2005 ROMS/TOMS Workshop2005 ROMS/TOMS WorkshopScripps Institution of OceanographyScripps Institution of Oceanography

La Jolla, CA, October 25, 2005La Jolla, CA, October 25, 2005

ean M od

earch C o m

r a i n - F o l l o w

M o d e l i n g

Generalized Stability Theory DriversGeneralized Stability Theory Drivers

Hernan G. ArangoIMCS, Rutgers

Andrew M. MoorePAOS, U. Colorado

Emanuele Di LorenzoGeorgia Tech

Bruce D. CornuelleSIO, UCSD

Arthur J. MillerSIO, UCSD

ObjectivesObjectives

• To explore the factors that limit the To explore the factors that limit the predictabilitypredictability of of

the circulation in regional models in a variety of the circulation in regional models in a variety of

dynamical regimes.dynamical regimes.

• To build a Generalized Stability Theory (GST) analysis To build a Generalized Stability Theory (GST) analysis

platforms: platforms: eigenmodeseigenmodes, , optimal perturbationsoptimal perturbations, , forcingforcing

singular vectorssingular vectors, , stochastic optimalsstochastic optimals,, balance balance

truncation vectorstruncation vectors,, EOF’s EOF’s ......

• To build an To build an ensemble predictionensemble prediction platform by platform by

perturbing forcing, initial, and boundary conditions perturbing forcing, initial, and boundary conditions

with GST singular vectors.with GST singular vectors.

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:

( ,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

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

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 ...

How To RunHow To Run

• Run nonlinear model (NLM) and save background state trajectory at regular intervals over Run nonlinear model (NLM) and save background state trajectory at regular intervals over

the desired analysis time window (the desired analysis time window (FWDnameFWDname))

Activate CPP options Activate CPP options FORWARD_WRITEFORWARD_WRITE, , FORWARD_RHSFORWARD_RHS, , andand OUT_DOUBLEOUT_DOUBLE

• Run any of the GST drivers by activating any of their associated CPP options:Run any of the GST drivers by activating any of their associated CPP options:

FT_EIGENMODESFT_EIGENMODES FORWARD_READFORWARD_READ

AFT_EIGENMODES FORWARD_MIXINGAFT_EIGENMODES FORWARD_MIXING

OPT_PERTURBATIONSOPT_PERTURBATIONS

FORCING_SVFORCING_SV

STOCHASTIC_OPTSTOCHASTIC_OPT

• Set input parameter in Set input parameter in ocean.in:ocean.in: NEVNEV Number of eigenvalues Number of eigenvalues

NCV NCV Lanczos vectors workspace (NCV ≥ 2*NEV+1) Lanczos vectors workspace (NCV ≥ 2*NEV+1)

LrstGSTLrstGST GST restart logical switch GST restart logical switch

MaxIterGSTMaxIterGST Maximum number of iterations Maximum number of iterations

NGST NGST Check-pointing interval Check-pointing interval

TTstrstr TTendend

ROMS/TOMS Framework

ADSEN_OCEAN

SANITY CHECK S

PERT_OCEAN

PICARD_OCEAN

GRAD_OCEAN

TLCHECK _OCEAN

RP_OCEAN

ESMF

AIR_OCEAN

M

AS

TE

RWAVE S _OCE AN

OCE AN IN IT IA L IZE

F IN A L IZE

RU N

S4DVAR_OCEAN

IS4DVAR_OCEAN

W4DVAR_OCEAN

ENSEMBLE_OCEAN

NL_OCEAN

TL_OCEAN

AD_OCEAN

PROPAGATOR

K ER NELNLM, T LM, RP M, ADM

phys ic sbiogeochemic al

sedimentsea ic e

Optimal pertubationsADM eigenmodes

TLM eigenmodes

Forc ing singular vectorsStochastic optimals

Balance Truncation vectors

EOF’s P seudospec tra

Finite Time Eigenmodes

n=1, N EV

DO WHI LE (.TRUE.)

