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Mathematics for the Fluid Earth

February 5-7, 2014

De Morgan House, LMS, London

List of Speakers and Abstracts

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Speaker Title Abstract
Gualtiero Badin On the role of shortwave instabilities in geostrophic turbulence A mathematical description for the emergence and the effects of instabilities in geophysical flows at scales smaller than the deformation radius in fully developed turbulence is still missing. Two different mechanisms, one balanced (making use of the quasi-geostrophic approximation) and one unbalanced (making use of the semi-geostrophic approximation), will be proposed and analyzed
Richard Blender Nambu Theory in Geophysical Fluid Dynamics The talk introduces Nambu’s (1973) extension of Hamiltonian mechanics by including several conservation laws in dynamical equations which satisfy the Liouville Theorem. Ideal hydrodynamics has been formulated in a Nambu representation in two and three dimensions using enstrophy and helicity as second conservation laws in addition to the total energy (Névir and Blender, 1993). The basis for these conservation laws is the particle relabeling symmetry in fluid dynamics. Noncanonical Hamiltonian mechanics is embedded in Nambu mechanics if a Casimir function can be incorporated as a conservation law. Salmon (1995) suggested the design of conservative numerical codes based on a Nambu formulation. The Nambu representations of the quasigeostrophic equations, the shallow water model, the Rayleigh-Bénard equations, and the baroclinic atmosphere are reviewed. These models demonstrate that the Nambu representation provides a natural description of geophysical fluid dynamics. Numerical simulations reveal realistic energy spectra. Possible future extensions are discussed.
Jean Bricmont Phase transitions in Probabilistic Cellular Automata

Cellular automata are deterministic dynamical systems whose phase space consists of a lattice of variables taking a finite set of values.

Probabilistic Cellular Automata (PCA) are stochastic perturbations of cellular automata that define Markov processes on the phase space of the cellular automaton.

PCA's provide simple models to study dynamical phase transitions: by varying the strength of the noise one can have one or several stationary measure for the Markov process.

In the talk, we will first define those notions and explain in which cases one can show the existence of a phase transition. There exists also mean field approximations to PCA that can be shown to have a phase transition. In a joint work with Hanne Van Den Bosch, we introduce a model interpolating between a class of PCA, called majority voters, and their corresponding mean field models. Using graphical methods, we prove that this model undergoes a phase transition.

Bin Cheng Time-averaging and error estimates for reduced fluid models. I will discuss the application of time-averaging in getting rigorous error estimates of some reduced fluid models, including the quasi-geostrophic approximation, incompressible approximation and zonal flows. The spatial boundary can be present as a non-penetrable solid wall. I will show a very recent (and somewhat surprising) result on the epsilon^2 accuracy of incompressible approximation of Euler equations, thanks to several decoupling properties.
Collin Cotter Compatible finite element methods for numerical weather prediction Within the "Gung Ho" UK dynamical core project to develop a dynamical core for the Met Office Unified Model that is scalable massively parallel processors, we have been developing finite element methods that extend the properties of the C-grid staggered finite difference methods by providing the flexibility to use arbitrary meshes, to control the ratio of degrees of freedom for different fields (which has an impact on the presence or otherwise of spurious modes), and to increase the orders of accuracy. These methods fit into the elegant framework of finite element exterior calculus. I will explain how properties of stability, accuracy and conservation laws all arise naturally from this framework.
Mike Cullen The semi-geostrophic model with variable rotation

The original derivation of what is now called the semi-geostrophic model by Eliassen included a variable Coriolis parameter, which is necessary for describing large-scale atmospheric flows. The model is second order accurate in the Phillips Type II limit, in which the Froude number is assumed to tend to zero as the square root of the Rossby number, which is consistent with the maintenance of geostrophic balance. The importance in this limit is that no approximations are required to the thermodynamics, which control large-scale atmospheric circulations, while the dynamics are filtered by the geostrophic momentum approximation.

While there are many rigorous results for this model with constant rotation, these all rely on the construction of 'dual space', in which the equations reduce to the conservation of mass under an divergence-free velocity field. The mass in dual space plays the role of potential vorticity, and is an approximation to the inverse of the Ertel potential vorticity accurate to the same degree as the evolution equations themselves. This transformation cannot be carried out with variable rotation. This reflects the same loss of symmetry as is present in the shallow water equations with variable rotation, and comes from the fact that the Coriolis parameter is no longer the curl of a vector. Under the geostrophic momentum approximation, this issue is more serious because the variable Coriolis parameter is used to define the geostrophic wind, which is then no longer non-divergent. In the less accurate planetary geostrophic model, the momentum is completely ignored in the definition of potential vorticity, so that the conservation is retained. This loss of symmetry is consistent with the observed qualitative difference between extratropical and tropical dynamics.

