Статті в журналах: "Nonlinear internal waves"

1

Holyer, Judith Y. "Nonlinear, periodic, internal waves." Fluid Dynamics Research 5, no. 4 (March 1990): 301–20. http://dx.doi.org/10.1016/0169-5983(90)90025-t.

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2

Helfrich, Karl R., and W. Kendall Melville. "LONG NONLINEAR INTERNAL WAVES." Annual Review of Fluid Mechanics 38, no. 1 (January 2006): 395–425. http://dx.doi.org/10.1146/annurev.fluid.38.050304.092129.

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3

Nakayama, Keisuke, Taro Kakinuma, Hidekazu Tsuji, and Masayuki Oikawa. "NONLINEAR OBLIQUE INTERACTION OF LARGE AMPLITUDE INTERNAL SOLITARY WAVES." Coastal Engineering Proceedings 1, no. 33 (October 9, 2012): 19. http://dx.doi.org/10.9753/icce.v33.waves.19.

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Solitary waves are typical nonlinear long waves in the ocean. The two-dimensional interaction of solitary waves has been shown to be essentially different from the one-dimensional case and can be related to generation of large amplitude waves (including ‘freak waves’). Concerning surface-water waves, Miles (1977) theoretically analyzed interaction of three solitary waves, which is called “resonant interaction” because of the relation among parameters of each wave. Weakly-nonlinear numerical study (Funakoshi, 1980) and fully-nonlinear one (Tanaka, 1993) both clarified the formation of large amplitude wave due to the interaction (“stem” wave) at the wall and its dependency of incident angle. For the case of internal waves, analyses using weakly nonlinear model equation (ex. Tsuji and Oikawa, 2006) suggest also qualitatively similar result. Therefore, the aim of this study is to investigate the strongly nonlinear interaction of internal solitary waves; especially whether the resonant behavior is found or not. As a result, it is found that the amplified internal wave amplitude becomes about three times as much as the original amplitude. In contrast, a "stem" was not found to occur when the incident wave angle was more than the critical angle, which has been demonstrated in the previous studies.

4

Helfrich, Karl R., and Roger H. J. Grimshaw. "Nonlinear Disintegration of the Internal Tide." Journal of Physical Oceanography 38, no. 3 (March 1, 2008): 686–701. http://dx.doi.org/10.1175/2007jpo3826.1.

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Abstract The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Since observations frequently show both tides and waves with amplitudes beyond the restrictions of weakly nonlinear theory, the evolution is studied using a fully nonlinear, weakly nonhydrostatic two-layer theory that includes rotation. In the hydrostatic limit, the governing equations have periodic, nonlinear inertia–gravity solutions that are explored as models of the nonlinear internal tide. These long waves are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initial sinusoidal linear internal tide is closely linked to the presence of these nonlinear waves. The initial tide steepens due to nonlinearity and sheds energy into short solitary waves. The disintegration is halted as the longwave part of the solution settles onto a state close to one of the nonlinear hydrostatic solutions, with the short solitary waves superimposed. The degree of disintegration is a function of initial amplitude of the tide and the properties of the underlying nonlinear hydrostatic solutions, which, depending on stratification and tidal frequency, exist only for a finite range of amplitudes (or energies). There is a lower threshold below which no short solitary waves are produced. However, for initial amplitudes above another threshold, given approximately by the energy of the limiting nonlinear hydrostatic inertia–gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations in the South China Sea are briefly discussed.

5

Moum, J. N., J. M. Klymak, J. D. Nash, A. Perlin, and W. D. Smyth. "Energy Transport by Nonlinear Internal Waves." Journal of Physical Oceanography 37, no. 7 (July 1, 2007): 1968–88. http://dx.doi.org/10.1175/jpo3094.1.

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Abstract Winter stratification on Oregon’s continental shelf often produces a near-bottom layer of dense fluid that acts as an internal waveguide upon which nonlinear internal waves propagate. Shipboard profiling and bottom lander observations capture disturbances that exhibit properties of internal solitary waves, bores, and gravity currents. Wavelike pulses are highly turbulent (instantaneous bed stresses are 1 N m−2), resuspending bottom sediments into the water column and raising them 30+ m above the seafloor. The wave cross-shelf transport of fluid often counters the time-averaged Ekman transport in the bottom boundary layer. In the nonlinear internal waves that were observed, the kinetic energy is roughly equal to the available potential energy and is O(0.1) megajoules per meter of coastline. The energy transported by these waves includes a nonlinear advection term 〈uE〉 that is negligible in linear internal waves. Unlike linear internal waves, the pressure–velocity energy flux 〈up〉 includes important contributions from nonhydrostatic effects and surface displacement. It is found that, statistically, 〈uE〉 ≃ 2〈up〉. Vertical profiles through these waves of elevation indicate that up(z) is more important in transporting energy near the seafloor while uE(z) dominates farther from the bottom. With the wave speed c estimated from weakly nonlinear wave theory, it is verified experimentally that the total energy transported by the waves is 〈up〉 + 〈uE〉 ≃ c〈E〉. The high but intermittent energy flux by the waves is, in an averaged sense, O(100) watts per meter of coastline. This is similar to independent estimates of the shoreward energy flux in the semidiurnal internal tide at the shelf break.

