Seiche modes in multi-armed lakes

Abstract

The effect of multiple arms on standing wave patterns, or seiches, present within a lake is not easy to predict. This study examines free seiche modes in fjord-type multi-armed lakes in order to generalize features of the response of those lakes. To do so, the study develops a simplified analytical model that predicts modal frequencies and associated mode-shapes based on idealized lake geometries. The model demonstrates that multi-armed lakes are subject to two different types of behavior: a full-lake response, in which all arms are active for all modes; and a decoupled response, in which seiching is constrained to only two arms of the lake for each mode. Which of the two behaviors is expressed depends on relative values of the travel time of a progressive shallow-water wave in each arm. In general, a decoupled response will exist if the mode-shape along a two-armed extent of the lake contains a node exactly at the confluence point between those two arms. We show that the period and associated mode-shape structure of the fundamental mode in multi-armed lakes conforms to that of a simple elongated lake as predicted by Merian's formula, but higher modes are highly impacted by lake geometry. Depth and width variation within the arms can lead to localization of mode-shapes, but this effect is distinct from the possible decoupled behavior. In some instances, it may be possible to apply the model to baroclinic modes which would then act as having a constant bottom depth equal to the surface layer depth.