scholarly journals Internal wave focusing by a horizontally oscillating torus

2017 ◽  
Vol 813 ◽  
pp. 695-715 ◽  
Author(s):  
E. V. Ermanyuk ◽  
N. D. Shmakova ◽  
J.-B. Flór
Keyword(s):  
Internal Wave ◽  
Wave Amplitude ◽  
Laser Sheet ◽  
Oscillation Amplitude ◽  
Nonlinear Effects ◽  
Direct Access ◽  
Focal Region ◽  
Linear Wave ◽  
Wave Regime ◽  
Wave Focusing

This paper presents an experimental study on internal waves emitted by a horizontally oscillating torus in a linearly stratified fluid. Two internal wave cones are generated with the kinetic energy focused at the apices of the cones above and below the torus where the wave amplitude is maximal. Their motion is measured via tracking of distortions of horizontal fluorescein dye planes created prior to the experiments and illuminated by a vertical laser sheet. The distortion of the dye planes gives a direct access to the Lagrangian displacement of local wave amplitudes and slopes, and in particular, allows us to calculate a local Richardson number. In addition particle image velocimetry measurements are used. Maximum wave slopes are found in the focal region and close to the surface of the torus. As the amplitude of oscillations of the torus increases, wave profiles in the regions of maximum wave slopes evolve nonlinearly toward local overturning. A theoretical approximation based on the theory of Hurley & Keady (J. Fluid Mech., vol. 351, 1997, pp. 119–138) is presented and shows, for small amplitudes of oscillation, a very reasonable agreement with the experimental data. For the focal region the internal wave amplitude is found to be overestimated by the theory. The wave breaking in the focal region is investigated as a function of the Keulegan–Carpenter number, $Ke=A/a$, with $A$ the oscillation amplitude and $a$ the short radius of the torus. A linear wave regime is found for $Ke<0.4$, nonlinear effects start at $Ke\approx 0.6$ and breaking for $Ke>0.8$. For large forcing, the measured wave amplitude normalized with the oscillation amplitude decreases almost everywhere in the wave field, but increases locally in the focal region due to nonlinear effects. Due to geometric focusing the amplitude of the wave increases with $\sqrt{\unicode[STIX]{x1D716}}$, with $\unicode[STIX]{x1D716}=b/a$ and $b$ is the mean radius of the torus. The relevance of wave focusing due to ocean topography is discussed.

scholarly journals Nonlinear aspects of focusing internal waves

2019 ◽  
Vol 862 ◽  
Author(s):  
Natalia D. Shmakova ◽  
Jan-Bert Flór
Keyword(s):  
Wave Amplitude ◽  
Oscillation Amplitude ◽  
Three Dimensional ◽  
Nonlinear Effects ◽  
Focal Region ◽  
Mean Flow ◽  
Wave Focusing ◽  
Tidal Waves ◽  
Two Dimensional Flow ◽  
Major Radius

When a torus oscillates horizontally in a linearly stratified fluid, the wave rays form a double cone, one upward and one downward, with two focal points where the wave amplitude has a maximum due to wave focusing. Following a former study on linear aspects of wave focusing (Ermanyuk et al., J. Fluid Mech., vol. 813, 2017, pp. 695–715), we here consider experimental results on the nonlinear aspects that occur in the focal region below the torus for higher-amplitude forcing. A new non-dimensional number that is based on heuristic arguments for the wave amplitude in the focal area is presented. This focusing number is defined as $Fo=(A/a)\unicode[STIX]{x1D716}^{-1/2}f(\unicode[STIX]{x1D703})$, with oscillation amplitude $A$, $f(\unicode[STIX]{x1D703})$ a function for the variation of the wave amplitude with wave angle $\unicode[STIX]{x1D703}$, and $\unicode[STIX]{x1D716}^{1/2}=\sqrt{b/a}$ the increase in amplitude due to the focusing, with $a$ and $b$, respectively, the minor and major radius of the torus. Nonlinear effects occur for $Fo\geqslant 0.1$, with the shear stress giving rise to a mean flow which results in the focal region in a central upward motion partially surrounded by a downward motion. With increasing $Fo$, the Richardson number $Ri$ measured from the wave steepness monotonically decreases. Wave breaking occurs at $Fo\approx 0.23$, corresponding to $Ri=0.25$. In this regime, the focal region is unstable due to triadic wave resonance. For the different tori sizes under consideration, the triadic resonant instability in these three-dimensional flows resembles closely the resonance observed by Bourget et al. (J. Fluid Mech., vol. 723, 2013, pp. 1–20) for a two-dimensional flow, with only minor differences. Application to internal tidal waves in the ocean are discussed.

