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.

scholarly journals Turbulence closure: turbulence, waves and the wave-turbulence transition – Part 1: Vanishing mean shear

2008 ◽  
Vol 5 (4) ◽  
pp. 545-580
Author(s):  
H. Z. Baumert ◽  
H. Peters
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Tidal Flow ◽  
Wave Turbulence ◽  
Mean Flow ◽  
Turbulent Length Scale ◽  
Turbulent Dissipation ◽  
New Model ◽  
Turbulence Transition ◽  
Ozmidov Scale

Abstract. A new two-equation, closure-like turbulence model for stably stratified flows is introduced which uses the turbulent kinetic energy (K) and the turbulent enstrophy (Ω) as primary variables. It accounts for mean shear – and internal wave-driven mixing in the two limits of mean shear and no waves and waves but no mean shear, respectively. The traditional TKE balance is augmented by an explicit energy transfer from internal waves to turbulence. A modification of the Ω-equation accounts for the effect of the waves on the turbulence time and space scales. The latter is based on the assumption of a non-zero constant flux Richardson number in the limit of vanishing mean-flow shear when turbulence is produced exclusively by internal waves. The new model reproduces the wave-turbulence transition analyzed by D'Asaro and Lien (2000). At small energy density E of the internal wave field, the turbulent dissipation rate (ε) scales like ε~E2. This is what is observed in the deep sea. With increasing E, after the wave-turbulence transition has been passed, the scaling changes to ε~E1. This is observed, for example, in the swift tidal flow near a sill in Knight Inlet. The new model further exhibits a turbulent length scale proportional to the Ozmidov scale, as observed in the ocean, and predicts the ratio between the turbulent Thorpe and Ozmidov length scales well within the range observed in the ocean.

scholarly journals Turbulence closure: turbulence, waves and the wave-turbulence transition – Part 1: Vanishing mean shear

Ocean Science ◽  
2009 ◽  
Vol 5 (1) ◽  
pp. 47-58 ◽  
Author(s):  
H. Z. Baumert ◽  
H. Peters
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Internal Gravity Waves ◽  
Turbulence Closure ◽  
Theoretical Development ◽  
Wave Turbulence ◽  
Turbulent Length Scale ◽  
Turbulent Dissipation ◽  
New Model ◽  
Turbulence Transition

Abstract. This paper extends a turbulence closure-like model for stably stratified flows into a new dynamic domain in which turbulence is generated by internal gravity waves rather than mean shear. The model turbulent kinetic energy (TKE, K) balance, its first equation, incorporates a term for the energy transfer from internal waves to turbulence. This energy source is in addition to the traditional shear production. The second variable of the new two-equation model is the turbulent enstrophy (Ω). Compared to the traditional shear-only case, the Ω-equation is modified to account for the effect of the waves on the turbulence time and space scales. This modification is based on the assumption of a non-zero constant flux Richardson number in the limit of vanishing mean shear when turbulence is produced exclusively by internal waves. This paper is part 1 of a continuing theoretical development. It accounts for mean shear- and internal wave-driven mixing only in the two limits of mean shear and no waves and waves but no mean shear, respectively. The new model reproduces the wave-turbulence transition analyzed by D'Asaro and Lien (2000b). At small energy density E of the internal wave field, the turbulent dissipation rate (ε) scales like ε~E2. This is what is observed in the deep sea. With increasing E, after the wave-turbulence transition has been passed, the scaling changes to ε~E1. This is observed, for example, in the highly energetic tidal flow near a sill in Knight Inlet. The new model further exhibits a turbulent length scale proportional to the Ozmidov scale, as observed in the ocean, and predicts the ratio between the turbulent Thorpe and Ozmidov length scales well within the range observed in the ocean.

scholarly journals The evolution of mode-2 nonlinear internal waves over the northern Heng-Chun Ridge south of Taiwan

2015 ◽  
Vol 22 (4) ◽  
pp. 413-431 ◽  
Author(s):  
S. R. Ramp ◽  
Y. J. Yang ◽  
D. B. Reeder ◽  
M. C. Buijsman ◽  
F. L. Bahr
Keyword(s):  
Internal Waves ◽  
High Frequency ◽  
Velocity Structure ◽  
Primary Study ◽  
Turbulent Dissipation ◽  
Nonlinear Internal Waves ◽  
Mode 2 ◽  
Sill Depth ◽  
The Waves ◽  
Very High

Abstract. Two research cruises were conducted from the R/V OCEAN RESEARCHER 3 during 05–16 August 2011 to study the generation and propagation of high-frequency nonlinear internal waves (NLIWs) over the northern Heng-Chun Ridge south of Taiwan. The primary study site was on top of a smaller ridge about 15 km wide by 400 m high atop the primary ridge, with a sill depth of approximately 600 m. A single mooring was used in conjunction with shipboard observations to sample the temperature, salinity and velocity structure over the ridge. All the sensors observed a profusion of mode-2 NLIWs. Some of the waves were solitary, while others had as many as seven evenly spaced waves per packet. The waves all exhibited classic mode-2 velocity structure with a core near 150–200 m and opposing velocities in the layers above and below. At least two and possibly three most common propagation directions emerged from the analysis, suggesting multiple generation sites near the eastern side of the ridge. The turbulent dissipation due to overturns in the wave cores was very high at order 10−4–10−3 W kg−1. The energy budget suggests that the waves cannot persist very far from the ridge and likely do not contribute to the South China Sea transbasin wave phenomenon.

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.

scholarly journals Scattering of Low-Mode Internal Waves at Finite Isolated Topography

2014 ◽  
Vol 44 (1) ◽  
pp. 359-383 ◽  
Author(s):  
Sonya Legg
Keyword(s):  
Internal Wave ◽  
Froude Number ◽  
Internal Waves ◽  
Wave Breaking ◽  
Internal Tides ◽  
Relative Height ◽  
Incoming Wave ◽  
Topographic Slope ◽  
Height Increase ◽  
Incoming Energy

Abstract A series of two-dimensional numerical simulations examine the breaking of first-mode internal waves at isolated ridges, independently varying the relative height of the topography compared to the depth of the ocean h0/H0; the relative steepness of the topographic slope compared to the slope of the internal wave group velocity γ; and the Froude number of the incoming internal wave Fr0. The fraction of the incoming wave energy, which is reflected back toward deep water, transmitted beyond the ridge, and lost to dissipation and mixing, is diagnosed from the simulations. For critical slopes, with γ = 1, the fraction of incoming energy lost at the slope scales approximately like h0/H0, independent of the incoming wave Froude number. For subcritical slopes, with γ < 1, waves break and lose a substantial proportion of their energy if the maximum Froude number, estimated as Frmax = Fr0/(1 − h0/H0)2, exceeds a critical value, found empirically to be about 0.3. The dissipation at subcritical slopes therefore increases as both incoming wave Froude number and topographic height increase. At critical slopes, the dissipation is enhanced along the slope facing the incoming wave. In contrast, at subcritical slopes, dissipation is small until the wave amplitude is sufficiently enhanced by the shoaling topography to exceed the critical Froude number; then large dissipation extends all the way to the surface. The results are shown to generalize to variable stratification and different topographies, including axisymmetric seamounts. The regimes for low-mode internal wave breaking at isolated critical and subcritical topography identified by these simulations provide guidance for the parameterization of the mixing due to radiated internal tides.

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.

scholarly journals The evolution of an internal bore at the Malin shelf break

1999 ◽  
Vol 17 (4) ◽  
pp. 547-565 ◽  
Author(s):  
J. Small ◽  
T. C. Sawyer ◽  
J. C. Scott
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Phase Speed ◽  
Shelf Break ◽  
Empirical Orthogonal Functions ◽  
Shelf Edge ◽  
Orthogonal Functions ◽  
Length Scales ◽  
The Waves ◽  
Internal Bore

Abstract. Observations of internal waves were made at the Malin shelf edge during SESAME (Shelf Edge Studies Acoustic Measurement Experiment), a part of the NERC LOIS-SES experiment, in August-September 1996. These measurements provide a high resolution dataset demonstrating internal wave generation and propagation. This note presents observations of the evolution of an internal bore. The process is shown clearly in a sequence of thermistor chain tows across the shelf break covering a complete tidal cycle, as the double-sided bore transforms into a group of undulations and eventually into more distinct solitary waveforms. Current structures associated with the bore and waves were also observed by ship-mounted ADCP. Analysis of the waveforms in terms of the linear modes and empirical orthogonal functions (EOFs) indicate the dominance of the first mode, which is typical of a shallow water seasonal thermocline environment. Determination of the phase speed of the waves from the consecutive ship surveys enabled the Doppler shift in the towed data to be removed, allowing analysis of the real length scales of the waves. The bore evolution has been modelled using a first order non-linear KdV model for the first mode, initialised with the waveform in the first survey. Comparison of the model and the observations show close agreement in the amplitudes, length scales, phase speeds and separations of the leading internal waves as they evolve. Finally, analysis of the observed internal wave shapes indicates that, within the uncertainties of measurement, the wave-lengths lie between those predicted by first and second order soliton theory.Key words. Oceanography: general (continental shelf processes; ocean prediction). Oceanography: physical (internal and inertial waves)

scholarly journals The evolution of Mode-2 nonlinear internal waves over the northern Heng-Chun ridge south of Taiwan

2015 ◽  
Vol 2 (1) ◽  
pp. 243-296 ◽  
Author(s):  
S. R. Ramp ◽  
Y. J. Yang ◽  
D. B. Reeder ◽  
M. C. Buijsman ◽  
F. L. Bahr
Keyword(s):  
Internal Waves ◽  
High Frequency ◽  
Velocity Structure ◽  
Primary Study ◽  
Turbulent Dissipation ◽  
Nonlinear Internal Waves ◽  
Mode 2 ◽  
Sill Depth ◽  
The Waves ◽  
Very High

Abstract. Two research cruises were conducted from the R/V OCEAN RESEARCHER 3 during 5–16 August 2011 to study the generation of high-frequency nonlinear internal waves (NLIW) over the northern Heng-Chun Ridge south of Taiwan. The primary study site, centered near 21°34' N, 120°54' E, was on top of a smaller ridge about 15 km wide by 400 m high atop the primary ridge, with a sill depth of approximately 600 m. The bottom slope was steep over both sides of the ridge, supercritical with respect to both diurnal and semidiurnal tides. The key result of the experiments is that a profusion of mode-2 NLIW were observed by all the sensors. Some of the waves were solitary while others had as many as seven evenly-spaced waves per packet. The waves all exhibited classic mode-2 velocity structure with a core near 150–200 m and opposing velocities in the layers above and below. At least two and possibly three most common propagation directions emerged from the analysis, suggesting multiple generation sites near the east side of the ridge. The turbulent dissipation due to overturns in the wave cores was very high at order 10−4–10−3 W kg−1. The energy budget suggests that the waves cannot persist very far from the ridge and likely do not contribute to the South China Sea transbasin wave phenomenon.

scholarly journals Wave-Breaking Turbulence in the Ocean Surface Layer

2016 ◽  
Vol 46 (6) ◽  
pp. 1857-1870 ◽  
Author(s):  
Jim Thomson ◽  
Michael S. Schwendeman ◽  
Seth F. Zippel ◽  
Saeed Moghimi ◽  
Johannes Gemmrich ◽  
...  
Keyword(s):  
Reference Frame ◽  
Wave Height ◽  
Wave Breaking ◽  
Significant Wave Height ◽  
Ocean Surface ◽  
Turbulent Dissipation ◽  
Significant Wave ◽  
Strong Surface ◽  
Fixed Reference ◽  
The Waves

AbstractObservations of winds, waves, and turbulence at the ocean surface are compared with several analytic formulations and a numerical model for the input of turbulent kinetic energy by wave breaking and the subsequent dissipation. The observations are generally consistent with all of the formulations, although some differences are notable at winds greater than 15 m s−1. The depth dependence of the turbulent dissipation rate beneath the waves is fit to a decay scale, which is sensitive to the choice of vertical reference frame. In the surface-following reference frame, the strongest turbulence is isolated within a shallow region of depths much less than one significant wave height. In a fixed reference frame, the strong turbulence penetrates to depths that are at least half of the significant wave height. This occurs because the turbulence of individual breakers persists longer than the dominant period of the waves and thus the strong surface turbulence is carried from crest to trough with the wave orbital motion.

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