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Seminars (MFE)

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Event When Speaker Title Presentation Material
MFE 23rd October 2013
11:00 to 12:00
S Lovejoy The weather: emergent laws and multi fractal cascades
MFE 23rd October 2013
15:00 to 16:00
V Lucarini Thermodynamic properties of the climate system
MFE 28th October 2013
11:00 to 12:00
Hurricane dynamics: on the role of Vortex Rossby Waves (VRWs)
Despite the fact that asymmetries in hurricanes, such as spiral rainbands, polygonal eyewalls and mesovortices, have long been observed in radar imagery, many aspects of their dynamics still remain unsolved, particularly in the formation of the secondary eyewall. To fill this gap, a simple 2D barotropic model (Martinez et al., 2010) and the high-resolution PSU-NCAR non-hydrostatic mesoscale model (MM5) are used to study hurricane asymmetries (Chen et al., 2003; Martinez et al., 2011).The Empirical Normal Mode (ENM) diagnostics (Brunet, 1994), together with the Eliassen-Palm (EP) flux calculations are used to isolate wave modes from the model datasets to investigate their impact on the changes in the structure and intensity of the simulated hurricanes (Chen et al., 2003). The ENMs are obtained in a similar manner as Empirical Orthogonal Functions (EOFs) but with the use of a quadratic form instead of Euclidean norm. The quadratic forms are global invariants, the pseudo-momentum and pseudo-energy wave activities, of the linearized equations about a basic state (Brunet and Vautard, 1996). ENM theory bridges two important diagnostic tools of geophysical fluid dynamics: principal component analysis and normal mode theory. The role of internal dynamics on concentric eyewall genesis is further evaluated using the full physics MM5 simulation. The leading modes of the ENM diagnostics exhibit mainly characteristics of VRWs and their contribution to the EP flux divergence induced two regions of maximum tangential wind acceleration; one inside the primary eyewall which accounts for eyewall contraction and the other outside the primary eyewall which explains the development of the secondary eyewall (Martinez et al., 2011). We will point out the expected implication of these results in the context of numerical weather prediction at different space-time resolutions for intensifying and mature hurricanes of different strengths.
MFEW01 29th October 2013
09:00 to 09:35
M Pollicott Geodesic flows: Mixing, zeta functions and resonances
Historically important examples of ``chaotic'' dynamical systems are Anosov flows, in particular, and geodesic flows on negatively curved manifolds. In particular, they provide a concrete setting to explore a wealth of interesting topics: (i) mixing rates (which can be studied using zeta function and resonances); (ii) large deviations and fluctuation theorems (Gallavotti-Cohen theorem in non-equilibrium statistical mechanics); and (iii) escape rates (the rate at which mass escapes from an open system) and extremes.
MFEW01 29th October 2013
09:35 to 10:10
Rigorous computation of invariant measures and Lyapunov exponents
Co-author: Isaia Nisoli (Universitade Federal Rio de Janeiro)

We will consider the problem of computation of invariant measures and other aspects related to the statistical behavior of the dynamics up to certified errors.

In this way the output of a computation represent some rigorous quantitative estimation on the behavior of the dynamics under study, going towards more reliable tools for the simulation of dynamical models.

We will show some general approach which can be applied in several cases of systems having some hyperbolic behavior, including maps with indifferent fixed points.

Time permitting we will also consider a class piecewise hyperbolic maps related to the Lorenz attractor.

MFEW01 29th October 2013
10:10 to 10:45
Paleo-climatic time series: statistics and dynamics
Co-authors: Arnaud Debussche (ENS Cachan), Jan Gairing (HU Berlin), Claudia Hein (HU Berlin), Michael Högele (U Potsdam), Ilya Pavlyukevich (U Jena)

Dynamical systems of the reaction-diffusion type with small noise have been instrumental to explain basic features of the dynamics of paleo-climate data. For instance, a spectral analysis of Greenland ice time series performed at the end of the 1990s representing average temperatures during the last ice age suggest an $\alpha-$stable noise component with an $\alpha\sim 1.75.$ We model the time series as a dynamical system perturbed by $\alpha$-stable noise, and develop an efficient testing method for the best fitting $\alpha$. The method is based on the observed $p$-variation of the residuals of the time series, and their asymptotic $\frac{\alpha}{p}$-stability established in local limit theorems.\par\smallskip

Generalizing the solution of this model selection problem, we are led to a class of reaction-diffusion equations with additive $\alpha$-stable L\'evy noise, a stochastic perturbation of the Chafee-Infante equation. We study exit and transition between meta-stable states of their solutions. Due to the heavy-tail nature of an $\alpha$-stable noise component, the results differ strongly from the well known case of purely Gaussian perturbations.

MFEW01 29th October 2013
11:10 to 11:45
T Kuna Typical behaviour of extremes of chaotic dynamical systems for general observables
In this talk we discuss the distribution of extreme events for a chaotic dynamical system for a general class of observables. More precisely, we link directly the distribution of events over threshold to the local geometrical structure on the surface of the attractor. It is shown how this can provide us with information about the local stable and unstable dimensions. Using Ruelle's response theory, we discuss the sensitivity of the parameters of the distribution under perturbations. This is a joint work with Vlaerio Lucarini, Davide Faranda and Jeroen Wouters.
MFEW01 29th October 2013
11:45 to 12:20
Interplay between Mathematics and Physics
Co-author: Tian Ma (Sichuan University)

In this talk, we shall present three first principles and a few examples, demonstrating the symbiotic interplay between theoretical physics and advanced mathematics.

We start with a general principle that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic and random. We shall illustrate this principle with a few examples in both equilibrium and non-equilibrium phase transitions, including the metastable oscillation mechanism of the El Nino Southern Oscillation (ENSO) and the existence of 3rd-order transitions beyond the Andrews critical point.

Then we present two basic principles: the principle of interaction dynamics (PID) and the principle of representation invariance (PRI), to study the nature's fundamental interactions/forces. Intuitively, PID takes the variation of the action functional under energy-momentum conservation constraint. PRI requires that physical laws be independent of representations of the gauge groups. These two principles give rise to a unified field model for four interactions, which can be naturally decoupled to study individual interactions. With PID, for example, we derive new gravitational field equations with a vector field $\Phi_\mu$, which can be considered as a spin-1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin-2 massless bosonic particle) and this spin-1 massless bosonic particle, leading to a unified theory for dark matter and dark energy. In addition, the PID offers a completely different and much simpler way of introducing Higgs fields.

MFEW01 29th October 2013
13:40 to 14:15
Quantifying uncertainty and improving reduced-order predictions of partially observed turbulent dynamical systems
Co-author: A. J. Majda (Courant Institute, NYU)

The issue of mitigating model error in reduced-order prediction of high-dimensional dynamics is particularly important when dealing with turbulent geophysical systems with rough energy spectra and intermittency near the resolution cut-off of the corresponding numerical models. I will discuss a new framework which allows for information-theoretic quantification of uncertainty and mitigation of model error in imperfect stochastic/statistical predictions of non-Gaussian, multi-scale dynamics. In particular, I will outline the utility of this framework in derivation of a sufficient condition for improving imperfect predictions via a popular but heuristic Multi Model Ensemble approach. Time permitting, the role and validity of 'fluctuation-dissipation' arguments for improving imperfect predictions of externally perturbed non-autonomous turbulent systems will also be addressed.
MFEW01 29th October 2013
14:15 to 14:50
Extreme value theory for randomly perturbed systems: getting the local dimensions
We present some new results for extreme values distributions in dynamical systems perturbed "via" random transformations and with observational noise. In both cases the linear scaling parameters of the Gumbel law will allow to get informations on the local behavior respectively of the stationary measure (random transformations), and of the invariant measure (observational noise). This collects work done with Aytac, Faranda, Freitas, Lucarini and Turchetti.
MFEW01 29th October 2013
14:50 to 15:25
A new recurrences based technique for detecting robust extrema in long temperature records
Co-author: Sandro Vaienti (University of Marseille)

By using new techniques originally developed for the analysis of extreme values of dynamical systems, several long records of temperatures at different locations are analysed by showing that they have the same recurrence time statistics of a chaotic dynamical system perturbed with dynamical noise and by instrument errors. The technique provides a criterion to discriminate whether the recurrence of a certain temperature belongs to the natural climate variability or can be considered as a real extreme event with respect to a specific time scale fixed as parameter. The method gives a self-consistent estimation of the convergence.

MFEW01 29th October 2013
15:50 to 16:25
Extreme sea waves in the coastal zone
MFEW01 29th October 2013
16:25 to 17:00
J Vollmer Dew droplets and cloud droplets: droplet growth, size distributions, and corrections to scaling
I present the results of comprehensive laboratory experiments and numerical studies addressing droplet growth and droplet size distributions in systems where droplets grow due to sustained supersaturation of their environment.

