Trapped internal waves over undular topography

1991 ◽  
Vol 226 ◽  
pp. 205-217 ◽  
Author(s):  
C. Kranenburg ◽  
J. D. Pietrzak ◽  
G. Abraham
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Large Amplitude ◽  
Bottom Topography ◽  
Stratified Flow ◽  
Wave Mode ◽  
Finite Amplitude ◽  
Trapped Waves ◽  
Acoustic Images ◽  
The Waves

We describe observations of slowly decelerating stratified flow over undular bottom topography in an estuary. The flow, which initially was supercritical with respect to the first internal wave mode, approached a resonance after it had become subcritical. A series of acoustic images showed large-amplitude first-mode trapped waves during this phase of the tide. We derive a criterion for quasi-steady response, and present an extension of Yih's class II linear finite-amplitude solutions that accounts for the waves observed.

Finite-amplitude internal wavepacket dispersion and breaking

2001 ◽  
Vol 429 ◽  
pp. 343-380 ◽  
Author(s):  
BRUCE R. SUTHERLAND
Keyword(s):  
Numerical Simulations ◽  
Internal Waves ◽  
Large Amplitude ◽  
Finite Amplitude ◽  
Critical Amplitude ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Stability ◽  
The Waves ◽  
Horizontal Wavelength

The evolution and stability of two-dimensional, large-amplitude, non-hydrostatic internal wavepackets are examined analytically and by numerical simulations. The weakly nonlinear dispersion relation for horizontally periodic, vertically compact internal waves is derived and the results are applied to assess the stability of weakly nonlinear wavepackets to vertical modulations. In terms of Θ, the angle that lines of constant phase make with the vertical, the wavepackets are predicted to be unstable if [mid ]Θ[mid ] < Θc, where Θc = cos−1 (2/3)1/2 ≃ 35.3° is the angle corresponding to internal waves with the fastest vertical group velocity. Fully nonlinear numerical simulations of finite-amplitude wavepackets confirm this prediction: the amplitude of wavepackets with [mid ]Θ[mid ] > Θc decreases over time; the amplitude of wavepackets with [mid ]Θ[mid ] < Θc increases initially, but then decreases as the wavepacket subdivides into a wave train, following the well-known Fermi–Pasta–Ulam recurrence phenomenon.If the initial wavepacket is of sufficiently large amplitude, it becomes unstable in the sense that eventually it convectively overturns. Two new analytic conditions for the stability of quasi-plane large-amplitude internal waves are proposed. These are qualitatively and quantitatively different from the parametric instability of plane periodic internal waves. The ‘breaking condition’ requires not only that the wave is statically unstable but that the convective instability growth rate is greater than the frequency of the waves. The critical amplitude for breaking to occur is found to be ACV = cot Θ (1 + cos2 Θ)/2π, where ACV is the ratio of the maximum vertical displacement of the wave to its horizontal wavelength. A second instability condition proposes that a statically stable wavepacket may evolve so that it becomes convectively unstable due to resonant interactions between the waves and the wave-induced mean flow. This hypothesis is based on the assumption that the resonant long wave–short wave interaction, which Grimshaw (1977) has shown amplifies the waves linearly in time, continues to amplify the waves in the fully nonlinear regime. Using linear theory estimates, the critical amplitude for instability is ASA = sin 2Θ/(8π2)1/2. The results of numerical simulations of horizontally periodic, vertically compact wavepackets show excellent agreement with this latter stability condition. However, for wavepackets with horizontal extent comparable with the horizontal wavelength, the wavepacket is found to be stable at larger amplitudes than predicted if Θ [lsim ] 45°. It is proposed that these results may explain why internal waves generated by turbulence in laboratory experiments are often observed to be excited within a narrow frequency band corresponding to Θ less than approximately 45°.

Internal wave excitation from stratified flow over a thin barrier

1998 ◽  
Vol 377 ◽  
pp. 223-252 ◽  
Author(s):  
BRUCE R. SUTHERLAND ◽  
PAUL F. LINDEN
Keyword(s):  
Internal Wave ◽  
Numerical Simulations ◽  
Internal Waves ◽  
Wavelet Transforms ◽  
Large Amplitude ◽  
Internal Gravity Waves ◽  
Wave Excitation ◽  
Mean Flow ◽  
Propagating Waves ◽  
The Waves

