Preferred spatio-temporal patterns

as non-equilibrium currents

Jeffrey B. Weiss

Atmospheric and Oceanic Sciences

University of Colorado, Boulder

Escher

Arin Nelson, CU

Baylor Fox-Kemper, Brown U

Royce Zia, Virginia Tech

Dibyendu Mandal, UC Berkeley

Planetary and stellar atmospheres

exhibit oscillations • Preferred spatio-temporal patterns of variability

• Earth:

• El-Niño Southern Oscillation (ENSO)

• Madden-Julien Oscillation (MJO)

• Pacific Decadal Oscillation (PDO)

• Atlantic Multidecadal Oscillation (AMO)

• Sun:

• Sunspot cycle

• Toroidal Oscillation

Planetary and stellar atmospheres

exhibit oscillations • Preferred spatio-temporal patterns of variability

• Earth:

• El-Niño Southern Oscillation (ENSO)

• Madden-Julien Oscillation (MJO)

• Pacific Decadal Oscillation (PDO)

• Atlantic Multidecadal Oscillation (AMO)

• Sun:

• Sunspot cycle

• Toroidal Oscillation

ENSO

Planetary and stellar atmospheres

exhibit oscillations • Preferred spatio-temporal patterns of variability

• Earth:

• El-Niño Southern Oscillation (ENSO)

• Madden-Julien Oscillation (MJO)

• Pacific Decadal Oscillation (PDO)

• Atlantic Multidecadal Oscillation (AMO)

• Sun:

• Sunspot cycle

• Toroidal Oscillation

MJO

Oscillations occur in subspaces of the

dynamics

• They have typical timescales

• Their dominant projection is onto many fewer degrees of

freedom than full dynamics

• Have smaller impact on many more degrees of freedom

• ENSO:

• Timescale: several months to years

• Projects onto tropical large scale SST, thermocline depth, Walker

circulation

• Affects rainfall and temperature across the globe

• MJO:

• Timescale: weeks to months

• Projects onto OLR and tropical convection

Oscillations characterized by indices

• Index is a low-dimensional empirically constructed filter of

the high-dimensional data

• Spatial averages of some carefully selected variable

• Temporally filtered to

• Designed to capture the important features of the

oscillation

• Different indices capture different aspects of an oscillation

• Often 1 dimension, sometimes higher

• ENSO

• Monthly data, NINO3 index, thermocline depth: d20

• MJO:

• Daily data, band pass filtered OLR, EOF amplitudes

Nonequilibrium Steady-States

• Turbulent fluids, planets and stars are in nonequilibrium

steady-states

• NOT thermodynamic equilibrium states

• Features of nonequilibrium steady-states

• Energy input distinct from energy dissipation

• Physical fluxes

• Violation of detailed balance

• Probability currents in phase space

Physics of nonequilibrium fluctuations

• Physics community has

made significant progress

on nonequilibrium

fluctuations.

• Mostly focused on

micro to nano scale

systems

• Does it apply to

climate and turbulence?

Yes: Applies to “small”

Subsystems.

Oscillations are

low dimensional

climate

Bustamante, et al 2005

“theory of the nonequilibrium

thermodynamics of small systems.”

Energy input distinct from energy

dissipation

• Two thermal reservoirs with different temperatures

• Kolmogorov 3d isotropic turbulence:

• energy input at large scales

• energy disipation at small scales

• Earth’s climate system:

• incoming short-wave solar radiation

• outgoing longwave to space

• Earth’s climate system:

• net energy input in tropics,

• net energy loss at poles

• Sun:

• energy input from nuclear fusion in the core

• energy radiated to space from the photosphere

Energy input distinct from energy

dissipation

• Two thermal reservoirs with different temperatures

• Kolmogorov 3d isotropic turbulence:

• energy input at large scales

• energy disipation at small scales

• Earth’s climate system:

• incoming short-wave solar radiation

• outgoing longwave to space

• Earth’s climate system:

• net energy input in tropics,

• net energy loss at poles

• Sun:

• energy input from nuclear fusion in the core

• energy radiated to space from the photosphere

Energy input distinct from energy

dissipation

• Two thermal reservoirs with different temperatures

• Kolmogorov 3d isotropic turbulence:

• energy input at large scales

• energy disipation at small scales

• Earth’s climate system:

• incoming short-wave solar radiation

• outgoing longwave to space

• Earth’s climate system:

