Saturation of the internal tide over the inner continental shelf. Part II: Parameterization

2021 ◽  
Author(s):  
Johannes Becherer ◽  
James N. Moum ◽  
Joseph Calantoni ◽  
John A. Colosi ◽  
John A. Barth ◽  
...  
Keyword(s):  
Continental Shelf ◽  
Energy Flux ◽  
Gravity Waves ◽  
Water Depth ◽  
Surf Zone ◽  
Internal Tide ◽  
Data Sets ◽  
Flux Divergence ◽  
Surface Gravity Waves ◽  
Inner Continental Shelf

AbstractHere, we develop a framework for understanding the observations presented in the accompanying paper (Part I) by Becherer et al. (2021). In this framework, the internal tide saturates as it shoals due to amplitude limitation with decreasing water depth (H). From this framework evolves estimates of averaged energetics of the internal tide; specifically, energy, 〈APE〉, energy flux, 〈FE〉, and energy flux divergence, ∂x 〈FE〉. Since we observe that 〈D〉 ≈ ∂x 〈FE〉, we also interpret our estimate of ∂x 〈FE〉 as 〈D〉. These estimates represent a parameterization of the energy in the internal tide as it saturates over the inner continental shelf. The parameterization depends solely on depth-mean stratification and bathymetry. A summary result is that the cross-shelf depth dependencies of 〈APE〉, 〈FE〉 and ∂x 〈FE〉 are analogous to those for shoaling surface gravity waves in the surf zone, suggesting that the inner shelf is the surf zone for the internal tide. A test of our simple parameterization against a range of data sets suggests that it is broadly applicable.

Saturation of the internal tide over the inner continental shelf. Part I: Observations

2021 ◽  
Author(s):  
Johannes Becherer ◽  
James N. Moum ◽  
Joseph Calantoni ◽  
John A. Colosi ◽  
John A. Barth ◽  
...  
Keyword(s):  
Energy Flux ◽  
Dissipation Rate ◽  
Internal Tide ◽  
Shallow Waters ◽  
Turbulence Energy ◽  
Flux Divergence ◽  
Central California ◽  
Turbulence Energy Dissipation ◽  
Wave Energy Flux ◽  
Inner Continental Shelf

AbstractBroadly-distributed measurements of velocity, density and turbulence spanning the inner shelf off central California indicate that (i) the average shoreward-directed internal tide energy flux (〈FE〉) decreases to near 0 at the 25 m isobath; (ii) the vertically-integrated turbulence dissipation rate (〈D〉) is approximately equal to the flux divergence of internal tide energy (∂x〈FE〉); (iii) the ratio of turbulence energy dissipation in the interior relative to the bottom boundary layer (BBL) decreases toward shallow waters; (iv) going inshore, 〈FE〉 becomes decorrelated with the incoming internal wave energy flux; and (v) 〈FE〉 becomes increasingly correlated with stratification toward shallower water.

Statistics of Internal Tide Bores and Internal Solitary Waves Observed on the Inner Continental Shelf off Point Sal, California

2018 ◽  
Vol 48 (1) ◽  
pp. 123-143 ◽  
Author(s):  
John A. Colosi ◽  
Nirnimesh Kumar ◽  
Sutara H. Suanda ◽  
Tucker M. Freismuth ◽  
Jamie H. MacMahan
Keyword(s):  
Continental Shelf ◽  
Energy Flux ◽  
Solitary Waves ◽  
Internal Tide ◽  
Propagation Loss ◽  
Internal Solitary Waves ◽  
Time Behavior ◽  
Time Duration ◽  
Match Filter ◽  
Inner Continental Shelf

AbstractMoored observations of temperature and current were collected on the inner continental shelf off Point Sal, California, between 9 June and 8 August 2015. The measurements consist of 10 moorings in total: 4 moorings each on the 50- and 30-m isobaths covering a 10-km along-shelf distance and an across-shelf section of moorings on the 50-, 40-, 30-, and 20-m isobaths covering a 5-km distance. Energetic, highly variable, and strongly dissipating transient wave events termed internal tide bores and internal solitary waves (ISWs) dominate the records. Simple models of the bore and ISW space–time behavior are implemented as a temperature match filter to detect events and estimate wave packet parameters as a function of time and mooring position. Wave-derived quantities include 1) group speed and direction; 2) time of arrival, time duration, vertical displacement amplitude, and waves per day; and 3) energy density, energy flux, and propagation loss. In total, over 1000 bore events and over 9000 ISW events were detected providing well-sampled statistical distributions. Statistics of the waves are rather insensitive to position along shelf but change markedly in the across-shelf direction. Two compelling results are 1) that the probability density functions for bore and ISW energy flux are nearly exponential, suggesting the importance of interference and 2) that wave propagation loss is proportional to energy flux, thus giving an exponential decay of energy flux toward shore with an e-folding scale of 2–2.4 km and average dissipation rates for bores and ISWs of 144 and 1.5 W m−1, respectively.

