Outline

1. Viewing Weather and Climate Extremes in a Probabilistic FrameworkPrashant. D. Sardeshmukh @ noaa.gov Climate Diagnostics Center, CIRES, University of Coloradoand Physical Sciences Division/ESRL/NOAAAMS Annual Meeting January 2009 Phoenix Outline • To understand the statistics of extreme anomalies and how they are altered in various situations, we need to understand the dynamics behind the associated probability density functions (PDFs). 2. We need to understand the PDF changes associated not only with shifts of the mean, but also with changes of Variance, and perhaps also with changes of higher moments such Skewness and Kurtosis. 3. If the PDFs are Gaussian, then one need be concerned only with changes of the mean and variance. Many large-scale and time-averaged variables in the climate system have roughly Gaussian PDFs. Their statistics and evolution are often surprisingly well captured even by low-dimensional linear models perturbed by Gaussian stochastic noise whose statistics do not depend on the system state (additive noise). Such models can be highly competitive with GCMs. However, they cannot represent non-Gaussian PDFs. Recently, it has been shown that if one allows the stochastic noise in such linear models to have both state-independent (additive) and linearly state-dependent (linear multiplicative) parts, then one can generate non-Gaussian PDFs with very realistic higher momentsand also Power-Law tails. Such extended linear models thus provide a dynamical basis for understanding how even non-Gaussian PDFs can change in various situations.

2. Even relatively minor changes of mean and variance can have large implications for the probabilities of extreme values Note that in the original PDF, the probabilities of both extreme positive and extreme negative values (defined here as having magnitudes of more than 1 sigma) are 16 %

3. See also Katz and Brown (Climatic Change, 1992) A change of variance has a large effect on the probability of extreme values In some cases it can even completely offset the effect of a shifted mean From Sardeshmukh, Compo and Penland (J. Climate 2000)

4. Estimating shifts of the 500 mb height PDFs in winter (JFM) due to anomalous SSTs during the 1987 El Nino and 1989 La Nina events Based on 180 winter simulations using the NCEP MRF9 atmospheric GCM Based on Observational El Nino and La Nina Composites The SST-forced height Anomalies are contoured Every 10 meters. Positive values are indicated by red and negative values by blue coloring. Note! The lower La Nina panel has the sign flipped ! This makes for an easier comparison with the upper El Nino panel. For the 1987 El Nino For the 1989 La Nina (sign flipped) From Sardeshmukh, Compo and Penland (J.Climate 2000)

5. Expected ratio of Subseasonal 500 mb height variance (2 to 90 day periods) in La Nina and El Nino winters Based on observations in 11 La Nina and 11 El Nino winters in 1948-2000 This is an estimate of what to expect in general Based on 60 simulations each of the 1989 La Nina and 1987 El Nino winters using the NCEP MRF9 atmospheric GCM This is an estimate of what to expect in specific instances From Compo, Sardeshmukh, and Penland (J. Climate 2001)

6. Changes expected in general from observations of 11 El Nino and 11 “Neutral” winters Changes expected in a specific instance such as the 1987 El Nino from NCEP GCM simulations Estimating expected changes of variance of 500 mb heights in winter (JFM) due to anomalous SSTs during an El Nino event The basic result here is that during El Nino . . . Storm-tracks are shifted south over the north Pacific and north America And Weekly variability associated with blocking activity is decreased over the north Pacific From Compo, Sardeshmukh, and Penland (J. Climate 2001)

7. The predictability of “storm tracks” is important for thepredictability of seasonal mean precipitation Changes of both the seasonal mean vertical velocity and of its variability on synoptic time scales determine the changes of seasonal mean precipitation. The figure shows the SST-forced signal to noise ratios of the three quantities in 60-member ensembles of NCEP atmospheric GCM runs with El Nino (JFM 1987) SST forcing. C.I. = 0.2

8. Knowledge of the first two statistical moments is sufficient to determine the probabilities of extreme values if the PDFs are Gaussian, which is true if the system dynamics are effectively Linear and Stochastically Forced x = N-component anomaly state vector  = M-component gaussian noise vector fext(t) = N-component external forcing vector A(t) = N x N matrix B(t) = N x M matrix Supporting Evidence for the Linear Stochastically Forced (LSF) Approximation - Linearity of coupled GCM responses to radiative forcings - Linearity of atmospheric GCM responses to tropical SST forcing - Linear dynamics of observed seasonal tropical SST anomalies - Competitiveness of linear seasonal forecast models with global coupled models - Linear dynamics of observed weekly-averaged circulation anomalies - Competitiveness of Week 2 and Week 3 linear forecast models with NWP models - Ability to represent observed second-order synoptic-eddy statistics

