A KdV Model for Multi-Modal Internal Wave Propagation in Confined Basins

The derivation of a reduced dynamics describing multimodal evolution of long internal waves in a confined basin is presented. The model is rationally extracted from the primitive equations of motion and preserves the essential physics of bi-directional propagation of non-hydrostatic, weakly nonlinear wave motions. Even though evolution in closed, one-dimensional domains necessarily involves fields possessing both left- and right-running characteristics for each mode, it is shown that a computationally-efficient reduction of the fundamental system to a set of KdV equations, without imposing any prejudice regarding the direction of propagation, can be accomplished. The specific case for two modes leading to just a pair of KdV equations, and which captures the leading-order effects of both co- and counter-propagating wave components for both self- and cross-modal interactions, is described. Both cases of free response from prescribed initial conditions and forced response via excitation by wind stresses at the exposed upper surface are equally accessible by the model. It is argued that the resulting model comprises an efficient, rapid-simulation tool that faithfully captures essential physics relating to the energy-containing scales of motion in lakes and reservoirs, a tool with considerable practical relevance toward the goal of providing a physics-based guide for resource management of lakes and reservoirs.