The ROMS TL and ADJ Models:The ROMS TL and ADJ Models:Tools for Generalized Stability Tools for Generalized Stability Analysis and Data AssimilationAnalysis and Data Assimilation

Hernan Arango, Rutgers UHernan Arango, Rutgers UEmanuele Di Lorenzo, GITEmanuele Di Lorenzo, GIT

Arthur Miller, Bruce Cornuelle, Arthur Miller, Bruce Cornuelle, Doug Neilson UCSD, Doug Neilson UCSD, Andrew Moore, CUAndrew Moore, CU

Major ObjectiveMajor Objective

• To provide the ocean modeling To provide the ocean modeling community with state-of-the-art community with state-of-the-art analysis, prediction and data analysis, prediction and data assimilation tools (currently used in assimilation tools (currently used in meteorology and NWP) using a meteorology and NWP) using a community OGCM (ROMS).community OGCM (ROMS).

• Generalized stability analysis.Generalized stability analysis.

• 4D Variational data assimilation.4D Variational data assimilation.

Tangent and Adjoint Models: An Tangent and Adjoint Models: An OverviewOverview

• NL ROMS:NL ROMS:

• TL ROMS:TL ROMS: 0

|Ss t S s As

• AD ROMS:AD ROMS: † †Ts t A s

( ) (0, ) (0)s t R t s

† †(0) ( ,0) ( )Ts R t s t

(TL1)

(AD)

0 0S t S

OverviewOverview

• Second TLM: Second TLM:

0 0( ) ( )S t S A S S (TL2)

• TL1= Representer ModelTL1= Representer Model

• TL2= Tangent Linear Model TL2= Tangent Linear Model

Current Status of ROMS TL and Current Status of ROMS TL and AD ModelsAD Models

• All advection schemesAll advection schemes

• Most mixing and diffusion schemesMost mixing and diffusion schemes

• All boundary conditionsAll boundary conditions

• Orthogonal curvilinear gridsOrthogonal curvilinear grids

• All equations of stateAll equations of state

• Coriolis, pressure gradient, etc.Coriolis, pressure gradient, etc.

Generalized Stability Generalized Stability AnalysisAnalysis• Explore growth of perturbations in the Explore growth of perturbations in the

ocean circulation.ocean circulation.• Dynamics/sensitivity/stability of flow Dynamics/sensitivity/stability of flow

to naturally occurring perturbations.to naturally occurring perturbations.• Dynamics/sensitivity/stability due to Dynamics/sensitivity/stability due to

error or uncertainties in forecast error or uncertainties in forecast system.system.

• Practical applications: ensemble Practical applications: ensemble prediction, adaptive observations, prediction, adaptive observations, array design...array design...

OverviewOverview

• NL ROMS: NL ROMS: 0 0S t S

• Perturbation: Perturbation: 0S S s

( , , , )TS u v S T

Available Drivers (TL1, AD)Available Drivers (TL1, AD)

• Singular vectors:Singular vectors:

andand• Eigenmodes of Eigenmodes of

• Forcing Singular vectors:Forcing Singular vectors:

• Stochastic Stochastic optimals: optimals:

• Pseudospectra: Pseudospectra: 1HI A I A

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

(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 Dynamics/sensitivity/stability of flow to naturally occurring perturbationsto naturally occurring perturbations

• Dynamics/sensitivity/stability due to Dynamics/sensitivity/stability due to error or uncertainties in forecast error or uncertainties in forecast systemsystem

ApplicationsApplications• Test problems (double gyre, etc)Test problems (double gyre, etc)• Southern California BightSouthern California Bight• NE North Atlantic NE North Atlantic (w/Wilkin)(w/Wilkin)

• Gulf of Mexico Gulf of Mexico (w/Sheinbaum)(w/Sheinbaum)

• Intra-Americas Sea Intra-Americas Sea (w/Sheinbaum)(w/Sheinbaum)

• East Australia Current East Australia Current (w/Wilkin)(w/Wilkin)• Moore, A.M., H.G Arango, E. Di Lorenzo, B.D. Moore, A.M., H.G Arango, E. Di Lorenzo, B.D.

Cornuelle, A.J. Miller and D. Neilson, 2003: A Cornuelle, A.J. Miller and D. Neilson, 2003: A comprehensive ocean prediction and analysis comprehensive ocean prediction and analysis system based on the tangent linear and adjoint of system based on the tangent linear and adjoint of a regional ocean model. a regional ocean model. Ocean Modelling,Ocean Modelling, 77, 227-, 227-258.258.

