Material dispersion in the ocean, such as the dispersion of natural and anthropogenic tracers (e.g. nutrients, dissolved gases, pollutants), is important in understanding processes at a variety of scales, ranging from plankton production to climate variability. Material dispersion is controlled by many dynamic processes; the present research focuses on the 3D material dispersion by ocean eddies and waves (inertial waves and internal gravity waves).;Ocean eddies may suffer various hydrodynamic instabilities, such as barotropic instability, inertial instability and 3D instability. In this work, I investigate how instabilities impact the 3D material dispersion by ocean eddies, using analytical methods and numerical simulations. I discover for the first time that material can be exchanged through 3D pathways which link a family of vortices generated by the instabilities of a single, initially unstable eddy. I also show that instabilities can increase the magnitude of vertical velocity and mixing rate. Besides, I find that instabilities can cause the kinetic energy wavenumber spectrum to have a power-law regime different with the classic regimes of k--5/3 and k --3, and propose a new energy spectrum to interpret the non-classic regime.;Inertial waves can arise in rotating homogeneous fluids. By numerically simulating an initially unstable eddy, I discover for the first time a special kind of inertial waves, which are emitted in a spiral manner from the eddies; I refer to these waves as spiral inertial waves (SIWs). SIWs appear at small Rossby numbers (0.01 ≤ Ro ≤ 1) according to our parameter sweep experiments; the amplitude, wavelength and frequency of SIWs are sensitive to Rossby numbers. I extend the theory of Lighthill-Ford radiation into inertial waves, and propose an indicator for the emission of inertial waves; this indicator may be adopted into general circulation models to parameterize inertial waves. Additionally, when releasing passive tracers into the wave field, SIWs organize tracers into spirals, and modify the tracer's local rate of change by advecting tracers vertically. Further, the spirals of SIWs resembles some spiral features observed in the ocean and atmosphere, for example, spiral ocean eddies and spiral hurricane rainbands; thus, SIWs may offer another mechanism to form spiral eddies and rainbands. Since no density anomaly is required to generate the spirals of SIWs, I infer that the density anomaly, hence the baroclinic or frontal instability, is unlikely to be the key factor in the formation of these spiral features.;Internal gravity waves are ubiquitous in the ocean; they can transport nutrients, pollutants, sediments, etc. Using numerical simulations of internal waves that are initialized with the Garrett-Munk spectrum, I investigate the material dispersion by internal waves; the dispersion regimes are identified in terms of two metrics, the relative dispersion and finite-scale Lyapunov exponent (FSLE). The metric of relative dispersion reveals that dispersion regime by internal waves is between ballistic and diffusive regimes; while, the metric of FSLE indicates that the regime is ballistic. Besides, I show that internal waves below an upper mixed layer can generate flows in the mixed layer, leading to material dispersion. The dispersion produced by both internal waves and mixed layer eddies is also examined.
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