• net energy input in tropics,

• net energy loss at poles

• Sun:

• energy input from nuclear fusion in the core

• energy radiated to space from the photosphere

Energy input distinct from energy

dissipation

• Two thermal reservoirs with different temperatures

• Kolmogorov 3d isotropic turbulence:

• energy input at large scales

• energy disipation at small scales

• Earth’s climate system:

• incoming short-wave solar radiation

• outgoing longwave to space

• Earth’s climate system:

• net energy input in tropics,

• net energy loss at poles

• Sun:

• energy input from nuclear fusion in the core

• energy radiated to space from the photosphere

Energy input distinct from energy

dissipation

• Two thermal reservoirs with different temperatures

• Kolmogorov 3d isotropic turbulence:

• energy input at large scales

• energy disipation at small scales

• Earth’s climate system:

• incoming short-wave solar radiation

• outgoing longwave to space

• Earth’s climate system:

• net energy input in tropics,

• net energy loss at poles

• Sun:

• energy input from nuclear fusion in the core

• energy radiated to space from the photosphere

violation of detailed balance

• Preferred transitions between states in phase space

• Probability currents in phase space

detailed balance

satisfied

no current

detailed balance

violated

nonzero current

thermodynamic non-equilibrium thermodynamic equilibrium

Equilibrium vs. Nonequilibrium Phase

Space Trajectories

• Nonequilibrium steady-states characterized by currents

equilibrium nonequilibrium

Climate oscillations in 2d phase space

• Phase space of indices

• Rotation apparent

ENSO MJO

Probability Angular Momentum

• Phase space rotation characterizes preferred transitions

• Probability rotates in phase space

• Introduce Probability Angular Momentum: L

• Analogue of mass angular momentum for a fluid

• Phase space position

• Phase space velocity

• Steady-state pdf

• Probability Angular Momentum is an antisymmetric matrix

Discrete Time Approximation

• Observations and models have discrete time

• Assume ergodicity in steady-state

• Probability angular momentum at time t

Easily calculated from Correlation Fn

• Time lagged correlation matrix

• Probability angular momentum is antisymmetric part

Linear Gaussian Models

• Perhaps simplest mathematical model of

nonequilibrium steady-state

• Deterministic dynamics: linear

• Stochastic: additive Gaussian white noise

• Crucial: multi-dimensional phase space

• Generalization of Langevin models

• Linear nature means many quantities can be

calculated analytically

• Multi-dimensional nature means must solve for

some quantities numerically

Linear Gaussian Models in Climate

• Used to model many climate phenomena

• El-Niño, Storm Tracks, Gulf Stream, … (Penland and Magorian, 1993; Farrell and Ioannou, 1993; Moore and Farrell, 1993)

• Dynamical argument from timescale separation

• Weather

• Timescales of days

• Chaotic

• Model as random noise on longer timescales

• Ocean or Large Scale Atmosphere

• Timescales of months and longer

• Model as deterministic

• Ridiculously simple

• Complex climate model: ~500,000 lines of code

• Linear Gaussian Model: ~10 lines of code

Constructing Linear Gaussian Models

• State vector:

• Temperature on a grid

x = (T1, T2, … TN)

• Reduce dimension

through EOF (principal

components, Karhunen-Loève)

truncation

• A point in phase space

is a pattern

• e.g. sea surface

temperature

• Fit dynamics to data

• Some work on obtaining

dynamics theoretically

T1 T2 T3

T4 T5 T6

Model Evaluation

SST Prediction (Saha, et al 2006)

stochastic

dynamical

older

dynamical

skill

time

Model Evaluation

stochastic model dynamical model

Storm Tracks (Newman, et al 2003)

Figure provided by the International Research

Institute (IRI) for Climate and Society

(updated 17 February 2016).

El-Niño Linear Gaussian Model Used in

Operational Forecasts

El-Niño/La-Niña

defined as 3 months

above/below ±0.5°C

Linear Gaussian model

Why do linear Gaussian models work?

• Linear Gaussian models CAN have skill similar to

complex dynamical models

• Success depends on fortuitously selecting phenomena

• Appropriate choice of spatio-temporal scales to capture oscillation

• Often turbulent flow self-organizes to marginal state

• Noise allows system to be modeled as stable with some small

eigenvalues

• e.g. noisy bifurcations

• These models succeed for phenomena where this occurs.