Southern California Bight Southern California Bight (SCB)(SCB)• Model grid Model grid

1200kmX1000km1200kmX1000km

• 10km resolution, 10km resolution, 20 levels20 levels

• Di Lorenzo et al. Di Lorenzo et al. (2003)(2003)

Southern California BightSouthern California Bight

EigenspectrumEigenspectrum

Eigenmodes Eigenmodes (coastally trapped (coastally trapped waves)waves)

Nonnormal SystemsNonnormal Systems

• Most if not all circulations of interest Most if not all circulations of interest are nonnormal in that they possess are nonnormal in that they possess nonorthogonal eigenmodes.nonorthogonal eigenmodes.

• Linear eigenmode interference can Linear eigenmode interference can produce can produce rapid produce can produce rapid perturbation growth, even in absence perturbation growth, even in absence of unstable modes.of unstable modes.

Nonmodal Growth and Nonmodal Growth and Eigenmode Interference: A Eigenmode Interference: A Simple ExampleSimple Example

1 2 1

2

( )cot

0

A

d dt=s As

4 5 1 0.05 2 110

Pseudospectra – Nonmodal Pseudospectra – Nonmodal GrowthGrowth

• ConsiderConsider ts t As he

• Response is proportional to Response is proportional to 1( ) ( )I A

• For a normal system For a normal system || ( ) || 1 ( , ( ))dist A

• For nonnormal systemFor nonnormal system

1 ( , ( )) || ( ) || ( ) ( , ( ))dist A E dist A

A PseudospectrumA Pseudospectrum

Singular VectorsSingular Vectors• The fastest growing of all nonmodal The fastest growing of all nonmodal

perturbations.perturbations.

• We measure perturbation amplitude as:We measure perturbation amplitude as:

( ) ( ) ( )TE t s t s t• Consider perturbation growth factor:Consider perturbation growth factor:

( ) (0) ( ,0) (0, ) (0)

(0) (0) (0)

T T

T

E s R XR s

E s s

Singular VectorsSingular Vectors

• Energy norm, 5 day growth timeEnergy norm, 5 day growth time

Confluence and diffluenceConfluence and diffluence

Boundary sensitivityBoundary sensitivity

Seasonal DependenceSeasonal Dependence

Forcing Singular VectorsForcing Singular Vectors

• Consider system subject to constant Consider system subject to constant forcing:forcing: s t As f

• Forcing singular vectors are Forcing singular vectors are eigenvectors of:eigenvectors of:

0 0

( , ) ( , )

T

R t dt X R t dt

Stochastic OptimalsStochastic Optimals

• Consider system subject to forcing Consider system subject to forcing that is stochastic in time:that is stochastic in time: ( )s t As f t

• Assume that: Assume that: | '|( ') ( ) ct t tTf t f t e C • Stochastic optimals are eigenvectors of:Stochastic optimals are eigenvectors of:

| '|/ '

0 0

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

Stochastic Optimals Stochastic Optimals (energy (energy norm)norm)

Optimal excitation for coastally trapped waves

Sensitivity Analysis – Forcing and Sensitivity Analysis – Forcing and transporttransport

Sensitivity Analysis – initial value Sensitivity Analysis – initial value problemproblem

SummarySummary

• Eigenmodes: natural modes of Eigenmodes: natural modes of variabilityvariability

• Adjoint eigenmodes: optimal Adjoint eigenmodes: optimal excitations for eigenmodesexcitations for eigenmodes

• Pseudospectra: response of system to Pseudospectra: response of system to forcing at different freqs, and forcing at different freqs, and reliability of eigenmode calculationsreliability of eigenmode calculations

• Singular vectors: stability analysis, Singular vectors: stability analysis, ensemble prediction (i.c. errors)ensemble prediction (i.c. errors)

Summary (cont’d)Summary (cont’d)

• Forcing Singular Vectors: ensemble Forcing Singular Vectors: ensemble prediction (systematic model errors)prediction (systematic model errors)

• Stochastic optimals: stochastic Stochastic optimals: stochastic excitation, ensemble prediction excitation, ensemble prediction (forcing errors)(forcing errors)

• 4-dimensional variational data 4-dimensional variational data assimilation (weak and strong assimilation (weak and strong constraints)constraints)

