Suppression of chaos at slow variables by rapidly mixing fast dynamics through linear energy-preserving coupling

Abstract

Abstract. Chaotic multiscale dynamical systems are common in many areas of science, one example being the interaction of the low-frequency dynamics in the atmosphere with the fast turbulent weather dynamics. One of the key questions about chaotic multiscale systems is how the fast dynamics affects chaos at the slow variables and, therefore, impacts uncertainty and predictability of the slow dynamics. Here we demonstrate that the linear slow-fast coupling with the total energy conservation property promotes the suppression of chaos at the slow variables through the rapid mixing at the fast variables, both theoretically and through numerical simulations. A suitable mathematical framework is developed, connecting the slow dynamics on the tangent subspaces to the infinite-time linear response of the mean state to a constant external forcing at the fast variables. Additionally, it is shown that the uncoupled dynamics for the slow variables may remain chaotic while the complete multiscale system loses chaos and becomes completely predictable at the slow variables through increasing chaos and turbulence at the fast variables. This result contradicts the common sense intuition where, naturally, one would think that coupling a slow weakly chaotic system with another much faster and much stronger mixing system would result in general increase of chaos at the slow variables.

Disentangling multi-level systems: averaging, correlations and memory

“…In [1] the averaging method is used to derive a simplified reduced model. The X dependent measure ρ Y |X is first replaced by the measure ρ Y |X * for a fixed value X * for X.…”

“…Such method allows to derive a renormalised dynamics for the X system whose trajectories converge on finite time scales to those of the original dynamics. An interesting use of the averaging method for deriving a simplified dynamics has been proposed in [1], where a sort of mean field ansatz is taken. In statical mechanics, projector operator techniques which allow to formally eliminate certain variables have found a great deal of success, as in the case of the Mori-Zwanzig equation [33].…”

Climate Change: Is It More Predictable Than We Think?

“…In this section we present the results of Abramov (2011c), where the chaotic behavior of slow variables is studied by applying the averaging formalism to the dynamics of the linearized model for the slow variables in a two-scale dynamical system with linear energy-preserving coupling. In particular, we consider a two-scale system of autonomous ordinary differential equations of the form…”

“…For simplicity of presentation, here we assume that the total energy is a weighted sum of squares of the components of x and y; the more general case with energy being an arbitrary positive-definite quadratic form is discussed in Abramov (2011c). Here note that f (x) and g(y) are not required to preserve the energy, as they might contain forcing and dissipation, which frequently happens in atmosphere/ocean dynamics.…”

“…In these variables, the energy emerges as a positive-definite quadratic form, and, therefore, the direct coupling between velocity/streamfunction fields of LFV and fast dynamics should preserve such a form to reflect the energy balance during its transfer between the large scale slow motion and fast small scale processes. In Section 2 of this chapter, we present the results of the recent study (Abramov, 2011c) how the fast small scale variables affect chaotic properties of the slow variables through linear energy-preserving coupling in a general nonlinear two-scale model with quadratic energy. White linear coupling is the simplest form of coupling between slow and fast variables, it is nonetheless common in interactions between velocity/streamfunction variables (see, for example, the model of mean flow -small scale interactions via topographic stress in Grote et al (1999), where the total energy conservation in the coupling between the zonal mean flow and small scale fluctuations is a key requirement).…”