Simulation of the Transformation of Internal Solitary Waves on Oceanic Shelves

2004 ◽  
Vol 34 (12) ◽  
pp. 2774-2791 ◽  
Author(s):  
Roger Grimshaw ◽  
Efim Pelinovsky ◽  
Tatiana Talipova ◽  
Audrey Kurkin
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Asymptotic Theory ◽  
Vries Equation ◽  
Internal Solitary Waves ◽  
Variable Depth ◽  
North West ◽  
Wave Parameters ◽  
De Vries

Abstract Internal solitary waves transform as they propagate shoreward over the continental shelf into the coastal zone, from a combination of the horizontal variability of the oceanic hydrology (density and current stratification) and the variable depth. If this background environment varies sufficiently slowly in comparison with an individual solitary wave, then that wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the framework of the variable-coefficient extended Korteweg–de Vries equation where the variation of the solitary wave parameters can be described analytically through an asymptotic description as a slowly varying solitary wave. Direct numerical simulation of the variable-coefficient extended Korteweg–de Vries equation is performed for several oceanic shelves (North West shelf of Australia, Malin shelf edge, and Arctic shelf) to demonstrate the applicability of the asymptotic theory. It is shown that the solitary wave may maintain its soliton-like form for large distances (up to 100 km), and this fact helps to explain why internal solitons are widely observed in the world's oceans. In some cases the background stratification contains critical points (where the coefficients of the nonlinear terms in the extended Korteweg–de Vries equation change sign), or does not vary sufficiently slowly; in such cases the solitary wave deforms into a group of secondary waves. This stage is studied numerically.

On an asymptotic solution of the Korteweg–de Vries equation with slowly varying coefficients

1973 ◽  
Vol 60 (4) ◽  
pp. 813-824 ◽  
Author(s):  
R. S. Johnson
Keyword(s):  
Solitary Wave ◽  
Asymptotic Solution ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Variable Depth ◽  
Exponential Form ◽  
Varying Coefficients ◽  
Numerical Integrations ◽  
De Vries ◽  
Depth Water

The variable-coefficient Korteweg–de Vries equation \[ H_X + {\textstyle\frac{3}{2}}d^{-\frac{7}{4}}HH_{\xi} + {\textstyle\frac{1}{6}}\kappa d^{\frac{1}{2}}H_{\xi\xi\xi} = 0 \] with d = d(εX) is discussed for solitary-wave initial profiles. A straightforward asymptotic solution for ε → 0 is constructed and is shown to be non-uniform both ahead of and behind the solitary wave. The behaviour ahead is rectified by matching to the appropriate exponential form and, together with the use of conservation laws for the equation, the nature of the solution behind the solitary wave is discussed. This leads to the formulation of the solution in the oscillatory ‘tail’, which is again matched directly.The results are applied to the development of the solitary wave into variable-depth water, and the predictions are compared with those obtained, for example, by Grimshaw (1970, 1971). Finally, the asymptotic behaviour of both the solitary wave and the oscillatory tail are assessed in the light of some numerical integrations of the equation.

Slowly varying solitary waves. I. Korteweg-de Vries equation

1979 ◽  
Vol 368 (1734) ◽  
pp. 359-375 ◽  
Keyword(s):  
Exact Solution ◽  
Solitary Wave ◽  
Asymptotic Solution ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Multiple Scale ◽  
Perturbation Parameter ◽  
Scale Method ◽  
De Vries

The slowly varying solitary wave is constructed as an asymptotic solution of the variable coefficient Korteweg-de Vries equation. A multiple scale method is used to determine the amplitude and phase of the wave to the second order in the perturbation parameter. The structure ahead and behind the solitary wave is also determined, and the results are interpreted by using conservation laws. Outer expansions are introduced to remove non-uniformities in the expansion. Finally, when the coefficients satisfy a certain constraint, an exact solution is constructed.

scholarly journals Painleve analysis for a forced Korteveg-de Vries equation arisen in fluid dynamics of internal solitary waves

