Outline: Background Tidal model & inverse Energy fluxes and dissipation

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

2. Background • 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 • Difficult to get everything right with conventional model • Need to incorporate observations • data assimilation

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

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

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

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

8. Elevation Amplitude & Major Semi-axis of a sample M2 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. M2 vertically-integrated energy flux (each full shaft in multi-shafted vector represents 100KW/m)

12. K1 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. M2 Dissipation from Bottom Friction (W/m2) • Mostly in Aleutian Passes & shallow regions like Bristol Bay • Bering Sea accounts for about 1% of global total of 2500GW

16. K1 Dissipation from Bottom Friction (W/m2) • K1 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 • 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 • Re-expressing gradient term gives • Customary to use continuity to replace 1st term on rhs

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

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

22. Energy Budget & Mass Conservation • Spurious rcterm can be significant

23. Energy Budget & Mass Conservation • With original FUNDY5SP solution for M2, energy associated with rc is 23% of bottom friction dissipation • assimilation of TOPEX/Poseidon harmonics can reduce this contribution to 9% • 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