INFLUENCE OF SURFACE TENSION AND BOTTOM TOPOGRAPHY ON INTERNAL WAVES

2009 ◽  
Vol 19 (12) ◽  
pp. 2145-2175 ◽  
Author(s):  
CUNG THE ANH
Keyword(s):  
Surface Tension ◽  
Internal Waves ◽  
Evolution Equations ◽  
Bottom Topography ◽  
Global Strategy ◽  
Nonlocal Operators ◽  
Asymptotic Models ◽  
Small Parameters ◽  
Euler Model ◽  
Tension Term

Following the global strategy introduced recently by Bona, Lannes and Saut in Ref. 7, we derive here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, the presence of surface tension and with uneven bottoms. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on ℝd, d = 1, 2, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators and the surface tension term with respect to suitable small parameters that depend variously on the amplitude, wavelengths and depth ratio of the two layers. Furthermore, the consistency of these asymptotic systems with the full Euler equations is established.

scholarly journals DECOUPLED AND UNIDIRECTIONAL ASYMPTOTIC MODELS FOR THE PROPAGATION OF INTERNAL WAVES

2013 ◽  
Vol 24 (01) ◽  
pp. 1-65 ◽  
Author(s):  
VINCENT DUCHÊNE
Keyword(s):  
Initial Data ◽  
Internal Waves ◽  
Evolution Equations ◽  
Approximate Solutions ◽  
Type Model ◽  
Fluid System ◽  
Asymptotic Models ◽  
Counterpropagating Waves ◽  
Scalar Equations ◽  
Two Fluid

We study the relevance of various scalar equations, such as inviscid Burgers', Korteweg–de Vries (KdV), extended KdV, and higher order equations, as asymptotic models for the propagation of internal waves in a two-fluid system. These scalar evolution equations may be justified in two ways. The first method consists in approximating the flow by two uncoupled, counterpropagating waves, each one satisfying such an equation. One also recovers these equations when focusing on a given direction of propagation, and seeking unidirectional approximate solutions. This second justification is more restrictive as for the admissible initial data, but yields greater accuracy. Additionally, we present several new coupled asymptotic models: a Green–Naghdi type model, its simplified version in the so-called Camassa–Holm regime, and a weakly decoupled model. All of the models are rigorously justified in the sense of consistency.

scholarly journals Comment on “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” [Nonlinear Dyn,  doi:10.1007/s11071-017-3938-7]

2021 ◽  
Author(s):  
Piotr Rozmej ◽  
Anna Karczewska
Keyword(s):  
Surface Tension ◽  
Shallow Water ◽  
Water Waves ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Two Dimensional ◽  
Shallow Water Waves ◽  
Small Parameters ◽  
Fifth Order

AbstractThe authors of the paper “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” Fokou et al. (Nonlinear Dyn 91:1177–1189, 2018) claim that they derived the equation which generalizes the KdV equation to two space dimensions both in first and second order in small parameters. Moreover, they claim to obtain soliton solution to the derived first-order (2+1)-dimensional equation. The equation has been obtained by applying the perturbation method Burde (J Phys A: Math Theor 46:075501, 2013) for small parameters of the same order. The results, if correct, would be significant. In this comment, it is shown that the derivation presented in Fokou et al. (Nonlinear Dyn 91:1177–1189, 2018) is inconsistent because it violates fundamental properties of the velocity potential. Therefore, the results, particularly the new evolution equation and the dynamics that it describes, bear no relation to the problem under consideration.

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.

Tidal Internal Waves in the Bransfield Strait, Antarctica

2021 ◽  
Author(s):  
Eugene Morozov ◽  
Dmitry Frey ◽  
Elizaveta Khimchenko
Keyword(s):  
Internal Waves ◽  
Bottom Topography ◽  
Pressure Sensors ◽  
Internal Tide ◽  
Internal Tides ◽  
South Shetland Islands ◽  
Flat Bottom ◽  
Bransfield Strait ◽  
Vertical Displacements ◽  
Rfbr Grant

<p>Observations of tidal internal waves in the Bransfield Strait, Antarctica, are analyzed. The measurements were carried out for 14 days on a moored station equipped with five autonomous temperature and pressure sensors. The mooring was deployed on the slope of Nelson Island (South Shetland Islands archipelago) over a depth of 70 m at point 62°21ꞌ S, 58°49ꞌ W. Analysis is based on the fluctuations of isotherms.  Vertical displacements of temperature revealed that strong internal vertical oscillations up to 30–40 m are caused by the diurnal internal tide. Spectral analysis of vertical displacements of the 0.9°C isotherm showed a clear peak at a period of 24 h. It is known that the tides in the Bransfield Strait are mostly mixed diurnal and semidiurnal, but during the Antarctic summer, diurnal tide component may intensify. The velocity ellipses of the barotropic tidal currents were estimated using the global tidal model TPXO9.0. It was found that tidal ellipses rotate clockwise with a period of 24 h and anticlockwise with a period of 12 h. The waves are forced due to the interaction of the barotropic tide with the bottom topography. Diurnal internal tides do not develop at latitudes higher than 30º over flat bottom. The research was supported by RFBR grant 20-08-00246.</p>

