Estimating Feedbacks from Natural Variability in the Global Energy Budget Cristian Proistosescu, Aaron Donohoe, Kyle Armour Malte Stuecker, Gerard Roe, Cecilia Bitz University of Washington
Energy budget at TOA T (K) Q = λ · T + F GISTEMP TOA (W/m 2 ) CERES
Zero forcing at TOA T (K) Q = λ · T + F GISTEMP TOA (W/m 2 ) Temperature driven by internal variability such as ENSO CERES
Regressed Feedbacks 2 (Forster 2016) 1 Q (W/m 2 ) Q = λ · T + 0 -1 λ = 1 . 2 (W/m 2 /K ) • Constraint on long-term feedbacks? -2 • Forster: Need to understand temporal -0.4 -0.2 0 0.2 0.4 structure of regression-based feedback T (K) (Forster & Gregory 2006, Murphy 2009, Trenberth et al 2010, Dessler 2010, Stevens & Schwartz 2012, Tsushima & Manabe 2013, Donohoe et al 2014)
Temporal Structure CERES v GISTEMP • What controls the lag structure? Q lags Q leads
Sampling rate CERES v GISTEMP • What controls the lag structure? • Why the dependence on sampling rate? Q lags Q leads
#goals CERES v GISTEMP • What controls the lag structure? • Why the dependence on sampling rate? • What is the source of temperature variability? Q lags Q leads
CESM 1 control run CERES v CESM 1 GISTEMP Q lags Q leads Q lags Q leads
Model Hierarchy CESM 1 • Energy Balance Model • CAM5 - fixed SST • CAM5 - slab ocean model • CESM1-CAM5 fully coupled Q lags Q leads
Fixed SSTs Spectral Energy (K 2 /s) T 1 : Surface air temperature variability atop fixed SSTs — CAM5 - FSST — EBM Global Hasselmann Model: C dT dt = λ · T + F T 1 + (stochastic atmospheric variability) Frequency (1/year)
Slab ocean model Spectral Energy (K 2 /s) T 1 : Surface air temperature variability atop fixed SSTs — CAM5 - SOM Good fit at high frequencies — EBM T 1 + Frequency (1/year)
Slab ocean model Spectral Energy (K 2 /s) T 1 : Surface air temperature variability atop fixed SSTs — CAM5 - SOM T 2 : Mixed-layer variability — EBM T 1 + T 2 Frequency (1/year)
Fully Coupled Spectral Energy (K 2 /s) T 1 : Surface air temperature variability atop fixed SSTs — CESM 1 T 2 : Mixed-layer variability — EBM Good fit except for ENSO band T 1 + T 2 T 1 + T 2 Frequency (1/year)
Fully Coupled Spectral Energy (K 2 /s) T 1 : Surface air temperature variability atop fixed SSTs — CESM 1 T 2 : Mixed-layer variability — EBM T 3: ENSO variability as damped oscillator (AR2) T 1 + T 2 + T 3 Frequency (1/year)
Forcing and Phase TOA TOA F rad = λ · T = λ · T | | F ocn Q = C dT Q = λ T dt (in phase) (in quadrature) (Murphy & Forster 2010, Dessler 2011) (Spencer & Braswell 2010,2011)
Fixed SST — CESM 1 — EBM Phase Zero phase lag indicates Oceanic source of forcing Q 1 = λ 1 · T 1 Forcing provided by air-sea fluxes T 1 + = Q 1 F ∝ U 0 ( T a − T o ) Frequency (1/year)
Slab Ocean — CESM 1 — EBM Phase Quadrature: Radiative forcing Q 2 = λ 2 · T 2 + F rad T 1 + T 2 = Q 1 + Q 2 + Frequency (1/year)
Fully Coupled — CESM 1 — EBM Phase Lag in the ENSO band Q 3 ( t ) = λ 3 · T 3 ( t + τ ) T 1 + T 2 + T 3 = Q 1 + Q 2 + Q 3 Frequency (1/year)
Lagged Regression Analytical spectral solution Cross Spectrum Wiener-Khinchin Theorem Lagged Covariance Lagged Regression
Fixed SST — CESM 1 — EBM Slope (W/m 2 /K) T 1 = Q 1 Lag (years)
Slab Ocean — CESM 1 — EBM Slope (W/m 2 /K) T 1 + T 2 + = Q 1 + Q 2 Lag (years)
Fully Coupled — CESM 1 — EBM Each mode has been fit individually to a Slope (W/m 2 /K) level in the CAM5 hierarchy. Their linear superposition reproduces the full lagged regression structure of the coupled model T 1 + T 2 + T 3 = Q 1 + Q 2 + Q 3 Lag (years)
