# Far-Field Evolution Of Turbulence-Emitted Internal Waves And Reynolds Number Effects On A Localized Stratified Turbulent Flow

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In this dissertation, internal waves (IWs) and turbulence in the stably stratified ocean are studied via a series of numerical simulations. First of all, internal wave beams that are representative of high-mode internal tide originated from the ocean topography and constituent element of turbulence-emitted IWs are studied via direct numerical simulations (DNS), with an emphasis on their reflection at the sea surface as modelled by a free-slip rigid lid. Nonlinear effects due to wave-wave interaction, such as mean flow and harmonics, are investigated; in particular, the amplitude of the wave-driven Eulerian mean flow is found to match the theoretical prediction of an inviscid weakly nonlinear theory. The IW beams can also degrade at late time of reflection due to parametric subharmonic instability. Subsequent particle tracking is performed based the DNS dataset, in an attempt to examine the mass transport driven by the reflecting IW beams. These fully nonlinear computations reveal a horizontal dispersion of ocean tracers with a dispersivity scaling with O(A4 ), where A is the steepness of the IW beam, while small-amplitude analysis accurate to O(A2 ) suggests an exact cancellation of Eulerian mean flow due to wave-wave interaction and the wave-driven Stokes drift. The second topic of the dissertation investigates the manifestation of submerged-turbulence-emitted IWs at the sea surface and the correlation between the IW characteristics to turbulent source of IW. The turbulent wake of a sphere of diameter D towed at speed U is investigated using three-dimensional implicit largeeddy simulations, in a linearly stratified Boussinesq fluid with buoyancy frequency N and kinematic viscosity [nu]. Six simulations are performed at Reynolds numbers Re ≡ U D/[nu] ∈ {5 x 103 , 105 } and Froude numbers Fr ≡ 2U/(N D) ∈ {4, 16, 64}, with the wave-emitting wake located at a fixed distance of 9D below the surface. As the wake evolves for up to O(300) units of buoyancy time scale 1/N , IW characteristics, such as horizontal wavelength [lamda]H and wave period T , are sampled at the sea surface via wavelet transforms of surface horizontal divergence signals. The statistics of amplitudes and orientations of IW-induced surface strains are also reported. The normalized mean observable wavelength [lamda]H /D at the sea surface decays in time as (N t)[-]1 , which is due to the waves' dispersion, the dominant process in the far-field, and is in agreement with a linear propagation model that is independent of the wake Re and Fr. This agreement suggests that, within the Re range considered, the most energetic waves impacting the surface originate from the early-time wake that is adjusting to buoyancy. Questions remain about the efficiency of late-time buoyancy driven stratified turbulence in radiating waves with ˆ considerable energy content. The most energetic wavelength [lamda]H , when normalized by D, is found to scale as Fr 1/3 and decrease with Re, which causes the arrival time (in N t units) of the strongest waves at the surface to scale as Fr [-]1/3 and ˆ increase with Re. This wavelength [lamda]H is also found to correlate with the vertical integral scale, V, of the wake turbulence. IW-driven phenomena at the surface that are of interest to an observer, such as the local enrichment of surfactant and the transport of ocean surface tracers, are also discussed. The local enrichment ratio of surfactant scales linearly with the steepness of IWs that reach the surface and often exceeds a possible visibility threshold. The nonlinear Lagrangian drifts of ocean tracers create a local divergence in lateral mass transport right above the wake centreline, an effect that intensifies strongly with increasing Fr. The final portion of the dissertation focuses on massively-parallel, implicit large-eddy simulations of stratified towed-sphere wakes at Re = 4 x 105 , a previously unattained Reynolds number of such flow. The analyses focus on the vortical structures within the wake, evolution of mean flow, turbulent length scales and turbulent viscosities. The key finding is that, the wake Reynolds number Re has a significant impact on the evolution of the dynamically critical buoyancy Reynolds number, R, i.e., as Re increases, the R at a given dimensionless time N t is higher, and thus the transition from the inviscid regime at R > 1 to the viscous regime at R < 1 is delayed to a higher value of N t. As R is found to be linked to the local Richardson number R ~ Ri[-]1 , the delay of R dropping below O(1) implies loc a delay of Riloc reaching beyond O(1), the latter being the requirement for the buoyancy-driven shear layers to restabilize. The prolongation of shear instabilities due to higher R in a higher-Re wake during the NEQ regime is confirmed through visual observations of the vortical structures and estimates of horizontally averaged Riloc . During both the inviscid and viscous regimes, the theoretical scalings of the vertical integral scale seem to hold. In order to improve upon an existing self-similarity model of the mean wake evolution, turbulent viscosities in both horizontal and vertical directions are estimated from the LES data set. The results question the validity of the constant turbulent viscosity in the horizontal direction and zero turbulent viscosity in the vertical (after N t = 2) that were assumed in the self-similarity model. When parameterized by the buoyancy Reynolds number R, the vertical turbulent viscosity, albeit insignificant after N t [ALMOST EQUAL TO] 10 in the average sense, seems to be comparable to the prediction by a canonical diapycnal diffusivity model for a range of R values.

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Philpot,William Douglas