Two dimensional normal modes in arbitrary enclosed basins on a rotating Earth: application to Lakes Ontario and Superiror

Abstract

A method for determining the free periods of oscillation of an arbitrary enclosed homogeneous water body on a rotating earth is considered. Bathymetry and shape of the water body are taken into account. The oscillations are quasi-static and horizontally two dimensional. Analytical foundation of the theory is based upon a method developed by Proudman (1916). The method requires the construction of two sets of orthogonal functions; one set satisfies a condition of vanishing normal derivative on the boundary and the other set of functions have a zero value on the boundary. These orthogonal functions are numerically constructed for two real water bodies. The numerical orthogonal functions are used as a basis for the expansion of velocity and height fields. The expansion coefficients are then determined so as to satisfy the dynamical equations. The coefficients appear as eigenvectors of a Hermitian matrix. The corresponding eigenvalues represent the frequencies of oscillation. Structures are determined by numerical evaluation of the velocity and height field expansions. Application of the above procedure to Lake Ontario gives for the lowest gravitational mode a period of 5.11 h and for Lake Superior, the period of the corresponding mode is 7.86 h. Periods of the lowest six gravitational modes and their structures in both lakes are presented. Comparison of Lake Superior calculations with the data analysis of Mortimer & Fee (1976, preceding paper) shows very good agreement. A few examples of rotational modes are also presented.