6

Yamashita, Kei, Taro Kakinuma, and Keisuke Nakayama. "NUMERICAL ANALYSES ON PROPAGATION OF NONLINEAR INTERNAL WAVES." Coastal Engineering Proceedings 1, no. 32 (February 1, 2011): 24. http://dx.doi.org/10.9753/icce.v32.waves.24.

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A set of nonlinear surface/internal-wave equations, which have been derived on the basis of the variational principle without any assumptions concerning wave nonlinearity and dispersion, is applied to compare numerical results with experimental data of surface/internal waves propagating through a shallow- or a deep-water region in a tank. Internal waves propagating over a submerged breakwater or a uniformly sloping beach are also simulated. The internal progressive wave shows remarkable shoaling when the interface reaches the critical level, after which physical variables including wave celerity become unstable near the wave-breaking point. In the case of the internal-wave trough reflecting at the vertical wall, the vertical velocities of water particles in the vicinity of the interface are different from that of the moving interface at the wall near the wave breaking, which means that the kinematic boundary condition on the interface of trough has been unsatisfied.

7

Jackson, Christopher, Jose da Silva, and Gus Jeans. "The Generation of Nonlinear Internal Waves." Oceanography 25, no. 2 (June 1, 2012): 108–23. http://dx.doi.org/10.5670/oceanog.2012.46.

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8

Müller, Peter, Greg Holloway, Frank Henyey, and Neil Pomphrey. "Nonlinear interactions among internal gravity waves." Reviews of Geophysics 24, no. 3 (August 1986): 493–536. http://dx.doi.org/10.1029/rg024i003p00493.

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9

Diebels, Stefan, Bernd Schuster, and Kolumban Hutter. "Nonlinear internal waves over variable topography." Geophysical & Astrophysical Fluid Dynamics 76, no. 1-4 (November 1994): 165–92. http://dx.doi.org/10.1080/03091929408203664.

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10

Kshevetskii, S. P., and S. B. Leble. "Nonlinear dispersion of long internal waves." Fluid Dynamics 23, no. 3 (1988): 448–52. http://dx.doi.org/10.1007/bf01054756.

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11

DAUXOIS, THIERRY, and W. R. YOUNG. "Near-critical reflection of internal waves." Journal of Fluid Mechanics 390 (July 10, 1999): 271–95. http://dx.doi.org/10.1017/s0022112099005108.

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Using a matched asymptotic expansion we analyse the two-dimensional, near-critical reflection of a weakly nonlinear internal gravity wave from a sloping boundary in a uniformly stratified fluid. Taking a distinguished limit in which the amplitude of the incident wave, the dissipation, and the departure from criticality are all small, we obtain a reduced description of the dynamics. This simplification shows how either dissipation or transience heals the singularity which is presented by the solution of Phillips (1966) in the precisely critical case. In the inviscid critical case, an explicit solution of the initial value problem shows that the buoyancy perturbation and the alongslope velocity both grow linearly with time, while the scale of the reflected disturbance is reduced as 1/t. During the course of this scale reduction, the stratification is ‘overturned’ and the Miles–Howard condition for stratified shear flow stability is violated. However, for all slope angles, the ‘overturning’ occurs before the Miles–Howard stability condition is violated and so we argue that the first instability is convective.Solutions of the simplified dynamics resemble certain experimental visualizations of the reflection process. In particular, the buoyancy field computed from the analytic solution is in good agreement with visualizations reported by Thorpe & Haines (1987).One curious aspect of the weakly nonlinear theory is that the final reduced description is a linear equation (at the solvability order in the expansion all of the apparently resonant nonlinear contributions cancel amongst themselves). However, the reconstructed fields do contain nonlinearly driven second harmonics which are responsible for an important symmetry breaking in which alternate vortices differ in strength and size from their immediate neighbours.

