Prediction of Ocean Circulation in the Prediction of Ocean Circulation in the Gulf of Mexico and Caribbean SeaGulf of Mexico and Caribbean Sea

An application of the ROMS/TOMS An application of the ROMS/TOMS Data Assimilation ModelsData Assimilation Models

Hernan G. Arango (IMCS, Rutgers University)Hernan G. Arango (IMCS, Rutgers University)

Emanuele Di Lorenzo (Georgia Institute of Technology)Emanuele Di Lorenzo (Georgia Institute of Technology)

Arthur J. Miller, Bruce D. CornuelleArthur J. Miller, Bruce D. Cornuelle (Scripps Institute of Oceanography, UCSD)(Scripps Institute of Oceanography, UCSD)

Andrew M. Moore (PAOS, Colorado University)Andrew M. Moore (PAOS, Colorado University)

Gulf of Mexico and Caribbean Seas

plus satellite data (SSH, SST) and radar

Ocean Observations

OCEAN INIT IALIZE

FINALIZE

RUN

S4DVAR_OCEAN

IS4DVAR_OCEAN

W4DVAR_OCEAN

ENSEMBLE_OCEAN

NL_OCEAN

TL_OCEAN

AD_OCEAN

PROPAGATOR

KERNELNLM, TLM, RPM, ADM

physicsbiogeochemicalsedimentsea ice

Optimal pertubations

ADM eigenmodes

TLM eigenmodes

Forcing singular vectors

Stochastic optimals

Pseudospectra

ADSEN_OCEAN

SANITY CHECK S

PERT_OCEAN

PICARD_OCEAN

GRAD_OCEAN

TLCHECK _OCEAN

RP_OCEAN

ESMF

AIR_OCEAN

MASTER

ROMS/TOMS

cean M odel

earch C o mm

Ocean Modeling Framework

Gulf of Mexico Ocean Model Grid

Ocean Modeling of North Atlantic

Ocean Model Surface Currents and Sea LevelOcean Model Surface Currents and Sea Level

Ocean Modeling Applications in

• Develop a real-time Develop a real-time data assimilationdata assimilation and and prediction prediction systemsystem for the Gulf of Mexico and Caribbean Seas for the Gulf of Mexico and Caribbean Seas based on a continuous upper ocean monitoring systembased on a continuous upper ocean monitoring system

• Demonstrate the utility of variational data assimilation Demonstrate the utility of variational data assimilation in a real-time, sea-going environmentin a real-time, sea-going environment

• Demonstrate the value of collecting routine ocean Demonstrate the value of collecting routine ocean observations from specially equipped ocean vessels observations from specially equipped ocean vessels ((Explorer of the SeasExplorer of the Seas))

• Develop much needed experience in both the Develop much needed experience in both the assimilation of disparate ocean data and ocean assimilation of disparate ocean data and ocean prediction in regional ocean models.prediction in regional ocean models.

• Add platform oceanic measurements (a possibility)Add platform oceanic measurements (a possibility)

Gulf of Mexico and Caribbean Seas

Ensemble PredictionEnsemble Prediction

t

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HighSpread

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Spread

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time

For an appropriate forecast skill measure, For an appropriate forecast skill measure, ss

Time=tN

Kelvin Wave Pattern

Maximum transport

Time=t0SSH SSHTime=tN

Kelvin Wave Pattern

Maximum transport

Time=t0SSH SSH

Example from the Caribbean model run, of sensitivity of the transport through the Yucatan Strait given a particular realization of the circulation. In this case the maximum transport at time tN, indicated by the strong

gradients in sea surface height (SSH), is sensitive to a pattern of Kelvin waves at previous time t0. These types of sensitivity, computed using the

non-linear and Adjoint models of ROMS, will be applied for the Florida Strait to explore how different topographic shapes affect the transport during different circulation regimes.

Ocean Adjoint Modeling Applications

4D Variational Data Assimilation Platforms 4D Variational Data Assimilation Platforms (4DVAR)(4DVAR)

• Strong Constraint (S4DVAR) drivers:Strong Constraint (S4DVAR) drivers: Conventional S4DVAR: outer loop, Conventional S4DVAR: outer loop, NLMNLM, , ADMADM Incremental S4DVAR: inner and outer loops, Incremental S4DVAR: inner and outer loops, NLMNLM, ,

TLMTLM, , ADMADM (Courtier et al., 1994) (Courtier et al., 1994) Efficient Incremental S4DVAR (Weaver et al., 2003)Efficient Incremental S4DVAR (Weaver et al., 2003)

• Weak Constraint (W4DVAR) - IOMWeak Constraint (W4DVAR) - IOM Indirect Representer Method: inner and outer loops, Indirect Representer Method: inner and outer loops,

NLMNLM, , TLMTLM, , RPMRPM, , ADM ADM (Egbert et al., 1994; Bennett (Egbert et al., 1994; Bennett et al, 1997)et al, 1997)

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misfit variance reduced 62%

1st guessIOM solution

2000 4000 60000 8000

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GUESS

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Free Surfac e Surface NS Veloc ity SST

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(Di Lorenzo et al., 2005)

NormalizedMisfit

Datum

Assimilated data:TS 0-500m Free surface Currents 0-150m

Strong Constraint

1st Guess

True Synthetic Data

SST

SST SST

SST

TS

V U

Weak Constraint

Strong and Weak Constraint 4DVAR(Southern California Bight)

0-500 mdata

CalCOFISampling

grid

AnnualClimatology

• Given the model state vector:Given the model state vector:

• Consider a Yucatan Strait transport index, , Consider a Yucatan Strait transport index, , defined in terms of space and/or time integrals of defined in terms of space and/or time integrals of : :

• Small changes in will lead to changes in Small changes in will lead to changes in where: where:

• We will define sensitivity as etc.We will define sensitivity as etc.

