Using stochastic physics to represent model error by Glenn Shutts

1. Using stochastic physics to represent model errorbyGlenn Shutts Predictability, Diagnostics and Extended-Range Forecasting Course, May 8-17 2013 with contributions from Martin Steinheimer, Martin Leutbecher and Alfons Callado Pallares

2. What are the symptoms of ‘model error’ ? 1. Systematic errors • e.g. excessive westerlies and climate drift. • - error pattern that persists and grows from • the start of a forecast (Jung and Tompkins, 2003) • (difficult to identify the cause although sometimes • obvious e.g. mountain/gravity wave drag) • large-scale atmospheric ‘blocking’ frequency too low for • horizontal resolution less than T511 • (transient eddy forcing versus ‘background flow’) • inability to sustain a Madden-Julian Oscillation

3. 2. Random errors • incorrect statistical fluctuation in sub-gridscale parametrization e.g. number of deep convective clouds per gridbox • in parametrized fluxes caused by sensitivity to uncertainty in model profiles (e.g. trapped lee wave drag, radiative energy flux) • chaotic upscale transfer of kinetic energy as a quasi-2D • turbulent process

4. ‘representing random model error’ If the model already has error, why put some in ?! • add random error and hope that some • systematic error improvements result e.g. from • nonlinear ‘rectification’ • added stochastic terms in the forecast model • equations may make them physically more accurate • (e.g. by counteracting numerical diffusivity)

5. Atmospheric processes that have a stochastic nature at the NWP grid-scale • deep mesoscale convection • mountain drag • radiative fluxes in the presence of cloud

6. Deep convection - a view from space

7. eddies shed from islands trapped lee waves generated by an island

8. thin cloud sheets

9. Practical motivation for stochastic physics • member forecasts too similar (even after inflating initial state perturbations) • Improve probability skill scores by increasing member • spread so as to match r.m.s forecast error • reduce mean climate errors (?) • improve average deterministic forecast skill (??)

10. Stochastic physics algorithms • represent random error using a random number generator to create a noise term in the model equations • random numbers need to pass through a space/time filter before application • create a ‘pattern generator’ with physically-appropriate space and time decorrelation scales

11. ‘Knowledge uncertainty’ and the Random Parameters method (scheme used at the Met Office) - uncertainty in the values of parameters that arise in physical parametrization e.g. convective entrainment rate, ice crystal fallspeed, gravity wave drag constant However, beware that: • this interpretation treats parametrization parameters as if they were well-defined constants whose uncertainty originates from measurement/estimation error. • independent variations of parametrization parameters can generate physically-implausible physics

12. Idealized model equations Stochastic parametrization at ECMWF Resolved scales = “dynamics” Horizontal diffusion Local tendency Stochastic perturbations Unresolved scales = “physics” (cloud microphysics, …) X = prognostic variable (e.g. u, v, T, q, …)

13. SPPT – Stochastic Perturbed Parametrization Tendency scheme • scheme implemented in Sept. 2009 (replacing the Buizza et al, 1999 scheme) • same random pattern r used for all variables (X=u, v, T, q): • r (l, f, t) composed of three independent patterns with: • spatial correlation scales 500, 1000 and 2000 km • temporal scales of 6 hours, 3 days and 30 days • pattern standard deviations 0.52, 0.18 and 0.06 • use m toreduce/remove perturbations in lowest 300m and above 50 hPa since:- • boundary layer eddies small compared to gridbox size • near-absence of cloud in stratosphere greatly reduces the uncertainty in radiative flux divergence

14. Spectral pattern generator The triangular spherical harmonic expansion of a grid point field is given by:

15. In the pattern generator the spectral coefficients are evolved in time with a first order auto-regressive process: • Settings used for the revised SPPT • Temporal correlation wave number independent • Spatial correlation • Scaling to provide intended variancein grid point space: (Weaver and Courtier, 2001) n

16. SPPT – random number pattern original Revised (L=500km)

17. Multi-scale SPPT 2000 km 30 d 500 km 6 h 1000 km 3 d

18. SKEB - Spectral Kinetic Energy Backscatter 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) Streamfunction forcing is given by: Total dissipation rate Streamfunction forcing Pattern generator Backscatter ratio

19. SKEB – spectral pattern Spectral pattern Decorrelation time (t)at the moment wave number independent (=7hr) Spatial correlation Vertical correlations are introduced by random phase shifts of the complex random numbers Scaling to normalize energy input implied by the pattern:

20. SKEB pattern Streamfunction forcing Vorticity forcing

21. SKEB – Dissipation rate Considered total dissipation rate is the sum of “Numerical” dissipation Loss of KE by numerical diffusion + interpolation in semi-Lagrangian advection Dissipation from gravity wave/orographic drag parametrization Deep convective KE production (estimated from mass detrainment and mean convective updraught speed) 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

22. Numerical dissipation Rate where z is the relative vorticity and K is the biharmonic diffusion coefficient. Dnum 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

23. Gravity wave/orographic drag and deep convection u and v increments from the orographic drag parametrization multiplied by u and v to give a KE increment i.e. N.B. recent work shows that this contribution creates unrealistic error patterns and makes only a small impact on spread. The Met Office SKEB scheme does not include such a term. Deep convection KE production Md is the mass detrainment rate; w is a mean convective updraught speed and r is the density