12

Carter, Glenn S., and Michael C. Gregg. "Persistent Near-Diurnal Internal Waves Observed above a Site of M2 Barotropic-to-Baroclinic Conversion." Journal of Physical Oceanography 36, no. 6 (June 1, 2006): 1136–47. http://dx.doi.org/10.1175/jpo2884.1.

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Abstract Near-diurnal internal waves were observed in velocity and shear measurements from a shipboard survey along a 35-km section of the Kaena Ridge, northwest of Oahu. Individual waves with upward phase propagation could be traced for almost 4 days even though the ship transited approximately 20 km. Depth–time maps of shear were dominated by near-diurnal waves, despite the fact that Kaena Ridge is a site of considerable M2 barotropic-to-baroclinic conversion. Guided by recent numerical and observational studies, it was found that a frequency of ½M2 (i.e., 24.84-h period) was consistent with these waves. Nonlinear processes are able to transfer energy within the internal wave spectrum. Bicoherence analysis, which can distinguish between nonlinearly coupled waves and waves that have been independently excited, suggested that the ½M2 waves were nonlinearly coupled with the dominant M2 internal tide only between 525- and 595-m depth. This narrow depth range corresponded to an observed M2 characteristic emanating from the northern edge of the ridge. The observations occurred in close proximity to the internal tide generation region, implying a rapid transfer of energy between frequencies. Strong nonlinear interactions seem a likely mechanism. Nonlinear transfers such as these could complicate attempts to close local single-constituent tidal energy budgets.

13

Dosser, Hayley V., and Bruce R. Sutherland. "Anelastic Internal Wave Packet Evolution and Stability." Journal of the Atmospheric Sciences 68, no. 12 (December 1, 2011): 2844–59. http://dx.doi.org/10.1175/jas-d-11-097.1.

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Abstract As upward-propagating anelastic internal gravity wave packets grow in amplitude, nonlinear effects develop as a result of interactions with the horizontal mean flow that they induce. This qualitatively alters the structure of the wave packet. The weakly nonlinear dynamics are well captured by the nonlinear Schrödinger equation, which is derived here for anelastic waves. In particular, this predicts that strongly nonhydrostatic waves are modulationally unstable and so the wave packet narrows and grows more rapidly in amplitude than the exponential anelastic growth rate. More hydrostatic waves are modulationally stable and so their amplitude grows less rapidly. The marginal case between stability and instability occurs for waves propagating at the fastest vertical group velocity. Extrapolating these results to waves propagating to higher altitudes (hence attaining larger amplitudes), it is anticipated that modulationally unstable waves should break at lower altitudes and modulationally stable waves should break at higher altitudes than predicted by linear theory. This prediction is borne out by fully nonlinear numerical simulations of the anelastic equations. A range of simulations is performed to quantify where overturning actually occurs.

14

VORONOVICH, VYACHESLAV V., DMITRY E. PELINOVSKY, and VICTOR I. SHRIRA. "On internal wave–shear flow resonance in shallow water." Journal of Fluid Mechanics 354 (January 10, 1998): 209–37. http://dx.doi.org/10.1017/s0022112097007593.

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The work is concerned with long nonlinear internal waves interacting with a shear flow localized near the sea surface. The study is focused on the most intense resonant interaction occurring when the phase velocity of internal waves matches the flow velocity at the surface. The perturbations of the shear flow are considered as ‘vorticity waves’, which enables us to treat the wave–flow resonance as the resonant wave–wave interaction between an internal gravity mode and the vorticity mode. Within the weakly nonlinear long-wave approximation a system of evolution equations governing the nonlinear dynamics of the waves in resonance is derived and an asymptotic solution to the basic equations is constructed. At resonance the nonlinearity of the internal wave dynamics is due to the interaction with the vorticity mode, while the wave's own nonlinearity proves to be negligible. The equations derived are found to possess solitary wave solutions of different polarities propagating slightly faster or slower than the surface velocity of the shear flow. The amplitudes of the ‘fast’ solitary waves are limited from above; the crest of the limiting wave forms a sharp corner. The solitary waves of amplitude smaller than a certain threshold are shown to be stable; ‘subcritical’ localized pulses tend to such solutions. The localized pulses of amplitude exceeding this threshold form infinite slopes in finite time, which indicates wave breaking.

15

Shroyer, E. L., J. N. Moum, and J. D. Nash. "Observations of Polarity Reversal in Shoaling Nonlinear Internal Waves." Journal of Physical Oceanography 39, no. 3 (March 1, 2009): 691–701. http://dx.doi.org/10.1175/2008jpo3953.1.