J

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Adjoint Sensitivity

+ …

PublicationsPublications

Arango, H.G., Moore, A.M., E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2003: The ROMS Tangent Linear and Arango, H.G., Moore, A.M., E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2003: The ROMS Tangent Linear and

Adjoint Models: A comprehensive ocean prediction and analysis system, Adjoint Models: A comprehensive ocean prediction and analysis system, Rutgers Tech. ReportRutgers Tech. Report..

http://marine.rutgers.edu/po/Papers/roms_adjoint.pdfhttp://marine.rutgers.edu/po/Papers/roms_adjoint.pdf

Di Lorenzo, E., A.M. Moore, H.G. Arango, B. Chua, B.D. Cornuelle, A.J. Miller and A. Bennett, 2005: The Inverse Regional Di Lorenzo, E., A.M. Moore, H.G. Arango, B. Chua, B.D. Cornuelle, A.J. Miller and A. Bennett, 2005: The Inverse Regional

Ocean Modeling System: Development and Application to Data Assimilation of Coastal Mesoscale Eddies, Ocean Modeling System: Development and Application to Data Assimilation of Coastal Mesoscale Eddies, Ocean Ocean

ModellingModelling, In preparation., In preparation.

Moore, A.M., H.G Arango, E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2004: A comprehensive ocean prediction Moore, A.M., H.G Arango, E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2004: A comprehensive ocean prediction

and analysis system based on the tangent linear and adjoint of a regional ocean model, and analysis system based on the tangent linear and adjoint of a regional ocean model, Ocean Modelling,Ocean Modelling, 7, 227-258. 7, 227-258.

http://marine.rutgers.edu/po/Papers/Moore_2004_om.pdfhttp://marine.rutgers.edu/po/Papers/Moore_2004_om.pdf

Moore, A.M., E. Di Lorenzo, H.G. Arango, C.V. Lewis, T.M. Powell, A.J. Miller and B.D. Cornuelle, 2005: An Adjoint Moore, A.M., E. Di Lorenzo, H.G. Arango, C.V. Lewis, T.M. Powell, A.J. Miller and B.D. Cornuelle, 2005: An Adjoint

Sensitivity Analysis of the Southern California Current Circulation and Ecosystem, Sensitivity Analysis of the Southern California Current Circulation and Ecosystem, J. Phys.J. Phys. Oceanogr.Oceanogr., In preparation., In preparation.

Wilkin, J.L., H.G. Arango, D.B. Haidvogel, C.S. Lichtenwalner, S.M.Durski, and K.S. Hedstrom, 2005: A Regional Modeling Wilkin, J.L., H.G. Arango, D.B. Haidvogel, C.S. Lichtenwalner, S.M.Durski, and K.S. Hedstrom, 2005: A Regional Modeling

System for the Long-term Ecosystem Observatory, System for the Long-term Ecosystem Observatory, J. Geophys. Res.J. Geophys. Res., 110, C06S91, doi:10.1029/2003JCC002218., 110, C06S91, doi:10.1029/2003JCC002218.

http://marine.rutgers.edu/po/Papers/Wilkin_2005_jgr.pdfhttp://marine.rutgers.edu/po/Papers/Wilkin_2005_jgr.pdf

Warner, J.C., C.R. Sherwood, H.G. Arango, and R.P. Signell, 2005: Performance of Four Turbulence Closure Methods Warner, J.C., C.R. Sherwood, H.G. Arango, and R.P. Signell, 2005: Performance of Four Turbulence Closure Methods

Implemented Using a Generic Length Scale Method, Implemented Using a Generic Length Scale Method, Ocean ModellingOcean Modelling, 8, 81-113., 8, 81-113.

http://marine.rutgers.edu/po/Papers/Warner_2004_om.pdfhttp://marine.rutgers.edu/po/Papers/Warner_2004_om.pdf

Background Material

OverviewOverview•Let’s represent Let’s represent NLMNLM ROMS as: ROMS as:

•The The TLMTLM ROMS is derived by considering a small perturbation ROMS is derived by considering a small perturbation ss to to SS. A first-order Taylor expansion yields:. A first-order Taylor expansion yields:

A is real, non-symmetricA is real, non-symmetric Propagator MatrixPropagator Matrix

•The The ADMADM ROMS is derived by taking the inner-product with an ROMS is derived by taking the inner-product with an

arbitrary vector , where the inner-product defines an arbitrary vector , where the inner-product defines an

appropriate norm (L2-norm):appropriate norm (L2-norm):

