Weak quadratic interactions of two-dimensional waves

1971 ◽  
Vol 50 (1) ◽  
pp. 107-132 ◽  
Author(s):  
Young Yuel Kim ◽  
Thomas J. Hanratty

This paper reports on weak quadratic interactions which can occur with two-dimensional waves on shallow water layers and in the capillary-gravity range on deep water layers. It supplies experimental support of theoretical predictions for resonant interactions, but, perhaps of more significance, it explores in detail interactions which occur under conditions near resonance.Waves of approximately sinusoidal form are introduced on the surface of water in a long rectangular tank. For deep water a rapid distortion in the sinusoidal wave and sometimes additional crests are observed because of energy exchange among the first, second and third harmonics at frequencies where both surface tension and gravity are important (7·5–13 c/s). An even greater exchange of energy can be observed on shallow water layers at low frequencies. For example, a wave train with seven secondary crests can be observed when the wave maker is operated at 3·04 c/s in a water layer of 0·65 cm.Measured amplitudes and phase angles of the Fourier components of the wave train are described by a system of equations using only quadratic interactions among participating harmonics. The exchange of energy among Fourier components under certain conditions is explained in terms of the rate of change of relative phase angles of the different harmonics.

2014 ◽  
Vol 70 (2) ◽  
pp. 198-201 ◽  
Author(s):  
Ji Qi ◽  
Xiang-Sheng Zhai ◽  
Hong-Lin Zhu ◽  
Jian-Li Lin

A tetranuclear CoIIIoxide complex with cubane topology, tetrakis(2,2′-bipyridine-κ2N,N′)di-μ2-carbonato-κ4O:O′-tetra-μ3-oxido-tetracobalt(III) pentadecahydrate, [Co4(CO3)2O4(C10H8N2)4]·15H2O, with an unbounded hydrogen-bonded water layer, has been synthesized by reaction of CoCO3and 2,2′-bipyridine. The solvent water molecules form a hydrogen-bonded net with tetrameric and pentameric water clusters as subunits. The Co4O4cubane-like cores are sandwiched between the water layers, which are further stacked into a three-dimensional metallo-supramolecular network.


Author(s):  
Bernard Deconinck ◽  
Dmitry E Pelinovsky ◽  
John D Carter

The dynamics of a one-dimensional slowly modulated, nearly monochromatic localized wave train in deep water is described by a one-dimensional soliton solution of a two-dimensional nonlinear Schrödinger (NLS) equation. In this paper, the instability of such a wave train with respect to transverse perturbations is examined numerically in the context of the NLS equation, using Hill's method. A variety of instabilities are obtained and discussed. Among these, we show that the solitary wave is susceptible to an oscillatory instability (complex growth rate) due to perturbations with arbitrarily short wavelength. Further, there is a cut-off on the instability with real growth rates. We show analytically that the nature of this cut-off is different from what is claimed in previous works.


2021 ◽  
Vol 9 (12) ◽  
pp. 1369
Author(s):  
Weida Xia ◽  
Yuxiang Ma ◽  
Guohai Dong ◽  
Jie Zhang ◽  
Xiaozhou Ma

Numerical simulations were performed to study the long-distance evolution of irregular waves in deep water. It was observed that some solitons, which are the theoretical solutions of the nonlinear Schrödinger equation, emerged spontaneously as irregular wave trains propagated in deep water. The solitons propagated approximately at a speed of the linear group velocity. All the solitons had a relatively large amplitude and one detected soliton’s height was two times larger than the significant wave height of the wave train, therefore satisfying the rogue wave definition. The numerical results showed that solitons can persist for a long distance, reaching about 65 times the peak wavelength. By analyzing the spatial variations of these solitons in both time and spectral domains, it is found that the third-and higher-order resonant interactions and dispersion effects played significant roles in the formation of solitons.


