Preferred spatio-temporal patterns as non-equilibrium currents

Preferred spatio-temporal patterns as non-equilibrium currents Escher Jeffrey B. Weiss Atmospheric and Oceanic Sciences University of Colorado, Boulder 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

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

violation of detailed balance Preferred transitions between states in phase space Probability currents in phase space thermodynamic equilibrium thermodynamic non-equilibrium detailed balance satisfied no current detailed balance violated nonzero current

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 = (T 1, T 2, T N ) 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 T 1 T 2 T 3 T 4 T 5 T 6

Model Evaluation SST Prediction (Saha, et al 2006) skill dynamical older dynamical stochastic time

Model Evaluation Storm Tracks (Newman, et al 2003) dynamical model stochastic model

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 Figure provided by the International Research Institute (IRI) for Climate and Society (updated 17 February 2016).

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 onedimensional 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 L τ (t): discrete time probability angular momentum following a trajectory at time t L τ 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 2 nd 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?