I describe how the problem can nevertheless be formulated and solved in principle in physical space using a Lagrangian formulation. This illustrates that the particle relabelling symmetry is still present, it is the rotational symmetry that is lost. The method relies on the characterisation of geostrophic balance as a local energy minimiser, the global minimisation property used in the constant rotation case is no longer well-defined. This allows the problem to be written as an optimal transportation problem, but this now has to be solved within a time-stepping scheme in order to ensure that the solution of the optimal transport problem corresponds to the correct equation. The solution then gives a geostrophic wind which is BV, though it can no longer be written in terms of the gradient of a convex function. Recent results by Feldman and Tudorascu then suggest that a relaxed version of the evolution equations can be solved.

H. A. Dijkstra Interaction of Noise and Nonlinear Dynamics in the Wind-Driven Ocean Circulation

Results will be presented of a study on the interaction of noise and nonlinear dynamics in a quasi-geostrophic model of the wind-driven ocean circulation. The recently developed framework of dynamically orthogonal field theory is used to determine the statistics of the flows which arise through successive bifurcations of the system as the ratio of forcing to friction is increased. Focus will be on the understanding of the role of the spatial and temporal coherence of the noise in the wind-stress forcing on the variability of the flows.

Darryl Holm Enforcing Selective Decay in Fluid Flows, by Darryl D Holm

A general theory of selective decay in fluids will be developed by using the Lie-Poisson structure of the ideal fluid theory in 2D and 3D. Several examples will be given and the relation of selective decay by Casimir dissipation to anticipated vorticity schemes will be discussed. Some trade secrets will be revealed about selective decay and anticipated vorticity methods for numerical simulations. The talk will report on published work with François Gay-Balmaz.

Parameterizing interaction of disparate scales: Selective decay by Casimir dissipation in fluids. F Gay-Balmaz and DD Holm Nonlinearity 26(2):495–524, http://arxiv.org/abs/1206.2607

Rustam Ibragimov Heavy-tailedness, dependence and diversification failures: Implications for models in economics, finance, econometrics and statistics The talk will focus on several modern approaches to modeling extremes, crises, heavy-tailedness and dependence in econometrics, statistics, economics and finance and discuss their applications. We will present several results on the effects of the interplay between heavy-tailedness and dependence on (non-)robustness of key models in these fields, focusing, in particular, on the important problems of diversification (sub-)optimality and problems for inference. The results provide further motivation for development and applications of econometric and statistical inference procedures that are robust to heterogeneity, dependence and heavy-tailedness; and the talk will discuss some of recent new developments in this direction.
Rupert Klein Atmospheric flows from a multiple-scales asymptotics perspective

The first part of this contribution will summarize the main concepts and results of my recent Annual Reviews report, [1], which maps out much of the landscape of known reduced atmospheric flow models from an asymptotics perspective. An interesting conclusion of that report was that practically all well-established reduced flow models can be understood through asymptotics for the compressible flow equations based on one and the same distinguished limit for the Mach, Rossby, and internal wave Froude numbers, with differences between models captured solely by suitable space-time scalings.

Most of the established models are single-scale in an asymptotic sense, however, in that they capture only one characteristic time scale and one characteristic scale in each of the canonical space coordinate directions. As a consequence, the model map from [1] can only serve as a starting point for studying more complex flow situations involving multiple scales. The second part of my lecture will provide several examples of recent multiple scales analyses of atmospheric flow phenomena and flow models, such as internal wave--cloud column interactions [2], intense large-scale atmospheric vortices [3], and the asymptotics and numerics of sound-proof models for the troposphere [4,5].

  • [1] R. Klein (2010), Scale-dependent models for atmospheric flows, Annu. Rev. Fluid Mech., vol. 42, 249--274
  • [2] D. Ruprecht, R. Klein, A.J. Majda (2010), Modulation of internal gravity waves in a multi-scale model for deep convection on mesoscales, J. Atmos. Sci., vol. 67, 2504--2519
  • [3] E.Paeschke, P. Marschalik, A.Z. Owinoh , R. Klein (2012), Motion and structure of atmospheric mesoscale baroclinic vortices: dry air and weak environmental shear, J. Fluid Mech., vol. 701, 137--170
  • [4] R. Klein, U. Achatz, D. Bresch, O.M. Knio, P.K. Smolarkiewicz (2010), Regime of validity of sound-proof atmospheric flow models, J. Atmos. Sci., vol. 67, 3226--3237
  • [5] T. Benacchio, W. O'Neill, R. Klein (2014), A blended soundproof-to-compressible numerical model for atmospheric dynamics, Mon. Wea. Rev., under review
Tobias Kuna Realizability problem In order to describe complex systems effectively, one often concentrate on a few characteristics of the system, like moments and their correlations. The large number of remaining degrees of freedom one want to supress as subscale effects, that is, by deriving effective equations just in terms of these characteristics, as for example the so-called moment closure. A related question is if an effective description of a particular type exists for a give complex system. The different characteristics are not independent of each other and not all putative choices may correspond to a complex system of a particular type. This circle of questions has been most extensively investigate for systems consisting of localizable objects like particles, eddies, trees etc. A general introduction to this circle of problems is given.
Valerio 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.
Paul Manneville An overview of the transition to/from turbulence in plane Couette flow as a paradigm of complex behavior in nonlinear distributed systems

