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A weakly nonlinear evolution model that accounts for multi-modal interaction in a small, continuously stratified lake of variable depth is derived. In particular, an evolution model for the first two vertical modes in a lake that is sub- ject to wind stress forcing is numerically simulated. Defining modal energies, energy transfer between the first and the sec- ond vertical modes is calculated for several different forms of the density stratification. Modal energy transfer mainly occurs during reflection of mode-one waves at the vertical end walls, and it is shown that the amount of energy trans- fer from the first to the second mode is greatly dependent on the shape of the stratification profile. Also, the initial modal energy partition at the wind setup is shown to depend signif- icantly on the penetration depth of the internal shear stress induced by the wind stress, especially if the stress distribu- tion extends into the upper levels of the metalimnion.

A weakly nonlinear model for multi-modal evolution of wind-generated long internal waves in a closed basin