For a slow mode amplitude representing an intense high pressure area, there is no corresponding fast mode amplitude, so the slow manifold is not global. For the full nonlinear shallow-water model the question remains if the slow manifold represents all states in superbalance. The constraint manifold is the part of phase space of the balanced model that evolves on a slow timescale. The constraint manifold is given by the balance conditions which give a algebraic constraint. The constraint manifold is an exact manifold.
Balanced motion is motion on the constraint manifold with fast modes slaved to slow modes.
Unbalanced motion is motion off the constraint manifold which is now understood to be related to Lighthill radiation or spontaneous emission of fast modes. In a linear balanced model, balanced motion is along the slow mode axis, and unbalanced motion is along the fast mode axis. The Hamiltonian balanced models are based on a variational formulation of the parent model that satisfies particle relabeling symmetry. The balanced dynamics automatically satisfies energy, angular momentum and potential vorticity conservation. The balanced conditions are not given explicity.