2. A method for the assimilation of Lagrangian data C.K.R.T. Jones and L. Kuznetsov, Lefschetz Center for Dynamical Systems, Brown University K. Ide, Atmospheric Sciences, UCLA

3. Data assimilation

4. Discrete ocean model t = i  t x = k  x, k=1,n y = l  y, l=1,m t = t i k x l y k l k i Model state vector x = {v, T, S, ,…} x  R (N = 5 n m) i N Ocean model: M – model’s dynamics operator

5. Observations True ocean: Covariance of the model residual: Covariance of the observation error:

6. Extended Kalman Filter

7. Ocean vs Atmosphere Ocean: Drifters/floats Slow (weeks) Horizontal coverage Atmosphere: Balloons Fast (days) Vertical structure Difference in scales and cost ($drifters & floats>>$balloons)

8. Lagrangian information y i-1,1 Lagrangian observations from drifters and floats do not give the data in terms of model variables. They are rarely used for assimilation into ocean models Solution: Include drifter coordinates into the model y i-1,2 y i,1 y i,2

9. Methodology

10. Point vortex systems Point vortex systems provide a rough model of 2d flows dominated by strong coherent vortices Simple dynamics (small number of degrees of freedom) makes them an attractive testing ground We consider flows due to N vortices. L tracer particles are observed. Tracers do not influence the flow.

13. Two point vortices

14. Simple example: N=1, L=1

15. Two vortices, N=2, one tracer, L=1  z 1 z 2 It works!

16. Two vortices, N=2, one tracer, L=1  z 1 z 2 Or does it?

17. The overall performance of EKF is represented by tr P Efficiency of tracking of individual vortices is measured by |  z| a

18. N=2, L=2

19. Assessment of method When does the assimilation works and when does it not? How does the filter fail? What is the role of Lagrangian structures Compare with the assimilation of velocity data directly

20. N=2, L=1 (N e =100 noise realizations) 0.75 1.0 1.25 1.5 1.75 2.0 0 0 2 2 8 18 0.090.100.120.190.350.51 0 1 7 19 22 29 0.120.130.220.400.430.67 4 7 16 25 - - 0.200.290.510.78 - -

21. Ensembles of different system noise realizations, N=2, L=1

22. Divergence of the filter Corotating reference frame Exponential separation of trajectories near the saddle causes the divergence of the filter

23. Ocean Atmosphere Drifters/floats Balloons Launching strategy Targeted observations

24. Dependence on initial tracer location 0.3-0.6i1-0.6i1-i2.4-2.4i-1.75i 0.555.50.5150 0.121.900.110.290.11

25. Vortices of different strength

26. Four vortices, N=4 Vortices: (-4,-1), (-4,1), (4,-1), (4,1) (regular motion) (5.24,3.61),(-2.12,0),(2.12,0),(0,-3) (chaotic motion) Tracers: (-0.5,1) and (0.5,0), L =2  T=0.5,  = 0.005,  = 0.02 z 5 z 2 z 3 z 4 z 1 z 6 z 5 z 2 z 3 z 1 z 6 z 4

27. N=4, L=2 Regular Chaotic

28. Comparison with assimilation of velocity 0.4 0.5 0.6 0.7 0.8 0 4 6 46 76 0.18 0.24 0.31 0.58 0.71

29. Conclusions Assimilation of the tracer data into the point vortex system affords successful tracking of the flow (N ~ L, both small) EKF fails for large  T, ,  : TLM is not a good approximation Passage next to (Lagrangian) saddle can cause filter divergence – efficiency of the method depends on the launch location relative to the Lagrangian flow structures Basis for launch strategies of floats and/or drifters Extend to layered models Future work: analyze filter behavior near the saddle; move to gridded ocean models increase N use other methods for model error propagation