Both, for droplets condensing on a substrate (like dew) and droplets entrained in an external flow (like in clouds), we observe remarkable shortcomings of classical scaling theories addressing these growth processes. The origins of the discrepancies are identified, and appropriate extensions of the theories are discussed.
MFEW01 30th October 2013
09:00 to 09:35
On thermodynamics of stationary states of diffusive systems Coauthors L. Bertini, A. De Sole, D. Gabrielli, C. Landim
Thermodynamic transformations connecting nonequilibrium stationary states have the peculiarity of dissipating, to keep the system out of equilibrium, an amount of energy which diverges for a quasi static transformation. By subtracting the divergent part one can define a renormalized work that satisfies a Clausius type inequality and with respect to which quasi static transformations are optimal. A different way of analyzing the energy balance and optimality criteria is to consider transformations over a long but finite time T developing the total work and the dissipated energy in powers of 1/T. The diverging terms cancel and one obtains relations among finite quantities.
MFEW01 30th October 2013
09:35 to 10:10
Environmental superstatistics
Complex systems in driven nonequilibrium situations often consist of a superposition of several dynamics on well-separated time scales. Sometimes the parameters of the system fluctuate as well, on a much larger time scale than the local dynamics. The resulting marginal distributions typically have fat tails, which can be understood by superstatistical techniques. After a short review of the field I will concentrate on some examples relevant for planet earth: The dynamics of tracer particles in turbulent flows, the surface temperature statistics at various locations on planet earth, and the dynamics of sea levels.
MFEW01 30th October 2013
10:10 to 10:45
Data-driven model reduction and climate prediction: nonlinear stochastic, energy-conserving models with memory effects
Co-authors: Mickael D. Chekroun (University of California, Los Angeles), Michael Ghil (University of California, Los Angeles)

This talk will focus on theoretical understanding and climate applications of a data-driven reduction strategy that leads to low-order stochastic-dynamical models with energy-conserving nonlinearities and conveying memory effects. New opportunities for climate prediction will be illustrated in the framework of "Past Noise Forecasting", by utilizing on the one hand estimated history of the driving noise by the low-order model, and on the other hand the phase of low-frequency variability estimated by advanced time series analysis.

MFEW01 30th October 2013
11:10 to 11:45
Efficient sampling of rare events by splitting
Standard (or crude) Monte Carlo (MC) simulation is known to be inefficient for simulating rare events. For events with low probability, the squared relative error on estimates obtained from straightforward MC simulation is inversely proportional to the number of samples, so that an excessively large number of samples may be required to reach a desired accuracy for the estimation of rare event probabilities.

To improve the efficiency of MC sampling for rare events, various techniques have been developed in the past, for applications in e.g. communication networks and reliability analysis. Such techniques can be of interest for studying extremes in geophysical models. I will discuss a technique called multilevel splitting, in which model sample paths are split into multiple copies each time they cross thresholds (or levels) that lead closer to the rare event set.

MFEW01 30th October 2013
11:45 to 12:20
Regime-dependent modelling of extremes in the extra-tropical atmospheric circulation
The talk discusses data-based statistical-dynamical modelling of vorticity and wind speed extremes in the extra-tropical atmospheric circulation. The extreme model is conditional on the large-scale flow, consisting of a collection of local generalised Pareto distributions, each associated with a cluster or regime in the space of large-scale flow variables. The clusters and the parameters of the extreme models are estimated from data, either separately or simultaneously. The large-scale flow is represented by the leading empirical orthogonal functions (EOFs). Also temporal clustering of extremes in the different large-scale regimes is investigated using an inhomogeneous Poisson process model whose rate parameter is conditional on the large-scale flow. The study is performed in the framework of an intermediate complexity atmospheric model with realistic mean state, variability and teleconnection patterns. The methodology can also be applied to data from GCM scenario simulations, predicting future extremes.
MFEW01 30th October 2013
13:40 to 14:15
The modified second fluctuation-dissipation theorem
Baths produce friction and random forcing on particles suspended in them. The relation between noise and friction in (generalized) Langevin equations is usually referred to as the second fluctuation-dissipation theorem. We show, beyond formalities, what is the proper nonequilibrium extension, to be applied when the environment is itself active and driven.
MFEW01 30th October 2013
14:15 to 14:50
On the use of Ruelle's formalism in response theory
We use Ruelle’s formalism to express the response of a generic observable to a certain perturbation in terms of correlation functions computed with respect to the unperturbed invariant measure, for deterministic as well as stochastic dynamics. We discuss the onset of two relevant terms for the entropy production, comment on the Hamiltonian version of the resulting formulae and also propose a connection with similar results, reported in the literature, allowing to extend the Fluctuation-Dissipation formalism to nonequilibrium steady states.
MFEW01 30th October 2013
14:50 to 15:25
Anomalous fluctuation relations
Co-authors: Aleksei V. Chechkin (Institute for Theoretical Physics NSC KIPT, Kharkov, Ukraine), Peter Dieterich (Institut fuer Physiologie, Medizinische Fakultaet Carl Gustav Carus, Dresden, Germany), Friedrich Lenz (Queen Mary University of London, School of Mathematical Sciences, London, UK)

We study Fluctuation Relations (FRs) for Gaussian stochastic systems that are anomalous, in the sense that the diffusive properties strongly deviate from the ones of Brownian motion. For this purpose we use a Langevin approach: We first briefly review the concept of transient work FRs for simple Langevin dynamics generating normal diffusion [1]. We then consider two different types of additive, power law correlated Gaussian noise [2,3]: (1) internal noise with a fluctuation-dissipation relation of the second type (FDR2), and (2) external noise without FDR2. For internal noise we find that FDR2 leads to conventional (normal) forms of transient work FRs. For external noise we obtain various forms of violations of normal FRs, which we call anomalous FRs. We show that our theory is important for understanding experimental results on fluctuations in systems with long-time correlations, such as glassy dynamics [1].

[1] R.Klages, A.V.Chechkin, P.Dieterich, Anomalous fluctuation relations, book chapter in: R.Klages, W.Just, C.Jarzynski (Eds.), Nonequilibrium Statistical Physics of Small Systems, Wiley-VCH, Weinheim (2013) [2] A.V.Chechkin, F.Lenz, R.Klages, J.Stat.Mech. L11001 (2012) [3] A.V.Chechkin, R.Klages, J.Stat.Mech. L03002 (2009)
MFEW01 30th October 2013
15:55 to 16:30
A large-deviation approach to passive scalar advection, diffusion and reaction
Co-authors: Peter H. Haynes (University of Cambridge), Alexandra Tzella (University of Birmingham)

The dispersion of a passive scalar in a fluid through the combined action of advection and molecular diffusion can often be described as a diffusive process, with an effective diffusivity that is enhanced compared to the molecular value. This description fails to capture the tails of the scalar concentration in initial-value problems, however. This talk addresses this issue and shows how the theory of large deviation can be applied to capture the concentration tails by solving a family of eigenvalue problems. Two types of flows are considered: shear flows and cellular flows. In both cases, large deviation is shown to generalise classical results (Taylor dispersion for shear flows, homogenisation results for cellular flows). Explicit asymptotic results are obtained in the limit of large Péclet number corresponding to small molecular diffusivity. The implications of the results for the problem of front propagation in reacting flows are also discussed.

MFEW01 31st October 2013
09:00 to 09:35
Extreme Events and Coupled Climate-Economics Modeling
In this talk, I will review some recent work on extreme events, their causes and consequences. The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models. The applications include climatic, seismic and socio-economic events, along with their prediction. Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems. Both these results have implications for the study and prediction of natural hazards and their human impacts. A substantial part of the talk will deal with an endogenous business cycle (EnBC) model and with the way that EnBCs affect the impact of natural hazards on a dynamic economy. An out-of-equilibrium fluctuation-dissipation result for macroeconomics is inferred from the model and confirmed by the analysis of US economic data.
MFEW01 31st October 2013
09:35 to 10:10
RC Dewar Kinetic energy dissipation and the stability of stationary turbulent flows
Variational principles of fluid turbulence offer an attractive alternative to numerical solution of the Navier-Stokes equation, especially for global climate studies. I discuss the principle (Max-D) that certain stationary turbulent flows maximize the rate of kinetic energy dissipation of the mean flow. Following its conjecture as an organizational principle for atmospheric circulation [1], Max-D has gained numerical support from global climate model simulations [2]. Max-D has also been derived for turbulent shear flow in a channel from considerations of dynamic stability, and yields realistic predictions for the mean velocity profile at all Reynolds numbers [3]. Further theoretical support for Max-D in channel flow has emerged from the statistical principle of maximum entropy [4]. Tying these threads together may lead to a clearer understanding of the theoretical basis and range of validity of Max-D for global climate studies. I outline possible approaches to doing this.

