2007/05/31 by Poul Olesen, Mogens H. Jensen, Mogens H Jensen
Engineering · Physics and Astronomy · #Fluid Dynamics and Turbulent Flows #Fluid dynamics and aerodynamics studies #Quantum chaos and dynamical systems #hep-th #nlin.CD
paper · pdf · doi:10.1088/0951-7715/20/10/004
published as Nonlinearity 20:2333-2352,2007 · 25 pages, 7 figures
arxiv created 2007/06/07 · openalex publication_date 2007/08/28 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
We derive exact analytical solutions of the GOY shell model of turbulence. In the absence of forcing and viscosity we obtain closed form solutions in terms of Jacobi elliptic functions. With three shells the model is integrable. In the case of many shells, we derive exact recursion relations for the amplitudes of the Jacobi functions relating the different shells and we obtain a Kolmogorov solution in the limit of infinitely many shells. For the special case of six and nine shells, these recursion relations are solved giving specific analytic solutions. Some of these solutions are stable whereas others are unstable. All our predictions are substantiated by numerical simulations of the GOY shell model. From these simulations we also identify cases where the models exhibit transitions to chaotic states lying on strange attractors or ergodic energy surfaces.