Meeting 1:

On Tuesday, 8th February, the Lagrangian data assimilation group met between 10:30am and 12:00pm to lay out some of the key questions concerned with the assimilation of Lagrangian data. The following list summarises the questions raised in the meeting:
 It was planned that over the next three meetings, background material on Lagrangian data will be discussed. The planned schedule is listed below:
 Meeting 2:

On 15th of February, background material an introduction on dynamical systems approaches to study chaotic transport in unsteady flows was presented. These well understood transport mechanisms are intrinsic of unsteady flows and have direct implications on the trajectories described by Lagrangian instruments.
A basic understanding of these mechanisms is an essential first step for understanding Lagrangian data assimilation.

Meeting 3:

On 1st of March, background material was presented on Lagrangian instruments and their use in ocean measurements. Examples of trajectories obtained from measurements in the ocean clearly demonstrated the chaotic nature of these instruments. Different kinds of floats were also discussed, those that follow isopycnal surfaces (surfaces of constant density) and those that follow isobars (surfaces of constant pressure).

Meeting 4:

On 8th of March, background material was presented on how to assimilate Lagrangian data into idealised point vortex models. The augmented approach was discussed whereby equations governing the motion of the drifters are augmented to the prognostic flow equations to allow Lagrangian data to be assimilated into our models. The formulation was presented using the Extended Kalman Filter and results for a number of point vortex systems were presented. The discussion will continue on 22nd of March in which the generelisation of the approach to a more realistic ocean model, namely the shallow-water equations will be discussed.

Meeting 5:

On 22nd of March, presentation of the augmented approach for the assimilation of Lagrangian data into the shallow-water system of equations was discussed. The extension of the method to the Ensemble Kalman Filter was laid out and results for several parametric studies were presented. The key advantage of the method over existing approaches is in its ability to converge for relatively large assimilation time scales (of the order of the Lagrangian integral timescale).

Meeting 6:

On 29th of March, a review of alternative methods for the assimilation of Lagrangian data was presented. These methods are based on the work of Molcard et al. and Özgökmen et al. in which a velocity is reconstructed from drifter observations which is then assimilated into the model velocity field. An attempt to relate this approach to the method employed by Kuznetsov et al. was attempted but was not possible due to the fundamentally different nature of the two approaches. The papers by the above authors can be downloaded from the links listed below.

Meeting 7:

On 5th of April, an overview of Lagrangian measurements of mid-depth circulation in the North Atlantic was discussed by Susan Lozier. The review will continue in two weeks following the Lagrangian data assimilation workshop to be held on 12th of April. In the mean time, background reading on this topic is
recommended and a link to the relevant article can be found in the list of references below (Bower et al., Nature, 2002).

Meeting 8:

On 19th of April, the group reviewed the work of Carter (see reference below), and McWilliams et al. and Hua et al. (references given below). The latter two references on objective analysis contain an interesting approach on how to reconstruct velocity fields (stream functions) from measurements and dynamical constraints. The next meeting will be on Monday, 2nd of May at Duke University.

Meeting 9:

On 2nd of May, the group met at Duke University to discuss how to assimilate drifter data into a quasi-geostrophic ocean model by reconstructing the stream function field. Three sub-groups were created to lay out seperate strategies for reconstructing the Eulerian flow field. An issue that arose from the discussion was how to treat boundary conditions in all the methods proposed. The next meeting will be on Wednesday, 11th of May back at SAMSI.
(Note: a recently published paper by Molcard et al. regarding Lagrangian data assimilation in a multi-layer ocean model has been added to the references below.)

