The internal gravity wave field emitted by a stably stratified turbulent wake

Abstract

The internal gravity wave (IGW) field emitted by a stably stratified, initially turbulent, wake of a towed sphere in a linearly stratified fluid is studied using fully nonlinear numerical simulations. A wide range of Reynolds numbers, $\mathit{Re}= UD/ \nu \in [5\times 1{0}^{3} , 1{0}^{5} ] $ and internal Froude numbers, $\mathit{Fr}= 2U/ (ND)\in [4, 16, 64] $ ($U$, $D$ are characteristic body velocity and length scales, and $N$ is the buoyancy frequency) is examined. At the higher $\mathit{Re}$ examined, secondary Kelvin–Helmholtz instabilities and the resulting turbulent events, directly linked to a prolonged non-equilibrium (NEQ) regime in wake evolution, are responsible for IGW emission that persists up to $Nt\approx 100$. In contrast, IGW emission at the lower $\mathit{Re}$ investigated does not continue beyond $Nt\approx 50$ for the three $\mathit{Fr}$ values considered. The horizontal wavelengths of the most energetic IGWs, obtained by continuous wavelet transforms, increase with $\mathit{Fr}$ and appear to be smaller at the higher $\mathit{Re}$, especially at late times. The initial value of these wavelengths is set by the wake height at the beginning of the NEQ regime. At the lower $\mathit{Re}$, consistent with a recently proposed model, the waves propagate over a narrow range of angles that minimize viscous decay along their path. At the higher $\mathit{Re}$, wave motion is much less affected by viscosity, at least initially, and early-time wave propagation angles extend over a broader range of values which are linked to increased efficiency in momentum extraction from the turbulent wake source.

Unsteady regimes of internal gravity wave generation in the ocean

“…The steady-state regime of wave oscillations is considered. In the linear formulation, assuming the Boussinesq approximation, we have the following equation, for example, for the vertical displacement of the isopycnic lines ( , , ) (lines of equal density with the same harmonic time-dependence) [Bulatov and Vladimirov, 2012, 2015a, 2015b …”

“…The problem of adequate and physically justified modeling of natural stratified environment, i.e., the ocean, has been studied in numerous works, because, on the one hand, the widely used models of stratified media described in the scientific literature ( ( ) =const, the linear or another model distribution of the buoyancy frequency) significantly simplify the mathematical solution of the problem, but on the other hand, problems appear of the adequacy and physical justification of the model representations. The model used here is one of the most widely used models of wave motions in the ocean, because, in certain regions of the World Ocean (Arctic basin), this approximation ( ( ) =const) gives a good description of the real hydrology and it is one of the basic approximations in real oceanographic and hydro-physical numerical simulations Vladimirov, 2012, 2015a;Massel, 2015;Morozov, 1995;Pisarev, 1996]. Function ( , , ) is related to the vertical component of velocity ( , , ) as follows:…”

“…These methods allow to carry out a qualitative analysis and express estimations of the solutions. In the framework of the linear theory, the asymptotic methods for integral representations and approximate methods of geometric optics are used to investigate the wave perturbations in natural stratified media analytically Vladimirov, 2012, 2015a;Massel, 2015;Mei et al, 2017;Morozov, 2018, Pedlosky, 2010Sutherland, 2010].…”

“…Moreover, the use of powerful numerical algorithms requires verification and comparison with the solutions of model problems [Abdilghanie et al, 2013;Rees et al, 2012;Tiugin et al, 2012;Vlasenko et al, 2005].…”

“…In the linear approximation, the existing approaches to the description of the wave pattern of excited fields of internal waves are based on the representation of the wave fields by Fourier integrals. As a rule, only the first wave mode is considered, because this mode makes the basic contribution to the complete wave field Vladimirov, 2012, 2015a;Morozov, 2018].…”