1. A Multistep Flux-Corrected Transport (MFCT) Jin Lee Earth System Research Laboratory (ESRL) NOAA

2. Continuity equation Monotonic advection Conservative and monotonic tracer transport scheme Conservation & Monotonicity Montonoticity: no new maximum/ no new minimum Consistency: q=1, then tracer eq -> continuity eq.

3. Two approaches for monotonic transports : Flux limiters: - which constraint the fluxes based on the monotonic subgrid distributions. Flux Corrected transports: - which include the diffusive and antidiffusive stages designed to limit fluxes in strong gradient regions and over smooth parts of solution, the maximum allowable high-order flux is restored to the low-order solution without violating the monotonicity.

4. Two-time level.vs. Multistep schemes :

5. Brief review of Zalesk two-time level FCT scheme :

6. Extension of Zalesak FCT to AB3-MFCT scheme Low-order monotone sol: discontinuous function / strong gradients. High-order AB3-sol: smooth function.

7. FIM icos plotting – Ning Wang – 240km res

13. Analytical test of “Cosine Bell Advection over poles” in one of Williamson et. al. (1992) standard test cases (dx~240 km).

14. “Slotted-Cylinder Advection over poles” (dx~120 km)

15. Cosine Bell Advection:(dx~120 km) Curve A: analytic Curve B: MFCT Slotted cylinder:(dx~120 km) Curve A: analytic Curve B: MFCT

17. No-FCTMFCTSmooth Functon

18. Discontinuous Functon No-FCTMFCT

19. FIM model/ system -Contributors FIM DESIGN Jin Lee Sandy MacDonald Rainer Bleck Jian-Wen Bao John M. Brown Jacques Middlecoff Ning Wang Stan Benjamin Tom Henderson Chris Harrop Bill Moninger Brian Jamison Susan Sahm Ed Szoke Georg Grell Jeff Whitaker, SOS project: Bob Lipschutz David Himes Beth Russell Steve Albers Tom Kent + dozens more… ESMF, Subversion WorkFlow Manager, xml real-time scripts Verification, Web page, Evaluation FIM- chemistry, aerosols EnKF data assimilation

20. Results of FIM medium-range weather forecasts FIM real time forecasts initialized with GFS initial condition at 00 and 12 UTC ( comparisons of FIM and GFS 500 mb ACC ) Interpolate GFS initial data to Icosahedral grid. Perform hydrostatic initialization. Perform 7-day fcst with dx~30km. Same initial condition, terrain & sfc parameters, physics package runing at similar model resolutions (dx~30 km)

22. Same IC, terrain & sfc parameters, physics package, runing at similar model resolutions (dx~30 km) Effect of different numerics, e.g., spectral v.s. finite-volume, on tracer transports.

25. A general MFCT tracer algorithm is developed to achieve conservative and monotonic transports in any multistep dynamical cores which provide mass fluxes by which tracers are transported. AB3-MFCT has been implemented on our Icosahedral SWM and tested with continuous and discontinuous analytical test cases as well as real time medium range weather forecasts. The advection accuracy to the monotonic constraint ? Final remarks