On the existence of solutions to Machenhauer's non-linear normal mode initialization

The existence or solutions to the non-linear normal mode initialization proposed by Machenhauer is examined in a low-order version of a shallow water model on an equatorial β-plane. The model contains only three modes: one Rossby mode and two gravity type modes. Simple physical forcing is included in the model. The analysis shows that generally there is more than one state that satisfies the initial conditions. Only one, however, can be accepted as a realistic initial state. Furthermore, in the case without non-adiabatic forcing, the iterative non-linear normal mode procedure can converge only to the realistic initial state. When the Rossby amplitude is increased beyond a critical value, the realistic initial state ceases to exist. The critical value of the Rossby amplitude decreases when the fluid becomes more shallow. Non-adiabatic forcing may also violate the existence of the realistic initial state. The critical forcing necessary to do this, decreases with decreasing depth of the fluid.