Unstable Periodic Orbits (UPOs) and Attractor
of the Barotropic Ocean Model.

Numerical method for detection of unstable periodic orbits on attractors of nonlinear models is proposed. This method requires the similar techniques as the data assimilation does. This fact facilitates its implementation for geophysical models. This method was used to find numerically several low-period orbits for the barotropic ocean model in a square. Some numerical particularities of application of this method are discussed.

The knowledge of periodic orbits of the model helps to explain some of these features like clear bimodality of PDFs of principal parameters. These PDFs have been reconstructed as weighted averages of periodic orbits with weights proportional to the period of the orbit and inversely proportional to its sum of positive Lyapunov exponents.

The fraction of time spent in the vicinity of each periodic orbit has been compared with its instability characteristics. The relationship between these values shows the 93\% correlation. The attractor dimension of the model has also been approximated as a weighted average of local attractor dimensions in vicinities of periodic orbits.

Three UPOs are presented below: one always skew-symmetric orbit wit respect to the middle line with T=38 days; one orbit with skew-symmetric average with T=41 days; and one orbit with no symmetry with T=216 days.

Some pictures
Click images to see the streamfunction evolution (animated gif files, about 500K)
Energy vs Enstrophy for UPO T=38 days Energy vs Enstrophy for UPO T=41 days
Energy vs Enstrophy for UPO T=216 days
Click the image to see the streamfunction evolution (animated gif file, about 2300K)

For more information,
see E. Kazantsev. Unstable Periodic Orbits and Attractor of the Barotropic Ocean Model. Nonlinear Processes in Geophysics, 1998, 5, pp.193-208, (pdf file) .
and E. Kazantsev. Unstable periodic orbits and Attractors of Non Linear Dynamical Models. XXIII General Assembly of the European Geophysical Society. Nice, April 1998. Slides ( .ps.gz file).


Last modified the 10 of July 2000