The predictability of a flow which possesses many scales of motion

TLDR

The study proposes that deterministic fluid systems with many scales of motion become observationally indistinguishable from indeterministic systems, because small initial errors grow to random‑like differences within a finite time that cannot be extended by reducing the error magnitude. The authors employ a simple mathematical model derived from the two‑dimensional incompressible vorticity equation, numerically integrating an error‑energy equation for scales comparable to the Earth's atmosphere to evaluate predictability. Each scale of motion has a finite predictability range: cumulus‑scale motions about one hour, synoptic‑scale a few days, and the largest scales a few weeks. DOI: 10.1111/j.2153-3490.1969.tb00444.x.

Abstract

It is proposed that certain formally deterministic fluid systems which possess many scales of motion are observationally indistinguishable from indeterministic systems; specifically, that two states of the system differing initially by a small “observational error” will evolve into two states differing as greatly as randomly chosen states of the system within a finite time interval, which cannot be lengthened by reducing the amplitude of the initial error. The hypothesis is investigated with a simple mathematical model. An equation whose dependent variables are ensemble averages of the “error energy” in separate scales of motion is derived from the vorticity equation which governs two-dimensional incompressible flow. Solutions of the equation are determined by numerical integration, for cases where the horizontal extent and total energy of the system are comparable to those of the earth's atomsphere. It is found that each scale of motion possesses an intrinsic finite range of predictability, provided that the total energy of the system does not fall off too rapidly with decreasing wave length. With the chosen values of the constants, “cumulus-scale” motions can be predicted about one hour, “synoptic-scale” motions a few days, and the largest scales a few weeks in advance. The applicability of the model to real physical systems, including the earth's atmosphere, is considered. DOI: 10.1111/j.2153-3490.1969.tb00444.x