[1] Lorenz EN (1955) Generation of available potential energy and the intensity of the general circulation. Scientific Report No. 1, UCLA Large Scale Synoptic Processes Project.

[2] Pascale S, Gregory JM, Ambaum MHP, Tailleux R (2012) A parametric sensitivity study of entropy production and kinetic energy dissipation using the FAMOUS AOGCM. Clim. Dyn. 38, 1211-1227 and references therein.

[3] Malkus WVR (2003) Borders of disorders: in turbulent channel flow. J. Fluid Mech. 489, 185-198.

[4] Dewar RC, Maritan A (2013) A theoretical basis for maximum entropy production. In Beyond the Second Law: Entropy Production and Non-equilibrium Systems (eds. RC Dewar, CH Lineweaver, RK Niven, K Regenauer-Lieb), Springer, in press.
MFEW01 31st October 2013
10:10 to 10:45
N Glatt-Holtz Inviscid Limits for the Stochastic Navier Stokes Equations and Related Systems
One of the original motivations for the development of stochastic partial differential equations traces it's origins to the study of turbulence. In particular, invariant measures provide a canonical mathematical object connecting the basic equations of fluid dynamics to the statistical properties of turbulent flows. In this talk we discuss some recent results concerning inviscid limits in this class of measures for the stochastic Navier-Stokes equations and other related systems arising in geophysical and numerical settings. This is joint work with Peter Constantin, Vladimir Sverak and Vlad Vicol.
MFEW01 31st October 2013
11:10 to 11:45
J Wouters A statistical mechanics approach to stochastic parametrizations
Co-author: Valerio Lucarini (University of Hamburg)

Current computer simulations of climate and weather prediction models can only take into account a limited number of the relevant degrees of freedom of the climate system. Therefore the physical dynamical equations need to be reduced to a smaller subset of variables.

The reduction of the number of degrees of freedom (also known as parametrization in the modeling community) is a central task in statistical mechanics and it is therefore not surprising that many techniques used in this field are also used in the derivation of stochastic parametrizations.

In this talk I will discuss some of these techniques and how they have been applied to climate modeling. I will then discuss how we have used the Mori-Zwanzig formalism and response theory to derive parametrizations for weakly coupled dynamical systems.

MFEW01 31st October 2013
11:45 to 12:20
PM Cox Emergent Constraints on Earth System Sensitivities
Co-author: Chris Huntingford (Centre for Ecology and Hydrology)

Climate and Earth System Models are designed to project changes in the climate-carbon cycle system over the coming centuries. These models have ever higher spatial resolution and are based on an improving understanding of key processes. However, climate modelling still suffers from a significant timescale problem – we need to find constraints on the huge range of projected changes in the climate-carbon system over the next century, but the observational data that we have relates to much shorter timescales. This talk will summarise one promising way around the timescale problem - the use of “Emergent Constraints”. An Emergent Constraint is a relationship between some climate system sensitivity to anthropogenic forcing and an observable (or already observed) feature of the climate system. We call it emergent because it emerges from the ensemble of models, and it is described as a constraint because it enables an observation to constrain the estimate of the cli mate system sensitivity in the real world. As an example, I will describe an emergent constraint on the projected loss of tropical land carbon under climate change, which has a huge range amongst climate-carbon cycle projections for the 21st century. We have recently identified an emergent linear relationship across the ensemble of models between the sensitivity of tropical land-carbon storage to warming and the sensitivity of the annual growth-rate in atmospheric CO2 to tropical temperature anomalies. When combined with contemporary observations of the atmospheric CO2 concentration and the tropical temperature, this relationship provides a tight constraint on the sensitivity of tropical land carbon to warming in the real climate system (Cox et al., Nature, 2013). The talk will conclude by hypothesising how such emergent constraints may relate to (a) the Fluctuation-Dissipation Theorem and (b) Time-series Precursors of Tipping Points.

MFEW01 31st October 2013
13:40 to 14:15
The modeling of rare events: from methodology to practice and back
In this talk I give a brief overview of the historical development of Extreme Value Theory (EVT), discuss some applications, highlighting EVT's strengths and weaknesses, and indicate relevant research themes going forward.
MFEW01 31st October 2013
14:15 to 14:50
Bayesian approaches for wind gust and quantitative precipitation forecasting
Co-author: Sabrina Bentzien (Meteorological Institute, University of Bonn, Germany)

Due to large uncertainties, predictions of high-impact weather on the atmospheric mesoscale are probabilistic in nature. Mesoscale weather ensemble prediction systems (EPS) are developed to obtain probabilistic guidance for high impact weather. An EPS not only issues a deterministic future state of the atmosphere but a sample of possible future states. Ensemble postprocessing then translates such a sample of forecasts into probabilistic measures.

We discuss Bayesian approaches for wind gust and quantitative precipitation forecasting. The Bayesian hierarchical model (BHM) for wind gusts uses extreme value theory, namely a generalized extreme value distribution (GEV), in the data level. A process level for the parameters is introduced which, on the one hand, models the spatial response surfaces of the GEV parameters as Gaussian random fields, and, on the other hand, incorporate the information of the COSMO-DE forecasts. The spatial BHM provides area wide forecasts of wind gusts in terms of a conditional GEV. It models the marginal distribution of the spatial gust process and provides not only forecasts of the conditional GEV at locations without observations, but also uncertainty information about the estimates. At this stage, the BHM ignores the conditional dependence between gusts at neighboring locations. However, an outline is given how this will be incorporated in a subsequent study using max-stable random fields.

For quantitative precipitation forecasting we use Bayesian quantile regression and its spatially adaptive extension together with a variable selection based on a Bayesian LASSO. All this is illustrated for the German-focused mesoscale weather prediction ensemble COSMO-DE-EPS, which runs operationally since December 2010 at the German Meteorological Service (DWD). We further discuss the issue of objective out-of-sample verification, where performance is measured using proper scoring rules and their decomposition.

MFEW01 31st October 2013
14:50 to 15:25
V Chavez-Demoulin Generalized additive modelling of hydrological sample extremes
Co-authors: Anthony Davison (EPFL, Lausanne), Marius Hofert (ETHZ, Zurich)

Estimation of flood frequencies and severities is important for many water management issues. We present a smoothing extreme value method fitted by penalized loglikelihood. Spline smoothing is used to estimate the parameters of the frequency and size distributions of extremes, depending on covariates in a non- or semiparametric way. The frequency process of high level extremes is modelled by a Poisson process, either homogeneous or non-homogeneous. The extreme sizes are considered to follow a generalized Pareto distribution. Being given by two parameters, the method of spline smoothing is not straightforward to apply. An efficient fitting algorithm based on orthogonal reparametrisation is developed to achieve this task. The method is applied to the daily maximum flows of an hydrological station in Switzerland and is used to estimate 20-year return levels.
MFEW01 31st October 2013
15:50 to 16:25
S Lovejoy Extreme events and the multifractal butterfly effect
Scaling processes abound in geophysics and this has important consequences for the probability distributions of the corresponding intensive and extensive geophysical variables. Classical scaling processes – such as in classical turbulence – are self-similar, they are characterized by exponents which are invariant under isotropic scale changes. However, the atmosphere and lithosphere are strongly stratified so that we must generalize the notion of scale allowing for invariance under anisotropic zooms. When this is done, it is often found that scaling can apply over huge ranges, up to planetary in extent. It is now clear that the generic scaling process is the multifractal cascade in which a scale invariant dynamical mechanism repeats (multiplicatively) from scale to scale; anisotropic scaling – and multifractal universality classes - imply that multifractals are widely relevant in the earth sciences. General (canonical) multifractal processes developed over finite ranges of scale and analyzed at their smallest scale (the “bare” process), have “long-tailed” distributions (e.g. the lognormal). However the small scale cascade limit is singular so that the integration/averaging of cascades developed down to their small scale limits leads to “dressed” properties characterized notably by “fat-tailed” power law probability distributions Pr(x>s)=s**-qD where x is a random value, s a threshold and qD the critical exponent implying that the moments for q>qD diverge. For cascades averaged over scales larger than the inne r cascade scale, the moments q>qD are no longer determined by the large scale finite by the small scale details: the “multifractal butterfly effect”. The sampling properties of such processes can be understood with “multifractal phase transitions”; we review this as well as evidence for the divergence of moments in laboratory, atmospheric and climatological series, and in data from the solid earth and discuss implications (abrupt changes, etc.).
MFEW01 31st October 2013
16:25 to 17:00
J Kurths Complex networks identify spatial patterns of extreme rainfall events of the Indian and the South American monsoon system
Co-authors: Niklas Boers (Potsdam Institute for Climate Impact Research ), Veronika Stolbova (Potsdam Institute for Climate Impact Research ), Bodo Bookhagen (UC Santa Barbara, Geography Department)

We investigate the spatial characteristics of extreme rainfall synchronicity of the Indian and South American Summer Monsoon System by means of Complex Networks (CN). By introducing a new combination of CN measures and interpreting it in a climatic context, we investigate climatic linkages and classify the spatial characteristics of extreme rainfall synchronicity. Although our approach is based on only one variable (rainfall), it reveals the most important features of the Monsoon Systems, such as the main moisture pathways, areas with frequent development of Mesoscale Convective Systems, and the major convergence zones. In addition, our results reveal substantial differences between the spatial structures of rainfall synchronicity above the 90th and above the 95th percentiles.

