Terence J. O’Kane Collaborator: Jorgen S. Frederiksen

1. A comparison of statistical dynamical and ensemble prediction methods during the formation of large-scale coherent structures in the atmosphere Terence J. O’Kane Collaborator: Jorgen S. Frederiksen Bureau of Meteorology Research Centre CSIRO Marine & Atmospheric Research Melbourne Australia

2. Motivation • Inhomogeneous statistical closure theory • Flow regimes • Ensemble prediction • Random and bred initial forecast perturbations • Atmospheric blocking transitions

3. Atmospheric turbulence Global Atmospheric spectra CSIRO MK3 Global Climate Model T63 January • Atmospheric spectra nearly 2-D • large scale Rossby waves • large scale flow instabilities • Inhomogeneous large scales • small scale turbulent eddies • homogeneous small scales • Quasi 2-D at the large scales • complex (emergence/coherent structures/instabilities) ~200km resolution Ensemble weather prediction NWP → ensemble forecasting. Vast computational cost → very small ensembles (<100) Insufficient to accurately resolve the forecast error covariances. Hence a variety of deficiencies including spurious long range correlations and grossly underestimated error variances requiring heuristic approximation methods such as covariance localization and inflation. Recent studies (Denholm-Price 2003) have suggested that ensemble NWP schemes have little capacity to produce anything beyond Gaussian statistics. Higher order cumulants have been shown to be necessary to track regime transitions in low dimensional (Miller et al 1994) & atmospheric data assimilation (O’Kane & Frederiksen 2006) studies and are of no less importance in the accurate determination of the predictability of atmospheric flows.

4. Obstacles to an accurate tractable inhomogeneous non-Markovian statistical closure • Generalize two-point two-time homogeneous closure theory to general 2-D flow over topography. • Tractable representations of the two- and three-point cumulants. Generalize special case of Kraichnan (1964): Boussinesq convection: diagonalizing closure for a mean horizontally averaged temperature field with zero fluctuations to general 2-D flow over topography. • Incorporate large scale Rossby waves (β-plane). • Long integrations of time-history integrals (non-Markovian integro-differential equations requiring large storage on super computers) • Vertex renormalization problem→ Regularization • Ensemble averaged DNS code for comparison

5. 2-D barotropic vorticity on a generalized β-plane Small scales Large scales Invariants

6. Mean and transient evolution equations We have written spectral BVE with differential rotation describing small scales and large scales using the same compact form as for f-plane through specification and extension of the interaction coefficients from  to 0. Mathematically elegant and avoids massive re-writing of codes.

7. The two-time cumulant equation

8. Functional Forms The closure problem (homogeneous isotropic turbulence) Response functions

9. 3-point → DIA Closure (+ correction?) Nonlinear noise } Nonlinear damping Kraichnan, J. Fluid Mech.,1959

10. 2 point terms → Renormalized perturbation theory (1) To zeroth order (2) (3) Solution in terms of Greens function To order l (4)

11. Similarly expand the two-time two-point cumulant 2 point terms → Renormalized perturbation theory Thus (5) Where formal soln to (4) is (6) (7)

12. Renormalized perturbation theory cont. Assume initially multivariate Gaussian To Zeroth Order Diagonal dominance So to first order Substituting in solution Eq. (6) gives Renormalize Sufficient conditions for diagonal dominance are that h & <> be sufficiently small. At canonical equilibrium the off-diagonal elements of the equal time covariance vanish regardless of the size of h & <>.

13. Thus for And for Using a reduced notation and a re-ordering of the wavevectors

14. Two-time two-point cumulant equation with updates Here is the contribution to the off-diagonal covariance matrix at initial time t0 and in reduced notation In a similar manner we derive the two-time two-point response function

15. DIA Closure + 3-point cumulant update Generalization of H. Rose, Physica D, 1985 Here allows for non-Gaussian initial conditions. Thus the QDIA equations including off-diagonal and non-Gaussian initial conditions and Eqs. for the single time cumulants and response functions may be used to periodically truncate time history integrals to obtain a computationally efficient closure

