It is fascinating to learn that the analogy between gyroscopes and rotating shallow-water (RSW) dynamics extends so far. We were unaware of the original work on gyroscope–fluid dynamics analogies of Arnol'd or Obukhov or of the interesting recent developments by Gluhovsky, Tong, and their collaborators.
Our focus was on the analogies between gyroscopic motion (precession and nutation) and linear RSW modes dominated by Coriolis forces (geostrophic flow and inertial oscillation, respectively). We were aware that close similarities exist between the nonlinear Euler equations for the gyroscope [(S19) and (S20) in our supplement] and for the nonlinear equations describing a frictionless particle sliding on a rotating sphere [(S32) and (S33)]. However, we had no idea that the equations for a system of coupled Volterra gyrostats provide a generic basis to construct nonlinear low-order models (LOMs) of fluid dynamics.
Such LOMs with quadratic nonlinearity are very common in geophysical fluid dynamics. They apply to a vast range of phenomena because the ubiquitous convective acceleration is a quadratic nonlinearity. Some important oceanic and atmospheric processes involve other nonlinearities, however, like radiative transfer, phase changes, or the seawater equation of state. It is unclear if coupled Volterra gyrostats extend to include LOMs of these cases.
Our initial suspicion that a link exists between RSW dynamics and gyroscopes was based on intuition and play. Perhaps the coupled Volterra gyrostat–fluids analogy can be exploited further to construct real physical machines that demonstrate other principles of ocean/atmosphere fluid dynamics—that would be a thought-provoking application of the important discoveries that Gluhovsky and Tong describe.