Quasi-Uniform Rectangular Grids for Globalization of Regional Models of the Atmosphere: Application for Hurricane Predic

1. Quasi-Uniform Rectangular Grids as a Framework for Globalization of Regional Models of the Atmosphere: Possible Application for Prediction of Hurricanes Miodrag Rancic and Hai Zhang UMBC/JCET mrancic@umbc.edu

2. Content • Review of the project • Description of the global model • Inclusion of variable resolution • Vector projected schemes • Performance of 3D version • Possible applications • Concluding Remarks

3. The general objective of this project was to explore alternative solutions to application of longitude-latitude grid in numerical modeling of atmosphere Review of the Project

4. What is wrong with the long-lat grid ? • There is need to use polar filtering, which may seriously draw back the computing performance, especially on distributed memory computers • The areas around poles are first over resolved, wasting memory • Then the effect of this resolution is removed, wasting computing time • Randal (1989) described close zonal spacing of long-lat grid as • an “excessive spatial resolution” • which “…does not effectively exploit computer resources.”

5. History of rectangular QU grids Gnomonic cube (Sadourny 1972) • Problems • Angular discontinuity along the edges • Eight singular corners Consequence: Lot of noise!

6. Rancic et al. (1996) Purser and Rancic (1997;1998) Cubic grid Octagonal Grid

7. Globalization project* • The goal was to expand a regional atmospheric model to global coverage using a framework of QU grids • Regional Eta model as a prototype • General curvilinear coordinates • A single code may run on different grid topologies • Increasing computational efficiency of integration *NSF grant in 2001

8. Momentum Thermo. Continuity Description of the Global Model • General curvilinear formulation

9. Vertical velocity Surface pressure Hydrostatic Transport

10. Covariant winds • Contravariant winds • Metric tensor • Jacobian of coordinate transformation

11. y y V h h V h h h h h h x x • Horizontal dynamics of the Eta model had to be adjusted to vector invariant (covariant-contravariant) formulation • Horizontal grid replacement (following Janjić 1984) E-grid B-grid

12. Inclusion of variable resolution (Rancic and Zhang 2006, to appear in MAP) Cubic grid with the stretched variable resolution Basic octagonal grid with the stretched variable resolution

13. Schmidt Transformation (Purser and Rancic 1997) A locally stretched grid – potentially of interest for simulation and tracking the hurricanes Rancic and Zhang (2006) have additionally investigated various other techniques for local enhancement of resolution, such as grid overlapping The local enhancement of resolution has several clear disadvantages

14. 1) A ‘banana’ effect

15. 2) A ‘bottle neck’ effect

16. Example with linear advection tests in the plane Stretched grid Locally overlapped grid

17. Test integrations with the wedge Solution after 5 translation of the wedge Locally overlapped grid Stretched grid

18. “Globally overlapped grid” Depending on method for coupling the solutions, this method is potentially the best, but involves programming difficulties

19. Rotational test on the octagonal grid • A modified Takacs 3rd order scheme • Bell-shaped perturbation • Uniform resolution • Stretched grid (stretch factor is σ=2) • Stretched grid (σ=8) • Shmidt locally stretched grid (σ=2)

20. Vector projected schemes(Rancic and Zhang 2006, submitted to MWR) • One of the major problems with the vector invariant dynamics was inability to apply Janjic (1984) and Rancic (1988) (forth-order accurate) nonlinear advection schemes, which are written in the advective form • These schemes are capable to very realistically describe nonlinear cascade of energy, critically important for the long-terms (seasonal and climate) simulations

21. x 1 What was the problem ? • One of the major problems on the QU grids is how to deal with the curvature of the coordinates close to the singular points

22. In the case of horizontal diffusion, the problem is solved by a projection of the winds to the base vectors of the same coordinate system, in order to avoid variation • In the case of vector invariant formulation, the problem is to some extent addressed by elimination of explicit curvature terms • However, what to do in the case of advective formulation needed for application of Janjić (1984) scheme?