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Abstract Observations off the New Jersey coast document the shoaling of three groups of nonlinear internal waves of depression over 35 km across the shelf. Each wave group experienced changing background conditions along its shoreward transit. Despite different wave environments, a clear pattern emerges. Nearly symmetric waves propagating into shallow water develop an asymmetric shape; in the wave reference frame, the leading edge accelerates causing the front face to broaden while the trailing face remains steep. This trend continues until the front edge and face of the leading depression wave become unidentifiable and a near-bottom elevation wave emerges, formed from the trailing face of the initial depression wave and the leading face of the following wave. The transition from depression to elevation waves is diagnosed by the integrated wave vorticity, which changes sign as the wave’s polarity changes sign. This transition is predicted by the sign change of the coefficient of the nonlinear term in the KdV equation, when evaluated using observed profiles of stratification and velocity.

16

Yamashita, Kei, Taro Kakinuma, and Keisuke Nakayama. "SHOALING OF NONLINEAR INTERNAL WAVES ON A UNIFORMLY SLOPING BEACH." Coastal Engineering Proceedings 1, no. 33 (December 15, 2012): 72. http://dx.doi.org/10.9753/icce.v33.waves.72.

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The internal waves in the two-layer systems have been numerically simulated by solving the set of nonlinear equations in consideration of both strong nonlinearity and strong dispersion of waves. After the comparison between the numerical results and the BO solitons, as well as the experimental data, the internal waves propagating over the uniformly sloping beach are simulated including the cases of the mild and long slopes. The internal waves show remarkable shoaling after the interface touches the critical level. In the lower layer, the horizontal velocity becomes larger than the local linear celerity of internal waves in shallow water just before the crest peak and the position is defined as the wave-breaking point when the ratio of nonlinear parameter to beach slope is large. The ratio of initial wave height to wave-breaking depth becomes larger as the slope is milder and the wave nonlinearity is stronger. The wave height does not increase so much before wave-breaking on the mildest slope.

17

Ruprecht, Daniel, and Rupert Klein. "A model for nonlinear interactions of internal gravity waves with saturated regions." Meteorologische Zeitschrift 20, no. 2 (April 1, 2011): 243–52. http://dx.doi.org/10.1127/0941-2948/2011/0213.

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18

Whitfield, A. J., and E. R. Johnson. "Rotation-induced nonlinear wavepackets in internal waves." Physics of Fluids 26, no. 5 (May 2014): 056606. http://dx.doi.org/10.1063/1.4879075.

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19

Chao, Y. ‐H, M. ‐K Hsu, H. ‐W Chen, Y. ‐H Wang, G. ‐Y Chen, and C. ‐T Liu. "Sieving nonlinear internal waves through path prediction." International Journal of Remote Sensing 29, no. 21 (October 23, 2008): 6391–402. http://dx.doi.org/10.1080/01431160802175413.

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20

Moum, J. N., and J. D. Nash. "Seafloor Pressure Measurements of Nonlinear Internal Waves." Journal of Physical Oceanography 38, no. 2 (February 1, 2008): 481–91. http://dx.doi.org/10.1175/2007jpo3736.1.

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Abstract Highly resolved pressure measurements on the seafloor over New Jersey’s continental shelf reveal the pressure signature of nonlinear internal waves of depression as negative pressure perturbations. The sign of the perturbation is determined by the dominance of the internal hydrostatic pressure (p0Wh) due to isopycnal displacement over the contributions of external hydrostatic pressure (ρ0gηH; ηH is surface displacement) and nonhydrostatic pressure (p0nh), each of opposite sign to p0Wh. This measurement represents experimental confirmation of the wave-induced pressure signal inferred in a previous study by Moum and Smyth.

21

Ostrovsky, Lev A., and John Grue. "Evolution equations for strongly nonlinear internal waves." Physics of Fluids 15, no. 10 (2003): 2934. http://dx.doi.org/10.1063/1.1604133.

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22

Caillol *, P. "Nonlinear internal waves in the upper atmosphere." Geophysical & Astrophysical Fluid Dynamics 99, no. 4 (August 2005): 271–308. http://dx.doi.org/10.1080/03091920500213585.

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23

Liu, Cho-Teng, Robert Pinkel, Jody Klymak, Ming-Kuang Hsu, Hsien-Wen Chen, and Cesar Villanoy. "Nonlinear internal waves from the Luzon Strait." Eos, Transactions American Geophysical Union 87, no. 42 (2006): 449. http://dx.doi.org/10.1029/2006eo420002.