References Arenas, A., A. Diaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Phys. Reports 2008, 469, 93. Boers, N., B. Bookhagen, N. Marwan, J. Kurths, and J. Marengo, Geophys. Res. Lett. 2013, DOI: 10.1002/grl.50681 (online). Donges, J., Y. Zou, N. Marwan, and J. Kurths, Europhys. Lett. 2009, 87, 48007. Malik, N., B. Bookhagen, N. Marwan, and J. Kurths, Climate Dynamics, 2012, 39, 971.
MFEW01 31st October 2013
17:00 to 17:35
Statistical stability arguments for maximum kinetic energy dissipation
The hypothesis that stationary turbulent flows have maximal mean-flow kinetic energy dissipation (Max-D) is intriguing because mean-flow properties can be predicted without modelling the turbulent component of the flow. Our knowledge of Max-D is largely restricted to relatively simple laboratory flows. Measured Poiseuille flow profiles match Max-D predictions closely and, under these simplified conditions, Malkus's statistical stability argument provides some theoretical justification for Max-D [1]. However, it is not clear whether Max-D is applicable to more complicated fluid systems, like Earth's atmosphere [2]. Recent global climate model simulations have found that the calibrated values of important tunable parameters are indeed consistent with Max-D [3]. Furthermore, the maximum entropy framework [4], which naturally gives a Max-D principle in the case of simple laboratory flows, can be readily applied to more complicated systems. I will discuss attempts to gener alise the Malkus statistical stability argument and how this connects with maximum entropy arguments. In doing so I hope to compare the physical insights of statistical stability, which emphasises dynamical resilience to perturbations, with maximum entropy considerations, which ignore system dynamics.

[1] W. V. R. Malkus. Borders of disorder: In turbulent channel ow. Journal of Fluid Mechanics, 489:185{198, 2003. [2] Richard Goody. Maximum entropy production in climate theory. Journal of the atmospheric sciences, 64(7):2735-2739, 2007. [3] Salvatore Pascale, Jonathan M. Gregory, Maarten H.P. Ambaum, and Remi Tailleux. A parametric sensitivity study of entropy production and kinetic energy dissipation using the FAMOUS AOGCM. Climate Dynamics, 38(5-6):1211-1227, 2012. [4] Dewar R and Maritan A. A theoretical basis for maximum entropy production. 2013. In Beyond the Second Law: Entropy Production and Non-equilibrium Systems (eds. R Dewar, C Lineweaver, R Niven, K Regenauer-Lieb), Springer, In Press
MFEW01 31st October 2013
17:35 to 18:10
Statistics of eddy transport
Co-author: Florin Spineanu (National Institute of Laser, Plasma and Radiation Physics)

A semi-analytical method for the study of eddy transport in presented. It determines the statistics of particle trajectories using the decorrelation trajectories, which are determined from the Eulerian correlation of the velocity. The fraction of free trajectories that effectivy determines the transport decreases with the increase of the Kubo number. The statistical method is able to describe the transport in these conditions where it is produced by a minority of the events. The effect of particle collisions is analysed. We show that eddy diffusion is strongly amplified by weak collisions and that the effective diffusion coefficient can be much larger than both the collisional diffusion coefficient and the eddy diffusion coefficients.

MFEW01 1st November 2013
09:00 to 09:35
Hydrodynamic turbulence as a problem in non-equilibrium statistical mechanics
The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems which describe units of fluid of smaller and smaller spatial extent. These units are macroscopic but have few degrees of freedom, and can be studied by the methods of (microscopic) non-equilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Specifically, we obtain the formula

$$ \zeta_p={p\over3}-{1\over\ln\kappa}\ln\Gamma({p\over3}+1) $$

for the exponents of the structure functions ($\langle|\Delta_rv|^p\rangle\sim r^{\zeta_p}$). The meaning of the adjustable parameter $\kappa$ is that when an eddy of size $r$ has decayed to eddies of size $r/\kappa$ their energies have a thermal distribution. The above formula, with $(\ln\kappa)^{-1}=.32\pm.01$ is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture which can thus also be used in related problems.
MFEW01 1st November 2013
09:35 to 10:10
GM Buttazzo Optimal location problems with routing cost
Co-authors: Serena Guarino (University of Pisa (Italy)), Fabrizio Oliviero (University of Pisa (Italy))

A model problem for the location of a given number $N$ of points in a given region $\Omega$ and with a given resources density $\rho(x)$ is considered. The main difference between the usual location problems and the present one is that in addition to the location cost an extra {\it routing cost} is considered, that takes into account the fact that the resources have to travel between the locations on a point-to-point basis. The limit problem as $N\to\infty$ is characterized and some applications to airfreight systems are shown.
MFEW01 1st November 2013
10:10 to 10:45
Phase transitions and large deviations in geophysical fluid dynamics
Geophysical turbulent flows (atmosphere and climate dynamics, the Earth core dynamics) often undergo very rapid transitions. Those abrupt transitions change drastically the nature of the flow and are of paramount importance, for instance in climate. By contrast with most theoretical models of phase transitions, for turbulent flows it is difficult to characterize clearly the attractors (they are not simple fixed points of a deterministic dynamics or statistical equilibrium states) and the trajectories that lead to transitions from one attractor to the others.

I will review recent researches in this subject, including experimental and numerical studies of turbulent flows. Most of the talk will focus on theoretical works in the framework of the 2D stochastic quasi-geostrophic Navier-Stokes equations, the quasi-geostrophic equations, and the stochastic Vlasov equations. We will discuss predictions of phase transitions, validity of large deviation results of the Freidlin-Wentzell type, or more involved approaches when the Freidlin-Wentzell approach is not valid.