2015 ◽  
Vol 19 (4) ◽  
pp. 1223-1226 ◽  
Author(s):  
Sheng Zhang ◽  
Mei-Tong Chen ◽  
Wei-Yi Qian
Keyword(s):  
Fluid Dynamics ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Internal Solitary Waves ◽  
Painlevé Analysis ◽  
New Exact Solutions ◽  
De Vries ◽  
Painleve Analysis ◽  
Truncated Painlevé Expansion

In this paper, Painleve analysis is used to test the Painleve integrability of a forced variable-coefficient extended Korteveg-de Vries equation which can describe the weakly-non-linear long internal solitary waves in the fluid with continuous stratification on density. The obtained results show that the equation is integrable under certain conditions. By virtue of the truncated Painleve expansion, a pair of new exact solutions to the equation is obtained.

scholarly journals Combined Effect of Rotation and Topography on Shoaling Oceanic Internal Solitary Waves

2014 ◽  
Vol 44 (4) ◽  
pp. 1116-1132 ◽  
Author(s):  
Roger Grimshaw ◽  
Chuncheng Guo ◽  
Karl Helfrich ◽  
Vasiliy Vlasenko
Keyword(s):  
Solitary Wave ◽  
Wave Packet ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Combined Effect ◽  
Internal Solitary Wave ◽  
Internal Solitary Waves ◽  
De Vries ◽  
Ostrovsky Equation ◽  
Variable Topography

Abstract Internal solitary waves commonly observed in the coastal ocean are often modeled by a nonlinear evolution equation of the Korteweg–de Vries type. Because these waves often propagate for long distances over several inertial periods, the effect of Earth’s background rotation is potentially significant. The relevant extension of the Kortweg–de Vries is then the Ostrovsky equation, which for internal waves does not support a steady solitary wave solution. Recent studies using a combination of asymptotic theory, numerical simulations, and laboratory experiments have shown that the long time effect of rotation is the destruction of the initial internal solitary wave by the radiation of small-amplitude inertia–gravity waves, and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. However, in the ocean, internal solitary waves are often propagating over variable topography, and this alone can cause quite dramatic deformation and transformation of an internal solitary wave. Hence, the combined effects of background rotation and variable topography are examined. Then the Ostrovsky equation is replaced by a variable coefficient Ostrovsky equation whose coefficients depend explicitly on the spatial coordinate. Some numerical simulations of this equation, together with analogous simulations using the Massachusetts Institute of Technology General Circulation Model (MITgcm), for a certain cross section of the South China Sea are presented. These demonstrate that the combined effect of shoaling and rotation is to induce a secondary trailing wave packet, induced by enhanced radiation from the leading wave.

scholarly journals Hamiltonian formulation for the description of interfacial solitary waves

1998 ◽  
Vol 5 (1) ◽  
pp. 3-12 ◽  
Author(s):  
R. Grimshaw ◽  
S. R. Pudjaprasetya
Keyword(s):  
Solitary Waves ◽  
Variable Coefficient ◽  
Hamiltonian Structure ◽  
Vries Equation ◽  
Hamiltonian Formulation ◽  
Constant Density ◽  
Two Fluids ◽  
De Vries ◽  
Lower Fluid ◽  
New Equation

Abstract. We consider solitary waves propagating on the interface between two fluids, each of constant density, for the case when the upper fluid is bounded above by a rigid horizontal plane, but the lower fluid has a variable depth. It is well-known that in this situation, the solitary waves can be described by a variable-coefficient Korteweg-de Vries equation. Here we reconsider the derivation of this equation and present a formulation which preserves the Hamiltonian structure of the underlying system. The result is a new variable-coefficient Korteweg-de Vries equation, which conserves energy to a higher order than the more conventional well-known equation. The new equation is used to describe the transformation of an interfacial solitary wave which propagates into a region of decreasing depth.

scholarly journals Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

2002 ◽  
Vol 9 (3/4) ◽  
pp. 221-235 ◽  
Author(s):  
R. Grimshaw ◽  
E. Pelinovsky ◽  
O. Poloukhina
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Solitary Waves ◽  
Vries Equation ◽  
Wave Theory ◽  
Higher Order ◽  
Internal Solitary Waves ◽  
Long Wave ◽  
De Vries ◽  
Stratified Shear Flow

