Unity, fidelity, and simplicity in parameterizations: The example of momentum fluxes

1. Unity, fidelity, and simplicity in parameterizations: The example of momentum fluxes Vincent Larson, Steffen Domke, and Brian Griffin Paracon meeting Jul 2019

2. Outline of talk • Unity, fidelity, and simplicity are three desiderata of parameterizations. • A particular parameterization challenge: the physics of momentum fluxes in a shallow cumulus case. • Some momentum fluxes are upgradient. Where does this upgradient physics reside in the equations? • Upgradient momentum fluxes can be parameterized if momentum fluxes are prognosed. This improves global simulations a bit.

3. Unity What is a unified parameterization of subgrid variability? Here’s a restrictive definition: A unified parameterization uses a single equation set to parameterize all aspects of subgrid variability, rather than using separate schemes for separate emergent phenomena.

4. Examples of unified and non-unified models Examples of unified models: • Navier-Stokes equations, • a large-eddy simulation (LES) model, • a (unclosed) Reynolds-Averaged Navier-Stokes (RANS) model. Examples of non-unified models: • A subgrid cloud parameterization plus a gravity wave scheme. • Any model with if-then statements in the code.

5. Fidelity (traceability) By fidelity, I mean fidelity of the parameterized equation set to the governing equation set (not observations). One aspect of this: I prefer that a parameterization has the same structure (or “framework”) as the governing equation set. E.g., Reynolds-Averaged Navier-Stokes (RANS) inherits the structure of the Navier-Stokes equations. Similarity in structure helps indicate whether an important process has been inadvertently omitted. It provides traceability.

6. Achieving both unity and fidelity is difficult It means that every large term in every equation must be operating at all times, even if a term is large only in special circumstances. It is tempting to shut off terms and trigger them only when needed, but then the parameterization is no longer unified.

7. Simplicity By simplicity, I mean simplicity of the parameterized equation set(s). Which is simpler: 1) a unified parameterization or 2) a non-unified package of parameterizations? A unified scheme must be complex enough to model atmospheric physics generally. A package with separate schemes for separate processes places complexity in interactions.

8. Outline of talk • Unity, fidelity, and simplicity are three desiderata of parameterizations. • A particular parameterization challenge: the physics of momentum fluxes in a shallow cumulus case. • Some momentum fluxes are upgradient. Where does this upgradient physics reside in the equations? • Upgradient momentum fluxes can be parameterized if momentum fluxes are prognosed. This improves global simulations a bit.

9. Something is different about momentum fluxes . . . Water flux budget For water flux, the ← mass-flux part matches the ← total flux. Momentum flux budget For momentum, environmental --> contribution is large. x10-3 x10-3 m2/s2 Zhu (2015). See also Schlemmer et al. (2017)

10. Steffen Domke’s LES: The BOMEX shallow Cu case has a 3-layer structure of momentum fluxes: SAM-LES Figure created by Steffen Domke; Larson et al. (2019)

11. The cloud is brought toward the environmental wind at all altitudes, but the flux is upgradient in the middle layer because of overshooting SAM-LES Note: BOMEX has no mesoscale organization!

12. The existence of upgradient momentum fluxes in the BOMEX case appears to depend on two ingredients: • a mechanism, such as convective plumes, to transport momentum vertically with slow adjustment to the environmental wind, and • the presence of a jet within the layer of vertical transport.

13. What is the best way to parameterize an upgradient flux?? A simple approach, like downgradient diffusion, might need to be modified. What physics needs to be added? Can in a unified way while maintaining fidelity (i.e., traceability to N-S equations)?

14. In the upper layer, the momentum flux is weak, and yet the clouds are not merely drifting with the mean wind SAM LES

15. The momentum flux is weak aloft partly because it has opposite signs within cloud and outside cloud: ← The environmental momentum flux is upgradient!

16. The presence of gravity waves is suggested by the fact that the environmental heat flux is small: ← The environmental heat flux is small.

17. What is the simplest way to parameterize a mixture of cumuli and gravity waves?? A non-unified approach might need to carefully interface a cumulus scheme and a gravity wave scheme. A unified approach must treat both simultaneously with the same equation set.