2007 ◽  
Vol 37 (4) ◽  
pp. 1022-1036 ◽  
Author(s):  
Mitsuhiro Tanaka

Abstract The temporal evolution of the energy spectrum of a field of random surface gravity waves in deep water is investigated by means of direct numerical simulations of the deterministic primitive equations. The detected rate of change of the spectrum is shown to be proportional to the cubic power of the energy density and to agree very well with the nonlinear energy transfer Snl as predicted by Hasselmann. Despite the fact that use of various asymptotic relations that are valid only for t → ∞ or integration with respect to time over a time scale much longer than O[period × (ak)−2] is necessary in the derivation of Hasselmann’s Snl, it is clearly demonstrated that the rate of change of the spectrum given by the numerical simulation agrees very well with Hasselmann’s Snl at every instance of ordinary time scale comparable to the period. The result implies that the four-wave resonant interactions control the evolution of the spectrum at every instant of time, whereas nonresonant interactions do not make any significant contribution even in a short-term evolution. It is also pointed out that the result may call for a reexamination of the process of derivation of the kinetic equation for the spectrum.


Author(s):  
Jon R. Ineson ◽  
John S. Peel

NOTE: This article was published in a former series of GEUS Bulletin. Please use the original series name when citing this article, for example: Ineson, J. R., & Peel, J. S. (1997). Cambrian shelf stratigraphy of North Greenland. Geology of Greenland Survey Bulletin, 173, 1-120. https://doi.org/10.34194/ggub.v173.5024 _______________ The Lower Palaeozoic Franklinian Basin is extensively exposed in northern Greenland and the Canadian Arctic Islands. For much of the early Palaeozoic, the basin consisted of a southern shelf, bordering the craton, and a northern deep-water trough; the boundary between the shelf and the trough shifted southwards with time. In North Greenland, the evolution of the shelf during the Cambrian is recorded by the Skagen Group, the Portfjeld and Buen Formations and the Brønlund Fjord, Tavsens Iskappe and Ryder Gletscher Groups; the lithostratigraphy of these last three groups forms the main focus of this paper. The Skagen Group, a mixed carbonate-siliciclastic shelf succession of earliest Cambrian age was deposited prior to the development of a deep-water trough. The succeeding Portfjeld Formation represents an extensive shallow-water carbonate platform that covered much of the shelf; marked differentiation of the shelf and trough occurred at this time. Following exposure and karstification of this platform, the shelf was progressively transgressed and the siliciclastics of the Buen Formation were deposited. From the late Early Cambrian to the Early Ordovician, the shelf showed a terraced profile, with a flat-topped shallow-water carbonate platform in the south passing northwards via a carbonate slope apron into a deeper-water outer shelf region. The evolution of this platform and outer shelf system is recorded by the Brønlund Fjord, Tavsens Iskappe and Ryder Gletscher Groups. The dolomites, limestones and subordinate siliciclastics of the Brønlund Fjord and Tavsens Iskappe Groups represent platform margin to deep outer shelf environments. These groups are recognised in three discrete outcrop belts - the southern, northern and eastern outcrop belts. In the southern outcrop belt, from Warming Land to south-east Peary Land, the Brønlund Fjord Group (Lower-Middle Cambrian) is subdivided into eight formations while the Tavsens Iskappe Group (Middle Cambrian - lowermost Ordovician) comprises six formations. In the northern outcrop belt, from northern Nyeboe Land to north-west Peary Land, the Brønlund Fjord Group consists of two formations both defined in the southern outcrop belt, whereas a single formation makes up the Tavsens Iskappe Group. In the eastern outcrop area, a highly faulted terrane in north-east Peary Land, a dolomite-sandstone succession is referred to two formations of the Brønlund Fjord Group. The Ryder Gletscher Group is a thick succession of shallow-water, platform interior carbonates and siliciclastics that extends throughout North Greenland and ranges in age from latest Early Cambrian to Middle Ordovician. The Cambrian portion of this group between Warming Land and south-west Peary Land is formally subdivided into four formations.The Lower Palaeozoic Franklinian Basin is extensively exposed in northern Greenland and the Canadian Arctic Islands. For much of the early Palaeozoic, the basin consisted of a southern shelf, bordering the craton, and a northern deep-water trough; the boundary between the shelf and the trough shifted southwards with time. In North Greenland, the evolution of the shelf during the Cambrian is recorded by the Skagen Group, the Portfjeld and Buen Formations and the Brønlund Fjord, Tavsens Iskappe and Ryder Gletscher Groups; the lithostratigraphy of these last three groups forms the main focus of this paper. The Skagen Group, a mixed carbonate-siliciclastic shelf succession of earliest Cambrian age was deposited prior to the development of a deep-water trough. The succeeding Portfjeld Formation represents an extensive shallow-water carbonate platform that covered much of the shelf; marked differentiation of the shelf and trough occurred at this time. Following exposure and karstification of this platform, the shelf was progressively transgressed and the siliciclastics of the Buen Formation were deposited. From the late Early Cambrian to the Early Ordovician, the shelf showed a terraced profile, with a flat-topped shallow-water carbonate platform in the south passing northwards via a carbonate slope apron into a deeper-water outer shelf region. The evolution of this platform and outer shelf system is recorded by the Brønlund Fjord, Tavsens Iskappe and Ryder Gletscher Groups. The dolomites, limestones and subordinate siliciclastics of the Brønlund Fjord and Tavsens Iskappe Groups represent platform margin to deep outer shelf environments. These groups are recognised in three discrete outcrop belts - the southern, northern and eastern outcrop belts. In the southern outcrop belt, from Warming Land to south-east Peary Land, the Brønlund Fjord Group (Lower-Middle Cambrian) is subdivided into eight formations while the Tavsens Iskappe Group (Middle Cambrian - lowermost Ordovician) comprises six formations. In the northern outcrop belt, from northern Nyeboe Land to north-west Peary Land, the Brønlund Fjord Group consists of two formations both defined in the southern outcrop belt, whereas a single formation makes up the Tavsens Iskappe Group. In the eastern outcrop area, a highly faulted terrane in north-east Peary Land, a dolomite-sandstone succession is referred to two formations of the Brønlund Fjord Group. The Ryder Gletscher Group is a thick succession of shallow-water, platform interior carbonates and siliciclastics that extends throughout North Greenland and ranges in age from latest Early Cambrian to Middle Ordovician. The Cambrian portion of this group between Warming Land and south-west Peary Land is formally subdivided into four formations.