Tangent Linear and Adjoint Based GST Tangent Linear and Adjoint Based GST DriversDrivers

• Singular vectors:Singular vectors:

• Forcing Singular vectors:Forcing Singular vectors:

• Stochastic optimals:Stochastic optimals:

• Pseudospectra:Pseudospectra: 1HI A I A

( ,0) (0, )TR t XR t

andand• Eigenmodes ofEigenmodes of (0, )R t ( ,0)TR t

0 0

( , ) ( , )

T

R t dt X R t dt

| '|/ '

0 0

( , ) ( , ) 'ct t t Te R t XR t dt dt

Two InterpretationsTwo Interpretations

• Dynamics/sensitivity/stability of flow to Dynamics/sensitivity/stability of flow to

naturally occurring perturbationsnaturally occurring perturbations

• Dynamics/sensitivity/stability due to Dynamics/sensitivity/stability due to error error

or uncertainties in the forecast systemor uncertainties in the forecast system

• Practical applications:Practical applications:

Ensemble predictionEnsemble prediction

Adaptive observationsAdaptive observations

Array design ...Array design ...

GSA on the Southern California Bight (SCB)GSA on the Southern California Bight (SCB)

Free-SurfaceSST and Surfacecurrents

EigenmodesEigenmodes

SCB coastally trapped wavesSCB coastally trapped waves

• TLM TLM eigenvectors ( eigenvectors (AA): normal modes): normal modes• ADMADM eigenvectors ( eigenvectors (AATT): optimal excitations ): optimal excitations

Real Part Imag Part

diffluencediffluence

Optimal PerturbationsOptimal Perturbations

• A measurement of the fastest growing of all possible A measurement of the fastest growing of all possible perturbations over a given time intervalperturbations over a given time interval

SCB maximum growth of perturbation energy over 5 daysSCB maximum growth of perturbation energy over 5 days

confluenceconfluence

Stochastic OptimalsStochastic OptimalsProvide information about the influence of stochasticvariations (biases) in ocean forcing

SCB patterns of stochastic forcing that maximizes theperturbation energy variance for 5 days

Open Boundary Sensitivity: Open Boundary Sensitivity: errorserrors growth quickly and appear to growth quickly and appear to propagate through the model domain as coastally trapped waves.propagate through the model domain as coastally trapped waves.

Singular VectorsSingular Vectors

Ensemble PredictionEnsemble Prediction

• Optimal perturbations / singular vectors and Optimal perturbations / singular vectors and stochastic optimal can also be used to generate stochastic optimal can also be used to generate ensemble forecasts.ensemble forecasts.

• Perturbing the system along the most unstable Perturbing the system along the most unstable directions of the state space yields information directions of the state space yields information about the about the firstfirst and and secondsecond moments of the moments of the probability density function (PDF):probability density function (PDF):

ensemble meanensemble mean

ensemble spreadensemble spread

• Adjoint based perturbations excite the full spectrumAdjoint based perturbations excite the full spectrum

Data Assimilation OverviewData Assimilation Overview

•Cost Function:Cost Function:

wherewhere model,model, background,background, observations,observations,

inverse background error covariance,background error covariance,

inverse observations error covarianceinverse observations error covariance

•Model solution depends on initial conditions ( ), Model solution depends on initial conditions ( ), boundary conditions, and model parametersboundary conditions, and model parameters

•Minimize JMinimize J to produce a best fit between model and to produce a best fit between model and observations by adjusting initial conditions, and/or observations by adjusting initial conditions, and/or boundary conditions, and/or model parameters.boundary conditions, and/or model parameters.

MinimizationMinimization

• Perfect model constrained minimization (Lagrange Perfect model constrained minimization (Lagrange function):function):

We require the minimum of at which:We require the minimum of at which:

, , ,, , ,

yieldingyielding

• AATT is the transpose of is the transpose of AA, often called the adjoint , often called the adjoint operator. It can be shown that: operator. It can be shown that:

The adjoint equation solutionThe adjoint equation solutionprovides gradient informationprovides gradient information

4D Variational Data Assimilation Platforms 4D Variational Data Assimilation Platforms (4DVAR)(4DVAR)

• Strong Constraint (S4DVAR) drivers:Strong Constraint (S4DVAR) drivers: Conventional S4DVAR: outer loop, Conventional S4DVAR: outer loop, NLMNLM, , ADMADM Incremental S4DVAR: inner and outer loops, Incremental S4DVAR: inner and outer loops, NLMNLM, ,

TLMTLM, , ADMADM (Courtier et al., 1994) (Courtier et al., 1994) Efficient Incremental S4DVAR (Weaver et al., 2003)Efficient Incremental S4DVAR (Weaver et al., 2003)

• Weak Constraint (W4DVAR) - IOMWeak Constraint (W4DVAR) - IOM Indirect Representer Method: inner and outer loops, Indirect Representer Method: inner and outer loops,

NLMNLM, , TLMTLM, , RPMRPM, , ADM ADM (Egbert et al., 1994; Bennett (Egbert et al., 1994; Bennett et al, 1997)et al, 1997)

RP:RP:

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