23 October 2000 Ž . Physics Letters A 275 2000 435–446 www.elsevier.nlrlocaterpla A unified view of Kolmogorov and Lorenz systems Antonello Pasini a, ) , Vinicio Pelino b a ( ) CNR – Istituto sull’Inquinamento Atmosferico, Via Salaria Km. 29.300, I-00016 Monterotondo Stazione Roma , Italy b SerÕizio Meteorologico dell’Aeronautica, CNMCA – Aeroporto A De Bernardi B , Via di Pratica di Mare, ( ) I-00040 Pratica di Mare Roma , Italy Received 16 December 1999; received in revised form 28 June 2000; accepted 14 September 2000 Communicated by A.P. Fordy Abstract The discussion on the relation between the Kolmogorov system, considered as low-order approximation of Navier–Stokes equations, and the well-known Lorenz equations is still not completely understood. In this Letter, referring to the mathematical theory of motion on Lie groups, a particular class of Kolmogorov systems, largely studied in low-dimensional models of geophysical fluid dynamics, is extended and analysed in its geometric and dynamical features. The dynamical behaviour of this extended and unifying system generally shows chaos, contrarily to the original Kolmogorov one, and actually two well known Lorenz models, useful as toy-models in geophysical fluid dynamics, are included in it. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.40.Gc; 05.45 Keywords: Kolmogorov system; Lorenz attractors 1. Introduction The failure in forecasting the dynamics of the atmospheric motion in an extended temporal range is always ascribed by the meteorological community to two principal reasons: the great number of feedback due to the non-linear physics of the system described wx by a set of so-called primitive equations 1 , whose dynamics is driven by the Navier–Stokes equations, and the uncertainty of its initial conditions. Even for a simpler system, like a perfect fluid, these facts ) Corresponding author. Tel.: q39 06 90672274; fax: q39 06 90672660. Ž . E-mail address: pasini@iia.mlib.cnr.it A. Pasini . were established in a mile-stone theoretical article by wx Arnold 2 , who described the Euler’s equation for an incompressible fluid filled in a Riemannian manifold 2 w x D, which for a planet becomes S = 0,1 , as a geodesic equation on the group of volume-preserving diffeomorphisms of D. The negative sectional curva- ture of this group implies the instability of the motion and therefore the lack of its predictability and chaos. In meteorology, this behaviour is clearly visi- ble as in Fig. 1, where the ten-days evolution of a meteorological parameter is computed by adding small perturbations to the initial conditions, giving wx rise to an ensemble of possible forecasts 3 . Many attempts have been performed in order to reduce the complexity of the atmosphere to a low-di- 0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. Ž . PII: S0375-9601 00 00620-4