Improvements of NCEP Dynamics Core Hann-Ming Henry Juang March, 2007

1. Improvements of NCEP Dynamics CoreHann-Ming Henry JuangMarch, 2007 • Recent implementations • Possible near future improvements Juang

2. Introduction • For years, NCEP spectral models, GFS and RSM/MSM, have not improved its dynamics; hydrostatic for GFS, sigma, Eulerian, semi-implicit, spectral horizontal, and finite difference in vertical etc • Recent implementations start from generalized hybrid vertical coordinates, then better thermodynamics equation and semi-Lagrangian advection. Juang

3. Generalized vertical coordinates with multiple conservations Juang

4. For specific hybrid coordinate where variables with ^ are at model interfaces, P, T and Ps are function of 3-D space and time. A, B, and C are layer constants, only function of k Juang

5. Sigma-theta Sigma-p C=0 A=0 B=0 B=0 A=0 C=0 C=0 A=0 B=1 Juang

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7. Multi conservations • The derivation of multi conservation is in NCEP Office Note 445 (Juang 2005) • It conserves • angular momentum • total energy • potential temperature • mass Juang

8. Tests • Cases test in sigma, sigma-p and sigma-theta • Half year parallel run with statistical scores • 2006 hurricane seasons Juang

9. T382L64 Juang

10. T382L64 Juang

11. T382L64 Juang

12. Black s: operational GFS Red t: sigma-theta GFS Juang

13. Black s: operational GFS Red t: sigma-theta GFS Juang

14. Black s: operational GFS Red t: sigma-theta GFS Juang

15. Black s: operational GFS Red t: sigma-theta GFS Juang

16. Black s: operational GFS Red t: sigma-theta GFS Juang

17. Black s: operational GFS Red t: sigma-theta GFS Juang

18. sigma-theta parallel M operational parallel H 2005 hurricane season Juang

19. Summary of hybrid vertical • Generalized vertical coordinates • sigma, sigma-pressure, sigma-theta • Multi conservation • Good for seasonal forecast • Ready to do parallel run for next operational implementation • may be mixed sigma-theta-pressure Juang

20. Use enthalpy as thermodynamics prognostic variable Juang

21. The thermodynamics equation we are using is where with ideal-gas law of including only standard atmospheric dry air and vapor. Juang

22. Generalization and accuracy • Collaboration with SEC (Space Environmental Center), concerns more and different gases • Ideal gas law should include all gases • Thermodynamic equation should be reconsidered, because it is related to much more different gases Juang

23. Consider three dimensional R and Cp by tracers We need the values of all R and Cp Our current tracers are specific humidity, ozone and cloud water, thus Ntracers=3 But cloud water may not be considered as gas. Juang

24. The ideal-gas law should be The thermodynamic equation, derived from internal energy equation, it should be written as and let as enthalpy the above energy equation can be re-written as Juang

25. From horizontal pressure gradient We have from generalized coordinate transform, above can be written from hydrostatic and or the pressure gradient force and hydrostatic can be written as Juang

26. Put previous into generalized coordinate system Juang

27. if adiabatic, we have if no sink/source we have We can define potential enthalpy as following and we have under the conditions of Q=0 and no sink/source for tracers conservation of potential enthalpy Juang

28. Same discretization • Finite difference with conservations • Conservation of angular momentum • Conservation of total energy • Conservation of mass • Conservation of potential enthalpy • Implement into current generalized coordinate GFS Juang

29. Summary for finite difference for u,v,q where and for h Juang

30. The specific hybrid coordinate can be defined as The vertical flux can be obtained by then, again, separating horizontal and vertical terms after some arrangement, we have Juang

31. we have linearized system of D, h and Ps for semi-implicit time integration as where all matrixes are the same as those in NCEP office note 445, except Rd changes to Kappa0 and T0 change to h0. Juang

32. Preliminary results • One case only 2006050100 • 5 day forecast, T62 L64 • Test adiabatic without influencing of q to Temperature and enthalpy -> identical • Test adiabatic with q included • Compare enthalpy runs with hybrid sigma-p run and generalized hybrid. Juang

33. Sigma-p Juang

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39. Sigma-p Juang

40. Summary for enthalpy • R and Cp are considered all gases ==> more generalized • Thermodynamics equation becomes enthalpy form ==> more accurate • Results show enthalpy equation is better than virtual temperature • Enthalpy + sigma-theta reduces the cold bias over tropical atmosphere • Positive impact is encouraging. Juang

41. Splitting Semi-Lagrangian Advection without iteration and halo Juang

42. What is S.L.? • The common elements for semi-Lagrangian method are • Iteration to find departure and/or mid-point values • Interpolation from regular grid points to departure and/or mid-points • Require halo grids in MPI • Advantage of semi-Lagrangian Method • Allowable larger CFL, saving time • Easy to implement positive/conserved advection Juang

43. A time step n+1 M time step n D time step n-1 Juang

44. Starting from mid-point A time step n+1 M time step n D time step n-1 No guessing and no iteration but one 2-D interpolation and one 2-D remapping Juang

45. Proposed Method • Splitting semi-Lagrangian advection • advection in one direction first • then advection in another direction • spatial splitting • Advantages • no guessing and no iteration • 1-D interpolation and remapping • possible no halo (with transport) • incremental implementation Juang

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47. Interpolation relocation D M A remapping Juang

48. A relocation M remapping D Interpolation Juang

49. Ay Ay Ax Dx Mx My My Dx Mx Ax Dy Dy Juang

50. Test Case 1 • Solid rotation (tracer advection) • Constant angular velocity for entire domain • A artificial forcing to maintain all wind field no change • Given a trace with concentration maxima away of the center • Rotation period is 60 sec • DX=1 m, DT=1 sec Juang