Estimating the Dimensions of Weather and Climate Attractors

The dimensions of attractors are estimated from phase space trajectories of observed weather and climate variables (local surface pressure and relative sunshine duration, zonal wave amplitude; a δ18O-record). They provide primary information for descriptions of properties of the attractors of dynamical systems and give a lower limit to the number of the essential variables necessary to model the dynamics. These estimates are based on distance distributions of pairs of points on the single variable trajectory evolving in phase spaces which embed the attractor. One observes a low fractal dimensionality between three and four for the weather attractor, if interannual variability and seasonal changes are eliminated. The physical interpretation is based on the three dominating scales of cyclones, cyclone families and index-cycle; the irregularity of the flow and strong dependence on initial conditions amount for the fractal value. The climate variable also reveals a low dimensionality (between four and five) of the climate attractor. This is supported by an independent estimate based on eigenfunction expansion of the embedded phase space trajectory. These types of analyses suggest how to extend the standard data evaluation and model verification techniques to an analysis of the phase space behavior of observed and simulated dynamical systems.