The results involve several works that have been done in collaborations with J. Laurie, M. Mathur, C. Nardini, E. Simonnet, J. Sommeria, T. Tangarife, H. Touchette, and O. Zaboronski.
MFEW01 1st November 2013
11:10 to 11:45
Turbulence transition in shear flows: coherent structures, edge states and all that
Pipe flow, plane Couette flow and several other shear flows show a transition to turbulence for flow rates where the linear profile is still stable. The turbulent dynamics is transient, so that the transition is related to the formation of a chaotic saddle in the state space of the system. The saddle is supported by exact coherent states and their heteroclinic connections. I will summarize the common features that appear across all these shear flows, sketch the numerical techniques used to identify and track the relevant structures in the state space of the system and point out possible applications beyond fluid mechanics.
MFEW01 1st November 2013
11:45 to 12:20
V Lucarini Noise, Fluctuation, and Response in Geophysical Fluid Dynamics
Response theory provides formidable methods for addressing many problems in statistical mechanics. Recently, it has been proposed as a gateway for various challenges in geophysical fluid dynamics, such as the provision of a rigorous conceptual framework for computing climate response to a variety of forcings and for deriving effective equations for coarse-grained variables, thus paving the way for constructing accurate parametrization of unresolved processes in numerical models. In this contribution, we first would like to present some new results showing how one can use response theory to compute the impact of adding stochastic forcing to deterministic chaotic systems. Then, we will discuss the applicability of the fluctuation-dissipation theorem in the context of non-equilibrium systems, focusing on the role played by the choice of observable. Finally, we will present some applications of response theory in geophysical fluid dynamical systems, ranging from low-order models such as the Lorenz 63 and Lorenz 96 models to General Circulation Models of the atmosphere.
MFEW01 1st November 2013
12:20 to 12:55
Nonequilibrium statistical mechanics of climate variability: modelling issues and applications to data assimilation techniques
Stochastic models and computational tools for the study of transitions between different metastable states (or regimes) in climate system are discussed using the barotropic quasi-geostrophic (QG) equation as a test case. Specifically, a stochastic partial differential equation (SPDE) is obtained by adding appropriate forcing and damping terms to the QG equation to make this equation dynamically consistent with the predictions of equilibrium statistical mechanics, while allowing to study nonequilibrium phenomena such as transitions between different regimes. In the small noise regime, the most likely states of the invariant measure for this SPDE coincide with the selective decay states and we establish conditions under which these states are not unique, implying the existence of different climate regimes. We also analyze the mechanism and rate of the dynamical transitions between these regimes by computing the most likely paths connecting them. Finally we will discuss how the se results can be used in the context of data assimilation procedure based on Kalman or ensemble filters to improve the efficiency of these methods in the presence of regime shifts.
MFE 5th November 2013
10:00 to 11:00
Thermostats for bias correction and statistically consistent model reduction
In meteorology and climate science, fluid models are simulated on time scales very long compared to the characteristic Lyapunov time of chaotic growth, with the goal of generating a data set suitable for statistical analysis. The choice of a numerical discretization scheme for a problem carries with it a certain bias in the statistical data generated in long simulations. In this talk I will discuss research on the use of thermostat techniques, commonly used in molecular dynamics, to control the invariant measure of a discretized model, with the goals of correcting bias or effecting a statistically consistent model reduction. These will be illustrated for a point-vortex gas and a Burgers/KdV equation. Finally I will discuss new extensions of the approach to cases where observations are available.
MFE 6th November 2013
10:00 to 11:00
Analysis of some nonlinear PDEs from multi-scale geophysical applications
MFE 6th November 2013
11:00 to 12:00
S Lovejoy Scaling, macro weather and the climate
MFE 6th November 2013
14:00 to 15:00
Some basics in the asymptotic analysis of geophysical fluids
MFE 6th November 2013
15:00 to 16:00
Mathematical analysis of equatorial waves
MFE 7th November 2013
10:00 to 11:00
R Klages The thermostated dynamical systems approach to nonequilibrium steady states
In this talk I will outline a theory that aims at understanding the emergence of irreversible macroscopic transport starting from reversible microscopic dynamics. At the heart of this approach is to suitably model the interaction of a subsystem with a thermal reservoir. A simple example is a tracer particle in a fluid exhibiting Brownian motion for which there is the well-known description in terms of stochastic Langevin dynamics. Three decades ago scientists proposed a fully deterministic, time reversible modeling of thermalized motion by deriving a generalized Hamiltonian formalism yielding generalized friction coefficients in terms what is called Gaussian and Nos´e-Hoover thermostats. Surprisingly, in nonequilibrium situations such as, e.g., under an external electric field, this time reversible dissipative dynamics generates fractal attractors, exhibits an identity between phase space contraction and entropy production, and furnishes formulas that express transport coefficients in terms of Lyapunov exponents. In my talk I will show how this class of dynamical systems is constructed, will review its basic dynamical systems properties, and will critically discuss a conjectured universality of these properties. I will present a rather general summary of this approach, not much pre-knowledge about this particular field of research is required. [1] R.Klages, Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics (World Scientific, Singapore, 2007)
MFE 7th November 2013
11:00 to 12:00
Stochastic modeling and predictability: Analysis of a low-order coupled ocean atmosphere model
MFE 12th November 2013
10:00 to 11:00
Breakdown of linear response in the presence of bifurcations
(Joint with: M. Benedicks and D. Schnellmann) Many interesting dynamical systems possess a unique SRB ("physical") measure, which behaves well with respect to Lebesgue measure. Given a smooth one-parameter family of dynamical systems f_t, is natural to ask whether the SRB measure depends smoothly on the parameter t. If the f_t are smooth hyperbolic diffeomorphisms (which are structurally stable), the SRB measure depends differentiably on the parameter t, and its derivative is given by a "linear response" formula (Ruelle, 1997). When bifurcations are present and structural stability does not hold, linear response may break down. This was first observed for piecewise expanding interval maps, where linear response holds for tangential families, but where a modulus of continuity t log t may be attained for transversal families (Baladi-Smania, 2008). The case of smooth unimodal maps is much more delicate. Ruelle (Misiurewicz case, 2009) and Baladi-Smania (slow recurrence case, 2012) obtained linear response for fully tangential families (confined within a topological class). The talk will be nontechnical and most of it will be devoted to motivation and history. We also aim to present our new results on the transversal smooth unimodal case (including the quadratic family), where we obtain Holder upper and lower bounds (in the sense of Whitney, along suitable classes of parameters).
MFE 12th November 2013
11:00 to 12:00
Perturbations of the Lorentz Gas via Spectral Methods
MFE 12th November 2013
16:00 to 17:00
Nonequilibrium: from heat conduction, to turbulence (to life): Rothschild Distinguished Visiting Fellow lecture
MFE 13th November 2013
10:00 to 11:00
On the vortex-wave system
The vortex-wave system is the coupling of the 2D vorticity equation with the point-vortex system, and it is a model for the motion of sharply concentrated vórtices within a smooth(er) vorticity background. This system was introduced by Marchioro and Pulvirenti in 1991. In this talk we will give a fairly complete look at what is known and what is not known about this system.
MFE 13th November 2013
11:00 to 12:00
Efficiency, Irreversibility, and Tipping Points in Climate Dynamics
MFE 14th November 2013
10:00 to 11:00
R Klages Deterministic chaos and diffusion in maps and billiard
A fundamental problem of statistical mechanics and dynamical systems theory is to understand transport processes such as diffusion on the basis of deterministic chaos. In my talk I will discuss this issue for deterministic random walks in one and two dimensions generated by simple dynamical systems. For a class of piecewise linear maps lifted onto the whole real line the parameter-dependent diffusion coefficient can be calculated exactly analytically. It turns out that the response of these systems to parameter variations is non-trivial by displaying both linear and fractal parameter dependencies in the diffusion coefficient. Computer simulations predict analogous results for Hamiltonian particle billiards like the periodic Lorentz gas. These results are supported by systematic approximations based on a Taylor-Green-Kubo formula. [1] R.Klages, N.Korabel, Understanding deterministic diffusion by correlated random walks, J.Phys.A: Math. Gen.35, 4823 (2002) [2] R.Klages, Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics (World Scientific, Singapore, 2007)
MFE 14th November 2013
11:00 to 12:00
Speed of convergence to equilibrium for systems with contracting fibers and the logarithm law for Lorenz like flows.
A system satisfy a logarithm law if the time which is needed to hit a small target scales as the inverse of the measure of the target. This kind of laws, related to the occurrence of a rare event are also related to the arithmetical properties and to the speed of mixing of the system. I will present a short introduction to the subject and show how a general statement about the speed of convergence to equilibrium for systems with contracting fibers can be applied to deduce the logarithm law for a class of singular hyperbolic flows.
MFE 19th November 2013
10:00 to 11:00
N Glatt-Holtz An introduction to the ergodic theory of the stochastic Navier-Stokes equations and related systems in fluid dynamics.
MFE 19th November 2013
11:00 to 12:00
The nonlinear local Lyapunov exponent (NLLE) and its application to predictability study
MFE 20th November 2013
10:00 to 11:00
HJ Nussenzveig Lopes On bounded velocity/bounded vorticity solutions to the incompressible 2D Euler equations
In 1963 V. I. Yudovich proved the existence and uniqueness of weak solutions of the incompressible 2D Euler equations in a bounded domain assuming that the vorticity, which is the curl of velocity, is bounded. This result was later extended by A. Majda to vorticities which are bounded and integrable in the full plane. Further extensions of this result have been obtained, yet always assuming some decay of vorticity at infi nity. In a short note in 1995, Philippe Serfati gave an incomplete, yet brilliant, proof of existence and uniqueness of solutions to the 2D Euler equations in the whole plane when the initial vorticity and initial velocity are bounded, without the need for decay at in finity. In this talk I will report on work aimed at completing and extending Serfati's result to flows in a domain exterior to an obstacle. This is joint work with David Ambrose (Drexel University), James P. Kelliher (University of California, Riverside) and Milton C. Lopes Filho (Federal University of Rio de Janeiro).
MFE 20th November 2013
11:00 to 12:00
Coarse Graining and Entropy Production in a General Circulation Model
MFE 20th November 2013
16:00 to 17:00
Discussion: Probability in Operational Weather and Climate Forecasting
MFE 25th November 2013
10:00 to 11:00
Hamiltonian balance models: derivation, structure, and analysis
I will give an overview on balance models within the simplified > setting of the rotating shallow water equations, explain their Hamiltonian > structure, derivation via variational principles, and analytical properties.
MFE 25th November 2013
11:00 to 12:00
Extremes in Chaotic Dynamical Systems: Theory and Applications
MFE 25th November 2013
15:00 to 16:00
W Bahsoun Decay of correlation for random intermittent maps
MFE 26th November 2013
10:00 to 11:00
R Aimino Some statistical properties for random and sequential dynamical systems
MFE 26th November 2013
11:00 to 12:00
Nonequilibrium response, relaxation and the t-mixing condition
Twenty years ago a relation was proposed by Evans, Cohen and Morriss, which has been since known as Fluctuation Relation. This relation was initially intended to quantify the relative probability of positive and negative energy dissipation rates in non-equilibrium steady states. Later, it developed in many steady state as well transient relations. The search for the minimal ingredients which allow such steady state fluctuation relations to hold led to the formulation of the t-mixing condition. This condition, similarly to the standard mixing condition, requires a certain decay of correlations, but different from that concerning standard mixing. We illustrate how the notion of t-mixing arose in the context of fluctuation relations, and we compare it with standard mixing properties. Further, we note that besides yielding the fluctuation relations for dissipative, time reversal invariant deterministic systems, without need to explicitly invoke special properties for the phase space invariant measures, t-mixing leads to a number of other results. We will discuss, in particular, those concerning relaxation to equilibrium or non-equilibrium stationary states and those concerning a general Green-Kubo-like response formula, which may be applied arbitrarily far from equilibrium to large as well as small systems (with correspondingly different meanings).
MFE 27th November 2013
14:00 to 15:00
Phase Transitions in the Arctic Climate System
We will discuss phase transitions in permafrost and sea ice (as the key players in the Arctic Climate System). Questions to be discussed include: How does phase transition theory improve our knowledge of climate tipping points? How can the Ising model be applied to the study of Arctic melt ponds? How can the Ginzburg-Landau model be applied to the study of permafrost lakes? What is the connection between permafrost and superconductors? What is the connection between melt ponds and ferromagnetics?
MFE 28th November 2013
10:00 to 11:00
Y Pesin An approach for constructing SRB-measures for chaotic attractors
I will discuss a general approach for constructing SRB measures for diffeomorphisms possessing chaotic attractors (i.e., attractors with nonzero Lyapunov exponents). I introduce a certain recurrence condition on the iterates of Lebesgue measure called “effective hyperbolicity” and I will show that if the asymptotic rate of effective hyperbolicity is exponential on a set of positive Lebesgue measure, then the system has an SRB measure. Along the way a new notion of hyperbolicity -- "effective hyperbolicity'' will be introduced and a new example of a chaotic attractor will be presented. This is a joint work with V. Climenhaga and D. Dolgopyat.
MFEW02 2nd December 2013
09:15 to 10:15
Trapping of Rossby waves in the equatorial betaplane model
In this talk we shall report on joint works with Christophe Cheverry, Thierry Paul and Laure Saint-Raymond in which we study an equatorial shallow water system under the betaplane approximation. We prove that in some asymptotic regimes, Rossby waves are trapped around the equator while Poincaré waves disperse. This involves the use of microlocal techniques related to semiclassical analysis, such as Mourre estimates, and some ODE methods.
MFEW02 2nd December 2013
10:15 to 11:00
J Gibbon Rescaled vorticity moments in the 3D Navier-Stokes equations
Co-authors: D. D. Donzis (Texas A and M), A. Gupta (University of Rome Tor Vergata), R. M. Kerr (University of Warwick), R. Pandit (Indian Institute of Science Bangalore), D. Vincenzi (CNRS, Universite de Nice)

