Surface manifestation of internal waves emitted by submerged localized stratified turbulence

Abstract

The internal waves (IWs) radiated by the turbulent wake of a sphere of diameter $D$ towed at speed $U$ are investigated using three-dimensional fully nonlinear simulations performed in a linearly stratified Boussinesq fluid with buoyancy frequency $N$. The study focuses on a broad range of wave characteristics in the far field of the turbulent wave source, specifically at the sea surface (as modelled by a free-slip rigid lid) where the IWs reflect. Six simulations are performed at Reynolds number $Re\equiv UD/{\it\nu}\in \{5\times 10^{3},10^{5}\}$ and Froude number $Fr\equiv 2U/(ND)\in \{4,16,64\}$, where ${\it\nu}$ is viscosity. The wave-emitting wake is 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 ${\it\lambda}_{H}$ and wave period $T$, are sampled at the sea surface via wavelet transforms of horizontal divergence signals. The statistics of amplitudes and orientations of IW-induced surface strains are also reported. The mean dimensionless observable wavelength $\overline{{\it\lambda}}_{H}/D$ at the sea surface decays in time as $(Nt)^{-1}$, which is due to the waves’ dispersion. This observation is in agreement with a linear propagation model that is independent of the wake $Re$ and $Fr$. This agreement further suggests that the most energetic waves impacting the surface originate from the early-time wake that is adjusting to buoyancy. The most energetic dimensionless wavelength $\hat{{\it\lambda}}_{H}/D$ is found to scale as $Fr^{1/3}$ and decrease with $Re$, which causes the arrival time (in $Nt$ units) of the strongest waves at the surface to scale as $Fr^{-1/3}$ and increase with $Re$. This wavelength $\hat{{\it\lambda}}_{H}$ is also found to correlate with the vertical Taylor scale 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 examined. The local enrichment ratio of surface scalar scales linearly with the steepness of IWs that reach the surface, and the ratio often exceeds a possible visibility threshold. The Lagrangian drifts of ocean tracers, which are linked to the nonlinear interaction between incident and reflecting IW packets, create a local divergence in lateral mass transport right above the wake centreline, an effect that intensifies strongly with increasing $Fr$. The findings of this study may serve as a platform to investigate the generation and surface manifestation of IWs radiated by other canonical submerged stratified turbulent flows.

Large-scale characteristics of stratified wake turbulence at varying Reynolds number

“…Computational domain for implicit large-eddy simulation of a temporally evolving, stratified towed-sphere wake [9,10,16,18,19]. The centerline of the wake is at (y, z) = (0, 0).…”

“…Three values of Re are considered here: 5 × 10 3 , 10 5 , and 4 × 10 5 . For the two lower Re values, simulations are conducted for Fr = 4, 16, and 64; these simulations have been reported by Zhou and Diamessis [18] with a focus on the far-field evolution of turbulence-generated internal gravity waves. For the largest Re considered by Zhou [20], i.e., Re = 4 × 10 5 , the same set of Fr values was initially used to carry out the simulations.…”

“…As tabulated in Table I, which summarizes the ILESs performed, the R5 and R100 simulations are performed in wider and taller domains than the R400 simulations to allow for an investigation of wake-emitted internal waves in the wake's far field [18]. The domains for R400 simulations are made narrower (in y) and shorter (in z) to curb computational cost, as our focus is primarily on the localized wake turbulence near the wake centerline at (y, z) = (0, 0) ( Fig.…”

“…In contrast to the experiments of Spedding [5], numerical simulations by Diamessis et al [10] at Re = 10 5 , a value that is one order of magnitude larger than those obtained in the laboratory, did show qualitative differences in the flow evolution, such as a significant prolongation of the duration of the NEQ regime in higher-Re wakes, due primarily to the emergence of secondary shear instabilities within the large-scale pancake vortices. The effects of varying Re in wakes have also been seen in terms of the internal waves emitted by the wake turbulence [16][17][18] and the properties of the turbulent-nonturbulent interface [19]. A recent numerical data set [20] brings into play an additional, sufficiently removed data point for Reynolds number at Re = 4 × 10 5 .…”

Formation of Recirculating Cores in Convectively Breaking Internal Solitary Waves of Depression Shoaling over Gentle Slopes in the South China Sea

“…As the foundation of the numerical tool used in this study, the SMPM, originally developed by Diamessis et al (2005), has been successfully applied to the study of small-scale stratified flow processes (Diamessis et al 2011;Abdilghanie and Diamessis 2012;Zhou andDiamessis 2015, 2016), with minimal artificial dispersion and diffusion, including the propagation of ISWs in a uniform depth waveguide and two-dimensional studies of their interaction with a model no-slip sea floor (Diamessis and Redekopp 2006). Recently, Joshi et al (2016) adapted the method to efficiently account for the nonhydrostatic effects with deformed boundaries while preserving high-order accuracy, thereby allowing the incorporation of gentle bathymetry over long domains.…”