FTE_OCEAN INIT IALIZE

FINALIZE

RUN

PROPAGATOR

NRM2

PROPAGATORESMF

TL_UNPACK

TL_PACK

TL_MAIN3D

ARPACKDNAUPD

ARPACKDNEUPD

Arnoldi Iteration Loop

Compute ConvergedRitz Eigenvectors

Non-symmetricProblem

Eigenmodes of R(0,t):Normal Modes

R(0,t)uu

Adjoint Finite Time Eigenmodes

n=1, N EV

DO WHI LE (.TRUE.)

AFTE_OCEAN INIT IALIZE

FINALIZE

RUN

PROPAGATOR

NRM2

PROPAGATOR

AD_UNPACK

AD_PACK

AD_MAIN3D

ESMF

ARPACKDNAUPD

ARPACKDNEUPD

Compute ConvergedRitz Eigenvectors

Arnoldi Iteration Loop

Non-symmetricProblem

Eigenmodes of RT(0,t):Optimal Excitations

RT(0,t)uu

Optimal Perturbations

SymmetricProblem

DO n=1, N EV

DO WHI LE (.TRUE.)

NRM2

PROPAGATOR

ESMF

ARPACKDSAUPD

ARPACKDSEUPD AD_MAIN3D

TL_MAIN3D

AD_INI_PERTURB

AD_PACK

Compute ConvergedRitz Eigenvectors

OP_OCEAN

FINALIZE

RUN

PROPAGATOR

TL_UNPACK

INIT IALIZE

A measure of the fastestgrowing of all possibleperturbations over a giventime interval RT(t,0)XR(0,t)u

u

Forcing Singular Vectors

SymmetricProblem

DO n=1, N EV

DO WHI LE (.TRUE.)

NRM2

PROPAGATOR

ESMF

ARPACKDSAUPD

ARPACKDSEUPD AD_MAIN3D

TL_MAIN3D

AD_INI_PERTURB

AD_PACK

Compute ConvergedRitz Eigenvectors

FSV_OCEAN

FINALIZE

RUN

PROPAGATOR

TL_UNPACK

INIT IALIZE

0 0

( , ) ( , )

T

R t dt X R t dt

Stochastic Optimals

ARPACKDSAUPD

ARPACKDSEUPD

Compute ConvergedRitz Eigenvectors

Arnoldi Iteration Loop

SymmetricProblem

DO n=1, N EV

DO WHI LE (.TRUE.)

DO n=1,Nintervals

to t /3f 2*t /3f tf

(1)

(2)

(3)If = 3,(1) run TLM ( , ) and ADM ( , )(2) run TLM ( , ) and ADM ( , )(3) run TLM ( , ) and ADM ( , )

Nintervalst t t tt /3 t t t /3

2*t /3 t t 2*t /3

o f f o

f f f f

f f f f

SO_OCEAN INIT IALIZE

FINALIZE

RUN

PROPAGATOR

TL_UNPACK

NRM2

PROPAGATOR

AD_MAIN3D

TL_MAIN3D

AD_INI_PERTURB

AD_PACK

Provide information about the influence of stochasticvariations (biases) in ocean forcing

| '|/ '

0 0

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

Ensemble PredictionEnsemble Prediction

• Optimal perturbations / singular vectors and Optimal perturbations / singular vectors and stochastic optimals can also be used to generate stochastic optimals 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

• Excite with dominant basis vectorsExcite with dominant basis vectors

Ensemble PredictionEnsemble Prediction

t

s

HighSpread

U npredic table

timet

sLow

Spread

P redic table

time

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

• It is running in parallelIt is running in parallel

• Modified ARPACK to provide check-pointing but we are Modified ARPACK to provide check-pointing but we are still not satisfied and more work is requiredstill not satisfied and more work is required

• We continue updating and improving GST driversWe continue updating and improving GST drivers

• Balance truncation vectorsBalance truncation vectors

• EOFEOF

• Revisiting stochastic optimalsRevisiting stochastic optimals

• Coding a simpler solution routine for symmetric eigen-Coding a simpler solution routine for symmetric eigen-problemsproblems

Final RemarksFinal Remarks