# Period of Concentration: Stochastic climate models

## Abstracts for the talks

### Estimation of stochastic models from data and application to el nino data

**Markus Abel** *(Universität Potsdam)*

Wednesday, June 01, 2005

We analyze data from temperature measurements in the pacific in three
steps to obtain a stochastic model. 1) Public data are embedded using the
isomap algorithm to obtain an estimate of the dimension of the system and
the corresponding embedded time series. 2) These time series are treated
by nonparametric regression, a system of three coupled differential
equations is obtained. The unexplained deviance can be modeled by a
stochastic term, corresponding to observation. As a side result we find
a delay of 6 months in one of the variables, consistent with a
recently proposed model by Tsiperman, taking into account Rossby waves in
the pacific ocean. 3) The so-obtained model is integrated, an ensemble
prediction is performed to yield probabilities for el nino occurence.

### Geometric singular perturbation theory applied to stochastic climate models

**Nils Berglund** *(University of Toulon)*

Tuesday, May 31, 2005

Geometric singular perturbation theory offers an efficient
framework for the study of ordinary differential equations
with well-separated time scales. It combines the construction
of invariant manifolds, which allow a low-dimensional effective
description of the dynamics reduced to slow variables, with a
local analysis near bifurcation points. We present extensions
of this theory to systems of slow-fast stochastic differential
equations, constructing, in particular, neighbourhoods of invariant
manifolds in which sample paths concentrate. This approach will be
illustrated on a few simple models of the North-Atlantic Thermohaline
Circulation. Joint work with Barbara Gentz (WIAS, Berlin).

### Some Application of Generalized Stability Theory to Climate Dynamics I

**Brian Farrell** *(Harvard University)*

Tuesday, May 24, 2005

These lectures review a set of ideas and approaches to climate theory based broadly on stochastic methods and specifically on
the non-normality of the linear system underlying perturbation dynamics in both certain and uncertain
systems. Because the term non-normal is imprecise in its connotation, the term Generalized Stability
Theory is
used for this approach. Areas to be covered are i) The concept of stability of a statistical quantity with application to
sensitivity of storm track statistics. ii) The concept of stability of uncertain systems with application to climate forecast.
iii) The concept of structural stability of turbulent systems with application to jet vacillation. iv) The concept of the
intrinsic non-normality of time dependent systems with application to the statistical stability of dynamical systems.

### The Stochastic Parametric Mechanism for Generation of Surface Water Waves by Wind

**Brian Farrell** *(Harvard University)*

Wednesday, June 01, 2005

Synoptic scale eddy variance and associated fluxes of heat and momentum in
mid-latitude jets are sensitive to small alterations in mean jet velocity,
dissipation, and static stability. In this lecture the sensitivity of
variance and fluxes to such structured changes in the mean jet is examined.
In particular the structured change in the jet producing the greatest change
in disturbance variance or flux is obtained. The method used builds on
previous work in which storm track statistics were obtained using a
stochastic model of jet turbulence. This work extends generalized stability
theory from addressing stability of deterministic forecast to addressing
stability in the context of statistical forecast.

### Averaging Principle for Deterministic and Stochastic Perturbations I

**Mark Freidlin** *(University of Maryland)*

Monday, May 23, 2005

First, I will consider averaging in the simplest case: for systems
with one degree of freedom and the first integral without singularities.
Then I will introduse various regularizations for the system with one
degree of freedom and saddle points and show that in the general situation
one should consider random perturbations of the equation, not just the
initial condition, to regularize the averaging principle for deterministic
perturbations. I will describe the limiting slow motion as a stochastic
process on the corresponding graph. Next, I will give conditions for
averaging principle to be valid for perturbations of multifrequency
systems in the action-angle coordinates. I will consider some many-degrees
of-freedom systems with singularities in the first integrals. In this
case, an open book space should be considered as the phase space for
the limiting slow motion. I will consider some applications of this theory
to dynamics of incompressible fluid.

### Convection-Diffusion in Stationary Incompressible 3D-Flow which is Close to Planar Motion

**Mark Freidlin** *(University of Maryland)*

Monday, May 30, 2005

We will show that a pure deterministic motion should be approximated
in certain situation by a stochastic motion. We will introduce for the
deterministic system a relative entropy and describe the motion of the
wavefronts for a class of reaction-convection equations.