The issue of intermittency in numerical solutions of the 3D Navier-Stokes equations is addressed using a new set of variables whose evolution has been calculated through three sets of numerical simulations. These variables are defined on a periodic box $[0,\,L]^{3}$ such that $D_{m}(t) = \left(\varpi_{0}^{-1}\Omega_{m}\right)^{\alpha_{m}}$ where $\alpha_{m}= 2m/(4m-3)$ \& the set of frequencies $\Omega_{m}$ for $1 \leq m \leq \infty$ are defined by $\Omega_{m}(t) = \left(L^{-3}\I |\mbox{\boldmath$\omega$}|^{2m}dV\right)^{1/2m}$\,; the fixed frequency $\varpi_{0} = \nu L^{-2}$. All three simulations unexpectedly show that the $D_{m}$ are ordered for $m = 1\,,...,\,9$ such that $D_{m+1}

MFEW02 2nd December 2013
11:30 to 12:15
Recent observations of mesoscale structures in intense cyclones
The impact of intense cyclones is often associated with coherent mesoscale structures embedded within them. Although the formation of cold fronts is to some extent described by semi-geostrophic theory, the dynamics of many of these features are not well understood. For example, whether they could be described by a form balance appropriate on the mesoscale, or in the presence of non-conservative processes. Some recent detailed observations of cyclones and fronts are presented and some of the unknown aspects of their dynamics highlighted.
MFEW02 2nd December 2013
14:00 to 14:45
P Bartello Between quasigeostrophic and stratified turbulence
While it is well-established that the frequency disparity between vortical and wave motion is key to understanding the quasigeostrophic limit, i.e. strong rotation and stratification, the starting point for this contribution is that it has recently been established that there is no such frequency disparity in stratified turbulence without rotation. It remains to ask what happens in between these two limits, long held as the prevailing dynamics between deformation-scale eddies and the microscale where isotropy is recovered. To do this, ideas from numerical weather prediction were borrowed in order to explore numerically the nonhydrostatic Boussinesq equations starting from initial conditions that are close to our current fuzzy notions of balance for a variety of Rossby and Froude numbers. It is found that evolution is immediately away from this balance in the small scales, and from steep to much more shallow spectra. It will be argued that this conclusion is robust to unce rtainties in the definition of balance.
MFEW02 2nd December 2013
14:45 to 15:30
M Petcu Exponential Decay of the Power Spectrum and Finite Dimensionality for Solutions of the Three Dimensional Primitive Equations
The purpose of this talk is to estimate the number of modes, volumes and nodes, sufficient to describe well the solution of the three dimensional primitive equations; the physical meaning of these estimates is also discussed. We also study the exponential decay of the spatial power spectrum for the three dimensional primitive equations.
MFEW02 2nd December 2013
16:00 to 16:45
D Straub Energy cascades in the baroclinic ocean wind-driven double gyre problem
Co-author: David Straub

We consider the classic baroclinic quasigeostrophic wind-driven ocean double gyre problem over a range of deformation radii, wind stress amplitudes, and bottom friction coefficients with an aim of understanding transfer of energy across scales. In this beta-plane basin setting, we find significant differences from classic studies of gestrophic turbulence, which generally assume zonal periodicity. In a basin geometry, the beta term (related to a latitudinal dependence in the Corioils parameter) can play a key role. For example, it can be the dominant term allowing for energy transfer between the basin scale and the baroclinic mesoscale. We also find that barotropization of baroclinic mesoscale energy forces the barotropic mode at scales where the barotropic mode is most energetic. Related to this, the barotropic nonlinear inverse energy cascade does not extend between mesoscale injection and large scale dissipation wavenumbers, as is often assumed. Instead, it is part of a ``double cascade" of barotropic energy in which the nonlinear inverse cascade is nearly offset by a forward cascade associated with the beta term. This is particularly evident in weak bottom drag simulations, for which a time eddy-mean flow decomposition reveals the double cascade to beassociated with the eddy-only terms.