Density Wave Measurements in a Rectangular Clarifier

1983 ◽  
Vol 18 (1) ◽  
pp. 129-150 ◽  
Author(s):  
Mark K. Watson ◽  
R.R. Hudgins ◽  
P.L. Silveston
Keyword(s):  
Power Spectrum ◽  
Internal Wave ◽  
Internal Waves ◽  
Wave Motion ◽  
Wave Amplitude ◽  
Suspended Solids ◽  
Small Volume ◽  
Density Wave ◽  
Measure Concentration ◽  
Wave Measurements

Abstract Internal wave motion was studied in a laboratory rectangular, primary clarifier. A photo-extinction device was used as a turbidimeter to measure concentration fluctuations in a small volume within the clarifier as a function of time. The signal from this device was fed to a HP21MX minicomputer and the power spectrum plotted from data records lasting approximately 30 min. Results show large changes of wave amplitude as frequency increases. Two distinct regions occur: one with high amplitudes at frequencies below 0.03 Hz, the second with very small amplitudes appears for frequencies greater than 0.1 Hz. The former is associated with internal waves, the latter with flow-generated turbulence. Depth, velocity in the clarifier and inlet suspended solids influence wave amplitudes and the spectra. A variation with position or orientation of the probe was not detected. Contradictory results were found for the influence of flow contraction baffles on internal wave amplitude.

Nonlinear effects in reflecting and colliding internal wave beams

2005 ◽  
Vol 526 ◽  
pp. 217-243 ◽  
Author(s):  
ALI TABAEI ◽  
T. R. AKYLAS ◽  
KEVIN G. LAMB
Keyword(s):  
Internal Wave ◽  
Nonlinear Effects

Internal wave focusing revisited; a reanalysis and new theoretical links

2008 ◽  
Vol 40 (2) ◽  
pp. 95-122 ◽  
Author(s):  
Frans-Peter A Lam ◽  
Leo R M Maas
Keyword(s):  
Internal Wave ◽  
Wave Focusing

scholarly journals Multi-spacecraft determination of wave characteristics near the proton gyrofrequency in high-altitude cusp

2005 ◽  
Vol 23 (3) ◽  
pp. 983-995 ◽  
Author(s):  
D. Sundkvist ◽  
A. Vaivads ◽  
M. André ◽  
J.-E. Wahlund ◽  
Y. Hobara ◽  
...  
Keyword(s):  
Magnetic Field ◽  
High Altitude ◽  
Nonlinear Effects ◽  
Linear Wave ◽  
Wave Growth ◽  
Ion Cyclotron Waves ◽  
Background Magnetic Field ◽  
Ion Cyclotron ◽  
The Waves ◽  
Proton Gyrofrequency

