Stochastic Upper Bounds in the Lorenz Equations and Applications to Geophysical Flows and Data.

Abstract:

Many geophysical systems are described by chaotic nonlinear systems of equations. A natural question arising is the relative roles of chaos and stochastic fluctuations. We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh-B\'enard convection the upper bounds are for heat transport versus Rayleigh number. As might be expected the stochastic upper bounds are larger than the deterministic counterpart of Souza & Doering (2015). but their variation with noise amplitude exhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude. However, in the chaotic regime this monotonicity depends on the number of realizations in the ensemble; at a particular Rayleigh number the bound may increase or decrease with noise amplitude. The origin of this behavior is the coupling between the noise and unstable periodic orbits, the degree of which depends on the degree to which the ensemble represents the ergodic set. This is confirmed by examining the close returns plots of the full solutions to the stochastic equations. Finally, we note that these solutions demonstrate that the effect of noise is equivalent to the effect of chaos. Such results have a wide range of implications in geophysical fluid dynamics.