• What do the models need to get right to be useful?

• Nonequilibrium current loops?

• Entropy production?

Nonequilibrium complexity

• Chaos and complexity: • complexity in simple systems due to nonlinearity

• Three degrees of freedom gives chaos

• Linear Gaussian models described by two matrices • Deterministic matrix

• Noise (diffusion) matrix

• Nonequilibrium when matrices do not commute • If matrices commute, can reduce system to uncoupled one-

dimensional dynamics

• Complexity in linear stochastic systems due to • matrix non-commutativity

• multi-dimensionality

Linear Gaussian Models:

A Null Hypothesis for Climate Oscillations • Climate Models capture mean state of climate pretty well

• Models much worse at climate variability

• e.g. models disagree on how ENSO will change under climate

change; don’t capture MJO well

• Length of observational record is limiting

• Are fluctuations seen over last decade – century representative of

full range of possible fluctuations?

• e.g. evidence from models that El-Niño variability requires

centuries to stabilize statistics … but see above

• Even a not-terrible null hypothesis would be useful

• Linear Gaussian models may fill this role

• Skillful for certain phenomena

• Provide a bridge to nonequilibrium thermodynamics

PDF of Probability Angular Momentum

• Lτ(t): discrete time probability angular momentum following

a trajectory at time t

• Lτ fluctuates as trajectory evolves: pdf from data

• Fit Linear Gaussian Model to data, compute pdf from model

ENSO MJO

Trajectory Entropy

• Entropy is classically a system property of an ensemble.

• We only have one climate system, not an ensemble.

• Ensembles possible and common with models.

• Trajectory entropy applies to individual trajectories (e.g. Seifert, 2008)

• Defined in terms of probability of finding a trajectory x(t)

• Entropy production related to ratio of probabilities of

finding trajectory x(t) and it’s time-reversed counterpart

• Entropy production in a nonequilibrium steady-state

quantifies the irreversible character of the fluctuations

• Storms, El-Niño, etc., have lifecycles

• They look different when you play the movie backwards

Nonequilibrium Fluctuation Theorems

• many kinds related in various ways

• steady-state, transient, forced, …

• stochastic, discrete, chaotic nonlinear, Hamiltonian, quantum, …

• Steady-state fluctuation theorem:

• p(σ): probability of finding a fluctuation with entropy production σ

• Theorem: probability of finding fluctuations which reduce entropy

(σ < 0) is exponentially small

p(-σ) = p(σ) exp(-σ)

• Entropy reducing fluctuations “violate” the 2nd Law

• Because exponentially unlikely, thought to only be

observable in microscopic systems

• Also observable in climate oscillations

Entropy production of El-Niño events

• Linear Gaussian model from

50 yrs. three-month average

tropical sea surface

temperatures

• Calculate pdf of σ for

fluctuations two ways

• Theory from model matrices

• put individual fluctuations in bins

• El-Niño

• Global spatial scales

• Annual time scale

is thermodynamically small

and fast

Entropy reducing

fluctuations

(Weiss, 2009)

Entropy production timescales

• Linear Gaussian Model based on 3 month average ocean

data

• Noise assumed to be white: infinitely fast

• Entropy production gives timescale for thermal reservoir

producing the noise

• For El-Niño model this is the fast chaotic weather fluctuations

• Entropy production in chaotic system given by Lyapunov

timescale

• Linear Gaussian model says entropy production timescale

is 3.6 days

• Agrees with Lyapunov timescale of weather

• Is this why Linear Gaussian models work?

Summary

• Climate variability = preferred spatio-temporal oscillations

• = fluctuations within a nonequilibrium steady-state

• Phase space currents dictate form of oscillations

• Quantify currents with Probability Angular Momentum

• Oscillations are (sometimes?) thermodynamically small and fast despite being physically large and slow

• Linear Gaussian models provide a null hypothesis for oscillations.

• Climate datasets are sufficient to calculate statistical mechanical quantities.

• Recent and future progress in statistical mechanics has implications for climate variability

Questions

• Which aspects of nonequilibrium steady-states must

models capture to be useful?

• Entropy production?

• Probability angular momentum?

• Physical meaning of entropy production?

• More complex models

• Include seasonal cycle

• Include more complex noise: multiplicative and colored noise

• Are these ideas useful for other complex systems?

• Oscillations in stellar and planetary atmospheres?