Abstract. We present a detailed study of waves with frequencies near the proton gyrofrequency in the high-altitude cusp for northward IMF as observed by the Cluster spacecraft. Waves in this regime can be important for energization of ions and electrons and for energy transfer between different plasma populations. These waves are present in the entire cusp with the highest amplitudes being associated with localized regions of downward precipitating ions, most probably originating from the reconnection site at the magnetopause. The Poynting flux carried by these waves is downward/upward at frequencies below/above the proton gyrofrequency, which is consistent with the waves being generated near the local proton gyrofrequency in an extended region along the flux tube. We suggest that the waves can be generated by the precipitating ions that show shell-like distributions. There is no clear polarization of the perpendicular wave components with respect to the background magnetic field, while the waves are polarized in a parallel-perpendicular plane. The coherence length is of the order of one ion-gyroradius in the direction perpendicular to the ambient magnetic field and a few times larger or more in the parallel direction. The perpendicular phase velocity was found to be of the order of 100km/s, an order of magnitude lower than the local Alfvén speed. The perpendicular wavelength is of the order of a few proton gyroradius or less. Based on our multi-spacecraft observations we conclude that the waves cannot be ion-whistlers, while we suggest that the waves can belong to the kinetic Alfvén branch below the proton gyrofrequency fcp and be described as non-potential ion-cyclotron waves (electromagnetic ion-Bernstein waves) above. Linear wave growth calculations using kinetic code show considerable wave growth of non-potential ion cyclotron waves at wavelengths agreeing with observations. Inhomogeneities in the plasma on the order of the ion-gyroradius suggests that inhomogeneous (drift) or nonlinear effects or both of these should be taken into account.

A viscous internal wave in a stratified fluid whose buoyancy frequency varies with altitude

1975 ◽  
Vol 69 (3) ◽  
pp. 615-624 ◽  
Author(s):  
D. Gordon ◽  
U. R. Klement ◽  
T. N. Stevenson
Keyword(s):  
Internal Wave ◽  
Small Amplitude ◽  
Wave Amplitude ◽  
Stratified Fluid ◽  
Similarity Solutions ◽  
Buoyancy Frequency ◽  
Experimental Measurements ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Curved Paths

A viscous incompressible stably stratified fluid with a buoyancy frequency which varies slowly with altitude is considered. A simple harmonic localized disturbance generates an internal wave in which the energy propagates along curved paths. Small amplitude similarity solutions are obtained for two-dimensional and axisymmetric waves. It is found that under certain conditions the wave amplitude can increase with height. The two-dimensional theory compares quite well with experimental measurements.

Internal-Wave-Driven Mixing: Global Geography and Budgets

2017 ◽  
Vol 47 (6) ◽  
pp. 1325-1345 ◽  
Author(s):  
Eric Kunze
Keyword(s):  
Internal Wave ◽  
Internal Tide ◽  
Internal Tides ◽  
Tidal Mixing ◽  
Linear Wave ◽  
Power Sources ◽  
The North ◽  
Dissipation Rates ◽  
Thickness Variability ◽  
Significant Pathway

AbstractInternal-wave-driven dissipation rates ε and diapycnal diffusivities K are inferred globally using a finescale parameterization based on vertical strain applied to ~30 000 hydrographic casts. Global dissipations are 2.0 ± 0.6 TW, consistent with internal wave power sources of 2.1 ± 0.7 TW from tides and wind. Vertically integrated dissipation rates vary by three to four orders of magnitude with elevated values over abrupt topography in the western Indian and Pacific as well as midocean slow spreading ridges, consistent with internal tide sources. But dependence on bottom forcing is much weaker than linear wave generation theory, pointing to horizontal dispersion by internal waves and relatively little local dissipation when forcing is strong. Stratified turbulent bottom boundary layer thickness variability is not consistent with OGCM parameterizations of tidal mixing. Average diffusivities K = (0.3–0.4) × 10−4 m2 s−1 depend only weakly on depth, indicating that ε = KN2/γ scales as N2 such that the bulk of the dissipation is in the pycnocline and less than 0.08-TW dissipation below 2000-m depth. Average diffusivities K approach 10−4 m2 s−1 in the bottom 500 meters above bottom (mab) in height above bottom coordinates with a 2000-m e-folding scale. Average dissipation rates ε are 10−9 W kg−1 within 500 mab then diminish to background deep values of 0.15 × 10−9 W kg−1 by 1000 mab. No incontrovertible support is found for high dissipation rates in Antarctic Circumpolar Currents or parametric subharmonic instability being a significant pathway to elevated dissipation rates for semidiurnal or diurnal internal tides equatorward of 28° and 14° latitudes, respectively, although elevated K is found about 30° latitude in the North and South Pacific.

