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An Eddy Parameterization Challenge Suite: Methods for Diagnosing Diffusivity

An Eddy Parameterization Challenge Suite: Methods for Diagnosing Diffusivity. Scott Bachman With Baylor Fox-Kemper NSF OCE 0825614. Outline. Motivation Math and Extant Parameterizations The models in the suite Results: Eady Conclusion and Future Tasks.

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An Eddy Parameterization Challenge Suite: Methods for Diagnosing Diffusivity

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  1. An Eddy Parameterization Challenge Suite: Methods for Diagnosing Diffusivity Scott Bachman With Baylor Fox-Kemper NSF OCE 0825614

  2. Outline • Motivation • Math and Extant Parameterizations • The models in the suite • Results: Eady • Conclusion and Future Tasks

  3. Figure courtesy of Baylor Fox-Kemper

  4. The evolution of a Parameterization • Figure out that you need a parameterization (Bryan 1969)! • Discover your parameterization isn’t doing well… (Sarmiento 1982) • Improve theory: physical reasoning + mathematical form (Redi, 1982; Gent and McWilliams, 1990; Gent et al., 1995; Griffies, 1998) • Test the theory (Danabasoglu and McWilliams, 1995; Hirst and McDougall, 1996) • Validate against observations / improve theory

  5. 5. Validate against observations / improve theory PROBLEM: What observations? In real life, getting these observations is expensive, technically difficult, time-consuming, and you would need a LOT of them… To complicate matters, it is not even clear how to apply observations to the values we use in a model (Marshall and Shuckburgh, 2006)…

  6. But computers can help… • What if we were to “build” a suite of eddy-resolving models that could tell us how a parameterization should look? (i.e. act as “truth” (McClean et al., 2008) Images courtesy of Julie McClean, SIO

  7. An Eddy Parameterization Challenge Suite • Primitive equation model (MITgcm) • Lots of individual simulations • Explore parameter space • Vary Ro, Ri, Charney-Green, etc. • Lots of shear/strat configurations • Look at stirring tensor elements • Direction of diffusion • Dependence on parameters • Tracer fluxes What does this all mean?

  8. Outline Outline • Motivation • Math and Extant Parameterizations • The models in the suite • Results: Eady • (Early) Results: Exponential

  9. The Tracer Flux-Gradient Relationship Relates the subgridscale eddy fluxes to coarse-grid gradient Is the basis for most modern parameterization methods (GM90, Redi, etc.) GOAL: We want R.

  10. What do we know about the transport tensor R? • R varies in time and space • Ris 3 x 3 in three dimensions (2 x 2 in 2D) • The structure of R should be deterministic • Based on physics or the phenomenology of turbulence • Not stochastic (…)

  11. Lots of questions! • Is such a closure even possible? • Physical assumptions correct? • Boundary layer tapering • Advective and Diffusive components • Rotational fluxes? • How strong is kappa? • Vertical structure? And most importantly… How do these things change in different flow regimes?

  12. think we know What do we know about the transport tensor R? Three things: 1) Redi (1982) 2) Gent and McWilliams (1990) 3) Dukowicz and Smith (1997) and Griffies (1998)

  13. What do we about R? think we know 1) Redi (1982) Diffusion is not geodesic, it is isopycnal Small isopycnal slopes

  14. What do we about R? think we know 2) Gent and McWilliams (1990) QUESTION: What happens if we diffuse density (i.e. use Redi) along an isopycnal? ANSWER: Nothing! Then Redi alone is inadequate… there must be another piece to this.

  15. What do we about R? think we know 2) Gent and McWilliams (1990) Construct a parameterization that diffuses layer thickness - Amounts to an advection by a thickness-weighted velocity

  16. What do we about R? think we know 2) Gent and McWilliams (1990) - Properties of - Nondivergent: - No flow normal to boundaries: - Conserves all domain-averaged moments of density - Conserves all domain-averaged tracer moments - Skew flux (TEM): - Conserves tracer mean, reduces higher moments between isopycnal surfaces

  17. What do we about R? think we know 2) Gent and McWilliams (1990) – GM90 - Properties of (cont.) - Local sink of mean potential energy - Consistent with phenomenology of baroclinic turbulence (mesoscale eddies) - NOT necessarily equal to downgradient PV diffusion along isopycnals

  18. What do we about R? think we know 1) Redi (1982) – diffusion (mixing) 2) GM90 – advection (stirring)

  19. What do we about R? think we know Put Redi + GM90 into the mean tracer equation… (in isopycnal coordinates) GM90 Redi

  20. What do we about R? think we know 3) Dukowicz and Smith (1997) and Griffies (1998) Compare with…. GM90 Redi

  21. What do we about R? think we know 3) Dukowicz and Smith (1997) and Griffies (1998) The thickness diffusivity coefficient is the same as the isopycnal diffusivity coefficient (?!) What do we do with this information?

