Generalized Stability Theory Drivers Hernan G Arango IMCS

Generalized Stability Theory Drivers Hernan G. Arango IMCS, Rutgers Bruce D. Cornuelle SIO, UCSD Emanuele Di Lorenzo Georgia Tech Arthur J. Miller SIO, UCSD Andrew M. Moore PAOS, U. Colorado 2005 ROMS/TOMS Workshop Scripps Institution of Oceanography La Jolla, CA, October 25, 2005

Objectives • To explore the factors that limit the predictability of the circulation in regional models in a variety of dynamical regimes. • To build a Generalized Stability Theory (GST) analysis platforms: eigenmodes, optimal perturbations, forcing singular vectors, stochastic optimals, balance truncation vectors, EOF’s. . . • To build an ensemble prediction platform by perturbing forcing, initial, and boundary conditions with GST singular vectors.

Tangent Linear and Adjoint Based GST Drivers • Eigenmodes of and • Singular vectors: • Forcing Singular vectors: • Stochastic optimals: 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 Modelling, 7, 227 -258. http: //marine. rutgers. edu/po/Papers/Moore_2004_om. pdf

Two Interpretations • Dynamics/sensitivity/stability of flow to naturally occurring perturbations • Dynamics/sensitivity/stability due to error or uncertainties in the forecast system • Practical applications: § Ensemble prediction § Adaptive observations § Array design. . .

How To Run • Run nonlinear model (NLM) and save background state trajectory at regular intervals over the desired analysis time window (FWDname) Tend Tstr § • • Activate CPP options FORWARD_WRITE, FORWARD_RHS, and OUT_DOUBLE Run any of the GST drivers by activating any of their associated CPP options: § FT_EIGENMODES FORWARD_READ § AFT_EIGENMODES FORWARD_MIXING § OPT_PERTURBATIONS § FORCING_SV § STOCHASTIC_OPT Set input parameter in ocean. in: § NEV Number of eigenvalues § NCV Lanczos vectors workspace (NCV ≥ 2*NEV+1) § Lrst. GST restart logical switch § Max. Iter. GST Maximum number of iterations § NGST Check-pointing interval

ROMS/TOMS Framework

Finite Time Eigenmodes of R(0, t): Normal Modes u R(0, t)u

Adjoint Finite Time Eigenmodes of RT(0, t): Optimal Excitations u RT(0, t)u

Optimal Perturbations A measure of the fastest growing of all possible perturbations over a given time interval RT(t, 0)XR(0, t)u u

Forcing Singular Vectors

Stochastic Optimals Provide information about the influence of stochastic variations (biases) in ocean forcing

Ensemble Prediction • Optimal perturbations / singular vectors and stochastic optimals can also be used to generate ensemble forecasts. • Perturbing the system along the most unstable directions of the state space yields information about the first and second moments of the probability density function (PDF): § ensemble mean § ensemble spread • Excite with dominant basis vectors

Ensemble Prediction For an appropriate forecast skill measure, s

Final Remarks • It is running in parallel • Modified ARPACK to provide check-pointing but we are still not satisfied and more work is required • We continue updating and improving GST drivers • Balance truncation vectors • EOF • Revisiting stochastic optimals • Coding a simpler solution routine for symmetric eigen-problems