The Ed Lorenz Symposium


Energy-conserving and Hamiltonian extensions of the Lorenz model

Alexander Gluhovsky, Purdue University, West Lafayette, IN

Employing low-order models (LOMs), Edward Lorenz made fundamental contributions to nonlinear dynamics and its applications in atmospheric dynamics and turbulence. His 3-mode LOM of 2-D Rayleigh-Bénard convection (the celebrated Lorenz model) has radically changed our perception of these areas and has had many implications. However, attempts to extend the Lorenz model (by increasing the number of modes beyond those studied originally by Lorenz) sometimes resulted in LOMs with unphysical behavior, as in the widely studied Howard-Krishnamurti (1986) model of convection with shear that has trajectories going to infinity. The trouble is with ad hoc truncations of the Galerkin expansions that may cause violations (in the LOM) of fundamental conservation properties of the original equations (The importance of maintaining conservation properties in LOMs has long been recognized by Lorenz.) As a remedy, it was proposed to enhance the Galerkin procedure by constructing LOMs of atmospheric circulations and turbulence in the form of coupled 3-mode nonlinear systems known in mechanics as Volterra gyrostats, based on the fact that the Lorenz model is actually the simplest Volterra gyrostat with added forcing and friction (Gluhovsky 1982). Such gyrostatic LOMs have a number of features shared with the Navier-Stokes equations: quadratic nonlinearity; at least one quadratic integral of motion (interpreted as some form of energy) in the inviscid, unforced limit and conservation of state space volume; and bounded solutions. Another advantage is their modular structure, allowing to expand the order of approximation or to include new physical mechanisms by adding on or modifying gyrostats, respectively. Examples include an improved Howard-Krishnamurti (1986) model, shell models of turbulence including the Lorenz (1972) model, LOMs for 3D Rayleigh-Bénard convection, and a LOM for the quasi-geostrophic potential vorticity equation (Gluhovsky et al., Phys. Fluids 1999; J. Atmos. Sci. 2002; Phys. Rev. E 2002).

Since in every relevant model in atmospheric dynamics, the conservative part is Hamiltonian, it was realized (e.g., Salmon 1988, Shepherd 1990) that in dynamical simplifications, an effective way to retain the fundamental conservation properties of the original system is through maintaining the Hamiltonian structure. In this presentation, it will be demonstrated how gyrostatic LOMs, thanks to their structure, become instrumental in constructing Hamiltonian LOMs. Among examples of gyrostatic LOMs reduced to a Hamiltonian form are multi-mode models: the Treve and Manley (1982) LOM of 2-D Rayleigh – Bénard convection and LOMs of 3-D Rayleigh – Bénard convection including an analog of the Lorenz model.

Poster Session 1, Lorenz Symposium Posters
Thursday, 13 January 2005, 9:45 AM-9:45 AM

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