An Assimilating Tidal Model for the Bering Sea Mike Foreman, Josef Cherniawsky, Patrick Cummins Institute of Ocean Sciences, Sidney BC, Canada

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An Assimilating Tidal Model for the Bering Sea
Mike Foreman, Josef Cherniawsky, Patrick
CumminsInstitute of Ocean Sciences, Sidney BC,
Canada

• Outline
• Background
• Tidal model inverse
• Energy fluxes and dissipation
• Energy 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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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)

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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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Assimilation Details

• de-couple forward/adjoint equations by
  calculating representers
• Representers basis functions (error covariances
  or squares of Greens 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

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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
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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, ? is density, g is gravity,
  f is Coriolis, ? is surface elevation.

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Energy Budget Mass Conservation

• Re-expressing gradient term
• gives
• Customary to use continuity to replace 1st term
  on rhs

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Energy Budget Mass Conservation

• But finite element methods like QUODDY, FUNDY5,
  TIDE3D, ADCIRC dont conserve mass locally.
• need to include a residual term
• Making this substitution taking time averages
• eliminates the time derivatives
• Finally, taking spatial integrals using Gausss
  Theorem
• where is unit vector normal to
  boundary

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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

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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 (contd)
• 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