MFEW02 3rd December 2013
09:15 to 10:15
Mathematical Study of Certain Geophysical Models: Global Regularity and Finite-time Blowup Results
The basic problem faced in geophysical uid dynamics is that a mathematical description based only on fundamental physical principles, the so-called the \Primitive Equations", is often prohibitively expensive computationally, and hard to study analytically. In this talk I will discuss the main obstacles in proving the global regularity for the three-dimensional Navier-Stokes equations and their geophysical counterparts. However, taking advantage of certain geophysical balances and situations, such as geostrophic balance and the shallowness of the ocean and atmosphere, geophysicists derive more simplied and manageable models which are easier to study analytically. In particular, I will present the global well-posedness for the three-dimensional Benard convection problem in porous media, and the global regularity for a three-dimensional viscous planetary geostrophic models. Even though the primitive equations look as if they are more dicult to study analytically than the three-dimensional Navier-Stokes equations I will show, on the one hand, that the viscous primitive equations have a unique global (in time) regular solution for all initial data. On the other hand, I will show that in the non-viscous (inviscid) case there is a one-parameter family of initial data for which the corresponding smooth solutions develop nite-time singularities (blowup).
MFEW02 3rd December 2013
10:15 to 11:00
RM Temam Change of phase for the humid atmosphere
In this lecture we will recall the atmospheric equations of water vapor with saturation. In their simplest form, these equations form a nonlinear system of partial differential equations with discontinuities. We will address the issue of the modelling of the system in the presence of singularities, and some questions on the existence, uniqueness and regularity of these solutions.
MFEW02 3rd December 2013
11:30 to 12:15
L Chumakova Leaky lid: new dissipative modes in the troposphere
Much of our understanding of tropospheric dynamics is based on the concept of discrete internal modes. Internal gravity waves, such as those associated with convective systems, propagate at definite speeds, typically associated with the first to third baroclinic vertical modes. These waves are the dynamical backbone of the tropospheric dynamics, even though their nature and speed can be altered significantly by nonlinearity, moist convection, mean wind shear, etc. These discrete modes are a signature of systems of finite extent, and are derived in a case when the atmosphere is bounded above by a rigid lid. In reality, the atmosphere does not have a definite top, and, some argue, should be modeled as semi-infinite, leading to a continuous spectrum. Are the discrete rigid lid modes then just a fallacy of overly simplified theoretical models? We present a correction to the rigid lid by using a boundary condition at the top of the troposphere, that allows for a fraction of waves to escape to the stratosphere. The new discrete ``leaky” modes decay with characteristic time-scales, which are in the ballpark of many atmospheric phenomena. We present both the non-rotating and rotating cases.
MFEW02 3rd December 2013
14:00 to 14:45
JG Esler Adaptive stochastic trajectory modelling of transport in geophysical flows
Motivated by the goal of improving and augmenting stochastic Lagrangian models of particle dispersion in turbulent geophysical flows, techniques from the theory of stochastic processes are applied to a model transport problem. The aim is to find an efficient and accurate method to calculate the total tracer transport between a source and a receptor when the flow between the two locations is weak, rendering direct stochastic Lagrangian simulation prohibitively expensive. Two methods are found to be useful. The first is Milstein's `measure transformation method', which involves adding an artificial velocity to the trajectory equation, and simultaneously correcting for the weighting given to each particle under the new flow. Various difficulties associated with making an appropriate choice for the artificial velocity field are detailed and addressed. The second method is a variant of Grassberger's `go-with-the-winners' branching process, which acts to remove particles unlikely to contribute to the net transport, and reproduces those that will contribute. A simple solution to the problem of defining a `winner' for flows in a high Peclet number chaotic advection regime is proposed. It is demonstrated that, used independently or together, the two methods can act to reduce the variance of estimators of the total transport by several orders of magnitude compared with direct simulation.
MFEW02 3rd December 2013
14:45 to 15:30
Modelling the interactions of near-inertial waves and vortical motion in the ocean
Co-author: Eric Danioux (University of Edinburgh)

Wind forcing of the ocean generates a spectrum of inertia-gravity waves that is sharply peaked near the local inertial (or Coriolis) frequency. The corresponding near-inertial waves (NIWs) make a dominant contribution to the vertical velocity and vertical shear in the ocean; they therefore play an important role for mixing, biological productivity, pollutant dispersion and, arguably, the thermohaline circulation. An asymptotic model proposed by Young and Ben Jelloul describes the slow evolution of NIWs that results from weak dispersion and from their interactions with the quasi-two-dimensional vortical motion. We derive this YBJ model by applying a form of Whitham averaging to the variational formulation of the primitive equations for a rotating stratified fluid. This provides a direct route to the YBJ equation and elucidates its variational structure and conservation laws. We then consider the effect of turbulent vortical motion (modelled as a homogeneous random field) of a scale similar to that of the waves. Specifically, we derive a transport equation for NIWs that describes their scattering by the vortical motion and show how this scattering leads to an isotropization of the NIW field. Direct numerical simulations of the YBJ equations are used to test the predictions of the transport equation. Possible models of the two-way coupling between NIWs and vortical motion are also discussed.

MFEW02 3rd December 2013
16:00 to 16:45
Direct Statistical Simulation of Jet Formation in Local and Global Geometries
Co-author: Brad Marston (Brown University)

We present Direct Statistical Simulation (DSS) of jet formation. We consider the simplest barotropic model both on a spherical surface and a local beta-plane. DSS involves the direct solution of the low-order statistics via an expansion in cumulants. In both cases we compare the results of our DSS with statistics obtained from long DNS simulations. We discuss in what circumstances truncating the cumulant expansion at second order (thereby including eddy – mean-flow interaction but neglecting eddy-eddy interactions for the fluctuating fields) gives a good description of the dynamics of the flow. We demonstrate that this depends on the degree of lack of statistical equilibrium in the flow (as measured by the Zonostrophy parameter). We discuss briefly how to proceed to higher order to include eddy-eddy interactions and the possibility of forward and inverse cascades.

MFEW02 4th December 2013
09:15 to 10:15
D Holm Slice models
Co-author: Colin Cotter (Imperial College)

A variational framework is defined for vertical slice models with three dimensional velocity depending only on horizontal $x$ and vertical $z$. The models that result from this framework are Hamiltonian, and have a Kelvin-Noether circulation theorem that results in a conserved potential vorticity in the slice geometry. These results are demonstrated for the incompressible Euler--Boussinesq equations with a constant temperature gradient in the $y$-direction (the Eady--Boussinesq model), which is an idealised problem used to study the formation and subsequent evolution of weather fronts. We then introduce a new compressible extension of this model for testing compressible weather models running in a vertical slice configuration. (Joint work with CJ Cotter, Imperial College).

MFEW02 4th December 2013
10:15 to 11:00
From simple particle models to PDE dynamics
One often aims to describe systems out of equilibrium by the governing energy E and entropy S, as well as the corresponding evolution laws for E and S. How can we derive these ingredients of the macroscopic evolution from particle models? In recent years, a dynamic scale-bridging approach has been developed and applied to a number of problems; large deviation theory plays an important role. The talk will present some of these results, focussing on the derivation of the Wasserstein-entropy formulation of diffusion and the Vlasov-Fokker-Planck equation as a system driven by energy and entropy. Time permitting, an approach of deriving stochastic equations mimicking the fluctuations in underlying mesoscopic models will be sketched. This is joint work with Hong Duong, Rob Jack and Mark. A. Peletier.
MFEW02 4th December 2013
11:30 to 12:15
B Marston Direct Statistical Simulation of a Two-Layer Primitive Equation Model
Co-authors: Wanming Qi (Brown University), Steve Tobias (University of Leeds)

Low-order statistics of the large-scale circulation of planetary atmospheres may be directly accessed by solving the equations of motion for the equal-time statistics. We implement such Direct Statistical Simulation of a two-layer primitive equation model by systematic expansion in the cumulants. The first cumulant is the zonally averaged vorticity, divergence, and temperature as a function of latitude and level, and the second cumulant contains information about nonlocal teleconnections. At second order (CE2) the expansion retains the eddy – mean-flow interaction but neglects eddy-eddy interactions and is realizable. Eddy-eddy interactions appear at third (CE3) order, but care must be taken to maintain realizability with a non-negative probability distribution function. The cumulant expansion is conservative, order-by order, in the total angular momentum, total energy, and mean-squared potential temperature. An intermediate approximation, CE2.5, is related to the Edd y-Damped Quasi-Normal Markovian (EDQNM) approximation and maintains realizability at the expense of the introduction of a phenomenological timescale for eddy damping. First and second cumulants accumulated by time-integration of the two-layer primitive equations are compared with those obtained at the fixed points found at CE2, CE2.5, and CE3 levels of approximation, and against statistics obtained from reanalysis of the mid-level atmosphere of the Earth. CE2 reproduces qualitative features of the zonal mean general circulation such as the mid-latitude jets. CE2.5 and CE3 improve quantitative agreement in both the zonal means, and in the teleconnections.

MFEW02 4th December 2013
14:00 to 14:45
Navier-Stokes equations on a rotating sphere
We showed that, as the rotation rate $1/\epsilon$ increases, the solution of the 2d Navier-Stokes equations on a rotating sphere becomes zonal, in the sense that the non-zonal component of the energy becomes bounded by $\epsilon$. This is obtained by estimating near-resonant interactions in the nonlinear term. As a consequence, the global attractor reduces to a single stable steady state when the rotation is fast enough (but still finite).
MFEW02 4th December 2013
14:45 to 15:30
The influence of fast waves and fluctuations on the evolution of three slow solutions of the Boussinesq equations
Co-authors: Jared P. Whitehead (Brigham Young University), Terry Haut (Los Alamos National Laboratory)

We present results from a study of the impact of the non-slow (typically fast) components of a rotating, stratified flow on its slow dynamics. We examine three known slow limits of the rotating and stratified Boussinesq equations: strongly stratified flow ($Fr \rightarrow 0, Ro \approx O(1)$), strongly rotating flow ($Ro \rightarrow 0, Fr \approx O(1)$) and Quasi-Geostrophy ($Ro \rightarrow 0, Fr \rightarrow 0, Fr/Ro = f/N$ finite).

In order to understand how the flow approaches and interacts with the slow dynamics we decompose the full solution into a component that is projected onto the null space of the fast operator and everything else. We use this decomposition to find evolution equations for the flow (and corresponding energy) on and off the slow manifold.

