Local Lyapunov exponents of the QG ocean dynamics The predictability of the quasigeostrophic ocean model is considered. This is a simple dynamical model which assumes the ocean depth to be divided into N layers of different water density. This model is driven by the wind stress on the surface, it includes effects of lateral and bottom friction, of the Earth rotation and the nonlinear interaction between adjacent layers.
Internal instability of the system leads to the rapid divergence of its trajectories limiting time of deterministic prediction of the system. To estimate the divergence rate of trajectories we compute Lyapunov exponents of this system. Along with Lyapunov exponents, their generalization, local Lyapunov exponents, are computed as a measure of divergence rate on finite time scales. These exponents provide us with the information about principal mechanisms of local instability. They can be useful in the forecasting of the forecast skill in meteorology and oceanology.
Numerous studies are performed in order to understand the influence of physical (form of the basin, number of layers) and numerical (grid resolution) parameters on the predictability of the system. To study the geographical structure of the perturbation, the distribution of initial error with the fastest norm growth is distinguished. This distribution shows us the initial data structure the most sensible to errors and points out the regions where the most accuracy should be required. The structure of this function depends much on the time scale of prediction.
This function is presented below for 1 day and two weeks time scales.
The most unstable mode for a shorttime currents prediction in North Atlantics. It indicates the regions where initial error exhibit the most rapid growth during one day.  The most unstable mode for a longtime currents prediction in North Atlantics. It indicates the regions where initial error exhibit the most rapid growth during 15 days. 
Click the pictures to see their variability (animated gif files, about 500K)  

Eugene.Kazantsev@imag.fr
Christine.Kazantsev@imag.fr