On the Nonexistence of a Slow Manifold

We define the slow manifold S in the state space of a primitive-equation model as a hypothetical invariant manifold on which there is no gravity-wave activity. and on which unique velocity-potential and streamfunction fields correspond to each isobaric-height field. We introduce a five-variable forced damped model, and show that for this model the point H representing the Hadley circulation and the two orbits forming the unstable manifold of H must lie in S if S exists. We then show that in traveling along one of these orbits one eventually encounters gravity waves, whereupon it follows that S does not exist. A measure G of gravity-wave activity is found to decrease very rapidly as the external forcing F decreases. An approximate formula is derived for G as a function of F. We show that a particular nine-variable forced damped model with orography also fails to possess a slow manifold, and we speculate as to the existence of slow manifolds in larger and more realistic models.