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24

Bauer, Georg, Stefan Diebels, and Kolumban Hutter. "Nonlinear internal waves in ideal rotating basins." Geophysical & Astrophysical Fluid Dynamics 78, no. 1-4 (December 1994): 21–46. http://dx.doi.org/10.1080/03091929408226571.

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25

Liapidevskii, V. Yu, M. V. Turbin, F. F. Khrapchenkov, and V. F. Kukarin. "Nonlinear Internal Waves in Multilayer Shallow Water." Journal of Applied Mechanics and Technical Physics 61, no. 1 (January 2020): 45–53. http://dx.doi.org/10.1134/s0021894420010058.

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26

Filatov, N. N., R. E. Zdorovennov, A. Yu Terzhevik, and K. Hutter. "Nonlinear internal waves in a large lake." Doklady Earth Sciences 441, no. 2 (December 2011): 1715–18. http://dx.doi.org/10.1134/s1028334x11120130.

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27

Stechmann, Samuel N., Andrew J. Majda, and Boualem Khouider. "Nonlinear dynamics of hydrostatic internal gravity waves." Theoretical and Computational Fluid Dynamics 22, no. 6 (May 17, 2008): 407–32. http://dx.doi.org/10.1007/s00162-008-0080-7.

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28

Lapin, V. G. "Nonlinear total internal reflection of electromagnetic waves." Radiophysics and Quantum Electronics 40, no. 9 (September 1997): 757–65. http://dx.doi.org/10.1007/bf02676527.

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29

Valentine, Daniel T., and Radica Sipcic. "Nonlinear Internal Solitary Wave on a Pycnocline." Journal of Offshore Mechanics and Arctic Engineering 124, no. 3 (August 1, 2002): 120–24. http://dx.doi.org/10.1115/1.1490380.

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This paper describes the theory of nonlinear internal-solitary waves of the type observed in coastal seas. It also describes a numerical solution of an initial-value problem that leads to an internal solitary-like wave. The equations solved numerically are the Navier-Stokes, diffusion, and continuity equations. The computer solution illustrates that solitary-like waves are easily generated. A comparison with the theory illustrates that the wave is a KdV-like solitary wave. Hence, the computed wave is caused by a near balance between dispersive and nonlinear effects. However, the shape of the fully-nonlinear solitary wave is fore-aft asymmetric with a relatively long, somewhat elevated tail. This feature is characteristic of the computationally derived wave as compared with the fore-aft symmetry of the theoretical wave. (This work is motivated by the fact that internal solitary-like waves have practical importance in the design of offshore structures and on the acoustic properties of the sea, among other environmental consequences.)

30

HWUNG, HWUNG-HWENG, RAY-YENG YANG, and IGOR V. SHUGAN. "Exposure of internal waves on the sea surface." Journal of Fluid Mechanics 626 (May 10, 2009): 1–20. http://dx.doi.org/10.1017/s0022112008004758.

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We theoretically analyse the impact of subsurface currents induced by internal waves on nonlinear Stokes surface waves. We present analytical and numerical solutions of the modulation equations under conditions that are close to group velocity resonance. Our results show that smoothing of the downcurrent surface waves is accompanied by a relatively high-frequency modulation, while the profile of the opposing current is reproduced by the surface wave's envelope. We confirm the possibility of generating an internal wave forerunner that is a modulated surface wave packet. Long surface waves can create such a wave modulation forerunner ahead of the internal wave, while other relatively short surface waves comprise the trace of the internal wave itself. Modulation of surface waves by a periodic internal wavetrain may exhibit a characteristic period that is less than the internal wave period. This period can be non-uniform while the wave crosses the current zone. Our results confirm that surface wave excitation by means of internal waves, as observed at their group resonance frequencies, is efficient only in the context of opposing currents.

31

GRUE, JOHN, ATLE JENSEN, PER-OLAV RUSÅS, and J. KRISTIAN SVEEN. "Properties of large-amplitude internal waves." Journal of Fluid Mechanics 380 (February 10, 1999): 257–78. http://dx.doi.org/10.1017/s0022112098003528.

