Uncertainty in Ocean General Circulation Model Mixing Tensors

Transcription

1 Uncertainty in Ocean General Circulation Model Mixing Tensors Baylor Fox-Kemper, University of Colorado Boulder, Brown U. (>Jan. 2013) Cooperative Insititute for Research in Environmental Sciences and Dept. of Atmospheric and Oceanic Sciences with Frank Bryan (NCAR), John Dennis (NCAR), Scott Bachman (CIRES/ATOC), Jim McWilliams (UCLA), NCAR Oceanography Section Bayesian Confab 13:30 -??, Aug. 9, 2012; Boulder, Colorado Sponsors: NSF ; NASA NNX09AF38G TeraGRID, IBM, NCAR CISL

2 Mesoscale Parameterizations Researchers have already cast much darkness on this subject and if they continue their investigations we shall soon know nothing at all about it. --Mark Twain

3 Ocean Equations*: Boussinesq Fluid on Tangent Plane to a Rotating Sphere Buoyancy (or S, T): Re, Pe for an affordable gridscale are 10 6 to Numerics require O(1) *From Grooms, Julien, & F-K, 11

4 What is a parameterization? Express the coarse-grain averages of quantities (including the subgrid effects), e.g.: As a function of the resolved coarse-grain fields Note that nonlinear terms require special treatment And Couple different scales, small talks to large!

5 The Character of the Mesoscale 100 km (NASA GSFC Gallery) (Capet et al., 2008) Boundary Currents Eddies Ro=O(0.1) Ri=O(1000) Full Depth Eddies strain to produce Fronts 100km, months Eddy processes mainly baroclinic & barotropic instability. Parameterizations of baroclinic instability (GM, Visbeck...). Surface Temp. 200m Temp. Temp x-z Section

6 Mesoscale Eddy Parameterizations all have the form: u τ = M τ u τ v τ = M xx M xy M xz M yx M yy M yz τ x τ y w τ M zx M zy M zz τ z Count Degrees of Freedom: 3 tracer flux components 3 tracer gradient components 9 tensor elements!

7 Does this cover all the degrees of freedom? More tracers does provide a just-determined or overdetermined (Moore-Penrose/least squares) problem for M with a unique answer, but... Different tracers will have different fluxes as they feel the subgrid nooks and crannies of the mesoscale eddies!

8 Mesoscale Eddy Parameterizations all have the form: u τ = M τ u τ v τ = M xx M xy M xz M yx M yy M yz τ x τ y w τ M zx M zy M zz τ z With John Dennis & Frank Bryan, we took a POP0.1 Normal-Year forced model (yrs 16-20) Added 9 Passive tracers--restored 3 rates Kept all the eddy fluxes for passive & active Coarse-grained to 2, transient eddies, tracers to M

9 u τ = M τ Sym Part=Anisotropic* Redi u τ v τ w τ = K xx K xy ˆx K z K yx K yy ŷ K K z ˆx K z ŷ K K z z K K z K K τ x τ y AntiSym Part=Anisotropic* GM τ z u τ v τ = 0 0 ˆx K K z 0 0 ŷ K K z τ x τ y w τ ˆx K K z ŷ K K z 0 τ z Yellow K are horizontal stirring & mixing

10 Underdetermined uτ 1 ū τ 1 vτ 1 v τ 1 = M xx M xy M xz M yx M yy M yz τ 1,x τ 1,y wτ 1 w τ 1 M zx M zy M zz τ 1,z uτ 1 ū τ 1 uτ 2 ū τ 2 uτ 3 ū τ 3 vτ 1 v τ 1 vτ 2 v τ 2 vτ 3 v τ 3 wτ 1 w τ 1 wτ 2 w τ 2 wτ 3 w τ 3 Just-determined = M xx M xy M xz M yx M yy M yz M zx M zy M zz τ 1,x τ 2,x τ 3,x τ 1,y τ 2,y τ 3,y τ 1,z τ 2,z τ 3,z uτ 1 ū τ 1 uτ N ū τ N vτ 1 v τ 1... vτ N v τ N wτ 1 w τ 1 wτ N w τ N Overdetermined = M xx M xy M xz M yx M yy M yz M zx M zy M zz τ 1,x τ N,x τ 1,y... τ N,y τ 1,z τ N,z

11 Is Lagrangian Transport Unique? Taylor 1921/1953 says yes, in a decorrelation timescale sense If non-conservative tracers, then no Nearly conservative tracers--probably something that becomes identical in limit Active vs. passive tracers?

