1. Forward-in-time Methods in Ocean Modeling Matthew Hecht Los Alamos National Lab

2. Primitive Equation Ocean Models Incompressible Navier-Stokes Eqns, but with: –Hydrostatic pressure –Shallow depth, relative to rotationally-constrained horizontal scales Boussinesq approximation usually made –Ignore variations in density, , except in pressure gradient term

3. Ocean model grids Horizontal grid is usually fixed, orthogonal Vertical grid can be: –Fixed vertical, independent of x, y and t (Z-grid) –Density, as in a stacked shallow water (isopycnal, or layered model) –Arbitrary Lagrangian/Eulerian (ALE)

4. One more point: Modal decomposition Momentum equations separated into –vertically-averaged component (“barotropic”) –departure from the vertical average (“baroclinic” component) Fast surface gravity waves contained within equations for vertically-averaged flow –Time step for 3-D momentum equations can be something like 2 orders of magnitude larger than for the 2-D “barotropic” equations

5. Who cares about the oceans? Climate scientists, oceanographers, coastal residents of South Wales, … Climate ocean models typically Order(1º) –Most of turbulent spectrum parameterized, not resolved Atmosphere forces the oceans –through winds, fluxes of heat, fresh water Ocean responds –with transport of heat, salinity Ocean forces the atmosphere –through sea surface temperature So… an ocean model for climate had better produce the right sea surface temperatures –implying circulation must be very nearly correct

7. Stommel Gyre test problem Stommel (1948) found analytical solution for gyre circulation from balance of wind stress and bottom drag, when on a rotating sphere Passive tracer advective test problem in highly sheared flow field, from Hecht, Holland and Rasch (J. Geophys. Res. 1995)

8. Reference Solution We send the tracer through the western boundary current once. Reference solution is produced numerically.

9. Selected results As on previous slideCentered leapfrog O(3) upwind With 1 antidiffusive pass

10. Error measures

11. Centered cases with diffusion With “typical” value: With lower value, such that reduction in peak ~same as for MPDATA Looks very similar to MPDATA Looks like O(1) donor cell result (not shown)

12. Rotated Stommel Gyre In previous work, fast western boundary current was aligned with principal grid axis –Here it is at 45º, for a more discriminating test, in Hecht, Wingate and Kassis (Ocean Modelling, 2000)

13. Reference Solution Same reference solution (produced more elegantly by Beth Wingate).

14. Error measures

15. Results

16. Evaluation of advection schemes in a primitive equation ocean model Very simple –wind stress –heat flux –fresh water flux with simple flat-bottom configuration produce idealization of Atlantic circulation (Hecht, Bryan and Holland, J. Geophys. Res., 1998)

17. Vertically- integrated flow Vertically integrated flow is as predicted by Sverdrup –Magnitude of N/S flow proportional to curl of wind stress Think of northern-most gyre as being idealization of “sub-polar gyre” (Labrador Sea/Northern North Atlantic) Think of second northern-most gyre as “subtropical gyre” -- containing the Gulf Stream

18. Velocities at level 3 Even though forcing is simple, response is complex

19. Poleward transports Atlantic-like northern deep water formation –This was set up through asymmetry in heat flux forcing

20. Gravitational instability and mixing From a section through the western boundary. Centered-leapfrog produces 2 dz noise which creates spurious instabilities. MPDATA run without limiting

21. Some problems require strict limiting Here, looking at idealized age tracer (“ideal age”), set to 0 at surface, constant clock source elsewhere. Age shouldn’t be greater than the duration of the run (or less zero)!

22. Cost of limiting motivates “supercycling” In some simulations CFL for passive scalars is significantly less restrictive than time-step limitation for dynamical tracers (density) presented by internal waves Longer time steps can be taken on passive scalars, using either instantaneous or time-averaged mid-point velocity.

23. From supercycling of passive scalars to “Method of Averaging” Supercycling in ocean models hasn’t caught on, because of competition from another acceleration technique –This thinking led us into: Method of Averaging (MOA) –Use of time-averaged (low-pass filtered) advecting velocities in momentum equations, to low-pass filter wave motions and allow for stable integrations with a relatively long time step.

24. Does any of this matter? “Equatorial nutrient trapping” -- discussed for years -- was a consequence of advective error

25. …and it’s hard to maintain 7 orders of magnitude difference in mixing between quasi-horizontal and vertical directions Griffies shows how to diagnose spurious, numerical mixing

26. Special projects in ocean modeling with MPDATA Now back to our regular programming… Method of Averaging (MOA) –Use of time-averaged (low-pass filtered) advecting velocities in momentum equations, to low-pass filter wave motions and allow for stable integrations with a relatively long time step.

27. Method of Averaging If DT is CFL-limited time step and dt is internal-wave limited time step where DT = M * dt (M is some integer) 1)First, using O(1) donor cell advection, integrate through DT by taking M small steps of size dt. 2)Then, using a high-order method with the time-averaged advecting velocity from 1), repeat integratation through DT in one step.

28. Rossby to Kelvin Wave Problem Problem taken from Milliff & McWilliams (1994) Applied with Method of Averaging in Nadiga, Hecht, Margolin & Smolarkiewicz (1997).

29. Results with MOA MOA result, with 8x larger time step, compares well with explicit reference case

30. Kelvin Wave signal at boundary point Explicit reference case contains higher frequencies, MOA case is smoother line (ignore 3rd, offset curve). MOA accurately reproduces low- pass Kelvin Wave signal.

31. Use of MOA In forest fire code –Reisner et al., Monthly Weather Review, 2000.

32. DNS of rotating tank/boundary separation From Baines and Hughes, J. Physical Oceanography, 1996. DNS with EULAG code

33. Time stepping schemes used in ocean models Momentum equations 3D (slow) 2D (fast) Scalar eqns HIM and POSUM have Forward with PC or FB; other models listed only have options for Forward on tracers (scalars). From Griffies 2000

34. The Future: New models under development with 2-time-level dynamical cores Bob Higdon’s dy. core uses MPDATA Developed as alternative dy. core for –Miami Isopycnic Ocean Model (MICOM) – Hybrid Coordinate Ocean Model (HYCOM) MI/HYCOM use form of MPDATA developed by Drange & Bleck (‘97) for adv of temperature Los Alamos’ HYPOP using Dukowicz’s forward-in-time “remapping” Are finite elements in our future (really)?

35. Higdon’s dynamical core JCP, 2002

36. Time-evolution of interface

37. Asymptotic state

38. In Summary Forward-in-time methods are in common use for scalar transport, where problems with 2 grid point noise and spurious extrema are obvious. New models are being developed with fully forward-in-time dynamical cores. –HOME effort to share code in layered ocean models will be largely forward-in-time

39. Gulf Stream, North Atlantic Current into the “Northwest Corner”, seen in Sea Surface Height C=1 C=1/2 C=1/4 TOPEX Obs Models

40. Western boundary currents (boundary jets) Maltrud and McClean, Ocean Modelling, 2004