We perform laboratory experiments in a recirculating shear flow tank of non-uniform salt-stratified water to examine the excitation of internal gravity waves (IGW) in the wake of a tall, thin vertical barrier. The purpose of this study is to characterize and quantify the coupling between coherent structures shed in the wake and internal waves that radiate from the mixing region into the deep, stationary fluid. In agreement with numerical simulations, large-amplitude internal waves are generated when the mixing region is weakly stratified and the deep fluid is sufficiently strongly stratified. If the mixing region is unstratified, weak but continuous internal wave excitation occurs. In all cases, the tilt of the phase lines of propagating waves lies within a narrow range. Assuming the waves are spanwise uniform, their amplitude in space and time is measured non-intrusively using a recently developed ‘synthetic schlieren’ technique. Using wavelet transforms to measure consistently the width and duration of the observed wavepackets, the Reynolds stress is measured and, in particular, we estimate that when large-amplitude internal wave excitation occurs, approximately 7% of the average momentum across the shear depth and over the extent of the wavepacket is lost due to transport away from the mixing region by the waves.We propose that internal waves may act back upon the mean flow modifying it so that the excitation of waves of that frequency is enhanced. A narrow frequency spectrum of large-amplitude waves is observed because the feedback is largest for waves with phase tilt in a range near 45°. Numerical simulations and analytic theories are presented to further quantify this theory.

A multi-layer model for nonlinear internal wave propagation in shallow water

2012 ◽  
Vol 695 ◽  
pp. 341-365 ◽  
Author(s):  
Philip L.-F. Liu ◽  
Xiaoming Wang
Keyword(s):  
Wave Propagation ◽  
Internal Wave ◽  
Shallow Water ◽  
Internal Waves ◽  
Bottom Topography ◽  
Laboratory Data ◽  
Constant Density ◽  
Layer Model ◽  
Fluid System ◽  
Nonlinear Internal Wave

AbstractIn this paper, a multi-layer model is developed for the purpose of studying nonlinear internal wave propagation in shallow water. The methodology employed in constructing the multi-layer model is similar to that used in deriving Boussinesq-type equations for surface gravity waves. It can also be viewed as an extension of the two-layer model developed by Choi & Camassa. The multi-layer model approximates the continuous density stratification by an $N$-layer fluid system in which a constant density is assumed in each layer. This allows the model to investigate higher-mode internal waves. Furthermore, the model is capable of simulating large-amplitude internal waves up to the breaking point. However, the model is limited by the assumption that the total water depth is shallow in comparison with the wavelength of interest. Furthermore, the vertical vorticity must vanish, while the horizontal vorticity components are weak. Numerical examples for strongly nonlinear waves are compared with laboratory data and other numerical studies in a two-layer fluid system. Good agreement is observed. The generation and propagation of mode-1 and mode-2 internal waves and their interactions with bottom topography are also investigated.

scholarly journals Impact of Wave-Vortical Interactions on Oceanic Submesoscale Lateral Dispersion

2021 ◽  
Author(s):  
Gerardo Hernández-Dueñas ◽  
M.-Pascale Lelong ◽  
Leslie M. Smith
Keyword(s):  
Internal Wave ◽  
Wave Mode ◽  
Projection Technique ◽  
Lateral Transport ◽  
Transfer Energy ◽  
Shear Dispersion ◽  
Horizontal Flows ◽  
Order Of Magnitude ◽  
The Waves ◽  
Mode Energy

AbstractSubmesoscale lateral transport of Lagrangian particles in pycnocline conditions is investigated by means of idealized numerical simulations with reduced-interaction models. Using a projection technique, the models are formulated in terms of wave-mode and vortical-mode nonlinear interactions, and they range in complexity from full Boussinesq to waves-only and vortical-modes-only (QG) models. We find that, on these scales, most of the dispersion is done by vortical motions, but waves cannot be discounted because they play an important, albeit indirect, role. In particular, we show that waves are instrumental in filling out the spectra of vortical-mode energy at smaller scales through non-resonant vortex-wave-wave triad interactions. We demonstrate that a richer spectrum of vortical modes in the presence of waves enhances the effective lateral diffusivity, compared to QG. Waves also transfer energy upscale to vertically sheared horizontal flows which are a key ingredient for internal-wave shear dispersion. In the waves-only model, the dispersion rate is an order of magnitude smaller and is attributed entirely to internal-wave shear dispersion.

Spectra and Coherence of Wind-Generated Internal Waves

1976 ◽  
Vol 33 (10) ◽  
pp. 2323-2328 ◽  
Author(s):  
R. H. Käse ◽  
C. L. Tang
Keyword(s):  
Energy Density ◽  
Internal Wave ◽  
Internal Waves ◽  
Wind Stress ◽  
Wave Mode ◽  
Wave Number ◽  
Small Scale ◽  
Two Dimensional ◽  
Scale Turbulence ◽  
Main Thermocline

On the basis of a model for an internal wave field that is generated by a randomly varying isotropic wind stress and in which energy is transferred to small-scale turbulence, we derive the two-dimensional energy density function. The coherence scales are determined by the highest order internal wave mode that is not affected by virtual friction in the main thermocline, provided the curl of the wind stress has a white noise wave number spectrum. In general, this mode number scale is increasing monotonically with frequency. As a result of such a frequency dependent mode bandwidth, the vertical coherence drops with increasing frequency.

scholarly journals The Impact of Finite-Amplitude Bottom Topography on Internal Wave Generation in the Southern Ocean

2014 ◽  
Vol 44 (11) ◽  
pp. 2938-2950 ◽  
Author(s):  
Maxim Nikurashin ◽  
Raffaele Ferrari ◽  
Nicolas Grisouard ◽  
Kurt Polzin
Keyword(s):  
Southern Ocean ◽  
Internal Wave ◽  
Linear Theory ◽  
Bottom Topography ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Wave Generation ◽  
Two Dimensional ◽  
Lee Wave ◽  
Internal Wave Generation

Abstract Direct observations in the Southern Ocean report enhanced internal wave activity and turbulence in a kilometer-thick layer above rough bottom topography collocated with the deep-reaching fronts of the Antarctic Circumpolar Current. Linear theory, corrected for finite-amplitude topography based on idealized, two-dimensional numerical simulations, has been recently used to estimate the global distribution of internal wave generation by oceanic currents and eddies. The global estimate shows that the topographic wave generation is a significant sink of energy for geostrophic flows and a source of energy for turbulent mixing in the deep ocean. However, comparison with recent observations from the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean shows that the linear theory predictions and idealized two-dimensional simulations grossly overestimate the observed levels of turbulent energy dissipation. This study presents two- and three-dimensional, realistic topography simulations of internal lee-wave generation from a steady flow interacting with topography with parameters typical of Drake Passage. The results demonstrate that internal wave generation at three-dimensional, finite bottom topography is reduced compared to the two-dimensional case. The reduction is primarily associated with finite-amplitude bottom topography effects that suppress vertical motions and thus reduce the amplitude of the internal waves radiated from topography. The implication of these results for the global lee-wave generation is discussed.

scholarly journals Scattering of a Semidiurnal Barotropic Kelvin Wave into Internal Waves over Wide Continental Shelves

2017 ◽  
Vol 47 (10) ◽  
pp. 2545-2562 ◽  
Author(s):  
Alexander E. Yankovsky ◽  
Tianyi Zhang
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Energy Flux ◽  
Wave Structure ◽  
Wave Mode ◽  
Wave Radiation ◽  
Semidiurnal Tide ◽  
Kelvin Wave ◽  
Scattering Parameter ◽  
Continental Shelves

AbstractIn boundary areas of the World Ocean, a semidiurnal tide propagates in the form of a Kelvin wave mode trapped by the coastline. Over wide continental shelves, the semidiurnal tide is no longer a pure Kelvin wave but attains features of a zero-mode edge wave. As a result, the wave structure and the alongshore energy flux concentrate over the continental shelf and slope topography and become very sensitive to the variations of shelf geometry. When a semidiurnal Kelvin wave encounters alongshore changes of the shelf width, its energy scatters into other wave modes, including internal waves. A particularly strong scattering occurs on wide shelves, where Kelvin wave structure undergoes significant modifications over short alongshore distances. These dynamics are studied using the Regional Ocean Modeling System (ROMS). This study found that when the alongshore energy flux in the Kelvin wave mode converges on the shelf, the offshore wave radiation occurs through barotropic waves, while for the divergent alongshore energy flux, internal waves are generated. Under favorable conditions, more than 10% of the incident barotropic Kelvin wave energy flux can be scattered into internal waves. For the surface-intensified stratification mostly the first internal mode is generated, while for the uniform with depth stratification, multiple internal modes are present in the form of an internal wave beam. A nondimensional internal wave scattering parameter is derived based on the theoretical properties of a Kelvin wave mode, bottom topography, and stratification.

