title: Stochastic Chaos in a Turbulent Swirling Flow**

##### Authors: D. Faranda, Y. Sato, B. Saint-Michel, C. Wiertel, V. Padilla, B. Dubrulle, and F. Daviaud

**Abstract:**

We report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the flux of angular momentum imparted to the flow, we can first reconstruct the associated turbulent attractor and then follow its route towards chaos. We further show that the experimental attractor can be modeled by stochastic Duffing equations, that match the quantitative properties of the experimental flow, namely, the number of quasistationary states and transition rates among them, the effective dimensions, and the continuity of the first Lyapunov exponents. Such properties can be recovered neither using deterministic models nor using stochastic differential equations based on effective potentials obtained by inverting the probability distributions of the experimental global observables. Our findings open the way to low-dimensional modeling of systems featuring a large number of degrees of freedom and multiple quasistationary states.

**Ulrike Feudel, Carl von Ossietzky University Oldenburg**

**Title: Harmful algal blooms: combining excitability, competition and hydrodynamic flows**

**Abstract:**

Harmful algal blooms (HABs) are rare events which are characterized by a sudden large abundance of potentially toxic plankton species which can alter the dynamics of the whole ecosystem. Since as a consequence of climate change, the frequency of HABs is increasing, there is a strong need in understanding the possible causes for such bloom events.

We discuss a model, which is based on the idea of an excitable activator-inhibitor system in which two activators (toxic and non-toxic phytoplankton) compete with each other and the inhibitor (zooplankton) has a certain preference for a specific activator. We show how the interplay of the competition and environmental factors like increasing nutrient input due to upwelling result in a sudden growth of toxic species.

Hydrodynamic flows are also important determinants for the emergence and the spread of HABs in the real ocean. Analzying data from observations in the Southern California Bight we demonstrate, that particularly mesoscale hydrodynamic vortices are of crucial importance for the spread of HABs. Moreover, such vortices can lead to heterogeneous dominance patterns of different plankton species in the ocean. We illustrate the mechanism of the emergence of spatially localized HABs using a simplified kinematic flow. Furthermore we demonstrate the importance of the interplay between biological and hydrodynamic time scales for the formation of blooms.

**Andrey Gritsun, Russian Academy of Sciences, Moscow**

**Title: Instability characteristics of blocking regimes in a simple quasigeostrophic atmospheric model**

**Abstract:**

In this paper we study statistics and instability characteristics of blocking events in the three layer quasi-geostrophic model of atmosphere by Marshall and Molteni.

It is shown that the model is able to reproduce reasonable longitudinal distribution of blocking events as well as simulate blocking events with lifetime of up to 40 days. Using Lyapunov exponents and covariant Lyapunov vectors we analyze predictability of onset, duration and decay of blockings. It is shown that on the average blockings are less predictable than the system trajectory with the blocking onset and decay being the most unstable and unpredictable. We verify our findings by looking at unstable periodic orbits (UPOs) of the system representing blocking and nonblocking events. It was found that blocking UPOs have 20% more positive (unstable) Lyapunov exponents than the system trajectory, and 50% larger leading exponent.

**Wilco Hazelheger, Wageningen University**

**Title: Future Weather Extremes**

**Abstract:**

Extreme weather events expose society’s vulnerability to weather. Recent extreme events are placed more and more in the context of climate change. Event attribution can indicate whether the risk of extreme events changes due to climate change. Information on future climate, however, is often based on projections with relatively coarse resolution climate models that may not resolve such extremes well. Downscaling can remedy this partly since the driving of extremes are often of large scale nature, such as blocking, and events are often compounded. We place recent extremes in a future context. We use EC-Earth simulations to generate plausible simulations of future weather events, similar to those experienced in current climate. We enrich the model information with knowledge on the physics and on vulnerability and impacts of the events. Different cases will be presented, such as the wet winter of 2013/14 in the UK and extreme rainfall in the city of Amsterdam last summer. Finally, some technological and methodological challenges will be addressed which will help the climate research community to generate high resolution simulations.

**Christian Kuehn, Technical University of Munich**

**Title: Stochastic PDEs in Climate: Tipping and Numerical Continuation**

**Abstract:**

In my talk, I am going to give an introduction to several aspects of stochastic partial differential equations (SPDEs) in climate modelling with a view towards practical applications. First, I am going to show that the theory of early-warning signs for critical transitions (or tipping points) can be extended to cover spatially extended systems, i.e., it is possible to extract warning signs from spatial data in certain cases. Secondly, I am going to illustrate, how one can compute bifurcation diagrams - and hence locate certain tipping points - effectively using numerical continuation in models. I shall demonstrate a major extension of these methods to SPDEs. The numerical methods will be illustrated with an SPDE model for the AMOC. The talk is also intended as a showcase, how flexible and broad the applications of SPDEs could be in climate science in the future.

