Normal modes of an incompressible and stratified fluid model including the vertical and horizontal components of coriolis force

Numerical solutions of the normal modes of a linearized Boussinesq fluid model are studied with respect to the variable static stability in height z, the Brunt-Väisälä frequency N(z). The model includes both the vertical and horizontal components of Coriolis force. Our aim is to explore the characteristics of the little-known class of normal modes, referred to as the ‘boundary-induced inertial (BII)’ modes, in addition to the traditional inertio-gravity (IG) modes. Two kinds of finite-difference schemes are used to set up the eigenvalue-eigenvector matrix problem for crosscheck.

Numerical results of the characteristics of IG and BII modes are presented for the case of an exponential form of N(z), together with the cases of constant N, whose exact solutions are used to check the performance of the numerical models. Because the frequencies of BII modes are close to the inertial frequency, which is a singular point of the model, the eigenfunctions of BII modes become highly oscillatory in z for a large value of N. It is shown that, under a realistic profile of N(z) in the oceans, the BII modes can appear complementary to the IG modes throughout the domain, suggesting that the joint consideration of both the IG and BII modes is necessary to understand the near-inertial oscillations in the seas.