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Properties of solitary waves propagating in a two-layer fluid are investigated comparing experiments and theory. In the experiments the velocity field induced by the waves, the propagation speed and the wave shape are quite accurately measured using particle tracking velocimetry (PTV) and image analysis. The experiments are calibrated with a layer of fresh water above a layer of brine. The depth of the brine is 4.13 times the depth of the fresh water. Theoretical results are given for this depth ratio, and, in addition, in a few examples for larger ratios, up to 100[ratio ]1. The wave amplitudes in the experiments range from a small value up to almost maximal amplitude. The thickness of the pycnocline is in the range of approximately 0.13–0.26 times the depth of the thinner layer. Solitary waves are generated by releasing a volume of fresh water trapped behind a gate. By careful adjustment of the length and depth of the initial volume we always generate a single solitary wave, even for very large volumes. The experiments are very repeatable and the recording technique is very accurate. The error in the measured velocities non-dimensionalized by the linear long wave speed is less than about 7–8% in all cases. The experiments are compared with a fully nonlinear interface model and weakly nonlinear Korteweg–de Vries (KdV) theory. The fully nonlinear model compares excellently with the experiments for all quantities measured. This is true for the whole amplitude range, even for a pycnocline which is not very sharp. The KdV theory is relevant for small wave amplitude but exhibit a systematic deviation from the experiments and the fully nonlinear theory for wave amplitudes exceeding about 0.4 times the depth of the thinner layer. In the experiments with the largest waves, rolls develop behind the maximal displacement of the wave due to the Kelvin–Helmholtz instability. The recordings enable evaluation of the local Richardson number due to the flow in the pycnocline. We find that stability or instability of the flow occurs in approximate agreement with the theorem of Miles and Howard.

32

Crighton, D. G. "The Taylor internal structure of weak shock waves." Journal of Fluid Mechanics 173 (December 1986): 625–42. http://dx.doi.org/10.1017/s0022112086001295.

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G. I. Taylor's solution in 1910 for the interior structure of a weak shock wave is, with appropriate generalization, an essential component of weak-shock theory. The Taylor balance between nonlinear convection and thermoviscous diffusion is, however, endangered when other linear mechanisms - such as density stratification, geometrical spreading effects, tube wall attenuation and dispersion, etc. - are included. The ways in which some of these linear mechanisms cause the Taylor shock structure to break down when a weak shock has propagated over a large (and in some cases quite moderate) distance will be studied. Different forms of breakdown of the Taylor shock structure will be identified, both for quadratic (gasdynamic) nonlinearity and also for cubic nonlinearity appropriate to transverse waves in solid media or electromagnetic waves in nonlinear dielectrics. From this a description will be given of the fate of a nonlinear wave containing a pattern of weak shock waves, as it propagates over large ranges under the influence of linear and nonlinear mechanisms.

33

Stamp, Andrew P., and Marcus Jacka. "Deep-water internal solitaty waves." Journal of Fluid Mechanics 305 (December 25, 1995): 347–71. http://dx.doi.org/10.1017/s0022112095004654.

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An experimental investigation of mode-2 (’lump-Like’) Solitary waves propagaling on a thin interface between two deep layers of different densities is presented. Small-and large-amplitude waves behaved differently: small waves carried energy and momentum, whereas sufficiently large waves also carried mass. Weakly nonlinear theory anticipated the result for amplitudes a/h [les ] 0.5 but did not provide even a qualitative description of the large-amplitude waves. In particular, the prediction that for waves to maintain permanent form their wavelength must decrease with increasing amplitude failed; instead the wavelength of large waves was observed to increase with increasing amplitude. Furthermore, whilst the waves were expected to emerge from interactions along their precollision trajectories, the large waves actually suffered a backward shift.

34

Grimshaw, R., E. Pelinovsky, and T. Talipova. "The modified Korteweg - de Vries equation in the theory of large - amplitude internal waves." Nonlinear Processes in Geophysics 4, no. 4 (December 31, 1997): 237–50. http://dx.doi.org/10.5194/npg-4-237-1997.

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Abstract. The propagation of large- amplitude internal waves in the ocean is studied here for the case when the nonlinear effects are of cubic order, leading to the modified Korteweg - de Vries equation. The coefficients of this equation are calculated analytically for several models of the density stratification. It is shown that the coefficient of the cubic nonlinear term may have either sign (previously only cases of a negative cubic nonlinearity were known). Cubic nonlinear effects are more important for the high modes of the internal waves. The nonlinear evolution of long periodic (sine) waves is simulated for a three-layer model of the density stratification. The sign of the cubic nonlinear term influences the character of the solitary wave generation. It is shown that the solitary waves of both polarities can appear for either sign of the cubic nonlinear term; if it is positive the solitary waves have a zero pedestal, and if it is negative the solitary waves are generated on the crest and the trough of the long wave. The case of a localised impulsive initial disturbance is also simulated. Here, if the cubic nonlinear term is negative, there is no solitary wave generation at large times, but if it is positive solitary waves appear as the asymptotic solution of the nonlinear wave evolution.