24. Smoothed total ‘dissipation’ rate • is latitude and the factor multiplying DC represents the dependence of balanced flow production on background rotation. Dtot is smoothed to T30 resolution using a Gaussian spectral filter

25. Resolution dependence of dissipation rate 0.7 T159 T255 T319 0.5 dissipation rate (Wm-2) T511 T399 0.3 T799 T1279 T639 backscatter energy input 0.1 1400 1000 600 200 spectral truncation order (N)

26. Validation of stochastic schemes by coarse-graining • Using high resolution model data for stochastic parametrization calibration: • CRM data: • Calculate spatial average of state variables and compute tendencies implied by them • Compare these tendencies to spatial averages of the high resolution tendencies • High resolution FC data: • Coarse grain tendencies from high resolution FC (e.g. T1279) • Compare these to low resolution FC tendencies (e.g. T159) • During SKEB development used for: • Defining spectral power distribution (=spatial scale) • Defining vertical correlations For more details see, e.g. Shutts and Palmer (2004, 2007)

27. Coarse-graining IFS forecasts run 32 12-hr forecasts at T1279 and T159 resolutions coarse-grain parametrization temperature tendencies in time by averaging over 12 hrs with a triangular weighting function peaking at T+6 Coarse-grain in space using a quasi-Gaussian spectral filter with 250 km length scale compute the r.m.s difference between the coarse-grained tendencies in the T159 and T1279 forecasts for different ranges of temperature tendency in the coarse-grained T159 field

28. Model error if the T1279 is defined to be ‘truth’, and the overbar represents the coarse-graining operator, then will be the error field. This is likely to be a ‘lower bound’ estimate of the true model T159 error. Note: no coarse-graining in the vertical since forecasts use the same number of levels (91)

29. Error variance versus mean dT/dt SPPT

30. Coarse-graining inferences:- • ‘error in the total physics tendency is proportional to the mean’ assumption in SPPT is very crude • Convection and large-scale condensation dT/dt error variance is proportional to the mean dT/dt • Uncertainty in radiation temperature tendency is asymmetric about zero and is a minimum for clear sky • Clear sky radiative temperature tendency should be removed from SPPT • Total parametrized temperature tendency error is non-zero at zero mean tendency – additive error ?

31. Coarse-graining the vorticity equation in the ECMWF IFS • Use the IFS’s spectral-gridpoint transforms to compute terms in the vorticity equation at : • Full resolution (e.g. T1279) • A lower ‘target’ resolution (e.g. T159) • Define error in the target resolution to be the difference • Compute the KE input spectrum implied by the error (or residual) forcing function • Gives the KE input from scales for 159 < n < 1279

32. KE source at 250 hPa due to waves with wavenumbers in the range 160 to 1279 – vorticity flux divergence by rotational wind and biharmonic diffusion sink KE backscatter diffusion energy sink at T159 energy sink due to waves with n > 159

33. Now including the KE input from SKEB(using default settings for T159) SKEB SKEB input noisy at low wavenumbers +ve SKEB input

34. Inferences from coarse-graining the vorticity equation • Upscale KE injection into low wavenumbers exists but is a small term • At T159, bi-harmonic diffusion is not scale-selective in respect of KE dissipation • SKEB tends to offset the diffusion although fluctuates wildly at low wavenumber • The net effect of waves with wavenumbers > 159 is dissipative at 250 hPa.

35. Operational usage of SPPT and SKEB Usage of model error representation by stochastic parametrization in the forecasting system: EPS-System (medium range + monthly) 3-scale SPPT+SKEB (CY36R4) Ensemble data assimilation 3-scale SPPT (currently SKEB not included) Seasonal forecasting system 3-scale SPPT+SKEB (system 4)

36. Spread vs r.m.s error for Z500 (NH extra-tropics) T639 (see Shutts et al, 2011 ECMWF Newsletter Autumn issue) Solid lines represent r.m.s error Dashed lines represent spread

37. Future Directions Coarse-graining studies: targeted on the spatial/temporal structure of model error use of models that don’t need convection or gravity wave drag parametrization (grids finer than 2 km) Stochastic physics implementation Make SPPT dependent on individual physical process (e.g. convection with variance proportional to the mean tendency Simplify SKEB implementation (e.g. remove GWD contribution to the dissipation rate; remove truncation at n=159) try perturbing parametrization scheme input profiles Use cellular automata as an intelligent pattern generator coupled to the model flow Try the ‘Random Parameters’ approach where highly-uncertain parametrization parameters have their values modulated by the pattern generator Use ensemble data assimilation to better understand the nature of model error

38. Summary Stochastic terms that aim to account for real statistical uncertainty in physical and dynamical processes can improve EPS probability skill scores localized upscale energy transport (particularly by deep convection) is probably suppressed in NWP models by numerical diffusion and convection parametrization that relaxes towards radiative-convective equilibrium The perturbed physical parametrization tendency and stochastic backscatter methods help to improve EPS spread and probability skill scores The coarse-graining methodology provides a means of assessing pdfs and calibrating stochastic parametrization