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

Andrew Moore, CUAndrew Moore, CUHernan Arango, Rutgers UHernan Arango, Rutgers U

Arthur Miller, Bruce Cornuelle, Arthur Miller, Bruce Cornuelle, Emanuele Di Lorenzo, Doug Neilson Emanuele Di Lorenzo, Doug Neilson

UCSDUCSD

Major ObjectiveMajor Objective

• To provide the ocean modeling To provide the ocean modeling community with analysis and community with analysis and prediction tools that are available in prediction tools that are available in meteorology and NWP, using a meteorology and NWP, using a community OGCM (ROMS).community OGCM (ROMS).

OverviewOverview

• NL ROMS: NL ROMS: 0 0S t S

• Perturbation: Perturbation: 0S S s

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 TL and ADCurrent Status of TL and AD

• 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 Explore growth of perturbations in the ocean circulation.the ocean circulation.

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

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

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)

SCB ExamplesSCB Examples

EigenspectrumEigenspectrum

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

PseudospectrumPseudospectrum

• 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

PseudospectrumPseudospectrum

Singular VectorsSingular Vectors

• Consider the initial value problem.Consider the initial value problem.

• 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

SV 1

SV 5

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)

InterpretationInterpretation

• Optimal forcing for coastally-trapped Optimal forcing for coastally-trapped waves?waves?

• Optimal forcing for recirculating flow Optimal forcing for recirculating flow in the lee of Channel Islands?in the lee of Channel Islands?

Stochastic Optimals Stochastic Optimals (transport (transport norm)norm)

Transport Singular VectorTransport Singular Vector

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

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 (model errors)prediction (model errors)

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

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

SCB ExamplesSCB Examples

Confluence and diffluenceConfluence and diffluence

Boundary sensitivityBoundary sensitivity

Stochastic OptimalsStochastic Optimals