Regression Coefficient ✓ σ T i ◆ — CESM 1 X r (lag) = acf(lag) λ i — EBM σ total Slope (W/m 2 /K) (timescale) T 1 + T 2 + T 3 = Q 1 + Q 2 + Q 3 Lag (years)
Annual Averages ✓ σ T i ◆ — CESM 1 X r (lag) = acf(lag) λ i — EBM σ total Slope (W/m 2 /K) (timescale) Annual averaging preferentially eliminates fast, air-sea forced mode T 1 + T 2 + T 3 = Q 1 + Q 2 + Q 3 Lag (years)
So what does it mean? ✓ σ T i ◆ X r (lag) = acf(lag) λ i σ total Air-sea forced λ 1 = 1 . 2 λ 2 = 0 . 9 Each GCM will have di ff erent Radiatively forced - feedbacks ENSO λ 3 = 2 . 7 - variances - time scales r (0) = 0 . 8 Zero-lag Peak regression r ( τ ) = 1 . 1 Zero-lag feedback coincidentally similar λ GHG = 0 . 9 Global-warming to long term feedback. Going forward: how to connect mode feedbacks with long term warming
Summary Feedback structure: Average of individual feedbacks, weighted by relative variance and timescale Sampling rate Averaging preferentially smooths out fast modes of variability Forcing Radiative and Oceanic Oceanic dominates but composed of two independent sources Previously unidentified unforced air - sea fluxes important Ongoing: Spatio-temporal: Do any of the 3 modes constrain long-term sensitivity? Constraining parameters from the short observational record
Supplementary Slides
Emergent Constraints (Zhou et al. 2015) Cloud feedback in response to GHG forcing (W/m 2 ) • Strong inter-model correlations • Different spatial structure of T can engender different feedbacks • What about temporal structure? GCMs Cloud feedback from natural variability (W/m 2 )
Surface fluxes dominate on fast timescales H surf = cU ( T − T o ) H surf ≈ cU 0 ( T a − T o ) + cU ( T 0 − To 0 ) (fixed SST) H surf ≈ F surf + λ surf T 0
Energy is extracted by wind from the ocean thermostat F surf ≈ cU 0 ( T a − T o )
Spectra consistent with dominant surface forcing ω ⌧ λ /C λ /C ⌧ ω
Need to account for aliasing and averaging We need to account for aliasing and averaging Using averages over bins of width Δ t is equivalent to (a) smoothing the continuous signal with a moving average (b) sampling the smoothed process every Δ t (here taken as equal to the smoothing window, although this is not required) We can represent this by convolution with a rectangular window and multiplication by a Dirac Comb T obs ( t ) = III ∆ t · ( W ∆ t ? T ( t ))
Accounting for sampling issues explains bias in We need to account for aliasing and averaging transfer function N ˜ X S XY ( f + k/ ∆ t ) · sinc 2 ( f ∆ t + k ) S XY,N = k = − N
Whittle Likelihood We need to account for aliasing and averaging Consider the spectrum of temperature in the SOM-EBM: θ = { σ , λ , τ } Parameters S T T | θ ( f ) = Spectrum Whittle likelihood log S T T | θ ( f j ) + S obs T T ( f j ) X l ( S obs T T | θ ) = − S T T, θ ( f j ) f j
De-biassing the Whittle Likelihood We need to account for aliasing and averaging Consider the spectrum of temperature in the SOM-EBM: θ = { σ , λ , τ } Parameters ( σ / λ ) 2 S T T | θ ( f ) = Spectrum 1 + f 2 τ 2 Use a truncated version of the sampled spectrum N ˜ X S T T ( f + k/ ∆ t ) · sinc 2 ( f ∆ t + k ) S T T,N = k = − N de-biased Whittle likelihood S obs T T ( f j ) log ˜ X l ( S obs T T | θ ) = − S T T,N | θ ( f j ) + ˜ S T T,N | θ ( f j ) f j
Posterior distributions We need to account for aliasing and averaging
Dependence on record length We need to account for aliasing and averaging 100 years 30 years
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