scholarly journals Generation of mode 2 internal waves by the interaction of mode 1 waves with topography

2019 ◽  
Vol 880 ◽  
pp. 799-830 ◽  
Author(s):  
Zihua Liu ◽  
Roger Grimshaw ◽  
Edward Johnson
Keyword(s):  
Internal Waves ◽  
Mode Coupling ◽  
Wave Amplitude ◽  
Fluid Model ◽  
Wave Theory ◽  
Topographic Slope ◽  
Long Wave ◽  
Mode 2 ◽  
Generation Mechanisms ◽  
Infinite Set

Oceanic internal waves can be decomposed into an infinite set of modes, and the dominant internal mode 1 waves have been extensively investigated. Although mode 2 waves have been observed, they have not received comparable attention, especially the generation mechanisms. In this work, we examine the generation of mode 2 internal waves by the interaction of mode 1 waves with topography. We use a coupled linear long-wave theory with mode coupling through topography, combined with evolution using a Korteweg–de Vries model, to predict the mode 2 wave amplitude, in an ideal three-layer fluid model, in a smooth density stratification and in two realistic oceanic settings. We find that the mode 2 wave amplitude is usually much smaller than the incident mode 1 wave amplitude and is quite sensitive to the pycnocline thickness, topographic slope and background stratification.

scholarly journals On Internal Waves Propagating across a Geostrophic Front

2019 ◽  
Vol 49 (5) ◽  
pp. 1229-1248 ◽  
Author(s):  
Qiang Li ◽  
Xianzhong Mao ◽  
John Huthnance ◽  
Shuqun Cai ◽  
Samuel Kelly
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Wave Reflection ◽  
Wave Theory ◽  
Horizontal Shear ◽  
Mixing Rate ◽  
Reflection And Transmission ◽  
Topographic Slope ◽  
Virtual Boundary ◽  
The Impact

AbstractReflection and transmission of normally incident internal waves propagating across a geostrophic front, like the Kuroshio or Gulf Stream, are investigated using a modified linear internal wave equation. A transformation from depth to buoyancy coordinates converts the equation to a canonical partial differential equation, sharing properties with conventional internal wave theory in the absence of a front. The equation type is determined by a parameter Δ, which is a function of horizontal and vertical gradients of buoyancy, the intrinsic frequency of the wave, and the effective inertial frequency, which incorporates the horizontal shear of background geostrophic flow. In the Northern Hemisphere, positive vorticity of the front may produce Δ ≤ 0, that is, a “forbidden zone,” in which wave solutions are not permitted. Thus, Δ = 0 is a virtual boundary that causes wave reflection and refraction, although waves may tunnel through forbidden zones that are weak or narrow. The slope of the surface and bottom boundaries in buoyancy coordinates (or the slope of the virtual boundary if a forbidden zone is present) determine wave reflection and transmission. The reflection coefficient for normally incident internal waves depends on rotation, isopycnal slope, topographic slope, and incident mode number. The scattering rate to high vertical modes allows a bulk estimate of the mixing rate, although the impact of internal wave-driven mixing on the geostrophic front is neglected.

Solitary internal waves with oscillatory tails

1992 ◽  
Vol 242 ◽  
pp. 279-298 ◽  
Author(s):  
T. R. Akylas ◽  
R. H. J. Grimshaw
Keyword(s):  
Internal Waves ◽  
Solitary Waves ◽  
Wave Theory ◽  
Wave Mode ◽  
Lower Mode ◽  
Weakly Nonlinear ◽  
Long Wave ◽  
Special Cases ◽  
Theoretical Predictions ◽  
Solitary Internal Waves

Solitary internal waves in a density-stratified fluid of shallow depth are considered. According to the classical weakly nonlinear long-wave theory, the propagation of each long-wave mode is governed by the Korteweg–de Vries equation to leading order, and locally confined solitary waves with a ‘sech’ profile are possible. Using a singular-perturbation procedure, it is shown that, in general, solitary waves of mode n > 1 actually develop oscillatory tails of infinite extent, consisting of lower-mode short waves. The amplitude of these tails is exponentially small with respect to an amplitude parameter, and lies beyond all orders of the usual long-wave expansion. To illustrate the theory, two special cases of stratification are discussed in detail, and the amplitude of the oscillations at the solitary-wave tails is determined explicitly. The theoretical predictions are supported by experimental observations.

scholarly journals The lifecycle of axisymmetric internal solitary waves

2010 ◽  
Vol 17 (5) ◽  
pp. 443-453 ◽  
Author(s):  
J. M. McMillan ◽  
B. R. Sutherland
Keyword(s):  
Solitary Waves ◽  
Wave Amplitude ◽  
Lower Layer ◽  
Average Density ◽  
Constant Speed ◽  
Supporting Evidence ◽  
Linear Dispersion ◽  
Long Wave ◽  
Mode 2 ◽  
Head Height

Abstract. The generation and evolution of solitary waves by intrusive gravity currents in an approximate two-layer fluid with equal upper- and lower-layer depths is examined in a cylindrical geometry by way of theory and numerical simulations. The study is limited to vertically symmetric cases in which the density of the intruding fluid is equal to the average density of the ambient. We show that even though the head height of the intrusion decreases, it propagates at a constant speed well beyond 3 lock radii. This is because the strong stratification at the interface supports the formation of a mode-2 solitary wave that surrounds the intrusion head and carries it outwards at a constant speed. The wave and intrusion propagate faster than a linear long wave; therefore, there is strong supporting evidence that the wave is indeed nonlinear. Rectilinear Korteweg-de Vries theory is extended to allow the wave amplitude to decay as r-p with p=½ and the theory is compared to the observed waves to demonstrate that the width of the wave scales with its amplitude. After propagating beyond 7 lock radii the intrusion runs out of fluid. Thereafter, the wave continues to spread radially at a constant speed, however, the amplitude decreases sufficiently so that linear dispersion dominates and the amplitude decays with distance as r-1.

scholarly journals Resonant coupling of mode-1 and mode-2 internal waves by topography

2020 ◽  
Author(s):  
Zihua Liu ◽  
Roger Grimshaw ◽  
Edward Johnson
Keyword(s):  
Internal Waves ◽  
Fluid System ◽  
Large Wave ◽  
Resonant Coupling ◽  
Topographic Slope ◽  
Kdv Equations ◽  
Mode 2 ◽  
Polarity Change ◽  
Comparable Amplitude ◽  
Variable Topography

<p>We consider the resonant coupling of mode-1 and mode-2 internal waves by topography. The mode-2 wave is generated by a mode-1 internal solitary wave encountering variable topography in the framework of a pair of coupled Korteweg-de Vries (KdV) equations.  Three cases (A) weak resonant coupling, (B) moderate resonant coupling, (C) strong resonant coupling, are examined using a three-layer fluid system with fixed total depth but different  layer thicknesses, and each case has two  different topographic slopes, gentle and steep, respectively.  The criterion for the strength of the resonant coupling is  the ratio of the  linear phase speeds c<sub>2 </sub>for mode-2 and c<sub>1</sub> for mode-1 waves.  This ratio c<sub>2</sub>/c<sub>1</sub> varies  from 0.42-0.48 (A), 0.58-0.72 (B), to 0.44-0.92 (C). The simulations using the coupled KdV model are compared with a KdV model for the evolution of a mode-1 wave alone. In case (A) a convex mode-2 wave of small amplitude is generated by a depression incident mode-1 wave and the feedback on mode-1 wave is negligible. In case (B) a concave mode-2 wave of  comparable amplitude to the incident mode-1 wave is formed from a depression incident mode-1 wave; strong feedback enhances the polarity change process of the mode-1 wave. In (C) a concave mode-2 wave of large wave amplitude with wave fission is produced by an elevation incident mode-1 wave; strong feedback from the mode-2 wave suppresses the fission of the mode-1 wave. In all cases, the amplitudes of the generated mode-2 waves are proportional to the topographic slope.</p>

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.

scholarly journals On the generation of internal waves by river plumes in subcritical initial conditions

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
R. Mendes ◽  
J. C. B. da Silva ◽  
J. M. Magalhaes ◽  
B. St-Denis ◽  
D. Bourgault ◽  
...  
Keyword(s):  
Numerical Modeling ◽  
Internal Waves ◽  
Coastal Waters ◽  
Initial Conditions ◽  
Spatial Scales ◽  
River Plumes ◽  
Wide Range ◽  
Near Shore ◽  
Generation Mechanisms ◽  
Douro River

AbstractInternal waves (IWs) in the ocean span across a wide range of time and spatial scales and are now acknowledged as important sources of turbulence and mixing, with the largest observations having 200 m in amplitude and vertical velocities close to 0.5 m s−1. Their origin is mostly tidal, but an increasing number of non-tidal generation mechanisms have also been observed. For instance, river plumes provide horizontally propagating density fronts, which were observed to generate IWs when transitioning from supercritical to subcritical flow. In this study, satellite imagery and autonomous underwater measurements are combined with numerical modeling to investigate IW generation from an initial subcritical density front originating at the Douro River plume (western Iberian coast). These unprecedented results may have important implications in near-shore dynamics since that suggest that rivers of moderate flow may play an important role in IW generation between fresh riverine and coastal waters.

scholarly journals Gravity currents and internal waves in a stratified fluid

2008 ◽  
Vol 616 ◽  
pp. 327-356 ◽  
Author(s):  
BRIAN L. WHITE ◽  
KARL R. HELFRICH
Keyword(s):  
Internal Waves ◽  
Wave Speed ◽  
Numerical Solutions ◽  
Gravity Current ◽  
Gravity Currents ◽  
Stratified Fluid ◽  
Front Speed ◽  
Long Wave ◽  
Conjugate State ◽  
Energy Conserving

A steady theory is presented for gravity currents propagating with constant speed into a stratified fluid with a general density profile. Solution curves for front speed versus height have an energy-conserving upper bound (the conjugate state) and a lower bound marked by the onset of upstream influence. The conjugate state is the largest-amplitude nonlinear internal wave supported by the ambient stratification, and in the limit of weak stratification approaches Benjamin's energy-conserving gravity current solution. When the front speed becomes critical with respect to linear long waves generated above the current, steady solutions cannot be calculated, implying upstream influence. For non-uniform stratification, the critical long-wave speed exceeds the ambient long-wave speed, and the critical-Froude-number condition appropriate for uniform stratification must be generalized. The theoretical results demonstrate a clear connection between internal waves and gravity currents. The steady theory is also compared with non-hydrostatic numerical solutions of the full lock release initial-value problem. Some solutions resemble classic gravity currents with no upstream disturbance, but others show long internal waves propagating ahead of the gravity current. Wave generation generally occurs when the stratification and current speed are such that the steady gravity current theory fails. Thus the steady theory is consistent with the occurrence of either wave-generating or steady gravity solutions to the dam-break problem. When the available potential energy of the dam is large enough, the numerical simulations approach the energy-conserving conjugate state. Existing laboratory experiments for intrusions and gravity currents produced by full-depth lock exchange flows over a range of stratification profiles show excellent agreement with the conjugate state solutions.

scholarly journals Spatially Broad Observations of Internal Waves in the Upper Ocean at the Hawaiian Ridge

2006 ◽  
Vol 36 (6) ◽  
pp. 1085-1103 ◽  
Author(s):  
Joseph P. Martin ◽  
Daniel L. Rudnick ◽  
Robert Pinkel
Keyword(s):  
Numerical Model ◽  
Energy Density ◽  
Potential Energy ◽  
Internal Waves ◽  
Upper Ocean ◽  
Mean Square ◽  
The South ◽  
South Side ◽  
Topographic Slope ◽  
Hawaiian Ridge

Abstract The density and current structure at the Hawaiian Ridge was observed using SeaSoar and Doppler sonar during a survey extending from Oahu to Brooks Banks. Across- and along-ridge changes in internal wave statistics in the upper ocean within 200 km of the ridge are investigated. Internal waves with trough-to-crest amplitude as large as 60 m and horizontal wavelength of about 50 km are observed repeatedly in across-ridge sections of potential density. Within 150 km of the ridge, kinetic and potential energy density exceed open-ocean values with maxima about 10 times Garrett–Munk levels. In the Kauai Channel (KC), the kinetic energy density is largest along an M2 internal tide ray. The ray originates at the northern edge of the ridge peak at a large across-ridge change in topographic slope and terminates at the ocean surface about 30–40 km south of the ridge peak. Kinetic and potential energy density are larger on the south side of the ridge at KC, the side with larger topographic slope. Energy density is also larger on the south side of the ridge at KC in numerical model results and on the side of steeper topographic slope in analytical model results. Along the ridge, the largest observed values of mean-square shear and mean-square slope of isopycnal depth are collocated with the largest energy density in numerical model results. Mean-square shear and mean-square slope increase with decreasing bottom depth and with increasing M2 barotropic tidal forcing.

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