empirical models of seasonal to decadal variability and predictability matt newman and mike...
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EMPIRICAL MODELS OF SEASONAL TO DECADAL VARIABILITY AND PREDICTABILITY
Matt Newman and Mike Alexander
CIRES/University of Colorado and NOAA/ESRL/PSD
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2010-2060 “A1B” tropical trends, same model, different ensemble members
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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
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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
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“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
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“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?
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“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)
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“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.
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ENSO FLAVORSNewman, M., S.-I. Shin, and M. A. Alexander, 2011: Natural variation in ENSO flavors. Geophys. Res. Lett., L14705, doi:10.1029/2011GL047658.
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“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)
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Verifying Multivariate Red Noise: compare observed and LIM-predicted lag-covariances and spectra
Note that LIM entirely determined from one-season lag statistics
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Multivariate red noise captures “optimal” evolution of ENSO types
SST: shading Thermocline depth: contoursZonal wind stress: arrows
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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
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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
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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
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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.
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PDO depends on ENSO (Newman et al. 2003)
Forecast: PDO (this year) = .6PDO(last year) + .6ENSO(this year)
r=.74
“reddened ENSO”
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“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
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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.
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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
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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
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Dominant “internal” North Pacific SST mode
Compute new EOFs from covariance matrix determined from uncoupled LIM
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“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
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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
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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”
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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).
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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
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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
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PDO hindcast skill – something more?
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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
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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