12 What does mean mean? Temporal averaging -> 8yr average by season or overall Must preserve covariances... 20x increase in variables Spatial coarse-graining 10km->200km 20x20 gridpoints per coarse-gridpoint 400x reduction of variables Collect slow,coarse; slow,fine; fast,coarse; fast,fine

13 K M Could you have guessed it?

14 Result: Strong Anisotropy Along/Across Isopycnals Mixing direction Mixing: Stirring:

15 Are Diffusivity Values Resonable? u τ v τ = K xx K xy ˆx K z K K yx K yy ŷ K K z τ x τ y w τ ˆx K z ŷ K K z z K K z K τ z Trace(M) Histogram Hor. Diffusivity is roughly Trace(M)/2 Peak of Diffusivity near 250 m 2 /s Median Diffusivity near 1000m 2 /s <6% negative

16 But, how well does it work? Suppose we only plot values where different tracer sets agree... Not so many trustworthy values! Can t reject params!

17 dτ dt = λ ( τ τ ) 0 v b rec = M b!"#$#!!%&!'()*&'+,-!!.//!0!123!!4//!0!123!!5//!0!123!!.///!0!123! time (days) In idealized runs, can see the effect of restoring. Whatever we do, we need to get buoyancy right!

18 In idealized setting, can do better Reconstruction of eddy buoyancy fluxes Original fluxes Reconstructed fluxes v'b' w'b' Using specially-tailored non-restored tracers improves estimate (error is now < 10%) but not feasible in realistic diagnosis. In realistic diagnosis, we can improve the estimate a bit by approximating restoring effect

19 Uncertainty... How many tracers needed? Distribution?

20 Underdetermined uτ 1 ū τ 1 vτ 1 v τ 1 = M xx M xy M xz M yx M yy M yz τ 1,x τ 1,y wτ 1 w τ 1 M zx M zy M zz τ 1,z uτ 1 ū τ 1 uτ 2 ū τ 2 uτ 3 ū τ 3 vτ 1 v τ 1 vτ 2 v τ 2 vτ 3 v τ 3 wτ 1 w τ 1 wτ 2 w τ 2 wτ 3 w τ 3 Just-determined = M xx M xy M xz M yx M yy M yz M zx M zy M zz τ 1,x τ 2,x τ 3,x τ 1,y τ 2,y τ 3,y τ 1,z τ 2,z τ 3,z uτ 1 ū τ 1 uτ N ū τ N vτ 1 v τ 1... vτ N v τ N wτ 1 w τ 1 wτ N w τ N Overdetermined = M xx M xy M xz M yx M yy M yz M zx M zy M zz τ 1,x τ N,x τ 1,y... τ N,y τ 1,z τ N,z

21 Uncertainty... Coarse-grain? What is the mean & eddy? Flierl & McWilliams 77: ~30 yrs of Eulerian obs. needed for variances & co-variances Single snapshots are huge at Global 10km (> gridpoints, 5 state variables, 9 tensor elements, 3N fluxes, N tracers, N variances, etc) Spatial coarse-graining desirable to improve averaging and condense dataset

22 Underdetermined uτ 1 ū τ 1 vτ 1 v τ 1 = M xx M xy M xz M yx M yy M yz τ 1,x τ 1,y wτ 1 w τ 1 M zx M zy M zz τ 1,z uτ 1 ū τ 1 uτ 2 ū τ 2 uτ 3 ū τ 3 vτ 1 v τ 1 vτ 2 v τ 2 vτ 3 v τ 3 wτ 1 w τ 1 wτ 2 w τ 2 wτ 3 w τ 3 Just-determined = M xx M xy M xz M yx M yy M yz M zx M zy M zz τ 1,x τ 2,x τ 3,x τ 1,y τ 2,y τ 3,y τ 1,z τ 2,z τ 3,z uτ 1 ū τ 1 uτ N ū τ N vτ 1 v τ 1... vτ N v τ N wτ 1 w τ 1 wτ N w τ N Overdetermined = M xx M xy M xz M yx M yy M yz M zx M zy M zz τ 1,x τ N,x τ 1,y... τ N,y τ 1,z τ N,z Could INCLUDE AVERAGING in linear model: Then get distribution pre-coarse graining & time avg

23 Conclusions: None Prospects: Can we make a better measure of the uncertainty profile? Can we use the distribution in stochastic, rather than deterministic, parameterizations? Can we detect times when the flux-gradient relationship itself fails? Can we specify better averaging kernels? uτ u τ = M τ +

24 Result: Strong Anisotropy Along/Across PV Grads. Mixing direction Either along PV contours or across cosine between 1rst eigenvector and PV gradient x 104 cosine between 2nd eigenvector and PV gradient 3 2nd Eigenvector Across PV contours rst Eigenvector