16. Two-point two-time closures at Finite Resolution • For infinite resolution and moderate Reynolds numbers the DIA under predicts the inertial range kinetic energy. Other variants (McComb LET, Herring SCFT ) based on differing applications of the FDT i.e. Ck(t,t’) (t-t’) = Rk(t,t’) Ck(t,t) (Frederiksen & Davies 2004, McComb & Kiyani 2006) • For finite resolution (>C48) and moderate RL (>200) numbers all homogeneous two-point non-Markovian closures underestimate the evolved small-scale KE and dramatically underestimate the skewness. • This is due to the fact that while we renormalized the propagators the vertices are bare i.e. interaction coefficients and some information about the higher order information is absent. • What to do in a practical sense? How does the inhomogeneous closure perform given small scale topography may act to localize transfers? • Large scale Reynolds number in terms of transient energy and enstrophy dissipation Herring et al 1974 JFM

17. A regularized approach to vertex renormalization • The regularization procedure consists of zeroing the interaction coefficients K(k,p,q) if p < k/α1 or q < k/α1 and A(k,p,q) if p < k/α2 or q < k/α2 in the two-time cumulant and response function equations of the QDIA equations i.e where Θ is the heavyside step function. • The interaction coefficients are unchanged in the single time cumulant equations. • It was found that α1=α2=4 was a universal best choice for the strong turbulence weak mean field flow regime on an f-plane. • And further found that α1=α2=4 was again the universal best choice where both strong mean fields and Rossby waves where dominating the large scale dynamics. • Thus with the choice α1=α2=4 the regularized QDIA in cumulant update form offers a one parameter two-time renormalized non-Markovian closure for inhomogeneous turbulence that compares very closely to ensemble averaged DNS regardless of the relative strengths of mean flow, topography, eddies and where large-scale waves are present.

18. Flow Regimes • For homogeneous isotropic turbulence skewness is commonly used as a sensitive measure of the growth of non-Gaussian terms. • What happens in the Inhomogeneous case? • Does skewness grow at comparable rates when topography and mean fields are present? • What are the relative contributions of non-Gaussian terms (three-point) and Inhomogeneity (two-point off-diagonal covariances) to the growth of the transient field? • What proportion of error growth is due to each of these components in data assimilation / ensemble prediction regime? • Palinstrophy production measure

19. Inhomogeneity - 2 point Skewness - 3 point Note: Small scale measure

20. Flow regimes RL ~ small C48 RL ~ a few 100’s C64 RL ~ 4000 at C92

21. Ensemble Prediction • 2-D Navier-Stokes equation could be used to describe systems whose many scales of motion can be simultaneously excited and that such systems have a finite range of predictability. • Statistics of error prediction (pdf) ↔ statistical theory of turbulence (higher moments). • Forecast depends critically on random initial errors, model errors and observational error. • Ensemble / probabilistic predictions give specific information on the nature and extent of the uncertainty of the forecast. • Our closure methodology extends 1) the work of Epstein(1969) (3and higher moment discard) and Fleming (1971; MC, QN & EDQN) and Leith (1971, 1974 TFM) and 2) extends the homogeneous statistical closure methods used by Herring etal(1973; DIA) & Fleming (1979) to non-Gaussian and strongly inhomogeneous flows. • All the above studies based on random isotropic initial conditions.

22. Generating initial perturbations • In EP independent initial disturbances are generated as fast growing disturbances with structures and growth rates typical of the analysis errors. Random isotropic initial perturbations grow more slowly and lead to underestimated error variances. • Generate independently perturbed initial conditions such that the covariance of the ensemble perturbations ≈ initial analysis error covariance at the time of the forecast. • Breeding method (Toth & Kalnay, Mon. Wea. Rev., 1997) forecast perturbations are transformed into analysis perturbations in order to simulate the effect of obs by rescaling nonlinear perturbations (Toth and Kalnay 1993,1997). Sample subspace of most rapidly growing analysis errors • Extension of linear concept of Lyapunov Vectors into nonlinear environment • Fastest growing nonlinear perturbations (Toth et al 2004, Wei & Toth 2006)