Numerical simulations indicate that for the geometry considered (triply periodic) and the type of forcing applied, the fast waves act as a conduit, moving energy onto the slow manifold. This decomposition clarifies how the energy is exchanged when either the stratification or the rotation is weak. In the quasi-geostrophic limit the energetics are less clear, however it is observed that the energy off the slow manifold equilibrates to a quasi-steady value.

At the end I will discuss how greater understanding of flow/fast dynamics could impact emerging numerical algorithms designed for future computer architectures.

MFEW02 4th December 2013
16:00 to 16:45
Resonant phenomena in the wave dynamics in the presence of boundaries in GFD
Co-author: Grigory Reznik (In-t of Oceanography, Moscow)

I will be discussing resonant interactions of coastal and shelf waves with free inertia-gravity waves and mean coastal currents in GFD.

MFEW02 5th December 2013
09:15 to 10:15
Use of the semi-geostrophic model in understanding large-scale atmosphere and ocean flows
The semi-geostrophic model is an accurate approximation to the Navier-Stokes equations when the Lagrangian Rossby number is small and the aspect ratio is less than f/N (where f is the Coriolis parameter and N the Brunt-Vaisala frequency). In practice this leads to a horizontal scale of around 1000km in the atmosphere and 100km in the ocean. The approximation is second order accurate in the limit epsilon tends to zero where the Rossby number is O(epsilon) and the Froude number O(sqrt(epsilon)). This approximation has a big advantage over the quasi-geostrophic approximation because it allow O(1) variations of the static stability, Coriolis parameter and orographic height. These features are essential in describing large-scale flows. The stability of the solutions of this system is consistent with the observed persistence of large-scale anomalies in the atmosphere and ocean. The failure of the approximation on smaller scales is associated with the lack of such structures on smal ler scales in the observed system. The talk will demonstrate how the semi-geostrophic model can be used to validate numerical methods, and how it can be extended to include a realistic model of the atmospheric boundary layer.
MFEW02 5th December 2013
10:15 to 11:00
M Feldman Lagrangian solutions for semigeostrophic system in physical space
Co-author: Adrian Tudorascu

In this introductory talk, we review results on existence of solutions to semigeostrophic system in physical and dual spaces. In particular, we show that Lagrangian solutions in physical space can be constructed for initial data satisfying a strict convexity condition. We also briefly discuss the recent joint work with A. Tudorascu, in which we relax the notion of Lagrangian solution to obtain existence for all convex initial data in physical space.

MFEW02 5th December 2013
11:30 to 12:15
Renormalized relaxed Lagrangian solutions for SG in physical space
A new, relaxed notion of Lagrangian solutions for SG in physical space will be introduced. The main motivation is the search for physical space solutions in the case of singular data in dual space. Existence and a weak stability result will be proved, along with an energy conservation result which comes as a consequence of the renormalization property. This presentation is based on joint work with M. Feldman (U. Wisconsin-Madison).
MFEW02 5th December 2013
14:00 to 14:45
Balance relations for rotating fluid flow
We discuss different nonlinear elliptic balance relations for rotating shallow water flow and describe computational tests which compare their utility as an initialization or diagnostic tool in nearly geostrophic situations.
MFEW02 5th December 2013
14:45 to 15:30
Generalized Large Scale Semigeostrophic Equations: geometric structure and global well-posedness
Co-authors: Marcel Oliver (Jacobs University), Mahmut Calik (Jacobs University)

We derive and study a family of approximate Hamiltonian balance models (called GLSG) for rotating shallow water in the semigeostrophic limit with spatially varying Coriolis parameter and non-trivial bottom topology. The models can be formulated in terms of an advected potential vorticity with a nonlinear vorticity inversion relation and include L_1 and LSG models proposed by R. Salmon as special cases.

We prove existence and uniqueness of global classical solutions to the GLSG equations for certain members of the family and study the PV invertibility as a function of the parameters.

MFEW02 5th December 2013
16:00 to 16:45
B Pelloni The semi-geostrophic system for large-scale atmospheric flows
Co-author: Mike Cullen (Met Office)

I could present results on the existence of solutions in 3D, free boundary setting as well as some preliminary results on the validity of the model as a reduction of the Euler system.

MFEW02 6th December 2013
09:00 to 09:45
G Vallis The Range of Planetary Circulations Described by the Dry Primitive Equations
Co-authors: Jonathan Mitchell (UCLA), Sam Potter (Princeton University)

The dry primitive equations can, with appropriate forcing and dissipation, provide a reasonable simulation of the large-scale features of the Earth's atmosphere and ocean. In this talk I will describe the behaviour of these PDEs when they are taken out of the terrestrial parameter regime. In n particular, I will describe their behaviour when the thermal Rossby number, Ekman number and a radiative relaxation timescale are varied considerably, moving into a parameter regime more appropriate for Mars or Titan.

I will pay particular attention to the formation of zonal jets, and in particular of equatorial superrotation, which is a feature of some other planets. It is well-known that zonal jets robustly arise in rotating atmospheres if there is a wavemaker at a particular latitude. Rossby waves are then generated that propagate away, and eastward momentum converges on the source region producing a zonal jet. The Earth's jet stream is, in part, formed this way. However, on slowly rotating atmospheres it seems unlikely that superrotation is produced by that mechanism. Rather, simulations indicate that, at small thermal Rossby number, a mechanism involving equatorial Kelvin waves is involved.

MFEW02 6th December 2013
09:45 to 10:30
I Roulstone Differential Geometry of the Semi-Geostrophic and Euler Equations
The role of contact and symplectic geometry of the semi-geostophic (SG) equations, in describing their Legendrian and Hamiltonian properties, will be reviewed. Using the geometry of 2-forms in 4 dimensions and the geometry of 3-forms in 6 dimensions, we show that the incompressible Euler equations in 2 and 3 dimensions admit geometric structures akin to the those present in the SG equations.
MFEW02 6th December 2013
11:00 to 11:45
O Bokhove On Time Integration and the Use of Clebsch Variables in Shallow Water Equations
Two topics will be covered in this lecture. The shallow water equations will be used as a test bed to introduce the ideas.

(i) For forced variational systems such as the potential flow shallow water wave equations, variational and symplectic time integrators will be extended using a new finite element approach. Here, a standard variational finite element discretization will be applied in space.

(ii) The shallow water equations formulated in terms of Clebsch variables will be discussed. The advantage of Clebsch variables is that they lead to canonical Hamilton's equations for shallow water dynamics, in the Eulerian framework. A disadvantage is that the the system, is less compactly expressed in comparison to the usual formulation in terms of the velocity and fluid depth. I will make a link between a symmetry in the Hamiltonian and the associated mass weighted potential vorticity conservation law, also within the Eulerian framework. This will be done in two dimensions (2D) and in a quasi-2D symmetric form.

MFEW02 6th December 2013
11:45 to 12:30
Monge-Ampère equations: geometry, invariants, and applications in 3D meteorological models
I describe a symplectic geometric approach to Monge-Ampère equations, discuss some classical and modern geometric structures related to this class of non-linear equations, their invariants and their role in 3D meteorological models. My talk is based on joint works with B. Banos and I . Roulstone.
MFE 10th December 2013
11:00 to 12:00
Large-scale tropical atmospheric dynamics: asymptotic nondivergence?
MFE 11th December 2013
11:00 to 12:00
Asymmetric inertial instability
MFE 12th December 2013
10:00 to 11:00
Rare events, negative measure dimensions and return time statistics
MFE 18th December 2013
10:00 to 11:00
Predictability of extreme events in dynamical systems: A case study of the Lorenz 84 model
MFE 18th December 2013
11:00 to 12:00
Nonequilibrium statistical mechanics approach to turbulence
I will present an overview how superstatistical techniques (a method from nonequilbrium statistical mechanics) can be successfully used to model the statistical properties of turbulent flows. Examples treated are classical turbulence, quantum turbulence, environmental turbulence (wind gusts) and, if time remains, dark matter turbulence.
MFE 18th December 2013
14:00 to 15:00
JM Freitas Extreme values for deterministic and random dynamical systems
It is well known that the Extremal Index (EI) measures the intensity of clustering of extreme events in stationary processes. We sill see that for some certain uniformly expanding systems there exists a dichotomy based on whether the rare events correspond to the entrance in small balls around a periodic point or a non-periodic point. In fact, either there exists EI in $(0,1)$ around (repelling) periodic points or the EI is equal to $1$ at every non-periodic point. The main assumption is that the systems have sufficient decay of correlations of observables in some Banach space against all $L^1$-observables. Then we consider random perturbations of uniformly expanding systems, such as piecewise expanding maps of the circle. We will see that, in this context, for additive absolutely continuous noise (w.r.t. Lebesgue), the dichotomy vanishes and the EI is always 1.
MFE 19th December 2013
10:00 to 11:00
Extremes as indicators of critical transitions
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons