Radiation of internal waves from groups of surface gravity waves

2017 ◽  
Vol 829 ◽  
pp. 280-303 ◽  
Author(s):  
S. Haney ◽  
W. R. Young
Keyword(s):  
Internal Waves ◽  
Energy Flux ◽  
Gravity Waves ◽  
Three Dimensions ◽  
Stokes Drift ◽  
Surface Gravity ◽  
Fourth Power ◽  
Buoyancy Frequency ◽  
Surface Gravity Waves ◽  
Short Period

Groups of surface gravity waves induce horizontally varying Stokes drift that drives convergence of water ahead of the group and divergence behind. The mass flux divergence associated with spatially variable Stokes drift pumps water downwards in front of the group and upwards in the rear. This ‘Stokes pumping’ creates a deep Eulerian return flow that sets the isopycnals below the wave group in motion and generates a trailing wake of internal gravity waves. We compute the energy flux from surface to internal waves by finding solutions of the wave-averaged Boussinesq equations in two and three dimensions forced by Stokes pumping at the surface. The two-dimensional (2-D) case is distinct from the 3-D case in that the stratification must be very strong, or the surface waves very slow for any internal wave (IW) radiation at all. On the other hand, in three dimensions, IW radiation always occurs, but with a larger energy flux as the stratification and surface wave (SW) amplitude increase or as the SW period is shorter. Specifically, the energy flux from SWs to IWs varies as the fourth power of the SW amplitude and of the buoyancy frequency, and is inversely proportional to the fifth power of the SW period. Using parameters typical of short period swell (e.g. 8 s SW period with 1 m amplitude) we find that the energy flux is small compared to both the total energy in a typical SW group and compared to the total IW energy. Therefore this coupling between SWs and IWs is not a significant sink of energy for the SWs nor a source for IWs. In an extreme case (e.g. 4 m amplitude 20 s period SWs) this coupling is a significant source of energy for IWs with frequency close to the buoyancy frequency.

Internal waves coupled to surface gravity waves in three dimensions

2015 ◽  
Vol 13 (4) ◽  
pp. 893-910 ◽  
Author(s):  
Walter Craig ◽  
Philippe Guyenne ◽  
Catherine Sulem
Keyword(s):  
Internal Waves ◽  
Gravity Waves ◽  
Three Dimensions ◽  
Surface Gravity ◽  
Surface Gravity Waves

scholarly journals Observations and a Model of Undertow over the Inner Continental Shelf

2008 ◽  
Vol 38 (11) ◽  
pp. 2341-2357 ◽  
Author(s):  
Steven J. Lentz ◽  
Melanie Fewings ◽  
Peter Howd ◽  
Janet Fredericks ◽  
Kent Hathaway
Keyword(s):  
Gravity Waves ◽  
Vertical Structure ◽  
Surf Zone ◽  
Stokes Drift ◽  
Vertical Shear ◽  
Surface Gravity ◽  
Linear Wave ◽  
Surface Gravity Waves ◽  
Offshore Flow ◽  
Offshore Transport

Abstract Onshore volume transport (Stokes drift) due to surface gravity waves propagating toward the beach can result in a compensating Eulerian offshore flow in the surf zone referred to as undertow. Observed offshore flows indicate that wave-driven undertow extends well offshore of the surf zone, over the inner shelves of Martha’s Vineyard, Massachusetts, and North Carolina. Theoretical estimates of the wave-driven offshore transport from linear wave theory and observed wave characteristics account for 50% or more of the observed offshore transport variance in water depths between 5 and 12 m, and reproduce the observed dependence on wave height and water depth. During weak winds, wave-driven cross-shelf velocity profiles over the inner shelf have maximum offshore flow (1–6 cm s−1) and vertical shear near the surface and weak flow and shear in the lower half of the water column. The observed offshore flow profiles do not resemble the parabolic profiles with maximum flow at middepth observed within the surf zone. Instead, the vertical structure is similar to the Stokes drift velocity profile but with the opposite direction. This vertical structure is consistent with a dynamical balance between the Coriolis force associated with the offshore flow and an along-shelf “Hasselmann wave stress” due to the influence of the earth’s rotation on surface gravity waves. The close agreement between the observed and modeled profiles provides compelling evidence for the importance of the Hasselmann wave stress in forcing oceanic flows. Summer profiles are more vertically sheared than either winter profiles or model profiles, for reasons that remain unclear.

scholarly journals On the Lagrangian description of steady surface gravity waves

2007 ◽  
Vol 589 ◽  
pp. 433-454 ◽  
Author(s):  
DIDIER CLAMOND
Keyword(s):  
Gravity Waves ◽  
Order Approximation ◽  
Mathematical Formulation ◽  
Finite Depth ◽  
Stokes Drift ◽  
Surface Gravity ◽  
Lagrangian Description ◽  
Perturbation Scheme ◽  
Third Order ◽  
Surface Gravity Waves

