1. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data1 Control of Gravity Waves Gravity waves and divergent flow in the atmosphere Two noise removal approaches: filtering and initialization Normal mode initialization Digital filter Control of gravity waves in the ECMWF assimilation system Lars Isaksen Room 308, Data Assimilation, ECMWF

2. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data2 Processes and waves in the atmosphere Sound waves, synoptic scale waves, gravity waves, turbulence, Brownian motions.. The atmospheric flow is quasi-geostrophic and largely rotational (non-divergent) – mass/wind balance at extra- tropical latitudes The energy in the atmosphere is mainly associated with fairly slow moving large-scale and synoptic scale waves (Rossby waves) Energy associated with gravity waves is quickly dissipated/dispersed to larger scale Rossby waves: the quasi-geostrophic balance is reinstated

3. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data3 500 hPa Geopotential height and winds Approximate mass-wind balance

4. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data4 MSL pressure and 10 metre winds Approximate mass-wind balance

5. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data5 Which atmospheric processes/waves are important in data assimilation and NWP? Sound and gravity waves are generally NOT important, but can rather be considered a nuisance Fast waves in the NWP system require unnecessary short time steps – inefficient use of computer time Large amplitude gravity waves add high frequency noise to the assimilation system resulting in: –rejection of correct observations –noisy forecasts with e.g. unrealistic precipitation BUT certain gravity waves and divergent features should be retained in a realistic assimilation system. We will now present some examples.

6. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data6 Ageostrophic motion – Jet stream related An important unbalanced synoptic feature in the atmosphere Ageostrophic winds at 250 hPa Wind and height fields at 250 hPa

7. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data7 Mountain generated gravity waves should be retained Rocky Mountains

8. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data8 Temperature cross-section over Norway Gravity waves in the ECMWF analysis Acknowledgements to Agathe Untch Norway Pressure [hPa]

9. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data9 Analysis temperatures at 30 hPa Acknowledgements to Agathe Untch

10. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data10 Equatorial Walker circulation

11. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data11 Divergent winds at 150hPa: ERA-40 average March 1989 Acknowledgements to Per Kållberg

12. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data12 Semi-diurnal tidal signal

13. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data13 Observed Mean Sea-Level pressure - Tropics Semi-diurnal tidal signal for Seychelles (5N 56E)

14. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data14 Filtering the governing equations Quasi-geostrophic equations/ omega equation Primitive equations with hydrostatic balance Primitive equations with damping time-step like Eulerian backward Primitive equations with digital filter Goal: Use filtered model equations that do not allow high frequency solutions (“noise”) – but still retain the “signal”

15. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data15 Initialization Goal: Remove the components of the initial field that are responsible for the “noise” – but retain the “signal” Make the initial fields satisfy a balance equation, e.g. quasi-geostrophic balance or Set tendencies of gravity waves to zero in initial fields – Non-linear Normal Mode Initialization

16. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data16 Normal-mode initialization Linearize forecast model about a statically-stable state of rest: where represents linear terms represents the nonlinear terms and diabatic forcing Diagonalize by transforming to eigenvalue-mode - “Hough space”: whereis the diagonal eigenvalue matrix Split eigenvalues into slow Rossby modes and fast Gravity modes.

17. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data17 Non-dimensional wavenumber Frequency Rossby modes and Gravity modes The ‘critical frequency’ separating fast modes from slow. Mixed Rossby-Gravity Wave

18. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data18 Non-linear Normal-Mode Initialization The fast Gravity modes generally represent “noise” to be eliminated. for one eigenvalue, If N k is assumed constant (i.e. slowly varying compared to gravity waves): At initial time set The high frequency component is removed and will NOT reappear. Assumes that the slow N k forcing balances the oscillations at initial time. thenso

19. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data19 Non-linear NMI: USA Great Planes Surface pressure evolution Uninitialized field Non-linear NMI initialized field Temperton and Williamson (1981)

20. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data20 Optimal and approximate low-pass filter Consider a infinite sequence of a ‘noisy’ function values: {x (i) } We want to remove the high frequency ‘noise’. This is identical to multiplying {x (i) } by a weighting function: One method: perform direct Fourier transform; remove high-frequency Fourier components; perform inverse Fourier transform. is the cut-off frequency The finite approximation is:

21. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data21 Digital filter Consider a sequence of model values {x (i) } at consecutive adiabatic time-steps starting from an uninitialized analysis A digital filter adjusts values to remove high frequency ‘noise’ Adiabatic, non-recursive filter: Perform forward adiabatic model integration {x (0),x (1),…,x (N) } Perform backward adiabatic model integration {x (0),x (-1),…,x (-N) } The filtered initial conditions are: where

22. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data22 Fourier filter and Lanczos filter Damping factor for waves Wave frequency in hours Gibbs Phenomenon for Fourier filter Broader cut-off for Lanczos filter

23. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data23 Transfer function for Lanczos filter 6 hour window

24. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data24 Transfer function for Lanczos filter 12 hour window

25. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data25 Transfer function for Lanczos filter 6 and 12 hour window

26. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data26 Response to Lanczos filter with 6h cut-off

27. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data27 Incremental initialization (ECMWF, 1996-1999) Let x b denote background state, expected to be “noise free” x U the uninitialized analysis x I the initialized analysis and Init(x) the result of an adiabatic NMI initialization. Then x I = x b + Init(x U ) – Init(x b ) Diabatic non-linear normal mode initialization Full-field initialization (ECMWF, 1982-1996)

28. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data28 Control of gravity waves within the variational assimilation Primary control provided by J b (mass/wind balance) In 4D-Var J o provides additional balance Digital filter or NMI based J c contraint Diffusive properties of physics routines Minimize: J o + J b + J c

29. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data29 Control of gravity waves within the variational assimilation Primary control provided by J b (mass/wind balance)

30. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data30 NMI based J c constraint Still used at ECMWF in 3D-Var and until 2002 in 4D-Var I.Project “analysis” and background tendencies onto gravity modes. II.Minimize the difference. Noise is removed because background fields are balanced.

31. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data31 Weak constraint J c based on digital filter Implemented by Gustafsson (1992) in HIRLAM and Gauthier+Thépaut (2000) in ARPEGE/IFS at Meteo-France Removes high frequency noise as part of 12h 4D-Var window integration Apply 12h digital filter to the departures from the reference trajectory A spectral space energy norm is used to measure distance. –At Meteo-France all prognostic variables are included in the norm –At ECMWF only divergence is now included in the norm, with larger weight Obtain filtered departures in the middle of the assimilation period (6h) Propagate filtered increments valid at t=6h by the adjoint of the tangent- linear model back to initial time, t=0. Get and J c calculation is a virtually cost-free addition to J o calculations

32. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data32 Weak constraint J c based on digital filter Apply 12h digital filter to the departures from the reference trajectory and obtain filtered values in the middle (6h): Define penalty term using energy norm, E: The gradient of the penalty term is propagated by the adjoint, R *, of the tangent-linear model back to time, t=0: Use tangent-linear model, R, to get: J c calculation is a virtually cost-free addition to J o calculations for

33. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data33 Hurricane Alma – impact of Jc formulation Jc on divergence only with weight=100 versus Jc on all prognostic fields with weight=10 MSL pressure and 850hPa wind analysis differences

34. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data34 Impact of Jc formulation Jc on divergence only with weight=100 versus Jc on all prognostic fields with weight=10 Impact near dynamic systems and near orography. Fit to wind data improved. In general a small impact. MSL pressure and 850hPa wind analysis differences

35. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data35 Minimization of cost function in 4D-Var Value of J o, J b and J c terms

36. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data36 Minimization of cost function in 4D-Var Value of J o, J b and J c terms – logarithmic scale

37. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data37 Himalaya grid point in 3D-Var - No Jc

38. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data38 Himalaya grid point in 3D-Var

39. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data39 Himalaya grid point in 4D-Var

40. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data40 Himalaya grid point in 4D-Var

41. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data41 Seychelles (5S 56E) MSL observations plus 3D-Var First Guess and Analysis Observed value First guess value Analysis value 8 Feb 199714 Feb1997

42. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data42 Observations and first guess valuesObservations and analysis values 4D-Var handles tidal signal very well ! Seychelles (5S 56E) MSL observations plus 4D-Var First Guess and Analysis

43. Lars Isaksen, ECMWF, March 2006 Data assimilation and use of satellite data43 Gravity waves and divergent flow in the atmosphere Two noise removal approaches: filtering and initialization Normal mode initialization Digital filter Control of gravity waves in the ECMWF assimilation system We discussed these topics today