Subharmonic resonance of short internal standing waves by progressive surface waves

1996 ◽  
Vol 321 ◽  
pp. 217-233 ◽  
Author(s):  
D. F. Hill ◽  
M. A. Foda
Keyword(s):  
Surface Wave ◽  
Internal Wave ◽  
Surface Waves ◽  
Internal Waves ◽  
Growth Rates ◽  
Evolution Equations ◽  
Standing Waves ◽  
Second Order ◽  
Inviscid Limit ◽  
Initial Growth

Experimental evidence and a theoretical formulation describing the interaction between a progressive surface wave and a nearly standing subharmonic internal wave in a two-layer system are presented. Laboratory investigations into the dynamics of an interface between water and a fluidized sediment bed reveal that progressive surface waves can excite short standing waves at this interface. The corresponding theoretical analysis is second order and specifically considers the case where the internal wave, composed of two oppositely travelling harmonics, is much shorter than the surface wave. Furthermore, the analysis is limited to the case where the internal waves are small, so that only the initial growth is described. Approximate solution to the nonlinear boundary value problem is facilitated through a perturbation expansion in surface wave steepness. When certain resonance conditions are imposed, quadratic interactions between any two of the harmonics are in phase with the third, yielding a resonant triad. At the second order, evolution equations are derived for the internal wave amplitudes. Solution of these equations in the inviscid limit reveals that, at this order, the growth rates for the internal waves are purely imaginary. The introduction of viscosity into the analysis has the effect of modifying the evolution equations so that the growth rates are complex. As a result, the amplitudes of the internal waves are found to grow exponentially in time. Physically, the viscosity has the effect of adjusting the phase of the pressure so that there is net work done on the internal waves. The growth rates are, in addition, shown to be functions of the density ratio of the two fluids, the fluid layer depths, and the surface wave conditions.

The detergent stress-cracking of polyethylene

1977 ◽  
Vol 352 (1671) ◽  
pp. 505-521 ◽  
Keyword(s):  
Surface Tension ◽  
Molecular Mass ◽  
Strain Hardening ◽  
Element Analysis ◽  
Stress Cracking ◽  
Plastic Materials ◽  
Environmental Stress Cracking ◽  
Tension Term ◽  
Plain Strain ◽  
Two Parameters

Many plastic materials fail under plain strain conditions by a process of micro-voiding in sheet-like structures known as crazes. These crazes are generally weak though their strength and stability may be improved in materials where orientation (strain) hardening occurs. Previously we have used finite element analysis, with a model consisting of an assembly of cylindrical holes, to study this type of plastic failure. With simple plasticity such models fail soon after yield, though they also would be stabilized by strain hardening. In this study we have introduced a surface-tension term which increases the resistance to plastic deformation. The effect is greatest when the holes are small and the yield strength of the material low. It is shown that when values of these two parameters representative of polyethylene at 50 °C are used surface tension should have a marked stabilizing effect for holes of < 25 nm radius. Even above this size range the failure stress can be affected at the 5-20 % level. An electron micrographic study was then carried out on polyethylenes cracked in the presence of an aqueous detergent (detergent stress cracking). It is shown that numerous small cavities and fibrils are formed on a scale small enough for the effect of surface tension to be significant. The rôle of the detergent in reducing surface tension and so promoting the fracture of polyethylene may therefore be understood. The marked influence of molecular mass on environmental stress cracking behaviour may be correlated with the greater development of orientation hardening at high strains with high molecular mass polymers. This increases the amount of plastic work preceding fracture and tends to stabilize a craze-like system.

Generalized vortex methods for free-surface flow problems

1982 ◽  
Vol 123 ◽  
pp. 477-501 ◽  
Author(s):  
Gregory R. Baker ◽  
Daniel I. Meiron ◽  
Steven A. Orszag
Keyword(s):  
Water Waves ◽  
Evolution Equations ◽  
Bottom Topography ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Three Dimensions ◽  
Fredholm Integral Equations ◽  
Free Surfaces ◽  
Flow Problems ◽  
Dipole Vortex

The motion of free surfaces in incompressible, irrotational, inviscid layered flows is studied by evolution equations for the position of the free surfaces and appropriate dipole (vortex) and source strengths. The resulting Fredholm integral equations of the second kind may be solved efficiently in both storage and work by iteration in both two and three dimensions. Applications to breaking water waves over finite-bottom topography and interacting triads of surface and interfacial waves are given.