Modelling of an Internal Wave Gravity Current Using Eulers Equations

2002 ◽  
Author(s):  
Deborah J. Wood
Keyword(s):  
Internal Wave ◽  
Wave Speed ◽  
Gravity Current ◽  
Laboratory Scale ◽  
Navier Stokes ◽  
Linear Wave ◽  
Field Observations ◽  
Medium Density ◽  
Sloping Bottom

In nature where thermoclines exist an internal wave may form, and if a sloping bottom is also present then a gravity current may occur. In this study we use a Navier-Stokes solver to solve Eulers equations to simulate the generation and evolution of such a wave. The thermoclines used in this study are similar to those seen in nature except scaled down to the laboratory scale used by some ongoing experiments. We find that the Navier-Stokes solver generates and evolves a wave similar to experimental observations. The head of the gravity current is dominated by medium density fluid with the thermocline thickness growing and becoming thickest at the centre of the head. Maximum velocities of approximately 0.5 of the linear wave speed are found which are similar to experimental and field observations.

Nonlinear Effects in Resonantly Driven Surface Waves on Fluids

1972 ◽  
Vol 50 (19) ◽  
pp. 2235-2243
Author(s):  
F. L. Curzon ◽  
G. N. Ionides
Keyword(s):  
Surface Waves ◽  
Electric Fields ◽  
Driving Force ◽  
Wave Amplitude ◽  
Nonlinear Effects ◽  
Microwave Resonator ◽  
Electrode Geometry ◽  
Fluid Surface ◽  
Electric Stress ◽  
Time Periodic

The results presented in this paper show that fluid surface waves, resonantly driven by spatially nonuniform, time periodic electric fields, exhibit nonlinear effects when the wave amplitude ξ exceeds a significant fraction of the distance D between the driver electrode and the fluid surface. The phase difference between the surface wave and the driving force, as well as the dependence of wave amplitude on the electric stress are computed and compared with experimental results. For ξ/D exceeding ~0.7 (dependent on electrode geometry) the surface waves excited are unstable (also confirmed experimentally). The experiments are performed on surface waves on mercury contained in a cylindrical microwave resonator. Shifts in the microwave resonant frequency (caused by the surface waves) monitor the displacement of the fluid surface.

Ocean Swell: How Much Do We Know

2017 ◽  
Author(s):  
Alexander V. Babanin ◽  
Haoyu Jiang
Keyword(s):  
Wave Amplitude ◽  
Arrival Time ◽  
Frequency Dispersion ◽  
Nonlinear Effects ◽  
Joint Analysis ◽  
Wave Forecast ◽  
Model Predictions ◽  
Forecast Models ◽  
Maritime Operations ◽  
Definition Of

Swell waves are present in more than 80% of ocean seas, and provide significant adverse impact on maritime operations. Their prediction by wave-forecast models, however, is poor, both in terms of wave amplitude and, particularly, arrival time. The very definition of ocean swell is ambiguous: while it is usually perceived as former wind-generated waves, in fact it may reconnect with the local wind through nonlinear interactions. The paper will bring together an overview of the complex swell problem. The visible swell attenuation is driven by a number of dissipative and non-dissipative processes. The dissipative phenomena include interaction with turbulence on the water and air sides, with adverse winds or currents. Non-dissipative contributions to the gradual decline of wave amplitude come from frequency dispersion, directional spreading, refraction by currents, and lateral diffraction of wave energy. The interactions with local winds/waves can, on the contrary, cause swell growth. Swell arrival time is the least understood and the most uncertain problem. Joint analysis of buoy observations and model reanalysis shows that swell can be tens of hours early or late by comparison with model predictions. Linear and nonlinear effects which can contribute to such biases are discussed.

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