**Philippe Naveau, Laboratoire des Sciences du Climat et l’Environnement**

**Title: An entropy-based test for multivariate extreme value models**

Sebastian Engelke Ecole Polytechnique Fédérale de Lausanne (Switzerland)

Philippe Naveau Laboratoire des Sciences du Climat et l’Environnement (France)

Chen Zhou Erasmus Universiteit Rotterdam & De Nederlandsche Bank (The Netherlands)

**Abstract:**

Many effects of climate change seem to be reflected in the frequency and severity of the extreme events in the distributional tails. Detecting such changes requires a statistical methodology that efficiently uses the large observations in the sample. We propose a simple, non-parametric test that decides whether two multivariate distributions exhibit the same tail behaviour. The test is based on the entropy, namely Kullback-Leibler divergence, between exceedances over a high threshold of the two multivariate random vectors. We show that such a type of divergence is closely related to the divergence between Bernoulli random variables. We study the properties of the test and further explore its effectiveness for finite sample sizes. As an application we apply the method to precipitation data where we test whether the marginal tails and/or the extreme value dependence structure have changed over time.

Key Words: multivariate extreme value dependence, Kullback-Leibler divergence, asymptotics.

**Peter Read, University of Oxford**

**Title: Macroturbulent cascades of energy and enstrophy in observations and models of planetary atmospheres**

P. L. Read, F. Tabataba-Vakili, A. Valeanu, Y. Wang & R. M. B. Young

Atmospheric, Oceanic & Planetary Physics, Department of Physics, University of Oxford, UK.

**Abstract:**

Many features of the structure and evolution of the large-scale flows in the atmospheres (and oceans) of the Earth and other planets can be regarded as resulting from the nonlinear exchanges of energy (kinetic and potential) and vorticity (enstrophy) across widely varying length and time scales in processes described as “macroturbulent cascades”. The influence of planetary rotation can be very substantial on some scales, for example, leading to quasi-two-dimensional flows accompanied by upscale transfers of kinetic energy, the dominance of anisotropically propagating waves and the formation of zonal jets. The theory behind these processes is far from complete, however, and it is important, therefore, to explore the quantitative character of such macroturbulent cascades in models and (where possible) in observations over a wide range of conditions. We will present diagnostics of such cascades of kinetic (and, in some cases, potential) energy and enstrophy in new analyses of observations of the atmospheres of Jupiter and Mars, based on structure functions and spectral flux computations for comparison with the Earth’s atmosphere and oceans. These reveal some clear similarities and differences, with evidence of upscale cascades at large scales and downscale transfers at small and mesoscales, though the precise nature of these downscale cascades is still uncertain. Similar trends are found in a set of simplified global circulation model simulations of Earth-like planetary atmospheres across a wide range of planetary parameters. These results will be discussed in the context of an over-arching theoretical framework for planetary and mesoscale turbulence.

**Stéphane Vannitsem, Royal Meteorological Institute of Belgium**

**Title: A dynamical systems approach to investigating the low-frequency variability of the ocean-atmosphere coupled system.**

**Abstract:**

The low-frequency variability (LFV) of the atmosphere at mid-latitudes develops on a wide range of time scales. One particularly interesting indicator of this variability is the North Atlantic Oscillation index measuring the fluctuations of predominant weather patterns in the course of the years over the Atlantic and Western Europe. The source of variability is, however, controversial and several possibilities have been envisaged, including oceanic and coupled ocean-atmosphere variability and stratospheric warming, possibly related to ENSO in the tropical Pacific.

Recently we have demonstrated that genuinely coupled LFV can emerge in a very simple low-order, nonlinear, coupled ocean-atmosphere model. This LFV concentrates on and near a long-periodic, attracting orbit. This orbit combines atmospheric and oceanic modes, and it arises for large values of the meridional gradient of radiative input and of the frictional coupling. Chaotic behavior develops around this orbit as it loses its stability. This behavior is still dominated by the LFV on decadal and multi-decadal time scales that is typical of oceanic processes. Furthermore, this natural coupled mode is still present as the number of variables is increased in the model. This dynamics will be discussed in the first part of the talk.

In the second part, we will discuss the relevance of these results for the realistic coupled ocean-atmosphere dynamics emerging from reanalysis datasets, using advanced tools from nonlinear time series analysis. It is shown in particular that LFV is present in the real coupled dynamics over the Atlantic, bearing some resemblances with the one that could be emulated in the low-order system.

**Jeroen Wouters, University of Reading**

**Title: Beyond the limit of infinite time-scale separation: Edgeworth approximations and homogenisation**

Authors: Jeroen Wouters^{1} & Georg Gottwald^{2}

^{1}Mathematics and Statistics, University of Reading; ^{2}School of Mathematics and Statistics, University of Sydney

**Abstract:**

Homogenization has been widely used in stochastic model reduction of slow-fast systems, including geophysical and climate systems.

The theory relies on an infinite time scale separation. In this talk we present results for the realistic case of finite time scale separation. In particular, we employ Edgeworth expansions as finite size corrections to the central limit theorem and show improved performance of the reduced stochastic models in numerical simulations.