1. TRANSPORT ROUTES IN THE NORTH-WESTERN MEDITERRANEAN SEA: A DYNAMICAL SYSTEM PERSPECTIVE
O A. M. Mancho1
Instituto de Matemáticas y Física Fundamental, CSIC, Spain
O S. Wiggins2
School of Mathematics, University of Bristol, UK
O E. Hernández-García1
Instituto de Física de Sistemas Complejos, CSIC, Spain
O V. Fernández
Instituto Nazionale di Geofisica e Vulcanologia, Italy
1Supported by CSIC Grants OCEANTECH (PIF06-059) and PI200650I224
2 Supported by ONR Grant No. N

2. OUTLINE c
The Ocean Circulation Model
Transport: the dynamical system perspective
Computing hyperbolic trajectories and manifolds
Transport in time dependent flows: lobe dynamics
Conclusions

3. THE OCEAN CIRCULATION MODEL
OWe analyze velocity fields which are obtained from an ocean model, (DieCAST adapted to the Mediterranean Sea)
OThe spin-up phase of integration is carried out for 16 years. Each year is considered to have 12 months 30 days length each (i.e. 360 days).
OResolution is
= (1/8) degree
 = cos   ,
OThe model uses control volumes of thickness smoothly increased up to the deepest bottom. In particular we focus on velocity fields obtained at the second layer which has its center at a depth of m.
O For our purposes velocity field can be considered 2D

4. THE OCEAN CIRCULATION MODEL
How particles disperse?

5. THE OCEAN CIRCULATION MODEL
Can we locate particles that eventually come into the eddy?
It is not that we were lucky. We did that thanks to manifolds that provide us
with the necessary information to place those initial conditions

6. TRANSPORT: THE DYNAMICAL SYSTEM PERSPECTIVE
Lagrangian transport is that suffered by a particle (red dot) that is transported by the velocity field of a fluid. Its dynamics in 2D is given by:
Typically even simple flows v(x,t) either stationary or periodic may produce very complicated trajectories x(t). dx =
v(x,t), x in R2 dt

7. TRANSPORT: THE DYNAMICAL SYSTEM PERSPECTIVE
The Building blocks of the geometrical template for Lagrangian transport
Vector field
Objects of the geometrical template
Stationary
Hyperbolic fixed points and their stable and unstable manifolds
Periodic
Hyperbolic periodic trajectories and their stable and unstable manifolds
Aperiodic
Hyperbolic trajectories and their stable and unstable manifolds

8. COMPUTING HYPERBOLIC TRAJECTORIES AND MANIFOLDS IN TIME DEPENDENT FLOWScc
We study transport in geophysical flows which are aperiodic vector fields.
In the dynamical systems perspective transport is described with manifolds of hyperbolic trajectories
Some difficulties in computing those in aperiodic vector fields
O What are the reference trajectories with manifolds defining a geometrical template we are interested in? The concept of fixed point needs to be generalized.
O How to compute manifolds in aperiodic flows?
O The vector fields are provided as finite time data sets. Concepts in dynamical systems are for infinite time vector fields
References
Mancho, Small, Wiggins, Physics Reports. 437 (3-4) 2006 pp
Mancho, Small, Wiggins, Nonlinear Processes in Geophysics 11 (1)
Mancho, Small, Wiggins, Ide, Physica D 182 (3-4) (2003)
Wiggins, Annual Review of Fluid Mechanics 37 (2005)

9. COMPUTING HYPERBOLIC TRAJECTORIES AND MANIFOLDS IN TIME DEPENDENT FLOWS
Manifolds are made of trajectories which either in plus infinite time (stable) or minus infinite time (unstable) tend to the hyperbolic trajectory.
Manifolds are therefore barriers to transport as they are made of trajectories

10. TRANSPORT IN TIME DEPENDENT FLOWS: LOBE DYNAMICScc
TRANSPORT IN TIME DEPENDENT FLOWS: LOBE DYNAMICS
Lobes are formed from the intersection of a piece of stable and a piece of unstable manifold of the hyperbolic trajectories.
Lobes are boundaries that water masses cannot cross.
Manifolds and lobes are useful as they constitute a geometrical template useful to predict time evolution of sets of initial conditions
Lobes

11. TRANSPORT IN TIME DEPENDENT FLOWS: LOBE DYNAMICS
Manifolds and lobes help us to answer the question, do particles cross these barriers?
They determine the routes through which a barrier may be crossed

12. TRANSPORT IN TIME DEPENDENT FLOWS: LOBE DYNAMICS
Hyperbolic trajectory
Stable manifold
Unstable manifold
Day=682
Day=701
Day=687
Day=695

13. TRANSPORT IN TIME DEPENDENT FLOWS: LOBE DYNAMICS
Hyperbolic trajectory
Stable manifold
Unstable manifold
Day=664
Day=683
Day=675
Day=680

14. TRANSPORT IN TIME DEPENDENT FLOWS: LOBE DYNAMICS

15. TRANSPORT IN TIME DEPENDENT FLOWS: LOBE DYNAMICS
If particles transport heat, can this particle evolution be correlated to temperature fields such as those taken from satellites. Do manifolds trace these images?

16. CONCLUSIONS c
We have computed manifolds in a realistic turbulent velocity field of the Mediterranean Sea
Manifolds define Lagrangian barriers either for currents or eddies and they trace transport across those barriers.
Manifolds are related to either temperature or salinity scalars.