18. The CAM6 climate model parameterizes boundary-layer momentum fluxes using simple down-gradient diffusion: In CAM6’s atmosphere, the eddy diffusivity is active everywhere, including shallow convection.

19. To ensure that the momentum flux is downgradient everywhere, eddy diffusion distorts the wind profile all the way down to the surface BOMEX Cu Excessive surface wind will cause excessive wind stress on the ocean. <u> <u’w’> eddy diffusion ---> ← LES Jet max --> u’ w’ u’w’>0

20. Outline of talk • Unity, fidelity, and simplicity are three desiderata of parameterizations. • A particular parameterization challenge: the physics of momentum fluxes in a shallow cumulus case. • Some momentum fluxes are upgradient. Where does this upgradient physics reside in the equations? • Upgradient momentum fluxes can be parameterized if momentum fluxes are prognosed. This improves global simulations a bit.

21. Where does the physics of upgradient transport reside? pressure pushes parcel back to environment wind value, u’=0, with time scale tau turb transport is a flux of flux: w’2u’ = w’ (u’w’) z u

22. . . . in the buoyancy production and turbulent transport (flux-of-flux) terms: The turbulent production term leads to downgradient diffusion, with diffusivity K = ( τ/C4 )〈w’2〉. To see this, drop the time tendency term and re-arrange:

23. The buoyancy production and turbulent transport (flux-of-flux) terms are large in the BOMEX case: An eddy diffusivity approximation would retain only the brown and purple lines. SAM-LES <u’w’> budget turb transport buoyancy turb prod pressure

24. A major choice facing our field is how to handle the buoyancy and turbulent transport terms Once eddy diffusivity is in place, then the other two terms can be modeled with either a mass-flux approach or a higher-order closure approach. But a mass-flux scheme would probably model part of the eddy diffusivity (turbulent production) term; we would lose traceability.

25. Outline of talk • Unity, fidelity, and simplicity are three desiderata of parameterizations. • A particular parameterization challenge: the physics of momentum fluxes in a shallow cumulus case. • Some momentum fluxes are upgradient. Where does this upgradient physics reside in the equations? • Upgradient momentum fluxes can be parameterized if momentum fluxes are prognosed. This improves global simulations a bit.

26. We have implemented prognostic momentum fluxes in the Cloud Layers Unified By Binormals (CLUBB) parameterization. (CLUBB is a higher-order closure model of turbulence and clouds.) CLUBB strives to model turbulence and clouds in a unified way. To close the buoyancy and turbulent advection terms, it assumes that the subgrid PDF is a double Gaussian. It is implemented in the CAM6 and E3SMv1 climate models.

27. The extra cost of prognosing momentum fluxes is small . . . . . . because CLUBB already prognoses the scalar fluxes, and the LU-decomposition from that calculation can be re-used. = u’w’

28. Prognosing momentum fluxes is capable of producing a region of upgradient flux in the BOMEX shallow cumulus case: CLUBB single-column simulation <u> <u’w’>

29. The meridional wind is also improved . . . <v’w’> <v>

30. CLUBB can produce weak values of <u’w’> aloft because it can produce positive values of the buoyancy term, g〈u’ θv’〉 SAM-LES <u’w’> budget CLUBB <u’w’> budget turb transport turb transport buoyancy turb prod turb prod pressure pressure

31. Does prognosing momentum fluxes improve global simulations? We’ll show some 5-year, 2° CAM-CLUBB-SILHS simulations. The simulations have prescribed SST, and the Zhang-McFarlane deep convective scheme is shut off.

32. Prognosing momentum fluxes can improve surface stress and sea-level pressure: ed dfsn Sea-level pressure ed dfsn Surface stress prognostic momentum prognostic momentum

33. Parameterization is hard! 2. Momentum fluxes can be upgradient if there is non-local transport in the presence of a jet. 3. Momentum fluxes can be prognosed in CLUBB at little additional cost. Conclusions

34. Thanks for your time!