  22. What do we about R? think we know 3) Dukowicz and Smith (1997) and Griffies (1998) GM90 GM90 skew flux

  23. What do we about R? think we know 3) Dukowicz and Smith (1997) and Griffies (1998) Identity GM90 skew flux

  24. What do we about R? think we know 3) Dukowicz and Smith (1997) and Griffies (1998) Identity GM90 skew flux

  25. What do we about R? think we know 3) Dukowicz and Smith (1997) and Griffies (1998) GM90 Redi (in z-coordinates) This becomes

  26. What do we about R? think we know 3) Dukowicz and Smith (1997) and Griffies (1998)

  27. What do we about R? think we know 3) Dukowicz and Smith (1997) and Griffies (1998) FINALLY!

  28. What do we about R? think we know So the theory says this…. But we need to know that this form is correct!

  29. Outline • Motivation • Math and Extant Parameterizations • The models in the suite • Results: Eady • Conclusion and Future Tasks

  30. Mesoscale Characteristics • 50-100 km (ocean) • Boundary Currents • Eddies • Ro = O(.01) • Ri = O(1000) • QG-scaling OK Courtesy: K. Shafer Smith The eddies extend the full depth of the water column, and are dominated by baroclinic instability.

  31. We are going to look at properties of baroclinic instability alone. Why? http://www.coas.oregonstate.edu/research/po/research/chelton/index.html Baroclinic instabilities dominate the mesoscale. - Barotropic instabilities smaller than a deformation radius (too small). - Symmetric instabilities appear at Richardson numbers < 0.95 (Nakamura, 1993). - Kelvin-Helmholtz instabilities at Ri < 0.25. too low

  32. Construct models focusing only on baroclinic instability A front to make PE available for extraction 900 x 150 x 60 grid Large Richardson number ( anything >> 1 ) > deformation radius = minimal barotropic component

  33. How do we solve for R? What happens if we have only one tracer? 2 Equations… Take a zonal average, and write the system out in full: 4 Unknowns! Underdetermined! (not unique)

  34. How do we solve for R? Use multiple tracers: > 4 Equations… 4 Unknowns! Overdetermined! Moore-Penrose pseudoinverse(least-squares fit) Tracer gradients less aligned = better LS fit! Overdetermining the system is appropriate to reduce degrees of freedom in the zonal average.

  35. How do we know our solution for Ris any good? If Rreally is the same for every tracer, we should be able to reconstruct the flux of a tracer that was not involved in the pseudoinversion. How about buoyancy? Can we produce this with our R ?

  36. If we are careful in initializing our tracers…

  37. The reconstruction is excellent. Original fluxes Reconstructed fluxes Snapshot Snapshot Good! (sinusoids) Bad! (not sinusoids) Estimates of these buoyancy fluxes have improved substantially (error is now < 10%)… Used to be that getting error within a factor of two was the best we could do!

  38. The reconstruction is excellent.

  39. Outline • Motivation • Math and Extant Parameterizations • The models in the suite • Results: Eady • Conclusion and Future Tasks

  40. The Review before the Results We think R looks like this: In 2D (zonal average) So we are going to use a bunch of tracers that will tell us if we are right:

  41. The Results We run 69 simulations, looking for And what do we get?

  42. RAW OUTPUT

  43. To make sense of these results, we need scalings We are interested in finding scalings of the form Our scalings need to be put in terms of quantities that are present in the GCM, but where do we begin? We should start by considering the physics of R.

  44. Previous Work (Fox-Kemper et al., 2008) One could pursue naïve scalings based purely on dimensional analysis: So that: We find that this is inaccurate (see below).

  45. A Better Idea: Scale to the Process Diffusion Advection A diffusive tensor can be written in terms of Lagrangian parcel displacements and velocities: The antisymmetric tensor is responsible for the release of mean potential energy by baroclinic instability, so we can scale according to the release rate

  46. A Better Idea: Scale to the Process Now choose length and time scales to substitute:

  47. A Better Idea: Scale to the Process Then with Dukowicz and Smith (1997), a dimensional base scaling for R should be:

  48. RAW OUTPUT now becomes…

  49. SCALED OUTPUT

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