9. Observed and Simulated Spectra of Tropical SST Variability Spectra of the projection of tropical SST anomaly fields on the 1st EOF of observed monthly SST variability in 1950-1999. Observations (Purple) IPCC AR4 coupled GCMs (20th-century (20c3m) runs) (thin black, yellow, blue, and green) A linear inverse model (LIM) constructed from 1-week lag covariances of weekly-averaged tropical data in 1982-2005 (Thick Blue) Gray Shading : 95% confidence interval from the LIM, based on 100 model runs with different realizations of the stochastic forcing. From Newman, Sardeshmukh and Penland (J. Climate 2009)

10. Seasonal Predictions of Ocean Temperatures in the Eastern Tropical Pacific :Comparison of linear empirical and nonlinear GCM forecast skill(Saha et al, J. Climate 2006) Simple linearempiricalmodels areapparentlyjust as good at predicting ENSOas“state of the art” coupled GCMs

11. Decay of lag-covariances of weekly anomalies is consistent with linear dynamics

12. Equations for the first two moments (Applicable to both Marginal and Conditional Moments) <x > = ensemble mean anomaly C = covariance of departures from ensemble mean If A(t), B(t) , and fext(t) are constant, then First two Marginal moments First two Conditional moments Ensemble mean forecast Ensemble spread An attractive feature of the LSF Approximation If x is Gaussian, then these moment equations COMPLETELY characterize system variability and predictability

13. Observed Skew S and (excess) Kurtosis K of daily 300 mb Vorticity (DJF) But . . daily atmospheric circulation statistics are not Gaussian . . From Sardeshmukh and Sura (J. Climate 2008)

14. Skew Kurtosis Observed Skew S and (excess) Kurtosis K of daily SSTs (DJF) Daily Sea Surface Temperature statistics are also not Gaussian . . . From Sura and Sardeshmukh ( J. Climate 2008 )

15. Modified LSF Dynamics in which the amplitude of the stochastic noise depends linearly on the system state

16. A simple view of how additive and linear multiplicative noise can generate skewed PDFs even in a deterministically linear system Additive noise only Gaussian No skew Additive and correlated Multiplicative noise Asymmetric non-Gaussian Additive and uncorrelated Multiplicative noise Symmetric non-Gaussian

17. A 1-D system with Correlated Additive and Multiplicative (“CAM”) noise Stochastic Differential Equation : Fokker-Planck Equation : Moments : A simple relationship between Skew and Kurtosis :

18. Note the quadratic relationship between K and S : K> 3/2 S2 Observed Skew S and (excess) Kurtosis K of daily 300 mb Vorticity (DJF)

19. Skew Kurtosis Note the quadratic relationship between K and S : K> 3/2 S2 Observed Skew S and (excess) Kurtosis K of daily SSTs (DJF) From Sura and Sardeshmukh (J. Climate2008 )

20. A linear 1-D system with non-Gaussian statistics, forced by “CAM” noise

21. Observed (NCEP Reanalysis) Simulated by a simple dry adiabatic GCM with fixed forcing Observed and Simulated pdfs in the North Pacific (On a log-log plot, and with the negative half folded over into the positive half) 500 mb Height 300 mb Vorticity

22. Observed and Simulated pdfs in the North Pacific (On a log-log plot, and with the negative half folded over into the positive half) Observed (NCEP Reanalysis) Simulated by a simple dry adiabatic GCM with fixed forcing 500 mb Height 300 mb Vorticity

23. The most general linear 1-D system with non-Gaussian statistics, forced by “Radical” noise

24. Summary To understand the statistics of extreme anomalies and how they are altered in various situations, we need to understand the dynamics behind the associated probability density functions (PDFs). 2. We need to understand the PDF changes associated not only with shifts of the mean, but also with changes of Variance, and perhaps also with changes of higher moments such Skewness and Kurtosis. 3. If the PDFs are Gaussian, then one need be concerned only with changes of the mean and variance. Many large-scale and time-averaged variables in the climate system have roughly Gaussian PDFs. Their statistics and evolution are often surprisingly well captured even by low-dimensional linear models perturbed by Gaussian stochastic noise, whose statistics do not depend on the system state (additive noise). Such models can be highly competitive with GCMs. However, they cannot represent non-Gaussian PDFs. Recently, it has been shown that if one allows the stochastic noise in such linear models to have both state-independent (additive) and linearly state-dependent (linear multiplicative) parts, then one can generate non-Gaussian PDFs with very realistic higher momentsand also Power-Law tails. Such extended linear models thus provide a dynamical basis for understanding how even non-Gaussian PDFs can change in various situations.