### Residence-time distributions as a measure for stochastic resonance

**Barbara Gentz** *(WIAS Berlin)*

Wednesday, June 01, 2005

Stochastic resonance (SR) is believed to play an important role not only in numerous technological and physical applications, but also in biological and climate systems. Apart from spectral properties of the signal, residence-time distributions have been proposed as a measure for SR. For the paradigm of the motion of a periodically forced Brownian particle in a bistable potential, we explain the relation between first-passage-time and residence-time distributions. Going beyond exponential asymptotics, we are able to give rigorous expressions for the densities of these distributions. In a broad range of forcing frequencies and amplitudes, the distributions are found to be close to periodically modulated exponential ones, where the periodic modulations are governed by a universal function, depending on a single parameter related to the forcing period.
Joint work with Nils Berglund (CPT-CNRS Luminy, France).

### Large deviations for diffusions and stochastic resonance

**Samuel Herrmann** *(Université Nancy)*

Tuesday, May 31, 2005

We consider potential type dynamical systems in finite
dimensions with two meta-stable states. They are subject to two sources of
perturbation: a slow external periodic perturbation of period *T* and a small
Gaussian random perturbation of intensity , and therefore mathematically
described as weakly time inhomogeneous diffusion processes.
A system is in stochastic resonance provided the small noisy
perturbation is tuned in such a way that its random trajectories follow the
exterior periodic motion in an optimal fashion, i.e. for some optimal intensity
.
The physicists' favorite measures of quality of periodic tuning
- and thus stochastic resonance - such as spectral power amplification
or signal-to-noise ratio have proven to be defective. They are not robust
w.r.t. effective model reduction, i.e. for the passage to a simplified
finite state Markov chain model reducing the dynamics to a pure jumping
between the meta-stable states of the original system. An entirely
probabilistic notion of stochastic resonance based on the transition
dynamics between the domains of attraction of the meta-stable states -
and thus failing to suffer from this robustness defect - is
investigated by using extensions and refinements of the Freidlin-Wentzell
theory of large deviations for time homogeneous diffusions. Large
deviation principles developed for weakly time inhomogeneous diffusions
prove to be key tools for a treatment of the problem of diffusion exit
from a domain and thus for the approach of stochastic resonance
via transition probabilities between meta-stable sets.

### Probability density functions, ensembles and limits to statistical predictability

**Richard Kleeman** *(Courant Institute)*

Monday, May 30, 2005

Ensemble predictions are an integral part of routine weather and climate prediction because of the sensitivity of such projections to the specification of the initial state. In many discussions it is tacitly assumed that ensembles are equivalent to probability distribution functions (p.d.fs) of the random variables of interest. In general for vector valued random variables this is not the case (not even approximately) since practical ensembles do not adequately sample the high dimensional state spaces of dynamical systems of practical relevance. In this talk these ideas are placed on a rigorous footing using concepts derived from Bayesian analysis and information theory. In particular it is shown that ensembles must imply a coarse graining of state space and that this coarse graining implies loss of information relative to the converged p.d.f. To cope with the needed coarse graining in the context of practical applications, a heirarchy of entropic functionals is introduced. These measure the information content of multivariate marginal distributions of increasing order. For fully converged distributions (i.e. p.d.f.s) these functionals form a strictly ordered heirarchy. As one proceeds up the heirarchy however, increasingly coarser partitions are required by the functionals which implies that the strict ordering of the p.d.f. based functionals breaks down. This breakdown is symptomatic of the poor sampling by ensembles of high dimensional state spaces and is unavoidable for most practical applications.
In the second part of the talk the theoretical machinery developed above is applied to the practical problem of mid-latitude weather prediction. It is shown that the functionals derived in the first part all decline essentially linearly with time and there appears in fact to be a fairly well defined cut off time (roughly 45 days for the model analyzed) beyond which initial condition information is unimportant to statistical prediction.

### A Stochastic View of El Nino I

**Cecile Penland** *(Climate Diagnostics Center )*

Monday, May 23, 2005

In this presentation, I review the
paradigm of El Nino as a stochastically-
forced system. In particular, the inter-
play of stochastic forcing and non-normal
linear dynamics explains a large part of
El Nino variability. Since controlled
laboratory simulations of climate cannot
be performed (except on computers), care-
ful falsification, as opposed to verifi-
cation, of theory by data was required
to establish this paradigm. I shall des-
cribe this procedure and, in so doing,
show how complex a simple model can be.

## Date and Location

**May 23 - June 01, 2005**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Peter Imkeller**

Humboldt Universität zu Berlin

Berlin

Contact by Email

**Stefan Müller**

Max Planck Institute for Mathematics in the Sciences

Leipzig

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## Administrative Contact

**Katja Bieling**

Max Planck Institute for Mathematics in the Sciences

Leipzig

Contact by Email