Inertial (gyroscopic) waves

Inertial (gyroscopic) waves in homogeneous rotating fluids:

Homogeneous, rotating fluids support waves that are restored by the Coriolis force. These waves of frequency less than twice the rotation rate are similar to internal gravity waves in uniformly-stratified, non-rotating fluids in being also transverse, and in again being constrained in propagation direction, but now relative to the rotation axis. In particular, the waves will focus when reflecting from a sloping boundary onto wave attractors. However, because particle motion is circular instead of rectilinear, as for the internal gravity waves [which mathematically translates into a requirement that the pressure field satisfies oblique-derivative (Robin) boundary conditions], these waves generate spatially complicated patterns, even when just reflecting from vertical boundaries. See for instance the patterns of inertial wave energy distribution in a horizontal cube, shown in a plan view (horizontal mid-plane) in the left-hand corner of these web pages. Both stratified as well as rotating fluid systems (and also their oceanographically relevant combination) can accommodate waves of arbitrary frequency, thus denying the existence of eigenmodes. The spatial structure of these waves is determined by the Poincare equation, which is hyperbolic in spatial coordinates, and which is responsible for the unusual behaviour of these waves. See the spatial pattern of the observed current magnitudes (low/high presented as blue/red colors) and the redicted attractors (solid line) or standing mode (dashed line, upper right figure). These figures present side views taken from Astrid Manders' thesis, and the first reference below.
Some short movies reveal the time-dependent inertial wave. A mean flow is generated above the location where the waves approach an attractor. This can be discerned in the following movie (in Microsoft .WMV format) taken from above: Movie for Dial-up connection: 38Kbps (160120 / ~710 kB) Movie for ISDN connection: 48 Kbps (160120 / ~875 kB) Movie for Broadband connection: 150 Kbps (320*240 / ~2.5 MB) In the lower/upper half of this frame the bottom is flat/ sloping upwards. It displays a steady, cyclonic dye displacement, presumably driven by the inertial waves that are forced by the modulation of the rotation (which itself is responsible for the observed periodic part of the motion). The inertial waves are geometrically focused by the sloping bottom. The mean flow that results from the angular momentum mixing that ensues is particularly strong over the middle part of the sloping region (3/4 upwards from the lower boundary of this frame) where the theoretical attractor reflects from the slope (and focusing occurs).
Publications:
Manders, A.M.M., Maas, L.R.M. (2004) On the three-dimensional structure of the inertial wave field in a rectangular basin with one sloping boundary. Fluid Dynamics Research 35,1-21. ( Download PDF-file ) Manders, A.M.M., Maas, L.R.M. (2003) Observations of inertial waves in a rectangular basin with one sloping boundary. Journal of Fluid Mechanics 493, 59-88. ( Download PDF-file ) Maas, L.R.M. (2003) On the amphidromic structure of inertial waves in a rectangular parallelepiped. Fluid Dynamics Research, 33, 373-401. ( Download PDF-file ) Maas, L.R.M. (2001) Wave focusing and ensuing mean flow due to symmetry breaking in rotating fluids Journal of Fluid Mechanics, 437: 13-28. ( Download PDF-file )
User Report:
Coriolis turntable experiments - Mean flow generation by geometrically focused gyroscopic waves
Links to related sites:
Joel Sommeria at rotating lab Coriolis/LEGI, Grenoble Michel Rieutord Boris Dintrans