Lagrangian Data Assimilation: Eddy- Tracking in the Gulf of Mexico. - - PDF document

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Lagrangian Data Assimilation: Eddy- Tracking in the Gulf of Mexico. - - PDF document

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Lagrangian Data Assimilation: Eddy- Tracking in the Gulf of Mexico. Guillaume Vernieres, Kayo Ide, Christopher K. R. T. Jones BI RS, 2007 Outline Motivations Assimilation of Lagrangian Observations Model of the loop current

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Lagrangian Data Assimilation: Eddy- Tracking in the Gulf of Mexico.

Guillaume Vernieres, Kayo Ide, Christopher K. R. T. Jones

BI RS, 2007

Outline

  • Motivations
  • Assimilation of Lagrangian Observations
  • Model of the loop current
  • Twin Experiment
  • Correlation Functions (representers)
  • Summary

BI RS, 2007

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Motivations

  • Why is Lagrangian Data Assimilation so efficient?
  • Application to shedding of eddies in the GoM

Assimilation of Lagrangian Observations

Over view of DA: Estimate of the “True” state of the ocean

Data assimilation = Linear regression

  • Prior
  • r

Forecast = Linear combination of covariance/correlation functions, in the data space Kalman filter:

PHT

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Assimilation of Lagrangian Observations

Over view of DA: Kalman filter:

PHT

“Bare bone” ensemble Kalman filter parallelized on 128 cpu (topsail cluster from UNC-Chapel Hill) Up to ~5000 members

Assimilation of Lagrangian Observations

Problem: Most models are Eulerian ! 2 cases: Convert trajectories into Eulerian velocities Assimilate directly the trajectories

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Assimilation of Lagrangian Observations

Problem: Most models are Eulerian ! 2 cases: Convert trajectories into Eulerian velocities Assimilate directly the trajectories

Assimilation of Lagrangian Observations: Augmented State

Model State: xM=[u,…,v,…,h,…]T Drifter position: xD=[x y]T Augmented State: x=[xM xD]

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Assimilation of Lagrangian Observations

Model : xM(t+dt)=M( xM(t) ) Trajectory : xD(t+dt)= xD(t)+ ∫

+dt t t

dt t y x u ) , , ( r

Complexity of the observing operator shifted in the model equation Augmented model

Model of the loop current

3 active layer, reduced gravity

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Model of the loop current

Limited area curvilinear domain

9 6

  • W

92oW 88oW 8 4o W 8

  • W

18oN 21oN 24oN 27oN 30oN

Model of the loop current

Limited area curvilinear domain Resolution: 5-25 km

9 6

  • W

92oW 88oW 8 4o W 8

  • W

18oN 21oN 24oN 27oN 30oN

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Model of the loop current

Inflow/outflow: ~30 Sv

9 6

  • W

92oW 88oW 8 4o W 8

  • W

18oN 21oN 24oN 27oN 30oN

Twin Experiment: Control

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Synthetic observation:

  • Fixed stations(u,v)
  • Surface floats (x,y)
  • Isopycnal floats (x,y,z)

Synthetic observation: 4 Fixed stations (u,v)

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Synthetic observation: 4 Surface floats (x,y) Synthetic observation: 4 Isopycnal floats (x,y,z)

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Results: ssh field

Fixed stations Surface drifters Isopycnal floats

Results: ssh field

Fixed stations Surface drifters Isopycnal floats

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Results: rms(truth-analysis) of interface’s depths

Interface between layer 1 and 2 Interface between layer 2 and 3 Interface between layer 3 and 4 EuDA (u,v) LaDA (x,y) LaDA (x,y,z)

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Eddy “recovery”

We define the eddy center/strength/scale by locally fitting a Gaussian shaped function to the top layer thickness field. We minimize, LaDA (x,y) LaDA (x,y,h) EuDA

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Phase speed L(t) A(t)

Covariance and Correlation Functions: Structure and Interpretation

Correlation functions (~representers) Convergence as Ne increases Volume of influence

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Correlation Functions: 1 drifter

First column of RFD

=

Corr(State,Longitudinal position of drifter)

Convergence of RFD

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Convergence

Correlation function: Ne=32, r(h(1),Longitudinal position of the drifter)

Convergence

Correlation function: Ne=64, r(h(1),Longitudinal position of the drifter)

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Convergence

Correlation function: Ne=128, r(h(1),Longitudinal position of the drifter)

Convergence

Correlation function: Ne=256, r(h(1),Longitudinal position of the drifter)

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Convergence

Correlation function: Ne=384, r(h(1),Longitudinal position of the drifter)

Convergence

Correlation function: Ne=512, r(h(1),Longitudinal position of the drifter)

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Convergence

Correlation function: Ne=640, r(h(1),Longitudinal position of the drifter)

Convergence

Correlation function: Ne=1024, r(h(1),Longitudinal position of the drifter)

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Representers

Snap shot of the correlation function after 25 days

Representers

Snap shot of the correlation function after 25 days

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Representers

Snap shot of the correlation function after 25 days

Volume of influence

∂V={(x,y,”z”), |corr(state(x,y,z),longitude or latitude|=0.3}

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Volume of influence

∂V={(x,y,”z”), |corr(state(x,y,z),longitude or latitude|=0.3}

Similar size for the 3 types of observation

Volume of influence

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Volume of influence

LaDA3D LaDA2D EuDA

Volume of influence

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Summary

LaDA (x,y,z) > LaDA (x,y) > EuDA (u,v)

Summary

LaDA (x,y,z) > LaDA (x,y) > EuDA (u,v) Recover the eddy characteristics

1 drifter !

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Summary

LaDA (x,y,z) > LaDA (x,y) > EuDA (u,v) Recover the eddy characteristics Efficiency of LaDA through the structure

  • f the covariance functions (volume of influence)

Gliders

g

u t x u t x + = )) ( ( ) (

.