Regimes in Simple Systems

Edward N. Lorenz Massachusetts Institute of Technology, Cambridge, Massachusetts

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Abstract

Dynamical systems possessing regimes are identified with those where the state space possesses two or more regions such that transitions of the state from either region to the other are rare. Systems with regimes are compared to those where transitions are impossible.

A simple one-dimensional system where a variable is defined at N equally spaced points about a latitude circle, once thought not to possess regimes, is found to exhibit them when the external forcing F slightly exceeds its critical value F* for the appearance of chaos. Regimes are detected by examining extended time series of quantities such as total energy. A chain of k* fairly regular waves develops if F < F*, and F* is found to depend mainly upon the wavelength L* = N/k*, being greatest when L* is closest to a preferred length L0. A display of time series demonstrates how the existence and general properties of the regimes depend upon L*.

The barotropic vorticity equation, when applied to an elongated rectangular region, exhibits regimes much like those occurring with the one-dimensional system. A first-order piecewise-linear difference equation produces time series closely resembling some produced by the differential equations, and it permits explicit calculation of the expected duration time in either regime. Speculations as to the prevalence of regimes in dynamical systems in general, and to the applicability of the findings to atmospheric problems, are offered.

Corresponding author address: Edward N. Lorenz, Dept. of Earth, Atmospheric, and Planetary Sciences, Room 54-1622, Massachusetts Institute of Technology, Cambridge, MA 02139. Email: jmsloman@mit.edu

Abstract

Dynamical systems possessing regimes are identified with those where the state space possesses two or more regions such that transitions of the state from either region to the other are rare. Systems with regimes are compared to those where transitions are impossible.

A simple one-dimensional system where a variable is defined at N equally spaced points about a latitude circle, once thought not to possess regimes, is found to exhibit them when the external forcing F slightly exceeds its critical value F* for the appearance of chaos. Regimes are detected by examining extended time series of quantities such as total energy. A chain of k* fairly regular waves develops if F < F*, and F* is found to depend mainly upon the wavelength L* = N/k*, being greatest when L* is closest to a preferred length L0. A display of time series demonstrates how the existence and general properties of the regimes depend upon L*.

The barotropic vorticity equation, when applied to an elongated rectangular region, exhibits regimes much like those occurring with the one-dimensional system. A first-order piecewise-linear difference equation produces time series closely resembling some produced by the differential equations, and it permits explicit calculation of the expected duration time in either regime. Speculations as to the prevalence of regimes in dynamical systems in general, and to the applicability of the findings to atmospheric problems, are offered.

Corresponding author address: Edward N. Lorenz, Dept. of Earth, Atmospheric, and Planetary Sciences, Room 54-1622, Massachusetts Institute of Technology, Cambridge, MA 02139. Email: jmsloman@mit.edu

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