Eight definitions of the slow manifold: seiches, pseudoseiches and exponential smallness

Abstract

Abstract The slow manifold is the Holy Grail of initialization schemes for weather forecasting, a hypothetical subspace of the model's phase space which is free of high-frequency (and forecast-wrecking) gravity waves. Using new machinery, we analyze seven previous definitions of the slow manifold and one definition original to this work. One new tool is the mathematics of nonlocal solitary wave theory. The traditional definition of a slow manifold is too restrictive, but a generalization in the spirit of nonlocal soliton theory is computable. Another novelty is the conceptual distinction between ‘seiche’ and ‘pseudoseiche’. The former are free, unforced gravity waves, whereas pseudoseiches are forced gravity mode oscillations which mimic seiches over a finite time interval. We show through a linear model that it is possible to remove seiches by initialization, even though the usual algorithms are divergent. However, intermittent gravity waves, i.e. pseudoseiches, appears as inexorable as death and the tides.