Quantifying Uncertainty in Turbulent Dynamical Systemsqidi/turbulence16/introduction.pdfLarge-scale shear ﬂow 3 Conserved quantities Conservation of energy and enstrophy Variational

Quantifying Uncertainty in Turbulent DynamicalSystems

Andrew J. Majda

Courant Institute of Mathematical Sciences

Fall 2016 Advanced Topics in Applied Math

Andrew J. Majda (CIMS) Turbulent Dynamical Systems Sept. 8, 2016 1 / 50

Turbulent Dynamical Systems

Turbulent dynamical systems are ubiquitous complex systems in geoscience andengineering and are characterized by a large dimensional phase space and a largedimension of strong instabilities which transfer energy throughout the systembetween di↵erent spatiotemporal scales.

(a) Red spot on Jupiter (b) MJO (c) QG


Grand Challenge in Extremely Complex System

Important societal impacts: predicting long range weather forecasting (intraseasonalto interannual) and short term (decadal) climate change.

Turbulent dynamical system: huge phase space and large dimension of instabilities.

Other examples: engineering turbulence, neural science, material science.

Need statistical, stochastic, thinking combined with nonlinear dynamics ideas.

Central Applied Math/Science as Issues:

1 Accurate prediction and representation of suitable statistical mechanism forobservations from nature.

2 Model error: lack of physical understanding and inadequate resolution due to curseof ensemble size, computational overload in generating even small number ofensemble members is overwhelming.

3 Uncertainty quantification (UQ) accurate bounds for 1) and 2).

4 Low order models which achieve 1 ) and 3) while coping with 2) in an optimalfashion.

5 Information-theoretical framework for measuring imperfect model error in complexsystem.


Modern Applied Math Paradigm

Modern Applied Math Paradigm

Rigorous Math Theory

Qualitative orQuantitativeModels

Novel NumericalAlgorithm

Crucial Improved Understanding ofComplex System

3 / 51Andrew J. Majda (CIMS) Turbulent Dynamical Systems Sept. 8, 2016 4 / 50

Course Outline:

Barotropic geophysical flows and two-dimensional fluid flows

The response to large-scale forcing

Nonlinear stability of steady geophysical flows

Introduction to information theory and empirical statistical theory

Equilibrium statistical mechanics for systems of ODEs

Statistical mechanics for the truncated quasi-geostrophic equations


Major Course References:

Majda & Wang, Nonlinear Dynamics and Statistical Theories for BasicGeophysical Flow (majorly Chapter 1, 2, 4, 6, 7, 8), Cambridge Press 2006

Majda, Introduction to Turbulent Dynamical Systems for Complex Systems(85 pages), Frontiers in Applied Dynamical Systems, Springer 2016

Papers Available from Prof. Majda’s websitehttp://www.math.nyu.edu/faculty/majda/publicationrevised.html


Book Contents: Chapter 1 & 2

Contents

Preface page xi

1 Barotropic geophysical flows and two-dimensional fluid flows:elementary introduction 1

1.1 Introduction 11.2 Some special exact solutions 81.3 Conserved quantities 331.4 Barotropic geophysical flows in a channel domain – an important

physical model 441.5 Variational derivatives and an optimization principle for

elementary geophysical solutions 501.6 More equations for geophysical flows 52

References 58

2 The response to large-scale forcing 592.1 Introduction 592.2 Non-linear stability with Kolmogorov forcing 622.3 Stability of flows with generalized Kolmogorov forcing 76

References 79

3 The selective decay principle for basic geophysical flows 803.1 Introduction 803.2 Selective decay states and their invariance 823.3 Mathematical formulation of the selective decay principle 843.4 Energy–enstrophy decay 863.5 Bounds on the Dirichlet quotient, !"t# 883.6 Rigorous theory for selective decay 903.7 Numerical experiments demonstrating facets of selective decay 95

