Application and interpretation of adjoint-derived sensitivities in synoptic-case studies

1. Application and interpretation of adjoint-derived sensitivities in synoptic-case studies Michael C. Morgan University of Wisconsin-Madison

2. Acknowledgements Linda Keller Kate La Casse Dr. Hyun Mee Kim (KMA) Daryl T. Kleist (NCEP/NOAA)

3. Goals • Describe what an adjoint model is • Demonstrate adjoint applications to • Synoptic case studies • Diagnosis of ‘key’ analysis errors • Data assimilation • Discuss interesting research problems for which adjoint-based tools might have some utility

4. Goals • Provide synoptic interpretations for selected forecast sensitivity gradients • Describe the “evolution” of sensitivities with respect to the forecast trajectory • Present a useful technique to display sensitivities with respect to vector quantities • Discuss interesting research problems for which adjoint-based tools might have some utility

5. Relationship between the nonlinear model and its adjoint Nonlinear Model Linear Model Adjoint Model

6. How might adjoints be used? adjoint model input perturbation An adjoint model is useful in the estimation of a change in response function associated with arbitrary, but small changes in the input to the linearized model.

7. Application #1: Synoptic case studies Impact studies vs. Sensitivity studies

8. Impact studies or“what if?” experiments • Impact studies involve studying the effects a specific initial and/or boundary perturbation (x0) to an NWP model has on some aspect of a forecast. • While these perturbations are often chosen based on “synoptic intuition”, typically the precise choice of the location and structure of the imposed initial perturbations is not known. • The chosen perturbations may have very little impact on the weather system of interest. • As these studies are performed to assess the importance of a particular synoptic feature, many integrations are needed to yield useful results.

9. Modeling System Used • MM5 Adjoint Modeling System (Zou et al. 1997) with non-linear model state vector: • All sensitivities were calculated by integrating the adjoint model “backwards” using dry dynamics, about a moist basic state. • The corresponding adjoint model state vector is:

10. Description of Case 1 and response functions • Cold frontal passage over the upper midwest during the 36h period beginning 1200 UTC 10 April 2003 • Sensitivity gradients were calculated for the 36 hour MM5 forecast from Eta model initial conditions at 1200 UTC 10 April 2003 for three response functions: • 1) average temperature over WI • 2) average north-south temperature difference over northern WI • 3) average zonal wind over WI

11. Mean sea level pressure and temperature (s=0.85)

12. Sensitivity with respect to initial conditions at 1200 UTC 10 April 2003

13. 36h temperature sensitivity evolution

16. 700 hPa sensitivities with respect to u and v valid at 1200 UTC 11 April 2003 (f24)

17. 700 hPa sensitivities with respect to u and v valid at 1200 UTC 11 April 2003 (f24)

18. f -1 Adjoint off -1 Sensitivity with respect to derived variables Inversion Adjoint of Inversion

19. 700 hPa sensitivity gradients valid at 1200 UTC 11 April 2003 (f24)

20. Description of Case 2 and response function

21. Impact study of McTaggart-Cowan (2002)

22. Initial state (MSLP and 925hPa q)

23. Initial state (250:300 hPa PV)

24. Forecast evolution

25. Final state

26. Sensitivity of 48h KE to vorticity

27. Application #2: Identification of ‘key’ analysis errors If the response function chosen is a (quadratic) measure of forecast error, the output of the adjoint model provides a means of changing the initial conditions to determine an initial condition which will minimize the forecast error

28. 11 April 1994 ECMWF forecast bust VERIFYING ANALYSIS DAY-5 FORECAST Rabier et al. (1996)

29. Control and perturbed analyses

30. Evolution of ‘key’ analysis errors Rabier et al. (1996)

31. VERIFYING ANALYSIS DAY-5 FORECAST “OPTIMAL” FORECAST Rabier et al. (1996)

32. Application #3: 4DVAR data assimilation

33. Application #3: 4DVAR data assimilation

34. Application #3: 4DVAR data assimilation

35. 1200 UTC 13 February 2001 La CASsE STUDY NCEP final analysis (mslp) and ship and buoy observations of wind (ms-1) and mean sea level pressure NCEP final analysis (blue) and 36 hour MM5 forecast (red) mslp

36. Water vapor image andsatellite-derived wind vectors (ms-1) 0600 UTC 12 February 2001 300 hPa (yellow) and 400 hPa (blue)

37. Assimilation in sensitive regions 1200 UTC 13 February 2001 NCEP final analysis (blue) and 36 hour MM5 forecast (red) mslp All observations assimilated at 0600 UTC Observations in sensitive regions assimilated at 0600 UTC

38. Assimilation in insensitive regions 1200 UTC 13 February 2001 36 hour forecast mslp (cont. – assim.) 25,000 20,000 15,000 Number of observations 10,000 5,000 0 Observations in insensitive regions assimilated at 0600 UTC

39. Questions? Real-time forecast sensitivities may be found at http://helios.aos.wisc.edu