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?
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