An Assimilating Tidal Model for the Bering Sea

Mike Foreman, Josef Cherniawsky, Patrick Cummins

Institute of Ocean Sciences, Sidney BC, Canada

Outline:

BackgroundTidal model & inverse

Energy fluxes and dissipationEnergy budget & mass conservation

Summary

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

Barotropic finite element method FUNDY5SP (Greenberg, Lynch) : linear basis functions, triangular elements e-it 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)

The Numerical Techniques

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

Tidal Observations from 300 cycle harmonic analysis at TP crossover

sites (Cherniawsky et al., 2001)

de-couple forward/adjoint equations by calculating representers

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

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

Assimilation Details

Elevation Amplitude & Major Semi-axis

of a sample M2 Representer

(amplitude normalized to 1 cm)

these fields are used to correct initial model calculation

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

1/ 22 20 0 0 0( cos cos ) ( sin sin )m m m mD A g A g A g A g

Corrected Elevation

Amplitudes

M2 vertically-integrated energy flux(each full shaft in multi-shafted vector represents 100KW/m)

K1 vertically-integrated energy flux(each full shaft in multi-shafted vector represents 100KW/m)

Energy Flux Through the Aleutian Passes

Energy Flux Through the Aleutian Passes & Bering Strait

(Vertically integrated tidal power (GW) normal to transects)

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

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

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

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.

Hu

/ ( / ) 0bu t f u g k H u

,bu u

Energy Budget & Mass Conservation

Re-expressing gradient term

gives

Customary to use continuity to replace 1st term on rhs

( )g H u g Hu Hu

0.5 / bH u u t g Hu g Hu ku u

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 averageseliminates the time derivatives

Finally, taking spatial integrals & using Gauss’s Theorem

where is unit vector normal to boundary

/ ( ) ct Hu r

Hu x y Hu n s

n

Energy Budget & Mass Conservation

We get the energy budget

which has an additional term due to a lack of local mass conservation

c bg H u n s g r ku u x y

Energy Budget & Mass Conservation

Spurious rc term can be significant

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

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

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