Summer Research Program on Dynamics of Complex Systems
Contens or abstracts of the lectures of the summer school, 23 May  04 June, 2016.
 Jayanta K. Bhattacharjee, HRI, Allahabad: "Introduction to Turbulence"
 Homogeneous isotropic turbulence: the scales, Kolmogorov scaling, the dissipation anomaly, inertial range  short distance or large distance?
 Special features in two dimensions , and if possible a bit of intermittency
 Turbulence in a stratified fluid (stratification from temperature gradient)
 Scaling a la Kolmogorov or Bolgiano and Obukhov. Heat transport in convective turbulence, boundary layers  the Nusselt number scaling with Rayleigh number
 Rotating convective turbulence. Ekmann boundary layer
 Georg Gottwald, University of Sydney, "Stochastic Model Reduction in Climate Science"

Introduction to some stochastic concepts

Deterministic Model reduction:(i) relaxation of fast process to a fix point; cnetre manifold theory(ii) relation to periodic orbit, averaging(iii) relaxation to strange attractor, averaging(iv) no relaxation (Hamiltonian model reduction; variational asymptotic)

MoriZwanzig formalism

Stochastic model reduction in timescale separated systems(i) averaging(ii) homogenisation

Results for deterministic dynamical systems

Applications in climate: MTV, energy conservation, Markov processes

Stochastic model reduction in the weak coupling limit
 References:
** Lasota and Mackey “Chaos, Fractals and noise: Stochastic aspects of noise” or ”Probabilistic aspects of dynamical systems” (same book; just changed title)
** Bernd Oksendal “Stochastic Differential Equations"** Stephen Wiggins "Introduction to Applied Nonlinear Dynamical Systems and Chaos”
** Chorin & Hald “Stochastic Methods in Applied Mathematics and Physics”** Robert Zwanzig “Nonequilibrium Statistical Mechanics”** Pavliotis & Stuart “An Introduction to Multiscale Methods”** Horsthemke & Lefever “Noiseinduced Transitions”** Givon, Kupferman, & Stuart “Extracting macroscopic dynamics: model problems and algorithms” (PDF)

 Rama Govindarajan, TCIS/ICTSTIFR: "Introduction to hydrodynamic stability"
 Michael Hoegele, Universidad de losAndes, Bogotá: "Stochastic climate models with Lévy noise with applications in paleoclimate"
 Understanding the future evolution of current climate patterns based on current observations remains a highly challenging and risky task when the understanding of past climate phenomena is still extremely poor. The evidence of the past climate evolution, however, is typically indirect, scarce and charged with a lot of uncertainty. In some special cases, such as ice core data or sediment analysis, it is still possible to assemble time series of an considerable depth. Due to the complete lack of information it is impossible to understand those data as the projection of high dimensional models. An alternative strategy consists in the comparison of those time series with “typical” paths of models sufficiently rich classes of low dimensional stochastic models. A rich class of models, where this can be done are stochastic differential equations driven by Levy processes, a natural generalization of Brownian motion, which allow for many features typically found in those time series, such as (large) jumps, heavytails and asymmetric noise distributions. This class exhibits already a large clase of random dynamical phenomena, such as metastability, stochastic resonance, synchronization etc.
In this course we shall give an introduction to Levy processes and their stochastic differential equations. In the second week we shall understand “typical” paths of such evolutions, and the associated first exit problem as well as associated metastable behavior. Furthermore we shall introduce adequate distances in order to measure the distance between such models and data, which will be illustrated for a paleoclimate time series.
 Understanding the future evolution of current climate patterns based on current observations remains a highly challenging and risky task when the understanding of past climate phenomena is still extremely poor. The evidence of the past climate evolution, however, is typically indirect, scarce and charged with a lot of uncertainty. In some special cases, such as ice core data or sediment analysis, it is still possible to assemble time series of an considerable depth. Due to the complete lack of information it is impossible to understand those data as the projection of high dimensional models. An alternative strategy consists in the comparison of those time series with “typical” paths of models sufficiently rich classes of low dimensional stochastic models. A rich class of models, where this can be done are stochastic differential equations driven by Levy processes, a natural generalization of Brownian motion, which allow for many features typically found in those time series, such as (large) jumps, heavytails and asymmetric noise distributions. This class exhibits already a large clase of random dynamical phenomena, such as metastability, stochastic resonance, synchronization etc.
 Jai Sukhatme, CAOS, IISc, Bangalore: "Dynamics of rotating fluids"
 Ravi Nanjundiah, CAOS, IISc, Bangalore:
 J. Srinivasan, Divecha Centre for Climate Change and CAOS, IISc, Bangalore: "What can we learn about climate change from energy balance models?"
 Abstract: The climate of the earth is determined by the complex interaction between radiation , dynamics and thermodynamics. The role of radiation can be highlighted by looking at simple energy balance models. These simple models demonstrate the existence of multiple climate states and the stability of these states. They can be used to explain why Venus is much hotter than earth and the oscillation of the earth's climate between warm periods and ice ages. They are also useful to understand how climate will change in the future.
 JeanLuc Thiffeault, University of Wisconsin, Madison: "Stirring, mixing and transport"
 Lecture 1  Stirring and mixing
 Lecture 2  Filament models
 Lecture 3  Homogenization theory
 Lecture 4  Mixnorms