Water ◽  
2021 ◽  
Vol 13 (16) ◽  
pp. 2152
Author(s):  
Gonzalo García-Alén ◽  
Olalla García-Fonte ◽  
Luis Cea ◽  
Luís Pena ◽  
Jerónimo Puertas

2D models based on the shallow water equations are widely used in river hydraulics. However, these models can present deficiencies in those cases in which their intrinsic hypotheses are not fulfilled. One of these cases is in the presence of weirs. In this work we present an experimental dataset including 194 experiments in nine different weirs. The experimental data are compared to the numerical results obtained with a 2D shallow water model in order to quantify the discrepancies that exist due to the non-fulfillment of the hydrostatic pressure hypotheses. The experimental dataset presented can be used for the validation of other modelling approaches.


Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.


2021 ◽  
pp. 105152
Author(s):  
Victor Michel-Dansac ◽  
Christophe Berthon ◽  
Stéphane Clain ◽  
Françoise Foucher

Author(s):  
Elton J. B. Ribeiro ◽  
Zhimin Tan ◽  
Yucheng Hou ◽  
Yanqiu Zhang ◽  
Andre Iwane

Currently the oil and gas industry is focusing on challenging deep water projects, particularly in Campos Basin located coast off Brazil. However, there are a lot of prolific reservoirs located in shallow water, which need to be developed and they are located in area very far from the coast, where there aren’t pipelines facilities to export oil production, in this case is necessary to use a floating production unit able to storage produced oil, such as a FPSO. So, the riser system configuration should be able to absorb FPSO’s dynamic response due to wave load and avoid damage at touch down zone, in this case is recommended to use compliant riser configuration, such as Lazy Wave, Tethered Wave or Lazy S. In addition to, the proposed FPSO for Tubarão Martelo development is a type VLCC (Very Large Crude Carrier) using external turret moored system, which cause large vertical motion at riser connection and it presents large static offset. Also are expected to install 26 risers and umbilicals hanging off on the turret, this large number of risers and umbilicals has driven the main concerns to clashing and clearance requirement since Lazy-S configuration was adopted. In this paper, some numerical model details and recommendations will be presented, which became a feasible challenging risers system in shallow water. For instance, to solve clashing problem it is strictly recommended for modeling MWA (Mid Water Arch) gutter and bend stiffener at top I-tube interface, this recommendation doesn’t matter in deep water, but for shallow water problem is very important. Also is important to use ballast modules in order to solve clashing problems.


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