A Theory of Vertical Modes in Multilayer Stratified Fluids

Abstract

theory of vertical modes in multilayer stratified fluids is proposed. A matrix notation is introduced to generalize the theory to an arbitrary number of layers. The properties of vertical modes including their orthogonality are derived. The linearized shallow-water equations and total energy equation are then decomposed into vertical modal components in the presence of large horizontal variations in bottom elevation and stratification, which induce linear interactions among vertical modes. The theory is applied to the derivation of vertical modes in three-layer stratified fluids and an internal tide generation problem under two-layer stratification. It is shown that the proposed theory correctly approaches the continuous counterpart as the number of layers increases.