EMPIRICAL MODELS OF SEASONAL TO DECADAL VARIABILITY AND PREDICTABILITY

Matt Newman and Mike AlexanderCIRES/University of Colorado and NOAA/ESRL/PSD

2010-2060 “A1B” tropical trends, same model, different ensemble members

Outline of Talk• Multivariate red noise: a basic model of Pacific climate variability

• Applied to:• Tropics• Pacific Basin (PDO)• Decadal forecasts of global surface temperature anomalies

Some requirements for empirical climate models• Capture the evolution of anomalies

• Growth/decay, propagation• need anomaly tendency: dynamical model• Can relate to physics/processes and estimate predictability?

• Limited data + Occam’s razor = not too complex• How many model parameters are enough?• Problem: is model fitting signal or noise? • Test on independent data (or at least cross-validate)

• Testable• Is the underlying model justifiable?• Where does it fail?• Can we understand where/why it succeeds? (no black boxes)

Previous success of linear diagnosis/theory for climate suggest potential usefulness of linear empirical dynamical model

“Linearization” : amplitude of nonlinear term is small compared to amplitude of linear term Then ignore nonlinear term

“Coarse-grained” : time scale of nonlinear term is small compared to time scale of linear term Then parameterize nonlinear term as (second) linear term +

unpredictable white noise: N(x) ~ Tx + ξ

For example, surface heat fluxes due to rapidly varying weather driving the ocean might be approximated as

Two types of linear approximations

“Multivariate Red Noise” null hypothesis

• Noise/response is local (or an index)• For example, air temperature anomalies force SST• use univariate (“local”) red noise:

dx/dt = bx + fs where x(t) is a scalar time series, b<0,

and fs is white noise

• Noise/response is non-local: patterns matter• For example, SST sensitive to atmospheric gradient• use multivariate (“patterns-based”) red noise:

dx/dt = Bx + Fs where x(t) is a series of maps, B is stable,

and Fs is white noise (maps)

• Note that B is a matrix and x and Fs are vectors• If B is not symmetric* (*nonnormal), transient anomaly growth is possible even

though exponential growth is not• How can we determine B?

“Inverse method” – derive B from observed statistics

If the climate state x evolves as

dx/dt = Bx + FS

then τ0-lag and zero-lag covariance are related asC(τ0) = G(τ0) C(0) = exp(Bτ0) C(0) [where C(τ) = <x(t+τ)x(t)T>].

Linear inverse model (LIM)

LIM procedure:• Prefilter data in EOF space (since B = logm [C(τ0)C(0)-1]/τ0 )• Determine B from one training lag τ0.• Test for linearity

For much longer lags τ, is C(τ) = exp(Bτ) C(0) ? This “τ-test” is key to LIM.

• Cross validate hindcasts (withhold 10% of data)

“Inverse method” – derive B from observed statistics

If the climate state x evolves as

dx/dt = Bx + FS

then ensemble mean forecast at lead τ is

x(τ) = exp(Bτ) x(0) .

Eigenmodes of B are all damped but can be either stationary or propagating* (*Bei = λiei , where λi can be complex) & not orthogonal.

When B is “nonnormal” (dynamics are not symmetric) transient “optimal” anomaly growth can occur* (*DG(τ)vi = σiui, where D is a

norm), leading to greater predictability.

Linear inverse model (LIM), cont.

ENSO FLAVORSNewman, M., S.-I. Shin, and M. A. Alexander, 2011: Natural variation in ENSO flavors. Geophys. Res. Lett., L14705, doi:10.1029/2011GL047658.

“Multivariate Red Noise” null hypothesis

dx/dt = Bx + Fs where x(t) is a series of maps, B is stable,

and Fs is white noise (maps)

• Determine B and Fs using “Linear Inverse Model” (LIM)• x is SST/20 C depth/surface zonal wind stress seasonal

anomalies in Tropics, 1959-2000 (Newman et al. 2011, Climate Dynamics)

• prefiltered in reduced EOF space (23 dof)• LIM determined from specified lag (3 months) as in AR1 model• Extension of work by Penland and co-authors (e.g. Penland and

Sardeshmukh 1995)

Verifying Multivariate Red Noise: compare observed and LIM-predicted lag-covariances and spectra

Note that LIM entirely determined from one-season lag statistics

Multivariate red noise captures “optimal” evolution of ENSO types

SST: shading Thermocline depth: contoursZonal wind stress: arrows

Optimal structures are relevant to observed EP and CP ENSO events

Composite: Six months after a > ± 1 sigmaprojection (blue dots) on either the first or second optimal initial condition, constructed separately for warm and cold events

Green dots representmixed EP-CP events

Multidecadal variations of CP/EP ENSOs driven by noise

24000 yr LIM “model run”: dx/dt = Bx + Fs Values determined over 30-yr intervals spaced 10 years apart

“Increasing CP/EP Cases” : Adjacent 60-yr segments where1) CP/EP ratio increases2) r(Nino3,Nino4) decreases

LIM can provide realistic synthetic data

Nino 3.4 times series: DJF (gray) and 25-yr running mean (black)

Multi-proxy reconstruction (Emile-Geay 2012), one of 100 LIM realizations, forced CCSM4 show decadal signal, CCSM4 control does not

PACIFIC SST EMPIRICAL MODELSNewman, M., 2007: Interannual to decadal predictability of tropical and North Pacific sea surface temperatures. J. Climate, 20, 2333-2356.Alexander, M. A., L. Matrosova, C. Penland, J. D. Scott, and P. Chang, 2008: Forecasting Pacific SSTs: Linear Inverse Model Predictions of the PDO. J. Climate, 21, 385-402.Newman, M., D. Smirnov, and M. Alexander, 2012: Relative impacts of tropical forcing and extratropical air-sea coupling on air/sea surface temperature variability in the North Pacific. In preparation.