Spatio-temporal variability of internal waves in the Caspian Sea

2020 ◽  
Author(s):  
Olga Lavrova ◽  
Andrey Kostianoy
Keyword(s):  
Surface Temperature ◽  
Sea Surface Temperature ◽  
Internal Waves ◽  
Satellite Data ◽  
Caspian Sea ◽  
Bottom Topography ◽  
Sea Surface ◽  
Anthropogenic Factors ◽  
Landsat 8 ◽  
The Caspian Sea

&lt;p&gt;Internal waves (IWs) are an intrinsic feature of all density stratified water bodies: oceans, seas, lakes and reservoirs. IWs occur due to various causes. Among them are tides and inertial motions, variations in atmospheric pressure and wind, underwater earthquakes, water flows over bottom topography, anthropogenic factors, etc. In coastal areas of oceans and tidal seas, &amp;#160;IWs induced by tidal currents over shelf edge predominate. Such IWs are well-studied in multiple field, laboratory and numerical experiments. However, the data on IWs in non-tidal seas, such as the Black, Baltic and Caspian Seas, are scarce. Meanwhile, our multi-year satellite observations prove IWs to be quite a characteristic hydrophysical phenomenon of the Caspian Sea. The sea is considered non-tidal because tide height does not exceed 12 cm at the coastline. And yet surface manifestations of IWs are regularly observed in satellite data, both radar and visible. The goal of our study was to reveal spatial, seasonal and interannual variability of IW surface manifestations in the Caspian Sea in the periods of 1999-2012 and 2018-2019 from the analysis of satellite data. All available satellite radar and visible data were used, that is data from ERS1/2 SAR; Envisat ASAR; Sentinel-1A,1B SAR-C; Landsat-4,5 TM; Landsat-7 ETM+; Landsat-8 OLI; Sentinel-2A,2B MSI sensors. During the year, IWs were observed from the beginning of May to mid-September. In certain years, depending on hydrometeorological conditions, such as water heating, wind field, etc., no IWs could be seen in May or September. IWs regularly occur in the east of Middle Caspian and in the northeast of South Caspian. In North Caspian, due to its shallowness and absence of pronounced stratification, IWs are not generated, at least their surface signatures cannot be found in satellite data. In the west of the sea, IWs are scarcely observed, primarily at the beginning of the summer season. IW trains propagate toward the coast, their generation sites are mainly over the depths of 50-200 m.&lt;/p&gt;&lt;p&gt;According to the available data for the studied periods, the time of the first appearance of IW signatures differs significantly from year to year. For example, in 1999 and 2000 it happened only in July.&lt;/p&gt;&lt;p&gt;Since no in situ measurements were conducted in the sites of regular IW manifestations, an attempt&amp;#160; was made to establish the dependence of IW occurrence frequency&amp;#160; on seasonal and interannual variations of sea surface temperature, an indirect indicator of the depth of the diurnal or seasonal thermocline, that is where IW were generated. Sea surface temperature was also estimated from satellite data.&lt;/p&gt;&lt;p&gt;Another issue addressed in the work was the differentiation between the sea surface signatures of IWs in the atmosphere and the sea. The Caspian Sea is known for their close similarity in spatial characteristics.&lt;/p&gt;&lt;p&gt;The work was carried out with financial support of the Russian Science Foundation grant #19-77-20060.&amp;#160; Processing of satellite data was carried out by Center for Collective Use &amp;#8220;IKI-Monitoring&amp;#8221; with the use of &amp;#8220;See The Sea&amp;#8221; system, that was implemented in frame of Theme &amp;#8220;Monitoring&amp;#8221;, State register No. 01.20.0.2.00164.&lt;/p&gt;

Propagation and transformation of nonlinear internal waves of tidal origin observed in the northeastern East China Sea

2020 ◽  
Author(s):  
Seung-Woo Lee ◽  
SungHyun Nam
Keyword(s):  
East China Sea ◽  
Internal Waves ◽  
Bottom Topography ◽  
Wave Interaction ◽  
Propagation Speed ◽  
East China ◽  
Modal Interactions ◽  
Nonlinear Internal Waves ◽  
China Sea ◽  
Mode 2

&lt;p&gt;Oceanic nonlinear internal waves (NLIWs) play an important role in regional circulation, biogeochemistry, energetics, vertical mixing, and underwater acoustics, causing hazards to marine engineering and submarine navigation. Mainly generated by the interaction of the barotropic tides with the bottom topography, they propagate and transform due to wave-wave interaction process. Here, we present characteristics of first two modes of NLIWs observed using high-resolution spatiotemporal data collected in a relatively flat area in the northeastern East China Sea in May 2015. Six groups of NLIWs were identified from the observations: four groups of mode-1 and two groups of mode-2. The amplitude, propagation speed, and characteristic width of mode-1 NLIWs had ranges of 4&amp;#8211;16 m, 0.53&amp;#8211;0.56 m s&lt;sup&gt;-1&lt;/sup&gt;, and 310&amp;#8211;610 m, respectively. The mode-2 NLIWs propagate eastward slowly with a speed less than 0.37 m s&lt;sup&gt;-1&lt;/sup&gt; with a comparable amplitude of 4&amp;#8211;14 m and longer characteristic width of 540&amp;#8211;1920 m. Intermodal interactions may take a role in the evolution of mode-1 NLIWs west of the observational area. Our results characterizing the two modes of NLIWs highlight the significance of propagation and transformation of NLIWs and their modal interactions on a broad and shallow shelf.&lt;/p&gt;

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