North East North AtlanticNorth East North Atlantic

• 10 km resolution10 km resolution

• 30 levels in vertical30 levels in vertical

• Embedded in a model of N. AtlanticEmbedded in a model of N. Atlantic

• Wilkin, Arango and HaidvogelWilkin, Arango and Haidvogel

SSTSV t=0

SV t=5

Intra-Americas Sea and Gulf of MexicoIntra-Americas Sea and Gulf of Mexico(Julio Sheinbaum)(Julio Sheinbaum)

InitialInitial

FinalFinal

SV 1

Weak Constraint 4DVarWeak Constraint 4DVar

• NL model:NL model:

• Initial conditions: Initial conditions: (0)S I i

• Observations: Observations: d H S • For simplicity, assume error-free b.c.sFor simplicity, assume error-free b.c.s

• Cost func:Cost func: 1 1 1Tf iJ f C f i C i C

• Minimize J using indirect representer methodMinimize J using indirect representer method

• (Egbert et al., 1994; Bennett et al, 1997)(Egbert et al., 1994; Bennett et al, 1997)

( ) ( )S t S F t f t

OSU Inverse Ocean Model OSU Inverse Ocean Model System (IOM)System (IOM)

• Chua and Bennett (2001)Chua and Bennett (2001)

• Provides interface for TL1, TL2 and Provides interface for TL1, TL2 and AD for minimizing J using indirect AD for minimizing J using indirect representer methodrepresenter method

• Initial cond: Initial cond: (0)FS I• Outer loop, n Outer loop, n

1 1 1( ) ( ) ( )n n n n nF FS t S A S S F t

TL2• Inner loop, m Inner loop, m

AD† ( ) 0ms T † 1 † ;n T T nm m ms t A s H

1 † ;nm m f ms t A s C s †(0) (0)m i ms C s TL1

1T n T n

m m FH s C d H S

1 1 1 †( ) ( ) ( )n n n n nf mS t S A S S F t C s TL2†(0) (0)n

i mS I C s

Strong Constraint 4DVarStrong Constraint 4DVar• Assume f(t)=0Assume f(t)=0

• Outer loop, nOuter loop, n

( ) ( )n nS t S F t

1(0)S I

• Inner loop, mInner loop, mn

m ms t A s † † 1( )nT T nm m ms t A s H C d H S s

†

1(0) (0) (0)m m ms s s 1

1(0) (0) (0)n nmS S s

TL1

AD

Drivers under developmentDrivers under development

• Ensemble prediction (SVs, FSVs, SOs, Ensemble prediction (SVs, FSVs, SOs, following NWP)following NWP)

• 4D Variational Assimilation (4DVar)4D Variational Assimilation (4DVar)

• Greens function assimilationGreens function assimilation

• IOM interface (IROMS) (NL, TL1, TL2, IOM interface (IROMS) (NL, TL1, TL2, AD)AD)

PublicationsPublications

• Moore, A.M., H.G Arango, E. Di Lorenzo, B.D. Moore, A.M., H.G Arango, E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2003:Cornuelle, A.J. Miller and D. Neilson, 2003: A A comprehensive ocean prediction and analysis comprehensive ocean prediction and analysis system based on the tangent linear and adjoint of system based on the tangent linear and adjoint of a regional ocean modela regional ocean model. . Ocean Modelling,Ocean Modelling, Final Final revisions.revisions.

• H.G Arango, Moore, A.M., E. Di Lorenzo, B.D. H.G Arango, Moore, A.M., E. Di Lorenzo, B.D. Cornuelle, A.J. Miller and D. Neilson, 2003:Cornuelle, A.J. Miller and D. Neilson, 2003: The The ROMS tangent linear and adjoint models: A ROMS tangent linear and adjoint models: A comprehensive ocean prediction and analysis comprehensive ocean prediction and analysis system. system. Rutgers Tech. Report, Rutgers Tech. Report, In preparation.In preparation.

What next?What next?

• Complete 4DVar driverComplete 4DVar driver

• Interface barotropic ROMS to IOMInterface barotropic ROMS to IOM

• Complete 3D Picard iteration test Complete 3D Picard iteration test (TL2)(TL2)

• Interface 3D ROMS to IOMInterface 3D ROMS to IOM

SV 5

SCB ExamplesSCB Examples

Confluence and diffluenceConfluence and diffluence

Boundary sensitivityBoundary sensitivity

Stochastic OptimalsStochastic Optimals