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

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 1

Motivations • Why is Lagrangian Data Assimilation so efficient? • Application to shedding of eddies in the GoM Assimilation of Lagrangian Observations Data assimilation Over view of DA: = Linear regression Estimate of the “True” state of the ocean - Prior Kalman filter: or PH T Forecast = Linear combination of covariance/correlation functions, in the data space 2

Assimilation of Lagrangian Observations Over view of DA: Kalman filter: PH T “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 3

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: x M =[u,…,v,…,h,…] T Drifter position: x D =[x y] T Augmented State: x =[ x M x D ] 4

Assimilation of Lagrangian Observations Model : x M (t+dt)= M ( x M (t) ) Augmented model Trajectory : x D (t+dt)= x D (t)+ ∫ r + dt t u ( x , y , t ) dt t Complexity of the observing operator shifted in the model equation Model of the loop current � 3 active layer, reduced gravity 5

Model of the loop current � Limited area curvilinear domain 30 o N 27 o N 24 o N 21 o N 18 o N o o 8 0 W 9 6 W 92 o W 88 o W 4 o 8 W Model of the loop current � Limited area curvilinear domain 30 o N � Resolution: 5-25 km 27 o N 24 o N 21 o N 18 o N o o W 4 o 8 0 9 6 W 92 o W 88 o W W 8 6

Model of the loop current � Inflow/outflow: ~30 Sv 30 o N 27 o N 24 o N 21 o N 18 o N o o 8 0 W 9 6 W 92 o W 88 o W 4 o 8 W Twin Experiment: Control 7

Synthetic observation: • Fixed stations(u,v) • Surface floats (x,y) • Isopycnal floats (x,y,z) Synthetic observation: 4 Fixed stations (u,v) 8

Synthetic observation: 4 Surface floats (x,y) Synthetic observation: 4 Isopycnal floats (x,y,z) 9

Results: ssh field Fixed stations Isopycnal floats Surface drifters Results: ssh field Fixed stations Isopycnal floats Surface drifters 10

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) 11

Eddy “ recovery ” We define the eddy center/strength/scale by locally fitting a Gaussian shaped function to the top layer thickness field. We minimize, EuDA LaDA (x,y,h) LaDA (x,y) 12

Phase speed L(t) A(t) Covariance and Correlation Functions: Structure and Interpretation � Correlation functions (~representers) � Convergence as Ne increases � Volume of influence 13

Correlation Functions: 1 drifter First column of R FD = Corr(State,Longitudinal position of drifter) Convergence of R FD 14

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) 15

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) 16

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) 17

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) 18

Representers Snap shot of the correlation function after 25 days Representers Snap shot of the correlation function after 25 days 19

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} 20

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 21

Volume of influence EuDA LaDA2D LaDA3D Volume of influence 22

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 ! 23

Summary � LaDA (x,y,z) > LaDA (x,y) > EuDA (u,v) � Recover the eddy characteristics � Efficiency of LaDA through the structure of the covariance functions (volume of influence) Gliders . = + x ( t ) u ( x ( t )) u g 24