PDO depends on ENSO (Newman et al. 2003)

Forecast: PDO (this year) = .6PDO(last year) + .6ENSO(this year)

r=.74

“reddened ENSO”

“Multivariate Red Noise” null hypothesis

dx/dt = Bx + Fs where x(t) is a series of maps, B is stable,

and Fs is white noise (maps)

• Determine B and Fs using “Linear Inverse Model” (LIM)• x is SST seasonal anomalies in the Pacific (30°S-60°N), 1950-2000

(Alexander et al. 2008, J. Climate) • prefiltered in reduced EOF space (13 dof)• LIM determined from specified lag (3 months) as in AR1 model• Skill in predicting Nino3.4 and PDO > 0.6 for 1 year forecasts when

initialized in late winter

Diagnosing coupling• Use slightly different LIM by separating Tropics and North

Pacific:• Define xtropics =SST/20 C depth/surface zonal wind stress

TNorthPac =SST (20ºN-60ºN)

• Coupling effects are determined by zeroing out the appropriate submatrices within B.

Diagnosing coupling• Use slightly different LIM by splitting Tropics and North

Pacific:• Define xtropics =SST/20 C depth/surface zonal wind stress

TNorthPac =SST (20ºN-60ºN)

• Coupling effects are determined by zeroing out the appropriate submatrices within B.

Decouple Tropics from North Pacific, then recalculate statistics given same noise

East Pacific SST variability almost entirely due to tropical forcing.In WBC, most variability is independent of the Tropics.

Variance 6 month lag covariance

LIM

Uncoupled

Impact of tropical coupling on SST variability

Dominant “internal” North Pacific SST mode

Compute new EOFs from covariance matrix determined from uncoupled LIM

“Multivariate Red Noise” null hypothesis

dx/dt = Bx + Fs where x(t) is a series of maps, B is stable,

and Fs is white noise (maps)

• Determine B and Fs using “Linear Inverse Model” (LIM)• x is SST annual mean (July-June) anomalies in Tropics and North

Pacific, 1900-2001 (Newman 2007)• prefiltered in reduced EOF space (10 dof)• LIM determined from specified lag (1 year) as in AR1 model

Components of the PDO

Leading eigenmodes of B, with time series (1900-2001)• Eigenmodes

represent:• Trend• “Pacific Multidecadal

Oscillation” (PMO)• “Decadal ENSO”

• Almost all long range skill contained in first 2 eigenmodes

Constructing the PDO from a sum of three red noise processes

Time series show projection of each mode onto the PDO

PDO = PMO+Decadal ENSO

+Interannual ENSO

“PMO”

“Decadal ENSO”

“Interannual ENSO”

Reconstructed PDO

PDO

“Regime shifts”

DECADAL FORECASTS OF GLOBAL SURFACE TEMPERATURENewman, M., 2012: An empirical benchmark for decadal forecasts of global surface temperature anomalies. J. Climate, in review (minor revision).

Multivariate red noise surface temperatures

dx/dt = Bx + Fs

• Determine B and Fs using “Linear Inverse Model” (LIM)• x is SST/Land (2m) temperature, 12-month running mean

anomalies, 1900-2008 (Newman 2012)• prefiltered in reduced EOF space (20 dof)• LIM determined from specified lag (12 months) as in AR1 model

Decadal skill for forecasts initialized 1960-2000

LIM has clearly higher skill than damped persistence, comparable skill to CMIP5 CGCM decadal “hindcasts”

Years 2-5 Years 6-9

LIM

PDO hindcast skill – something more?

Leading eigenmodes of B, with time series (1900-2008)

Eigenmodes represent:• Trend• Atlantic Multidecadal

Oscillation (AMO)• Pacific Multidecadal

Oscillation (PMO)

Almost all skill contained in these 3 eigenmodes

Enhanced LIM PDO skill due to PMO

Conclusions• North Pacific Climate Variability

• Sum of “reddened” ENSO + northwest Pacific-based (KOE?) variability • Coupled GCMs may underpredict the second process

• LIM is a good model of climate• Captures statistics of anomaly evolution and makes forecasts• Serves as a benchmark for numerical models• Can diagnose dynamical relationships between different

variables/locations and how they provide/limit predictability• Can generate long runs of realistic synthetic “data”• Consistent with apparent “regime shifts” with limited predictability

• Uses of climate variability • for scenario building to test sensitivity of ecosystem• to make predictions of ecosystem