23. Bred Perturbations • Analysis cycle acts as a nonlinear perturbation model on the evolution of the real atmosphere (nudging) resulting in error growth associated with the evolving atmospheric state to develop within the analysis cycle and dominate forecast error growth. • Other approaches include mixed initial and evolved singular vectors, Ensemble square root filters (i.e. non-perturbed observations) etc. • Bred vectors are superpositions of the leading local time dependent Lyapunov vectors (LLV’s). • All random perturbations will assume the structure of the LLV given time thereby reducing the spread to 1

24. Atmospheric Regime transitionsNorthern Hemisphere blocking, Gulf of Alaska 6 Nov 1979 Transition from strong zonal to “wavy” flow and the emergence of a coherent high low blocking dipolar structure. Rapidly growing large scale flow instabilities and a loss of predictability i.e. rapid growth of the error field.

25. Comparison of DNS and RQDIA zonally asymmetric streamfunction in 5 day breeding / 5 day forecast experiment during block formation and maturation.

26. Ensemble prediction and error growth studies Breeding 1/11/79 Isotropic 26/10/79 • PM slope indicates growth rate. • Drop in PM corresponds to growth of instability vectors at the large scales. • When error KE growing PM largely determined by dynamics of large scale flow instabilities. • As errors saturate non-Gaussian terms become of increasing importance as is the case for increasing resolution. • For decaying homogeneous turbulence PM = SK and saturates at a nearly constant value. Day 2-5: rescaling of error variances Day 5-7: Error fields with LLV structures amplify rapidly Day1: growth of Ck,-l and organization of error structures Day 7 onwards reduced growth as errors saturate Forecast 5/11/79

27. Integral contributions to error growth.How important are memory effects? • QDIA: direct interactions only, inhomogeneous + non-Gaussian initial forecast perturbations • RQDIA: direct + indirect interactions, inhomogeneous + non-Gaussian initial forecast perturbations • ZQDIA as for RQDIA but with homogeneous initial forecast perturbations, neglect information from off-diagonal covariances and non-Gaussian terms at time of forecast • CD & QN variants , respectively.

28. Maintaining spread;EDQNM Stochastic Backscatter Forcing Incorporated.

29. Conclusions • The November 1979 Gulf of Alaska block is typical of a large scale coherent structure in the atmosphere and is associated with markedly increased flow instability and a corresponding loss of predictability. • Comparison of the DNS, QDIA closure and variants enabled quantification of the respective contributions of off-diagonal and non-Gaussian terms to error growth. • For ensemble prediction instantaneous error growth is largely due to the inhomogeneity and not the non-Gaussian terms which at any given time are small. However the cumulative effect of the non-Gaussian terms determines the correct amplitude of the evolved kinetic energy variances. • Instantaneous non-Gaussian terms only become important after the transients saturate the mean and the flow enters a regime where turbulence dominates. • Other applications include statistical dynamical data assimilation methods and subgrid-scale parameterizations. • Future work focussing on developing baroclinic QDIA and averaging over the large wavenumbers

30. References • J.S. Frederiksen (1999) Subgrid-scale parameterizations of eddy-topographic force, eddy viscosity, and stochastic backscatter for flow over topography, J. Atmos. Sci., 56, pp1481--1494 • T.J. O’Kane & J.S. Frederiksen (2004) The QDIA and regularized QDIA closures for inhomogeneous turbulence over topography, J. Fluid Mech., 504, pp133--165 • J.S. Frederiksen & T.J. O’Kane (2005) Inhomogeneous closure and statistical mechanics for Rossby wave turbulence over topography, J. Fluid Mech., 539, pp137—165 • T.J. O’Kane & J.S. Frederiksen (2007) A comparison of statistical dynamical and ensemble prediction methods during the formation of large-scale coherent structures in the atmosphere,J. Atmos. Sci. In Press • T.J. O’Kane & J.S. Frederiksen (2007) Comparison of statistical dynamical, square root and ensemble Kalman filters, Submitted Tellus • T.J. O’Kane & J.S. Frederiksen (2007) The structure and functional form of the subgrid scales for inhomogeneous barotropic flows in the atmosphere using the QDIA, To appear Physics Scripta