35

Zimmerman, W. B., and J. M. Rees. "Long solitary internal waves in stable stratifications." Nonlinear Processes in Geophysics 11, no. 2 (April 14, 2004): 165–80. http://dx.doi.org/10.5194/npg-11-165-2004.

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Abstract. Observations of internal solitary waves over an antarctic ice shelf (Rees and Rottman, 1994) demonstrate that even large amplitude disturbances have wavelengths that are bounded by simple heuristic arguments following from the Scorer parameter based on linear theory for wave trapping. Classical weak nonlinear theories that have been applied to stable stratifications all begin with perturbations of simple long waves, with corrections for weak nonlinearity and dispersion resulting in nonlinear wave equations (Korteweg-deVries (KdV) or Benjamin-Davis-Ono) that admit localized propagating solutions. It is shown that these theories are apparently inappropriate when the Scorer parameter, which gives the lowest wavenumber that does not radiate vertically, is positive. In this paper, a new nonlinear evolution equation is derived for an arbitrary wave packet thus including one bounded below by the Scorer parameter. The new theory shows that solitary internal waves excited in high Richardson number waveguides are predicted to have a halfwidth inversely proportional to the Scorer parameter, in agreement with atmospheric observations. A localized analytic solution for the new wave equation is demonstrated, and its soliton-like properties are demonstrated by numerical simulation.

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Макаренко, Николай Иванович, Валерий Юрьевич Ляпидевский, Данила Сергеевич Денисенко, and Дмитрий Евгеньевич Кукушкин. "Nonlinear internal wave packets in shelf zone." Вычислительные технологии, no. 2(24) (April 17, 2019): 90–98. http://dx.doi.org/10.25743/ict.2019.24.2.008.

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В рамках модели невязкой слабостратифицированной жидкости рассматривается длинноволновое приближение, описывающее нелинейные волновые пакеты типа кноидальных волн. Построены семейства асимптотических решений, одновременно описывающие периодические последовательности приповерхностных волн в форме впадин и придонных волн типа возвышений. Показано, что картины расчетных профилей качественно согласуются со структурами внутренних волн, наблюдавшихся авторами в натурных экспериментах в шельфовой зоне моря. The problem on nonlinear internal waves propagating permanently in shallow fluid is studied semi-analytically in comparison with the field data measured on the sea shelf. At present, the most studied in this context are nonlinear solitary-type waves generated due to the tidal activity over continental slope. This paper deals with periodic cnoidaltype wave packets considered in the framework of mathematical model of continuously stratified fluid. Basic model involves the Dubreil-Jacotin-Long equation for a stream function that results from stationary fully non-linear 2D Euler equations. The longwave approximate equation describing periodic non-harmonic waves is derived by means of scaling procedure using small Boussinesq parameter. This parameter characterizes slight stratification of the fluid layer with the density profile being close to the linear stratification. The fine-scale density plays important role here because it determines the non-linearity rate of model equation, so it permits to consider strongly non-linear dispersive waves of large amplitude. As a result, constructed asymptotic solutions can simulate periodic wave-trains of sub-surface depression coupled with near-bottom wavetrains of isopycnal elevation. It is demonstrated that calculated wave profiles are in good qualitative agreement with internal wave structures observed by the authors in the field experiments performed annually during 2011-2018 in expeditions on the shelf of the Japanese sea.

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Ryazanov, D. A., M. I. Providukhina, I. N. Sibgatullin, and E. V. Ermanyuk. "Biharmonic Attractors of Internal Gravity Waves." Fluid Dynamics 56, no. 3 (May 2021): 403–12. http://dx.doi.org/10.1134/s0015462821030046.

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Abstract— The hydrodynamic system that admits the development of internal wave attractors under biharmonic forcing is investigated. It is shown that in the case of low amplitude of external forcing the wave pattern consists of two attractors that interact between themselves only slightly: the total energy of the system is equal to the sum of energies of the components with high accuracy. In the nonlinear case the attractors interact in the more complex way which leads to the development of a cascade of triad interactions generating a rich set of time scales. In the case of closely adjacent frequencies of the components of a biharmonic perturbation, the nonlinear “beating” regime develops, namely, the mean energy of the system of coupled attractors performs oscillations at a large time scale that corresponds to the beating period. It is found that the high-frequency energy fluctuations corresponding to the same mean energy can differ by an order of magnitude depending on whether the envelope of the mean value increases or decreases.

