Nonlinear Dynamics

, Volume 61, Issue 1–2, pp 101–107 | Cite as

Symmetry justification of Lorenz’ maximum simplification

  • Alexander Bihlo
  • Roman O. Popovych
Original Paper


In 1960 Edward Lorenz (1917–2008) published a pioneering work on the ‘maximum simplification’ of the barotropic vorticity equation. He derived a coupled three-mode system and interpreted it as the minimum core of large-scale fluid mechanics on a ‘finite but unbounded’ domain. The model was obtained in a heuristic way, without giving a rigorous justification for the chosen selection of modes. In this paper, it is shown that one can legitimate Lorenz’ choice by using symmetry transformations of the spectral form of the vorticity equation. The Lorenz three-mode model arises as the final step in a hierarchy of models constructed via the component reduction by means of symmetries. In this sense, the Lorenz model is indeed the ‘maximum simplification’ of the vorticity equation.

Lie symmetries Point symmetries Discrete symmetries Induced symmetries Barotropic vorticity equation Lorenz model 


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.Institute of Mathematics of NAS of UkraineKyivUkraine

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