Andrew Poje (1), Anne Molcard (2,3), Tamay ÖzgÖkmen (4)

1 Dept of Mathematics, CSI-CUNY,USA2 LSEET - Universite de Toulon et du Var, France3 ISAC-CNR Torino, Italy4 RSMAS/MPO, Univ. of MIAMI, USA

Direct drifter launch strategies for Lagrangian data assimilation using hyperbolic trajectories

Problem

• Given a model assimilating Lagrangian data, determine initial launch locations which accelerate convergence of model to ‘truth’.

• Well defined optimization problem?

Xo -> Nonlinear ODE’s -> +Assimilation + Nonlinear PDE -> Model Output

• High dimensionality• Unique solutions?• ‘Simple Case’ - perfect model

•How long?• What domain?

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Properties of “Good” Lagrangian Data

• Span velocity space• Span physical space• Independent measurements

• Provide continuous velocity corrections• Capture phase and position of energetic structures in field

Approach

• Molcard et al. Lagrangian Assimilation + Lagrangian C.S. Launch

Strategy

•Target hyperbolic trajectories -

• maximize relative dispersion

• track coherent feature boundaries

The assimilation scheme (OI)

Ua=Ub+K (Yo-H(Ub))

K=BGT(GBGT+E)-1

Uaij=Ub

ij+-1 ijm(Vom-Vb

m)

=1 + o2/b

2

Oth order reduction:assimilation frequency which resolve the gradients;Observed and simulated variables non correlated in space and time; Gaussian distribution of the correction;Only 1 t or 2 successive positions.

Ua analysis variable Ub model calculated variableYo observed variableH(Ub) observation matrix that transforms the forecast data in the observed variable at the observed pointG=dH(Ub)/dUb sensitivity matrixB, E model and observation covariance matrices

A1 :Observation (t0)A2: Observation (t0+t)C1: First guess (t0+t)C2: Corrected forecast (t0+t)

A1

C1

C2

A2

Assimilation results for double-gyre, 1.5 layer MICOM

inflow

outflow

A direct launch strategy, based on tracking the Lagrangian manifolds emanating from strong hyperbolic regions in the flow

field is developed by computing the eigen-structure of the Eulerian fied at ti and tf and use it to initialize the manifold

calculation.

Optimal deployment strategy

Manifolds

Ws inflowing: attracted toward the hyperbolic trajectoryin forward time

Wu outflowing: repelled away from hyperbolic point

FSLE Finite Scale Lyapunov Exponents(measure of time required for a pair of particle trajectories to separate)

Deployments: Each 3 lines of 4 drifters:

•1 ‘Optimal’ (Directed) on Wu

•50 Randomly Choice

Analysis: 50 random deployments vs 1 optimal in terms of:overall coverage (or spreading), integrated Kinetic energy,

integrated distance between drifter observations and simulated trajectories.

Spreading KE

YO-YB Correction

The product of the overall coverage and the integrated distance difference provides a good proxy for both terms in the correction term in the assimilation scheme

ijm(Vom-Vb

m)

E=100% E=60% E=40%

Control

Assimilation

Free

DAY 0 DAY 30 DAY 0 DAY 30

Dependance on the length of the segment and on the “exact point” error

Dependance on the length of the segment and on the “exact point” error

Conclusion

Alignment of initial drifter positions along the out-flowing branch of identifiable Lagrangian boundaries is shown to

optimize both the relative dispersion of the drifters and the sampling of high kinetic energy features in the flow.

The convergence of the assimilation scheme is consistent and considerably improved by this optimal launch strategy.

•‘Real’ ocean? • Identifiable structures• Appropriate time/space scales? (Dart05?)

•Something slightly deeper here?• Fast convergence to attractor (Wu)• Knowing location of attractor => more information

Questions

Preprint available: Ocean Modeling, 2006