|Understanding the Equatorial Ocean: Theoretical and Observational Studies|Rabitti, A. (2016). Understanding the Equatorial Ocean: Theoretical and Observational Studies. PhD Thesis. [S.n.]: [s.l.]. ISBN 978-94-6299-262-7. 215 pp. hdl.handle.net/1874/330921
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Understanding the behaviour of internal waves in fully enclosed domains constitutesone of the big challenges in fluid dynamics, especially because of thenumerous possible applications both in astrophysical and geophysical fluid dynamics.Since analytical solutions for internal waves in arbitrarily shaped domainsare not available, numerical approaches or geometrical ray tracing techniqueshave been widely used in order to infer properties of the underlying wavefield and energy distribution in the container. Ray tracing, commonly used intwo-dimensional settings, has recently been extended as a three-dimensionaltool, in the direction of a more realistic representation of the correspondingwave field and of the source of perturbation. However, the significance of threedimensionalinternal wave rays has not been completely understood.In this work, a three dimensional algorithm of ray tracing has been appliedfor the first time to the full homogeneous rotating sphere, in order to investigatethe nature of three-dimensional inertial wave ray orbits. The sphere in fact isone of the few domains where analytical solutions are known for the linear,inviscid case, and where WKBJ theory has also been commonly applied.Three-dimensional periodic orbits have been then observed in the sphere.Moreover, two main frequency regimes can be distinguished, according to thedifferent behaviour of the orbits: !2 < 1/2 and !2 > 1/2.In the low frequency regime, three “families” of periodic orbits are found:polygon, star and flower trajectories, classified because of their appearance intop view. In this regime, the critical circles define an inscribed cylinder thatacts as a physical barrier for the orbits, confining the rays to the equatorial belt.This crucial role played in the ray dynamics by the critical latitudes finds itscounterpart in the model proposed by Wu [2005a], where the critical circlesare named “singularity belts” and right so.In the high frequency regime, no regular pattern but of flower type is observed.However, periodic orbits are still present, even if critical circles do notplay such a prominent role.Difficulties in relating the observed periodic trajectories to the simplest eigenfrequenciescomputed using equation (4.6) (the Bryan modes) could be due to128several reasons, discussed in section 4.4. Perhaps the most intriguing one is thefact that when a three-dimensional ray tracing algorithm is used, the absenceof any assumption on separability of the solutions could unveil the structure ofintrinsically three-dimensional modes in the sphere, which remained hidden sofar because of the particular form of solutions (equation (4.5)), and because ofthe requirement for the n and k indices to be integer. This perspective certainlyrequires further investigation.We are led to think, that the periodic orbits observed so far have been limitedby our poor understanding of the geometrical properties of these trajectories,and by the human tendency to prefer regular and repetitive patterns such as,indeed, polygons, stars, and flower shapes. We also know that the inertial wavespectrum in the sphere is dense [Bryan, 1889], therefore a complete catalogueof existing periodic orbits would be of no practical use. In this work only thefirst proof of the existence of such trajectories is given: further investigationswould contribute to a more general perspective on the geometrical features ofthese trajectories, and to a complete understanding of plots like the ones infigure 4.4 and of the emerging patterns.Nevertheless, enough and differently shaped closed patterns have been foundto deduce some general properties on the periodicity of the orbits and theobservation of other trajectories would not change the overall picture of thesystem. The existence, in fact, of inertial wave, periodic orbits in a regularthree-dimensional container such as the sphere allows to extend the usual twodimensionalcorrespondence between modes and periodic ray trajectories, previouslydiscarded [Dintrans et al., 1999], to a three-dimensional setting.Moreover, the presence of repelling periodic orbits, such as the ones hereobserved, and of invariant orbits presenting ergodic aspects, points out the interestingfact that the three-dimensional hyperbolic problem of inertial wavesin the homogeneous full sphere presents, in terms of quantum chaology [Berry,1987], some properties typical of elliptic systems. The three-dimensional orbitsfound so far in the sphere seem to be only of the periodic and of the invariantkind. However, the existence of two different orbit regimes (as in the ovalbilliard [Berry, 1981]), and of isolated orbits (isolated with respect to rotationaround an axis different from the z-axis), are suggestive of a more heterogeneousdynamics. A three-dimensional phase space study will possibly providefurther insight in the hybrid nature of these trajectories.