23. x 1 • In principle, the problem could be solved by a vector discretization of the momentum equation

24. However, the Janjic (1984) scheme is not locally isotropic • It turned out that there is an isotropic formulation, with similar properties as the original the Janjic (1984) scheme

25. Flat Square Earth test • J77 scheme (E-grid enstrophy) • J84 scheme (C-grid enstrophy) • J06 scheme (C+E –grid enstrophy) • Diagnostics

26. We can now discretize the momentum equation in the vector form • … and we derive the time tendency of covariant components by a scalar product of fluxes • … which virtually eliminates the described problem (and absorbs the curvature terms

27. In the shallow water benchmark tests of Williamson et al. (1996), the new technique clearly outperforms vector invariant formulation • The new version of Rancic (1988) 4th order scheme is also introduced following the same principle • It clearly outperforms the 2nd order scheme running at double resolution

28. l1

29. Results of the 3D model(Zhang and Rancic 2006, to appear in QJ) • Both cubic and octagonal grids are tested • Model passes the Held-Suarez (1996) benchmark test • A series of 10-day simulations with real data and various resolutions (250, 100 and 50 km)

30. Cubic North Hemisphere: 5-day forecast of 500 mb Analysis Octagonal

31. Cubic North Hemisphere: 10-day forecast Analysis Octagonal

32. The results compare favorably against Wyman (1996) results with a global version of Eta on standard long-lat grid • The average anomaly correlation of 500 mb surface of global Eta is better over the US from those of the regional Eta model after 3 days of integration

33. Anomaly correlation over US

34. The dynamics of Global Eta on QU grids is about 30% more efficient than GFDL Flexible Modeling System

35. Experiments (A series of 5-day forecasts) • Variable resolution cubic grid The low resolution area has a grid distance of 166 km and the high resolution area has a grid distance of about 40 km. The global stretch factor is 4.2. • Global Eta with uniform cubic gridat 166 km • Regional Eta at 40 km with boundary conditions from GFS that was run at about 110 km • Regional Eta at 40 km with boundary conditions from Global Eta run at 166 km

36. Do we need a larger area for grid stretching? • RMS errors of 500 mb geopotenitial surface over US

37. 14 tiles 68 tiles 82 tiles

38. Possible applications • Research of polar phenomena (sudden stratospheric warming, ozone depletion, atmosphere-ice interaction, …) • Research of tropical forcing • There is one proposal out on paleoclimate research • Several commercial applications are under consideration • Merging with NMM-WRF, and exploiting the variable resolution option for forecasting of hurricanes

39. Concluding Remarks • There is a sentiment in the global modeling community to replace standard long-latitude grids • A number of groups use the cubic grid suggested in Rancic at al (1996) and Purser and Rancic (1998) • A climate model at MIT (John Marshal) • A model of the Australian Weather Bureau (John McGregor) • At least two groups at Earth Simulator (Motohiko Tsugawa; Sung-Dea Kang) • An ocean model developed at NCEP (Dmitry Chalikov)

40. At UMBC we have: • Solved a series of problems related to application of rectangular QU grids • Developed a “globalization framework”, using as a prototype regional Eta model • This framework includes • Basic cubic and several octagonal grids • A series of higher cubic/octagonal grid topologies capable to handle variable resolution • One of the preferable future development may include • Incorporation of NMM-WRF infrastructure • Coupling with an oceanic component • Application in the long-term hurricane prediction

41. Modeling potentials at UMBC • At Physics Department of UMBC there is a potential and desire to • Expand numerical modeling research • Establish collaboration with NCEP • We have several graduate students working, or preparing to start work with NMM-WRF • Access to two Beowulf clusters at campus acquired through NSF grants (at Math and Computer Science Departments) • Collaborations with • Bowie State University (a powerful Macintosh cluster) • Howard University • We are publishing papers and writing proposals, but … • We need input from NCEP in organizing and engaging our students in meaningful offline research projects that may support the research and operations of NCEP

42. Acknowledgements • This project was sponsored by an NSF ATM -0113037 and by JCET internal funds • The authors are grateful to Prof. Ray Hoff, the Director of JCET, and Prof. Wallace McMillan for their support • Computation is being performed at a Beowulf cluster at Math Department of UMBC