Dynamics and energetics of nonlinear internal wave around a double-canyon system

2020 ◽  
Author(s):  
Qun Li
Keyword(s):  
Continental Shelf ◽  
Internal Wave ◽  
Energy Flux ◽  
Internal Tide ◽  
Internal Tides ◽  
Flux Divergence ◽  
Wave Regime ◽  
Nonlinear Internal Wave ◽  
The North ◽  
Shelf Slope

<p>The continental shelf/slope northeastern Taiwan is a ‘hotspot’ of nonlinear internal wave (NLIW). The complex spatial pattern of NLIW indicates the complexity of the source and the background conditions. In this talk, we investigated the dynamic and energetics of the internal tide (IT) and NLIW around this region based on a 3D high resolution nonhydrostatic numerical model. Special attention is paid on the role of two main topographic features-the Mien-Hua Canyon and the North Mien-Hua Canyon, which are the energetic sources for ITs and NLIW.</p><p>The complex IT field is excited by the double-Canyon system and the rotary tidal current. ITs from different sources and formation time interference with each other further strengthen the complexity. The area-integrated energy flux divergence (the area-integrated dissipation rate) is ~0.45GW (~0.28GW) and ~0.26 GW (~0.17 GW) over the Mien-Hua Canyon and the North Mien-Hua Canyon, respectively. Along with the energetic internal tides, large-amplitude NLIW and trains are also generated over the continental shelf and slope region. The amplitude of the NLIW can reach to about 30 m on the continental slope with a water depth of 130 m and shows similar spatial complexity, which is consistent with in situ and satellite observations. Further analysis shows that the dominant generation mechanism of the NLIW belongs to the mixed tidal-lee wave regime. In addition, the dynamic processes can be significantly modulated by the Kuroshio. With the present of Kuroshio, the energy flux of the M2 internal tide shows a distinct gyre pattern and strengthens over the double canyon system, which is more close to the mooring observations and previous study.</p>

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.

scholarly journals Energy transformations and dissipation of nonlinear internal waves over New Jersey's continental shelf

2010 ◽  
Vol 17 (4) ◽  
pp. 345-360 ◽  
Author(s):  
E. L. Shroyer ◽  
J. N. Moum ◽  
J. D. Nash
Keyword(s):  
Continental Shelf ◽  
Internal Waves ◽  
Turbulent Mixing ◽  
Internal Tide ◽  
Flux Divergence ◽  
Wave Interactions ◽  
Nonlinear Internal Waves ◽  
Dissipative Loss ◽  
Peak Energy

Abstract. The energetics of large amplitude, high-frequency nonlinear internal waves (NLIWs) observed over the New Jersey continental shelf are summarized from ship and mooring data acquired in August 2006. NLIW energy was typically on the order of 105 Jm−1, and the wave dissipative loss was near 50 W m−1. However, wave energies (dissipations) were ~10 (~2) times greater than these values during a particular week-long period. In general, the leading waves in a packet grew in energy across the outer shelf, reached peak values near 40 km inshore of the shelf break, and then lost energy to turbulent mixing. Wave growth was attributed to the bore-like nature of the internal tide, as wave groups that exhibited larger long-term (lasting for a few hours) displacements of the pycnocline offshore typically had greater energy inshore. For ship-observed NLIWs, the average dissipative loss over the region of decay scaled with the peak energy in waves; extending this scaling to mooring data produces estimates of NLIW dissipative loss consistent with those made using the flux divergence of wave energy. The decay time scale of the NLIWs was approximately 12 h corresponding to a length scale of 35 km (O(100) wavelengths). Imposed on these larger scale energetic trends, were short, rapid exchanges associated with wave interactions and shoaling on a localized topographic rise. Both of these events resulted in the onset of shear instabilities and large energy loss to turbulent mixing.

Bragg scattering of random surface gravity waves by irregular seabed topography

2002 ◽  
Vol 451 ◽  
pp. 1-33 ◽  
Author(s):  
FABRICE ARDHUIN ◽  
T. H. C. HERBERS
Keyword(s):  
Energy Balance ◽  
Continental Shelf ◽  
Gravity Waves ◽  
Balance Equation ◽  
Source Term ◽  
Energy Balance Equation ◽  
Surface Gravity ◽  
Spectral Energy ◽  
Bragg Scattering ◽  
Surface Gravity Waves

The Bragg scattering of random, non-stationary surface gravity waves by random topography on a gently sloping bottom is investigated. A correction is given of previously published expressions for the triad wave–wave–bottom interaction source term in the spectral energy balance equation, and the result is reconciled with deterministic theories for the reflection of waves from sinusoidal seabed undulations. For both normal and oblique incidence, the stochastic and deterministic theories are equivalent in the limit of long propagation distances. Even for relatively short distances (for example two bottom undulations), the reflected energy predicted by the stochastic source term formulation is generally within 15% of values predicted by deterministic theories. The detuning of Bragg resonance by refraction and shoaling is discussed, suggesting practical validity conditions for the stochastic theory. The effect of bottom scattering on swell propagation is illustrated with numerical model computations for the North Carolina continental shelf using high-resolution bathymetry and an efficient semi-implicit scheme to evaluate the bottom scattering source term and integrate the energy balance equation. Model results demonstrate the importance of forward scattering of waves that propagate at large oblique angles over bottom features with typical scales of one to several surface wavelengths. This process contributes significantly to the directional spread of swell on the continental shelf by diffusing energy, in the spectrum, around the mean wave direction. Back-scattering, caused by bottom features with crests parallel to those of the surface waves and wavelengths close to half the surface wavelength, is weak, owing to the sharp roll-off of the bottom elevation spectrum at high wavenumbers. Model predictions are consistent with field measurements.

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 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.

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