On the amphidromic structure of inertial waves in a rectangular parallelepiped

Enclosed rotating homogeneous fluids support waves that are restored by Coriolis forces. These waves propagate obliquely through the fluid under a fixed angle with respect to the rotation axis which is set by the ratio of the frequency of the wave and the inertial frequency (twice the angular frequency of the tank). In the particular case that boundaries are either parallel, or perpendicular to the rotation axis, eigenmodes may be expected. An example is Kelvin's (1880) axial can. However, the spatial structure of these waves can still be quite complicated, as shown here for waves in a 'horizontal' rectangular parallelepiped. The container's rigid horizontal top and bottom surfaces allow for modal expansion in the vertical, each mode being governed by the 'inertial' analogs of surface Poincaré and Kelvin waves. Combining these, as in the Taylor problem of reflecting surface Kelvin waves, one finds both the eigenfrequencies as well as the typical amphidromic structure (phase lines circling around nodal points) of the resulting inertial wave pattern in dependence of the remaining two parameters: the horizontal and vertical aspect ratios. This entails the approximative determination of eigenvalues of an infinite matrix equation, obtained with the Proudman–Rao method. It appears that certain specific frequencies persist as eigenfrequency for different aspect ratios, albeit belonging to different eigenmodes. Surprisingly, these correspond to the eigenfrequencies of an infinitely long channel of waves lacking along-channel structure. Inertial wave patterns are presented by means of the amplitude and phase of the internal elevation field (related to the axial velocity component) and of the two horizontal velocity components. These reveal predominant anticyclonic turning of phase and currents; maximum elevation and current amplitudes detached from the solid walls; and, occasionally, a nearly sloshing type of response (nodal lines, instead of points).