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Ostrovsky, L. A., and K. R. Helfrich. "Strongly nonlinear, simple internal waves in continuously-stratified, shallow fluids." Nonlinear Processes in Geophysics 18, no. 1 (February 14, 2011): 91–102. http://dx.doi.org/10.5194/npg-18-91-2011.

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Abstract. Strongly nonlinear internal waves in a layer with arbitrary stratification are considered in the hydrostatic approximation. It is shown that "simple waves" having a variable vertical structure can emerge from a wide class of initial conditions. The equations describing such waves have been obtained using the isopycnal coordinate as a variable. Emergence of simple waves from an initial Gaussian impulse is numerically investigated for different density profiles, from two- and three-layer structure to the continuous one. Besides the first mode, examples of second- and third-mode simple waves are given.

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Shiryaeva, S. O., N. A. Petrushov, A. I. Grigor’ev, and M. S. Fedorov. "Nonlinear internal resonance interaction between surface waves and internal waves in an inhomogeneous laminated liquid." Technical Physics 59, no. 9 (September 2014): 1291–99. http://dx.doi.org/10.1134/s106378421409028x.

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CHOI, WOOYOUNG, and ROBERTO CAMASSA. "Fully nonlinear internal waves in a two-fluid system." Journal of Fluid Mechanics 396 (October 10, 1999): 1–36. http://dx.doi.org/10.1017/s0022112099005820.

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Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg–de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

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Constantin, Adrian. "Some Nonlinear, Equatorially Trapped, Nonhydrostatic Internal Geophysical Waves." Journal of Physical Oceanography 44, no. 2 (February 1, 2014): 781–89. http://dx.doi.org/10.1175/jpo-d-13-0174.1.

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Abstract The author presents an explicit exact solution to the governing equations for geophysical equatorial waves in the β-plane setting. The solution describes equatorially trapped waves propagating eastward above the thermocline and beneath the near-surface layer where wind effects are confined. At great depths the water is still, while the transition toward the large-amplitude oscillation of the thermocline is accommodated by an eastward-flowing current. Above the thermocline a flow reversal occurs, with the underlying current flowing westward close to the layer where wind effects are confined.

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Jackson, Christopher, Yessy Arvelyna, and Ichio Asanuma. "High-Frequency Nonlinear Internal Waves Around the Philippines." Oceanography 24, no. 01 (March 1, 2011): 90–99. http://dx.doi.org/10.5670/oceanog.2011.06.

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Grimshaw, Roger. "Nonlinear Wave Equations for Oceanic Internal Solitary Waves." Studies in Applied Mathematics 136, no. 2 (November 16, 2015): 214–37. http://dx.doi.org/10.1111/sapm.12100.

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Hsu, M. K., and A. K. Liu. "Nonlinear Internal Waves in the South China Sea." Canadian Journal of Remote Sensing 26, no. 2 (April 2000): 72–81. http://dx.doi.org/10.1080/07038992.2000.10874757.

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Gerkema, T., and J. T. F. Zimmerman. "Generation of Nonlinear Internal Tides and Solitary Waves." Journal of Physical Oceanography 25, no. 6 (June 1995): 1081–94. http://dx.doi.org/10.1175/1520-0485(1995)025<1081:gonita>2.0.co;2.

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Dolin, L. S., and I. S. Dolina. "Model of lidar images of nonlinear internal waves." Izvestiya, Atmospheric and Oceanic Physics 50, no. 2 (March 2014): 196–203. http://dx.doi.org/10.1134/s0001433814020029.

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Lavery, Andone. "Broadband acoustic scattering from nonlinear internal solitary waves." Journal of the Acoustical Society of America 123, no. 5 (May 2008): 3898. http://dx.doi.org/10.1121/1.2935865.

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Howell, Thomas L., and Wendell S. Brown. "Nonlinear internal waves on the California continental shelf." Journal of Geophysical Research 90, no. C4 (1985): 7256. http://dx.doi.org/10.1029/jc090ic04p07256.

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Dolin, L. S., and I. S. Dolina. "Model of Lidar Images of Nonlinear Internal Waves." Известия Российской академии наук. Физика атмосферы и океана 50, no. 2 (2014): 224–31. http://dx.doi.org/10.7868/s0002351514020023.

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Pineda, Jesús, Sally Rouse, Victoria Starczak, Karl Helfrich, and David Wiley. "Response of small sharks to nonlinear internal waves." Limnology and Oceanography 65, no. 4 (October 22, 2019): 707–16. http://dx.doi.org/10.1002/lno.11341.

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