Outline: Background Tidal model & inverse Energy fluxes and dissipation
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Transcript of Outline: Background Tidal model & inverse Energy fluxes and dissipation
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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
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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
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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
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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
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Tidal Observations from 300 cycle harmonic analysis at TP crossover
sites (Cherniawsky et al., 2001)
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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
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Elevation Amplitude & Major Semi-axis
of a sample M2 Representer
(amplitude normalized to 1 cm)
these fields are used to correct initial model calculation
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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
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Corrected Elevation
Amplitudes
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M2 vertically-integrated energy flux(each full shaft in multi-shafted vector represents 100KW/m)
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K1 vertically-integrated energy flux(each full shaft in multi-shafted vector represents 100KW/m)
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Energy Flux Through the Aleutian Passes
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Energy Flux Through the Aleutian Passes & Bering Strait
(Vertically integrated tidal power (GW) normal to transects)
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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
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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
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Ratio of average tidal bottom friction dissipation: April 2006 vs April 1997.
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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
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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
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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
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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
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Energy Budget & Mass Conservation
Spurious rc term can be significant
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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
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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
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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