Abstract. A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation.

scholarly journals The stability of solitary waves

1972 ◽  
Vol 328 (1573) ◽  
pp. 153-183 ◽  
Keyword(s):  
Solitary Wave ◽  
Spectral Theory ◽  
Solitary Waves ◽  
Vries Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Long Time ◽  
Stability Of Solitary Waves ◽  
The Stability

The Korteweg-de Vries equation, which describes the unidirectional propagation of long waves in a wide class of nonlinear dispersive systems, is well known to have solutions representing solitary waves. The present analysis establishes that these solutions are stable, confirming a property that has for a long time been presumed. The demonstration of stability hinges on two nonlinear functionals which for solutions of the Korteweg-de Vries equation are invariant with time: these are introduced in § 2, where it is recalled that Boussinesq recognized their significance in relation to the stability of solitary waves. The principles upon which the stability theory is based are explained in § 3, being supported by a few elementary ideas from functional analysis. A proof that solitary wave solutions are stable is completed in § 4, the most exacting steps of which are accomplished by means of spectral theory. In appendix A a method deriving from the calculus of variations is presented, whereby results needed for the proof of stability may be obtained independently of spectral theory as used in § 4. In appendix B it is shown how the stability analysis may readily be adapted to solitary-wave solutions of the ‘regularized long-wave equation’ that has recently been advocated by Benjamin, Bona & Mahony as an alternative to the Korteweg-de Vries equation. In appendix C a variational principle is demonstrated relating to the exact boundaryvalue problem for solitary waves in water: this is a counterpart to a principle used in the present work (introduced in §2) and offers some prospect of proving the stability of exact solitary waves.

scholarly journals Internal solitary wave generation by tidal flow over topography

2018 ◽  
Vol 839 ◽  
pp. 387-407 ◽  
Author(s):  
R. Grimshaw ◽  
K. R. Helfrich
Keyword(s):  
Froude Number ◽  
Solitary Waves ◽  
Vries Equation ◽  
Tidal Flow ◽  
Phase Speed ◽  
Wave Generation ◽  
Equation Model ◽  
Internal Solitary Waves ◽  
Generation Process ◽  
De Vries

Oceanic internal solitary waves are typically generated by barotropic tidal flow over localised topography. Wave generation can be characterised by the Froude number $F=U/c_{0}$, where $U$ is the tidal flow amplitude and $c_{0}$ is the intrinsic linear long wave phase speed, that is the speed in the absence of the tidal current. For steady tidal flow in the resonant regime, $\unicode[STIX]{x1D6E5}_{m}<F-1<\unicode[STIX]{x1D6E5}_{M}$, a theory based on the forced Korteweg–de Vries equation shows that upstream and downstream propagating undular bores are produced. The bandwidth limits $\unicode[STIX]{x1D6E5}_{m,M}$ depend on the height (or depth) of the topographic forcing term, which can be either positive or negative depending on whether the topography is equivalent to a hole or a sill. Here the wave generation process is studied numerically using a forced Korteweg–de Vries equation model with time-dependent Froude number, $F(t)$, representative of realistic tidal flow. The response depends on $\unicode[STIX]{x1D6E5}_{max}=F_{max}-1$, where $F_{max}$ is the maximum of $F(t)$ over half of a tidal cycle. When $\unicode[STIX]{x1D6E5}_{max}<\unicode[STIX]{x1D6E5}_{m}$ the flow is always subcritical and internal solitary waves appear after release of the downstream disturbance. When $\unicode[STIX]{x1D6E5}_{m}<\unicode[STIX]{x1D6E5}_{max}<\unicode[STIX]{x1D6E5}_{M}$ the flow reaches criticality at its peak, producing upstream and downstream undular bores that are released as the tide slackens. When $\unicode[STIX]{x1D6E5}_{max}>\unicode[STIX]{x1D6E5}_{M}$ the tidal flow goes through the resonant regime twice, producing undular bores with each passage. The numerical simulations are for both symmetrical topography, and for asymmetric topography representative of Stellwagen Bank and Knight Inlet.

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