This paper concerns the mathematical formulation of two-dimensional steady surface gravity waves in a Lagrangian description of motion. It is demonstrated first that classical second-order Lagrangian Stokes-like approximations do not exactly represent a steady wave motion in the presence of net mass transport (Stokes drift). A general mathematically correct formulation is then derived. This derivation leads naturally to a Lagrangian Stokes-like perturbation scheme that is uniformly valid for all time – in other words, without secular terms. This scheme is illustrated, both for irrotational waves, with seventh-order and third-order approximations in deep water and finite depth, respectively, and for rotational waves with a third-order approximation of the Gerstner-like wave on finite depth. It is also shown that the Lagrangian approximations are more accurate than their Eulerian counterparts of the same order.

scholarly journals Numerical investigation of turbulence of surface gravity waves

2022 ◽  
Vol 933 ◽  
Author(s):  
Zhou Zhang ◽  
Yulin Pan
Keyword(s):  
Size Effect ◽  
Energy Flux ◽  
Gravity Waves ◽  
Narrow Band ◽  
Finite Size ◽  
Surface Gravity ◽  
Finite Size Effect ◽  
Wave Turbulence ◽  
Surface Gravity Waves ◽  
Free Decay

In this paper, we numerically study the wave turbulence of surface gravity waves in the framework of Euler equations of the free surface. The purpose is to understand the variation of the scaling of the spectra with wavenumber $k$ and energy flux $P$ at different nonlinearity levels under different forcing/free-decay conditions. For all conditions (free decay and narrow-band and broad-band forcing) that we consider, we find that the spectral forms approach the wave turbulence theory (WTT) solution $S_\eta \sim k^{-5/2}$ and $S_\eta \sim P^{1/3}$ at high nonlinearity levels. With a decrease of nonlinearity level, the spectra for all cases become steeper, with the narrow-band forcing case exhibiting the most rapid deviation from WTT. We investigate bound waves and the finite-size effect as possible mechanisms causing the spectral variations. Through a tri-coherence analysis, we find that the finite-size effect is present in all cases, which is responsible for the overall steepening of the spectra and the reduced capacity of energy flux at lower nonlinearity levels. The fraction of bound waves in the domain generally decreases with the decrease of nonlinearity level, except for the narrow-band case, which exhibits a transition at a critical nonlinearity level below which a rapid increase is observed. This increase serves as the main reason for the fastest deviation from WTT with the decrease of nonlinearity in the narrow-band forcing case.

scholarly journals Computation of Density Perturbation and Energy Flux of Internal Waves from Experimental Data

Fluids ◽  
2020 ◽  
Vol 5 (3) ◽  
pp. 119
Author(s):  
Lucie Bordois ◽  
Jonas Nycander ◽  
Alexandre Paci
Keyword(s):  
Experimental Data ◽  
Internal Waves ◽  
Energy Flux ◽  
Laboratory Experiments ◽  
Density Perturbation ◽  
Three Dimensions ◽  
Buoyancy Frequency ◽  
Periodic Field ◽  
Anomaly Field ◽  
Density Anomaly

We hereby present two different spectral methods for calculating the density anomaly and the vertical energy flux from synthetic Schlieren data, for a periodic field of linear internal waves (IW) in a density-stratified fluid with a uniform buoyancy frequency. The two approaches operate under different assumptions. The first method (hereafter Mxzt) relies on the assumption of a perfectly periodic IW field in the three dimensions (x, z, t), whereas the second method (hereafter MxtUp) assumes that the IW field is periodic in x and t and composed solely of wave components with downward phase velocity. The two methods have been applied to synthetic Schlieren data collected in the CNRM large stratified water flume. Both methods succeed in reconstructing the density anomaly field. We identify and quantify the source of errors of both methods. A new method mixing the two approaches and combining their respective advantages is then proposed for the upward energy flux. The work presented in this article opens new perspectives for density and energy flux estimates from laboratory experiments data.

On the interaction of internal waves and surface gravity waves

1974 ◽  
Vol 63 (4) ◽  
pp. 773-800 ◽  
Author(s):  
John E. Lewis ◽  
Bruce M. Lake ◽  
Denny R. S. Ko
Keyword(s):  
Surface Wave ◽  
Theoretical Analysis ◽  
Internal Wave ◽  
Internal Waves ◽  
Gravity Waves ◽  
Surface Gravity ◽  
One Dimensional ◽  
Surface Gravity Waves ◽  
Wide Range ◽  
The One

The perturbation of pre-existing surface gravity waves caused by the presence of an internal wave was studied both experimentally and analytically. An extensive series of experiments was performed, and quantitative results were obtained for the one-dimensional monochromatic interaction of internal waves and surface gravity waves. Internal wave-induced surface slope, amplitude and wavenumber modulations were measured for a wide range of interaction conditions. A complementary theoretical analysis, based on the conservation approach of Whitham (1962) and Longuet-Higgins & Stewart (1960,1961), was performed and a closed form solution obtained for the one-dimensional wave interaction. Both the theory and the experiment demonstrate that the effect increases with interaction distance. The maximum interaction effect is found to occur when the phase speed of the internal wave and the group velocity of the surface wave are matched. The phase of the internal wave at which maximum surface-wave modulation occurs is found to be a sensitive and continuous function of the relative wave speeds. The experimental data are in good agreement with the present theoretical analysis.

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