Models of energy loss from internal waves breaking in the ocean

2017 ◽  
Vol 836 ◽  
pp. 72-116 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Exact Solution ◽  
Internal Wave ◽  
Internal Waves ◽  
Wave Breaking ◽  
Convective Instability ◽  
Helmholtz Instability ◽  
Turbulent Dissipation ◽  
The Waves ◽  
Data Representative ◽  
Made In

The supply of energy to the internal wave field in the ocean is, in total, sufficient to support the mixing required to maintain the stratification of the ocean, but can the required rates of turbulent dissipation in mid-water be sustained by breaking internal waves? It is assumed that turbulence occurs in regions where the field of motion can be represented by an exact solution of the equations that describe waves propagating through a uniformly stratified fluid and becoming unstable. Two instabilities leading to wave breaking are examined, convective instability and shear-induced Kelvin–Helmholtz instability. Models are constrained by data representative of the mid-water ocean. Calculations of turbulent dissipation are first made on the assumption that all the waves representing local breaking have the same steepness, $s$, and frequency, $\unicode[STIX]{x1D70E}$. For some ranges of $s$ and $\unicode[STIX]{x1D70E}$, breaking can support the required transfer of energy to turbulence. For convective instability this proves possible for sufficiently large $s$, typically exceeding 2.0, over a range of $\unicode[STIX]{x1D70E}$, while for shear-induced instability near-inertial waves are required. Relaxation of the constraint that the model waves all have the same $s$ and $\unicode[STIX]{x1D70E}$ requires new assumptions about the nature and consequences of wave breaking. Examples predict an overall dissipation consistent with the observed rates. Further observations are, however, required to test the validity of the assumptions made in the models and, in particular, to determine the nature and frequency of internal wave breaking in the mid-water ocean.

The fate and impact of internal waves induced by strong shear current over a marginal ridge

2020 ◽  
Author(s):  
Peiwen Zhang ◽  
Wenjia Min
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Vertical Mixing ◽  
Internal Tide ◽  
Stratified Flow ◽  
Internal Tides ◽  
Energy Field ◽  
Nonlinear Internal Waves ◽  
Highly Nonlinear ◽  
Shear Current

&lt;p&gt;Internal waves with strong vertical mixing could be induced by stratified flow over seafloor obstacles. Noted that the stratified flow not only trigger internal tides, but also highly nonlinear internal waves like internal lee waves and internal solitary waves over steep topography features, and the highly nonlinear internal waves are suggested to play an important role in turbulence and mixing. As a typical seafloor obstacle, ridge could significantly modified the propagation of internal tide, internal lee wave and internal solitary wave. We focused on I-Lan ridge with asymmetrical topography feature in Kuroshio region. To the north of the I-Lan ridge, the depth of Philippine basin reached 4000m compared with the depth of 1500m in the south of the ridge, leading to different characteristics of internal wave energy field and ecological characteristics between two sides. Based on numerical simulations, we revealed the generation and propagation of internal waves over marginal ridge, causing by the shear current induced by Kuroshio. We also discussed the turbulence kinetic energy contributed by linear internal waves and nonlinear internal waves, providing the strength of vertical turbulent mixing around the I-Lan ridge. Then we demonstrated the characteristics of complex internal wave field in the strong background shear current over I-Lan ridge.&lt;/p&gt;

scholarly journals Amplification and Suppression of Internal Waves by Tides over Variable Bottom Topography

2006 ◽  
Vol 36 (10) ◽  
pp. 1959-1973 ◽  
Author(s):  
Vasiliy Vlasenko ◽  
Nataliya Stashchuk
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Energy Exchange ◽  
Bottom Topography ◽  
Tidal Flow ◽  
Wave Structure ◽  
Energy Balance Equation ◽  
Wave Flow ◽  
Irregular Wave ◽  
Flow Interaction

Abstract The energy exchange between internal waves and barotropic currents over inclined bottom topography is studied theoretically and in the framework of the numerical model. The energy balance equation derived for a continuously stratified fluid predicts that energy can either be transferred toward or away from the internal wave depending on the direction of propagation of both the wave and current. Four scenarios of wave–flow interaction over the inclined bottom were identified. An internal wave extracts energy from the background tidal flow during its propagation upslope–upstream or downslope–downstream and its amplitude grows. The wave loses energy propagating downslope–upstream or upslope–downstream and reduces in amplitude. This mechanism of suppression or amplification of internal waves by a current over an inclined bottom is verified numerically. When applied to the area of the Knight Inlet sill, a high-resolution fully nonlinear, nonhydrostatic model reproduces the packets of internal waves generated by supercritical tidal flow over the sill. Careful inspection of the wave fields revealed the presence of an irregular wave structure within wave packets—namely, internal waves are not arranged by amplitude. This phenomenon, obtained numerically and observed in situ, is treated in terms of the mechanism of wave–flow interaction: the energy exchange between the tidal current and generated internal waves over the inclined bottom topography is the reason for the absence of traditional rank-ordered waves in the packet.

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