Summaries of methods proposed:
 Meeting 10:

On 9th of May, the group met to discuss the issue of boundary conditions for the methods proposed in the previous meeting for the reconstruction of the velocity field.  The boundary conditions are needed for a kinematic (mass conserving) and a dynamic model of the stream-function field and needs to be prescribed from measurements or by assuming no-flow/ no-slip boundary conditions. It was also decided that we should classify the different methods for reconstructing the flow field into four categories:
  1. incompressible model
  2. prognostic model for stream-function employing assumption of geostrophy (needs more thought)
  3. QG model with assimilation
  4. shallow water model with assimilation

Useful Links:

References on Lagrangian data assimilation:
  1. L. Kuznetsov, K. Ide, and C.K.R.T. Jones: A Method for Assimilation of Lagrangian Data.
    Mon. Wea. Rev., 131, 2247-2260 (2003)
  2. K. Ide, L. Kuznetsov, and C.K.R.T. Jones: Lagrangian data assimilation for point vortex systems. J. Turbul., 3, 053 (2002)
  3. A.J. Mariano, A. Griffa, T.M. Özgökmen, and E. Zambianchi: Lagrangian Analysis and Predictability of Coastal and Ocean Dynamics. J. Atmos. Oceanic Technol., 19, 1114-1126 (2000)
  4. A. Molcard, L.I. Piterbarg, A. Griffa, T.M. Özgökmen, and A.J. Mariano: Assimilation of drifter observations for the reconstruction of the Eulerian circulation field. J. Geophys. Res., 108, C3, 1 1-1 21 (2003) (PDF - geostrophic equations)
  5. T.M. Özgökmen, A. Griffa, and A.J. Mariano: On the Predictability of Lagrangian Trajectories in the Ocean. J. Atmos. Oceanic Technol., 17, 366-383 (2000)
  6. T.M. Özgökmen, A. Molcard, T.M. Chin, L.I. Piterbarg, and A. Griffa: Assimilation of drifter observations in primitive equation models of midlatitude ocean circulation. J. Geophys. Res., 108, C7, 31 1-31 17 (2003) (PDF - shallow water equations)
  7. H. Salman, L. Kuznetsov, C.K.R.T. Jones, K. Ide: A method for assimilating Lagrangian data into a shallow-water equation ocean model. Monthly Weather Review, Submitted (PDF)
  8. A.S. Bower, B. Le Cann, T. Rossby, W. Zenk, J. Gould, K. Speer, P.L. Richardson, M.D. Prater, H.-M. Zhang: Directly measured mid-depth circulation in the northeastern North Atlantic Ocean. Nature, 419, 603-607, (2002) (PDF)
  9. E.F. Carter: Assimilation of Lagrangian data into a numerical-model. Dyn. Atmos. Oceans., 13, 335-348, (1989) (PDF)
  10. J.C. McWilliams, W.B. Owens, B.L. Hua: An objective analysis of the POLYMODE local dynamics experiment. Part I: General formalism and statistical model selection. J. Phys. Oceanogr., 15, 483-504, (1986) (PDF)
  11. B.L. Hua, J.C. McWilliams, W.B. Owens: An objective analysis of the POLYMODE local dynamics experiment. Part II: Streamfunction and potential vorticity fields during the intensive period. J. Phys. Oceanogr., 15, 506-522, (1986) (PDF)
  12. A. Molcard, A. Griffa, T.M. Özgökmen: Lagrangian Data Assimilation in Multilayer Primitive Equation Ocean Models. J. Atmos. Oceanic Technol., 22, 70-83, (2005) (PDF)

References on chaotic advection:
  1. V. Rom-Kedar, A. Leonard, and S. Wiggins: An analytical study of transport, mixing, and chaos in an unsteady vortical flow. J. Fluid Mech., 214, 347-358 (1990)
  2. J.M. Ottino: The kinematics of mixing: stretching, chaos, and transport. Cambridge University Press, Cambridge, UK (1989)
  3. A.M. Rogerson, P.D. Miller, L.J. Pratt, and C.K.R.T. Jones: Lagrangian motion and fluid exchange in a barotropic meandering jet. J. Phys. Oceanogr. 29, 2635 (1999)
  4. C.K.R.T. Jones, and S. Winkler: Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere. in Handbook of Dynamical Systems III: Towards Applications, edited by B. Fiedler, Vol. 2, North-Holland, 55-92 (2002)
  5. H. Aref: Stirring by chaotic advection. J. Fluid Mech. 143, 1-21 (1984)
  6. S. Wiggins: The dynamical systems approach to Lagrangian transport in oceanic flows. Ann. Rev. Fluid Mech., 37, 295-328 (2005)
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E-mail: hsalman@email.unc.edu