1. An Assimilating Tidal Model for the Bering Sea Mike Foreman, Josef Cherniawsky, Patrick Cummins Institute of Ocean Sciences, Sidney BC, Canada Outline: Background Tidal model & inverse Energy fluxes and dissipation Energy budget & mass conservation Summary

2. Background l complex tidal elevations & flows in the Bering Sea  Large elevation ranges in Bristol Bay  Large currents in the Aleutian Passes  both diurnal & semi-diurnal amphidromes  Large energy dissipation (Egbert & Ray, 2000)  Seasonal ice cover  Internal tide generation from Aleutian channels (Cummins et al., 2001)  Relatively large diurnal currents that will have 18.6 year modulations l Difficult to get everything right with conventional model l Need to incorporate observations  data assimilation

4. l Barotropic finite element method FUNDY5SP (Greenberg, Lynch) :  linear basis functions, triangular elements  e -i  t time dependency,  = constituent frequency  solutions ( ,u,v) have form Ae ig l FUNDY5SP adjoint model  development parallels Egbert & Erofeeva (2002), Foreman et al. (2004)  representers: Bennett (1992, 2002) The Numerical Techniques

5. Grid & Forcing l 29,645 nodes, 56,468 triangles l variable resolution: 50km to less than 1.5km l Tidal elevation boundary conditions from TP crossover analysis l Tidal potential, earth tide, SAL

6. Tidal Observations from 300 cycle harmonic analysis at TP crossover sites (Cherniawsky et al., 2001)

7. l de-couple forward/adjoint equations by calculating representers l Representers = basis functions (error covariances or squares of Green’s functions) that span the “data space” as opposed to “state space”  one representer associated with each observation l optimal solution is sum of prior model solution and linear combination of representers l Adjoint wave equation matrix is conjugate transpose of the forward wave equation matrix l covariance matrices assume 200km de-correlation scale Assimilation Details

8. Elevation Amplitude & Major Semi-axis of a sample M 2 Representer (amplitude normalized to 1 cm)   these fields are used to correct initial model calculation

9. Model Accuracy (cm): average D at 288 T/P crossover sites

10. Corrected Elevation Amplitudes

11. M 2 vertically-integrated energy flux (each full shaft in multi-shafted vector represents 100KW/m)

12. K 1 vertically-integrated energy flux (each full shaft in multi-shafted vector represents 100KW/m)

13. Energy Flux Through the Aleutian Passes

14. Energy Flux Through the Aleutian Passes & Bering Strait (Vertically integrated tidal power (GW) normal to transects)

15. M 2 Dissipation from Bottom Friction (W/m 2 )  Mostly in Aleutian Passes & shallow regions like Bristol Bay  Bering Sea accounts for about 1% of global total of 2500GW

16. K 1 Dissipation from Bottom Friction (W/m 2 )  K 1 dissipation accounts for about 7% of global total of 343GW  Mostly in Aleutian Passes, along shelf break, & in shallow regions Strong dissipation off Cape Navarin as shelf waves must turn corner enhances mixing and nutrient supply significant 18.6 year variations

17. Ratio of average tidal bottom friction dissipation: April 2006 vs April 1997.

18. Energy Budget & Mass Conservation l l Energy budget can be derived by taking dot product of with discrete version of 3D momentum equation (neglecting tidal potential, earth tide, SAL) where are bottom & vertically-integrated velocity, k is bottom friction, H is depth, ρ is density, g is gravity, f is Coriolis, η is surface elevation.

19. Energy Budget & Mass Conservation l l Re-expressing gradient term gives l l Customary to use continuity to replace 1 st term on rhs

20. Energy Budget & Mass Conservation l l But finite element methods like QUODDY, FUNDY5, TIDE3D, ADCIRC don’t conserve mass locally.  need to include a residual term l l Making this substitution & taking time averages eliminates the time derivatives l l Finally, taking spatial integrals & using Gauss’s Theorem where is unit vector normal to boundary

21. Energy Budget & Mass Conservation l l We get the energy budget which has an additional term due to a lack of local mass conservation

22. Energy Budget & Mass Conservation l l Spurious r c term can be significant

23. Energy Budget & Mass Conservation l l With original FUNDY5SP solution for M 2, energy associated with r c is 23% of bottom friction dissipation l l assimilation of TOPEX/Poseidon harmonics can reduce this contribution to 9% l l But it can never be eliminated unless mass is conserved locally

24. Summary many interesting physical & numerical problems associated with tides in the Bering Sea Adjoint has been developed for FUNDY5SP & applied to Bering Sea tides representer approach is instructive way to solve the inverse problem

25. Summary (cont’d): If mass is not conserved locally, there will be a spurious term in the energy budget  It will disrupt what should be a balance between incoming flux & dissipation  The imbalance can be significant Yet another reason that irregular-grid methods should conserve mass locally More details in Foreman et al., Journal of Marine Research, Nov 2006