1. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Stochastic representations of model uncertainty Glenn Shutts ECMWF/Met Office Acknowledgements : Judith Berner, Martin Leutbecher

2. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Outline Ensemble model spread The nature of model error The stochastic physics scheme (perturbing parametrized tendencies) The spectral stochastic backscatter scheme Calibrating the schemes by coarse-graining

3. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Ensemble Forecast for Thurs 15 th 2007

4. Representing initial state uncertainty by an ensemble of states analysis spread RMS error ensemble mean Represent initial uncertainty by ensemble of atmospheric flow states Flow-dependence: Predictable states should have small ensemble spread Unpredictable states should have large ensemble spread Ensemble spread should grow like RMS error

5. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Buizza et al., 2004 Systems Under-dispersion of the ensemble system ------- spread around ensemble mean RMS error of ensemble mean RMS error of ensemble mean The RMS error grows faster than the spread Ensemble is under-dispersive Ensemble forecast is over- confident Under-dispersion is a form of model error Forecast error = initial error + model error

6. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Manifestations of model error In medium-range: Under-dispersion of ensemble system (Over-confidence) Can extreme weather events be captured? On seasonal to climatic scales: Not enough internal variability To what degree do detection and attribution studies for climate change depend on a correct estimate of internal variability? Underestimation of the frequency of blocking Tropical variability, e.g. MJO, wave propagation Systematic error in T, Precip, …

7. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Causes of model error : Unrepresented processes in weather and climate models Systematic versus random error physical parametrization delivers ensemble-mean or most likely tendencies ? random model error can be associated with : (i) statistical fluctuations in sub-grid (or filter-scale) transport processes (e.g. convective mass flux) (ii) unrepresented statistical physical process e.g. turbulent backscatter different systematic errors associated with model framework (e.g. gridpoint vs spectral) and parametrization choices can be used to create an ensemble forecast system (e.g. multi-model ensemble; Hadley Centre QUMP)

8. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Kinetic energy spectra from aircraft Nastrom and Gage, 1985

9. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Kinetic Energy spectrum in the ECMWF IFS Wavelength ~ 600 km Missing mesoscale energy

10. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Representing Uncertainty within conventional parameterization schemes Stochastic parameterizations (Buizza et al, 1999, Lin and Neelin, 2000) Multi-parameterizations approaches (Houtekamer, 1996) Multi-parameter approaches (e.g. Murphy et al,, 2004; Stainforth et al, 2004) Multi-models (e.g. DEMETER, ENSEMBLES, TIGGE, Krishnamurti) outside conventional parameterisation schemes Nonlocal parameterizations, e.g., cellular automata pattern generator (Palmer, 1997, 2001) Stochastic kinetic energy backscatter (Shutts and Palmer 2004, Shutts 2005; Bowler et al, 2009)

11. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Stochastic parameterizations have the potential to reduce model error Weak noise Multi-modal Strong noise Unimodal Stochastic parameterizations can change the mean and variance of a PDF Impacts variability of model (e.g. internal variability of the atmosphere) Impacts systematic error (e.g. blocking, precipitation error) Potential PDF

12. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Spectral stochastic perturbed tendency scheme (New Stochastic Physics) Revised form of the scheme due to Buizza et al (1999) use a spectral pattern generator based on triangularly-truncated spherical harmonic expansions to represent a global field multiplier at any spatial point the multiplier has a mean of 1 and prescribed variance the field has Gaussian horizontal auto-correlation function with an adjustable correlation scale (e.g. 500 km) Each spectral component in the pattern evolves in time according to a first-order autoregressive process with prescribed decay time (e.g. 6 model steps) model parametrization tendencies are multiplied by the pattern field (excluding boundary layer and stratosphere)

13. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 New stochastic physics pattern generator

14. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009

15. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Decrease in ensemble mean error x Ensemble members x Ensemble mean error Analysis x Ensemble mean

16. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Continuous Ranked Probability Skill Score

17. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 r.m.s. error of 850 hPa temperature in the tropics versus spread for the ensemble-mean (Crosses are for r.m.s. error) Under-dispersion Spread increased with new Stochastic physics

18. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Continuous Ranked Probability Skill Score

19. Spectral Backscatter Scheme Rationale: A fraction of the dissipated energy is scattered upscale and acts as streamfunction forcing for the resolved-scale flow (LES, CASBS: Shutts and Palmer 2004, Shutts 2005); New: spectral pattern generator Total Dissipation rate from numerical dissipation, convection, gravity/mountain wave drag. Forcing pattern: temporal and spatial correlations prescribed

20. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Spectral Backscatter Scheme (SPBS) Spectral pattern generator: where and f j m,n are the complex spectral amplitudes at step j and are associated Legendre functions ( * denotes the complex conjugate) Rationale: A fraction of the dissipated energy is backscattered upscale and acts as streamfunction forcing for the resolved-scale flow ( Shutts and Palmer 2004, Shutts 2005, Berner et al (2009)

21. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 1 st -order autoregressive process for horizontal pattern generation (n) is a scale-dependent parameter that sets the decorrelation time Currently ~ 0.07 for all n and is chosen so that is s. g(n) sets the amplitude of the random number noise r j m,n based on coarse-graining calculations using a big-domain cloud-resolving model g(n) is (1+n) where j is the step number

22. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Power spectrum of coarse-grained streamfunction forcing at z=11.5 km computed from a cloud-resolving model k -1.54 Log(E) Log(k) g(n) ~ k -1.27 E~ n g(n) 2

23. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Streamfunction forcing Backscatter ratio Total KE dissipation rate Pattern generator D tot = numerical dissipation + gravity/mountain wave drag dissipation + deep convective production of KE

24. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Smoothed total dissipation rate D *tot

25. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Numerical dissipation Rate where is the relative vorticity and K is the biharmonic diffusion coefficient. D num is augmented by a factor of 3 to account for the kinetic energy loss that occurs as a result of interpolation of winds to the departure point in the semi-Lagrangian advection step

26. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Gravity wave/orographic drag u and v increments from the orographic drag parametrization multiplied by u and v to give a KE increment i.e. Deep convection KE production M d is the mass detrainment rate; w is a mean convective updraught speed and is the density

27. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Smoothed total dissipation rate D tot is smoothed to T30 using a tapered spectral filter Boundary layer dissipation is omitted on the assumption that turbulent eddies of scale < 1 km will not project sufficiently on quasi-balanced, meso->synoptic scale motions The bracketed term multiplying D c is the absolute vorticity normalized by twice the Earths angular rotation rate. This represents the dependence of balanced flow production on background rotation.

28. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Impacts on probability skill scores Continuous Ranked Probability Skill Score for temperature at 850 hPa (20-90 degrees N)

29. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Continuous Ranked Probability Skill Score for temperature at 850 hPa (Tropics)

30. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Continuous Ranked Probability Skill Score for u at 850 hPa (20 – 90 degrees N)

31. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Continuous Ranked Probability Skill Score for u at 850 hPa (Tropics)

32. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Continuous Ranked Probability Skill Score for u at 200 hPa (20 – 90 N)

33. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Continuous Ranked Probability Skill Score for u at 200 hPa (Tropics)

34. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Continuous Ranked Probability Skill Score for geopotential height at 850 hPa (20 – 90 degs N)

35. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Rms error of the ensemble mean versus spread about the ensemble mean for T at 850 hPa (20-90 N)

36. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Rms error of the ensemble mean versus spread about the ensemble mean of T at 850 hPa (tropics)

37. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Rms error of the ensemble mean versus spread about the ensemble mean of u at 200 hPa (20-90 N)

38. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Rms error of the ensemble mean versus spread about the ensemble mean of u at 200 hPa (tropics)

39. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Experimental Setup for Seasonal Runs Seasonal runs: Atmosphere only Atmosphere only, observed SSTs 40 start dates between 1962 – 2001 (Nov 1) 5-month integrations One set of integrations with stochastic backscatter, one without Model runs are compared to ERA40 reanalysis (truth)

40. No StochasticBackscatter Stochastic Backscatter Reduction of systematic error of z500 over North Pacific and North Atlantic

41. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Increase in occurrence of Atlantic and Pacific blocking ERA40 + confidence interval No StochasticBackscatter Stochastic Backscatter

42. Wavenumber-Frequency Spectrum Symmetric part, background removed (after Wheeler and Kiladis, 1999) No Stochastic Backscatter Observations (NOAA)

43. Improvement in Wavenumber-Frequency Spectrum Stochastic Backscatter Observations (NOAA) Backscatter scheme reduces erroneous westward propagating modes

44. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Coarse-graining as a method of computing model error Cloud-Resolving Model (CRM) approach 1)Spatially-average model fields and tendencies to a coarse grid 2)Compute tendencies implied by the coarse-grained model fields 3)Subtract the tendencies computed in 2) from the coarse-grained CRM tendendies Forecast model method 1.Run a very high resolution forecast model e.g. IFS at T1279 2.Coarse-grain the tendency fields to a lower resolution e.g. T159 3.Run a forecast at the lower resolution and subtract tendency field early in the forecast from the tendency field computed in 2)

45. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Computing the streamfunction forcing 1)Run a T1279 forecast for 2 hours and compute the total vorticity tendency from increments of u and v. 2)Smooth to T159 and take the inverse Laplacian to obtain streamfunction tendency 3)Run a T159 forecast for 2 hours. 4)Repeat 1) and 2) without smoothing 5)Compute the difference in the two streamfunction forcing functions

46. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Vertical section of the difference in u between T1279 run and T159 run at t+8 hrs

47. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009 Streamfunction forcing estimated by coarse-graining approach

48. ECMWF Stochastic representations of model uncertainty: Glenn Shutts March 2009Summary Insufficient ensemble model spread indicates the need to account for the statistical aspects of model error The true nature of this model error is not fully understood ! Statistical fluctuations ignored in conventional physical parametrization maybe included Energy backscatter from unresolved flow structures may perturb the balanced flow dynamics The coarse-graining methodology provides a method for calibrating/validating assumptions inherent in stochastic parametrization