An extended channel model for the prediction of motion in elongated homogeneous lakes. Part 1. Theoretical introduction

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Abstract
Taking account of slenderness of many lakes, a hydrodynamic model is developed. The three-dimensional differential equations are formulated in a curvilinear Co-ordinate system along the ‘long’ axis of the lake. Applying the method of weighted residuals and expanding the field variables with shape functions over the cross-sections, approximate equations for the fluid motion are derived. The emerging equations form a cross-sectionally discretized set of spatially one-dimensional partial differential equations in the longitudinal lake direction. At first, these channel equations are presented for unspecified fluid properties and arbitrary shape functions, leaving applications possible for inviscid or viscous fluids with arbitrary closure conditions. The channel equations are subsequently specialized for Cauchy series as shape functions. For the free oscillation the simplest channel model is shown to reduce to the classical Chrystal equation. A first-order linear channel model is deduced. It exhibits the essential features of gravitational oscillations in rotating basins, in that it provides wave-type solutions with the characteristics of Kelvin and Poincaré waves. This paper presents the derivation of the equations. Their application to ideal and real basins is deferred to several further papers.

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