Wall-bounded flow, and especially plane Couette flow, the flow between counter-translating parallel plates, exhibit coexistence of laminar and turbulent flow in a range of Reynolds number called `transitional'. I shall present some recent experimental, numerical and theoretical findings about that coexistence and discuss the extent to which they can illustrate some of the topics addressed during the MFE13 program.

Antonin Novotny Thermodynamic stability conditions and weak solutions in the compressible fluid dynamics. We will talk about the role of the thermodynamic stability conditions in the investigation of the structural stability of weak solutions to the complete Navier-Stokes-Fourier system describing flows of viscous, compressible and heat conducting fluids. We will derive the so called relative energy inequality and discuss some of its applications that range from weak-strong uniqueness through the investigation of singular limits to the derivation of error estimates to the numerical schemes for compressible flows.
Giovanni Gallavotti Random matrices and Lyapunov exponents analyticity relationship between Lyapunov exponents for products of matrices preserving cones and the cluster expansion for one dimensional spin systems.
Mark Holland Speed of convergence to an extreme value distribution. We consider an ergodic dynamical system together with an observation function having a unique maximum at a (generic) point in the phase space. Under this observable, we consider the time series of successive maxima along typical orbits. Recent works have focused on the distributional convergence of such maxima (under suitable normalization) to an extreme value distribution. For certain dynamical systems, we establish convergence rates to the limiting distribution. In contrast to the case of i.i.d random variables, the convergence rates depend on the rate of mixing and the recurrence time statistics. For a range of applications, including uniformly expanding maps, quadratic maps, and intermittent maps, we establish corresponding convergence rates. We also establish convergence rates for certain hyperbolic systems such Anosov systems, and discuss convergence rates for non-uniformly hyperbolic systems, such as H\'enon maps. Work is joint with M. Nicol.
Laure Saint-Raymond About the role of boundary layers in the large-scale oceanic circulation

The exchanges of energy in boundary layers are crucial to explain the global oceanic circulation. We will present here some mathematical tools to study these mechanisms, and illustrate them on the example of Munk layers. A challenging open question is to understand how general are these methods.

Richard Sharp Fluctuation theorems with shrinking intervals.

Fluctuation theorems arise in statistical mechanics and describe systems away from equilibrium. They have been rigorously proved for hyperbolic dynamical systems and are related to large deviations of ergodic averages. We will discuss a version where averages lie in intervals that shrink (at an appropriately slow rate) as the systems evolves. (This is joint work with Mark Pollicott.)>

Mike Todd The impact of dependence on Extreme Value Laws

For an iid system there are three classic types of Extreme Value Laws (EVLs), which from an abstract point of view, where different probability distribution functions can be treated a simple change of variables, boil down to some exponential recurrence law. Time series derived from dynamical systems can be viewed as dependent systems, and many recent works have shown that if the system mixes fast enough then they follow standard EVLs. Starting with simple periodicity, I’ll discuss what can happen when there is so much dependence that the EVLs are qualitatively different.

Giorgio Turchetti Randomness in dynamical models: comparison of round off errors and noise

The orbits of a dynamical system change due to the uncertainty in the initial conditions, to intrinsic random perturbations and to finite precision of instruments when they are measured. When the system is simulated in a digital computer the orbits are affected by finite precision arithmetic which introduces a sort of noise. The round off errors and noise cause detectable effects with a time threshold which depends on their amplitude and the nature of the dynamics. We first compare the decay of correlations and fidelity for three prototype models, the translations on torus, the skew map on the cylinder and the Bernoulli shifts on the torus. Additive random noise and multiplicative observational noise on regular maps cause the decay of correlations and fidelity, though with a different law, whereas round off errors and additional observational noise do not. For chaotic orbits the effect of round off errors and random noise is almost the same.

The spectrum of recurrences , the extreme value laws and the related convergence rates provide additional information. For reversible maps the error in the return to the initial point allows to investigate in a simple way the uncertainty introduced by round off or random noise.

Sandro Vaienti Sequential dynamical systems: loss of memory and extreme value theory We introduce a class of intermittent sequential dynamical systems and we prove their polynomial loss of memory. Moreover we adress (and solve for some sequential system), the problem of extreme values distributions for non-stationary processes.