On the Lorenz '96 Model and Some Generalizations

John Kerin, Hans Engler

Published 2020-05-15Version 1

In 1996, Edward Lorenz introduced a system of ordinary differential equations that describes a single scalar quantity as it evolves on a circular array of sites, undergoing forcing, dissipation, and rotation invariant advection. Lorenz constructed the system as a test problem for numerical weather prediction. Since then, the system has also found widespread use as a test case in data assimilation. Mathematically, it belongs to a class of systems with a single bifurcation parameter (rescaled forcing) that undergoes multiple bifurcations and exhibits chaotic behavior for large forcing. In this paper, the main characteristics of the advection term in the model are identified and used to describe and classify a number of possible generalizations of the system. A graphical method to study the bifurcation behavior of constant solutions is introduced, and it is shown how to use the rotation invariance to compute normal forms of the system analytically. In addition, visualization approaches to depict the global bifurcation behavior are proposed. Problems with site-dependent forcing, dissipation, or advection are considered and basic existence and stability results are proved for these extensions. For some advection-only systems, explicit solutions are found. It is demonstrated how to use such versions to assess numerical schemes for the Lorenz '96 system.