References 102

v


Lecture 1 & 2: Elementary Introduction

1 Introduction

2 Some special exact solutionsBeta-e↵ect and generalized Kolmogorov forcingRossby wavesTopographic e↵ectLarge-scale shear flow

3 Conserved quantitiesConservation of energy and enstrophyVariational derivatives and an optimization principle

4 More equations for geophysical flows


Outline

1 Introduction

2 Some special exact solutionsBeta-e↵ect and generalized Kolmogorov forcingRossby wavesTopographic e↵ectLarge-scale shear flow

3 Conserved quantitiesConservation of energy and enstrophyVariational derivatives and an optimization principle

4 More equations for geophysical flows


Dynamical processes involved in the description ofgeophysical flows

Physical variables:

I the velocity, the pressure, the density;I in addition, the humidity in the case of atmospheric motions or the salinity in

the case of oceanic motions.

Physical processes that determine the evolution of the geophysical flows:

I the Coriolis force due to the earth’s rotation;I the sun’s radiation;I the topographical barriers, e.g. mountain ranges in the case of atmospheric

flows and the ocean floor and the continental masses in the case of oceanicflows;

I The dissipative energy mechanisms, for example due to eddy di↵usivity orEkman drag;

Spatial and temporal scales:

I The space scales: from a few hundred meters to thousands of kilometers.I The time scales: from as short as minutes to as long as days, months, or even

years.


The simplest set of equations that meaningfully describes the motion oftwo-dimensional geophysical flows under these circumstances is given by the:

Barotropic Quasi-Geostrophic Equations:

Dq

Dt=@q

@t+ v ·rq = D (�) + F (x, t)

whereq = ! + �y + h (x , y) , ! = � , v = (�@

y

, @x

)T .

q: the potential vorticity; : the stream function;

v = r? : the horizontal velocity field; ! = � : the relative vorticity;

�y : the beta-plane e↵ect from the Coriolis force;

h (x): the bottom floor topography, given by ocean floor or mountain range;

D (�): various possible dissipation mechanisms

F (x, t): additional external forcing.


Choices of dissipation operators

Newtonian (eddy) viscosity: D (�) = ⌫�2

Ekman drag dissipation: D (�) = �d�

Hyper-viscosity dissipation: D (�) = (�1)j dj

�j

Ekman drag dissipation + Hyper-viscosity: D (�) = �d� + (�1)j dj

�j

Radiative damping: D (�) = d

General dissipation operator: D (�) =P

l

j=0 (�1)j dj

�j

Generalized Kolmogorov forcing

F (x, t) =X

|k|=⇤

F̂k

(t) e ix·k + c.c.


The origin of the barotropic quasi-geostrophic equations

Barotropic Rotational Equations:

I The slow mode of propagation corresponds to themotion of the bulk of the fluid by advection (theweather patterns in the atmosphere, evolving on a timescale of days)

I The fast mode of propagation corresponds to gravitywaves (do not contribute to the bulk motion of thefluid, evolving on a short time scale of the order ofseveral minutes)

The barotropic quasi-geostrophic equations: the resultof “filtering out” the fast gravity waves from the rotatingbarotropic equations.

Charney (1949) first formulated the quasi-geostrophicequations, and thus opened the modern era of numericalweather prediction (Charney 1949, Fj

¨

ortoft, and von

Neumann, 1950).


Two-dimensional classical fluid flow equations

D!

Dt= ⌫�! + F (x, t) , ! = � , v = r?

and in the case without dissipation we have the classical Euler equations withforcing

D!

Dt= F (x, t) , ! = � , v = r?

One of our objectives of this course is to compare and contrast the barotropicquasi-geostrophic equations and the Navier–Stokes equations to better understandthe role of the beta-plane e↵ect and the topography on the behavior ofgeophysical flows.


Documents

Quantifying Uncertainty in Turbulent Dynamical Systemsqidi/turbulence16/introduction.pdfLarge-scale shear ﬂow 3 Conserved quantities Conservation of energy and enstrophy Variational