Topographic waves in a stratified elliptical basin, with application to the Lake of Lugano

Abstract

A theory is presented for the wind generation of topographic waves in a stratified, elongated lake. The stratification is modelled by a two-layer fluid with upper layer thickness D 1 , < D 2 , the characteristic lower layer depth. It is shown by means of scale analysis that to O(D 1 / D 2 ), the governing pair of equations for the barotropic stream function ]s and interfacial displacement £ decouple, with Jr satisfying the linearized potential vorticity equation, which includes wind stress forcing. The solution of this equation is then substituted into the governing equation for £, which can now be solved in terms of the known forcing x]s. The solutions for Jr and £ are next used to find u 1 and u 2 , the upper and lower layer velocities respectively. To O(D 1 / D 2 ) the velocity field u 2 is barotropic, whereas u 1 consists of a barotropic part, a baroclinic part, and a directly forced wind component. The horizontal currents are generally surface intensified. The general theory described is applied to an elliptically shaped basin in which the shoreline and depth contours form a family ofconfocal ellipses. For an exponential depth profile, simple solutions for the radial and azimuthal barotropic modes describing free, elliptically travelling waves are obtained by using a variational method. The corresponding solutions for £, u 1 and u 2 can be expressed in terms of elementary functions provided r 2 1 < L 2 , where r 1 is the two-layer internal Rossby radius of deformation and L is the half-length of the lake. The theory is applied to the Lake of Lugano, Switzerland, where a distinct 74 h oscillation in both the temperature and current records was observed during summer 1979. It is shown that the period, direction of phase propagation, isotherm-depth amplitude distribution along the lake, and horizontal current motion associated with this signal are consistent with the model of a free, fundamental-mode topographic wave that travels anticlockwise around an elliptical basin representative of the Lake of Lugano. From scaling arguments and cross-spectral analysis it is also demonstrated that the energy source for this wave is probably the wind along the shore at the southern end of the lake. However, a complete solution to the forced problem with the observed wind spectrum as input is not given. The existence of topographic waves (rotational modes) in the Lake of Lugano is surprising, considering the small horizontal dimensions of this Alpine lake (approximately 17 km x 1.5 km). Thus it would be of considerable interest to apply the theory developed here to other intermontane lakes to see whether low-frequency topographic waves of this scale are a common occurrence in nature.