Shoaling solitary internal waves: on a criterion for the formation of waves with trapped cores

2003 ◽  
Vol 478 ◽  
pp. 81-100 ◽  
Author(s):  
KEVIN G. LAMB
Keyword(s):  
Internal Waves ◽  
Solitary Waves ◽  
Large Amplitude ◽  
Propagation Speed ◽  
Background Current ◽  
Near Surface ◽  
Numerical Evidence ◽  
Flow Limit ◽  
Limiting Behaviour ◽  
Solitary Internal Waves

Shoaling solitary internal waves are ubiquitous features in the coastal regions of the world's oceans where waves with a core of recirculating fluid (trapped cores) can provide an effective transport mechanism. Here, numerical evidence is presented which suggests that there is a close connection between the limiting behaviour of large-amplitude solitary waves and the formation of such waves via shoaling. For some background states, large-amplitude waves are broad, having a nearly horizontal flow in their centre. The flow in the centre of such waves is called a conjugate flow. For other background states, large-amplitude waves can reach the breaking limit, at which the maximum current in the wave is equal to the wave's propagation speed. The presence of a background current with near-surface vorticity of the same sign as that induced by the wave can change the limiting behaviour from the conjugate-flow limit to the breaking limit. Numerical evidence is presented here which suggests that if large solitary waves cannot reach the breaking limit in the shallow water, that is if the background flow has a conjugate flow, then waves with trapped cores will not be formed via shoaling. It is also shown that, due to a change in the limiting behaviour of large waves, an appropriate background current can enable the formation of waves with trapped cores in stratifications for which such waves are not formed in the absence of a background current.

A numerical investigation of solitary internal waves with trapped cores formed via shoaling

2002 ◽  
Vol 451 ◽  
pp. 109-144 ◽  
Author(s):  
KEVIN G. LAMB
Keyword(s):  
Internal Waves ◽  
Mixed Layer ◽  
Propagation Speed ◽  
Horizontal Velocity ◽  
Buoyancy Frequency ◽  
Surface Mixed Layer ◽  
Near Surface ◽  
Flow Limit ◽  
Wave Propagation Speed ◽  
Solitary Internal Waves

The formation of solitary internal waves with trapped cores via shoaling is investigated numerically. For density fields for which the buoyancy frequency increases monotonically towards the surface, sufficiently large solitary waves break as they shoal and form solitary-like waves with trapped fluid cores. Properties of large-amplitude waves are shown to be sensitive to the near-surface stratification. For the monotonic stratifications considered, waves with open streamlines are limited in amplitude by the breaking limit (maximum horizontal velocity equals wave propagation speed). When an exponential density stratification is modified to include a thin surface mixed layer, wave amplitudes are limited by the conjugate flow limit, in which case waves become long and horizontally uniform in the centre. The maximum horizontal velocity in the limiting wave is much less than the wave's propagation speed and as a consequence, waves with trapped cores are not formed in the presence of the surface mixed layer.

scholarly journals Mass and momentum transfer by solitary internal waves in a shelf zone

2012 ◽  
Vol 19 (2) ◽  
pp. 265-272 ◽  
Author(s):  
N. Gavrilov ◽  
V. Liapidevskii ◽  
K. Gavrilova
Keyword(s):  
Mathematical Model ◽  
Internal Waves ◽  
Solitary Waves ◽  
Large Amplitude ◽  
Shelf Zone ◽  
Swash Zone ◽  
Second Mode ◽  
Before And After ◽  
The Mathematical Model ◽  
Solitary Internal Waves

Abstract. The evolution of large amplitude internal waves propagating towards the shore and more specifically the run up phase over the "swash" zone is considered. The mathematical model describing the generation, interaction, and decaying of solitary internal waves of the second mode in the interlayer is proposed. The exact solution specifying the shape of solitary waves symmetric with respect to the unperturbed interface is constructed. It is shown that, taking into account the friction on interfaces in the mathematical model, it is possible to describe adequately the change in the phase and amplitude characteristics of two solitary waves moving towards each other before and after their interaction. It is demonstrated that propagation of large amplitude solitary internal waves of depression over a shelf could be simulated in laboratory experiments by internal symmetric solitary waves of the second mode.

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 Advection of pollutants by internal solitary waves in oceanic and atmospheric stable stratifications

1998 ◽  
Vol 5 (4) ◽  
pp. 209-217 ◽  
Author(s):  
G. W. Haarlemmer ◽  
W. B. Zimmerman
Keyword(s):  
Internal Waves ◽  
Solitary Waves ◽  
Physical Simulation ◽  
Stable Stratification ◽  
Internal Solitary Waves ◽  
High Concentration ◽  
Large Eddies ◽  
The Waves ◽  
Solitary Internal Waves

Abstract. When a pollutant is released into the ocean or atmosphere under turbulent conditions, even a steady release is captured by large eddies resulting in localized patches of high concentration of the pollutant. If such a cloud of pollutant subsequently enters a stable stratification-either a pycnocline or thermocline-then internal waves are excited. Since large solitary internal waves have a recirculating core, pollutants may be trapped in the sclitary wave, and advected large distances through the waveguide provided by the stratification. This paper addresses the mechanisms, through computer and physical simulation, by which a localized release of a dense pollutant results in solitary waves that trap the pollutant or disperse the pollutant faster than in the absence of the waves.

Internal solitary waves with subsurface cores

2019 ◽  
Vol 873 ◽  
pp. 1-17 ◽  
Author(s):  
Yangxin He ◽  
Kevin G. Lamb ◽  
Ren-Chieh Lien
Keyword(s):  
Numerical Simulation ◽  
South China Sea ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Internal Solitary Waves ◽  
Two Dimensional ◽  
Background Current ◽  
Near Surface ◽  
Nonlinear Solutions ◽  
Surface Background

Large internal solitary waves with subsurface cores have recently been observed in the South China Sea. Here fully nonlinear solutions of the Dubreil–Jacotin–Long equation are used to study the conditions under which such cores exist. We find that the location of the cores, either at the surface or below the surface, is largely determined by the sign of the vorticity of the near-surface background current. The results of a numerical simulation of a two-dimensional shoaling internal solitary wave are presented which illustrate the formation of a subsurface core.

scholarly journals The modified Korteweg - de Vries equation in the theory of large - amplitude internal waves

1997 ◽  
Vol 4 (4) ◽  
pp. 237-250 ◽  
Author(s):  
R. Grimshaw ◽  
E. Pelinovsky ◽  
T. Talipova
Keyword(s):  
Solitary Wave ◽  
Internal Waves ◽  
Solitary Waves ◽  
Large Amplitude ◽  
Nonlinear Term ◽  
Vries Equation ◽  
Nonlinear Effects ◽  
Wave Generation ◽  
Density Stratification ◽  
De Vries

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.

scholarly journals Large amplitude internal solitary waves over a shelf

2011 ◽  
Vol 11 (1) ◽  
pp. 17-25 ◽  
Author(s):  
N. Gavrilov ◽  
V. Liapidevskii ◽  
K. Gavrilova
Keyword(s):  
Internal Waves ◽  
Solitary Waves ◽  
Large Amplitude ◽  
Numerical Scheme ◽  
Laboratory Experiments ◽  
Open Channel ◽  
Internal Solitary Waves ◽  
Open Channel Flows ◽  
Mode 2 ◽  
The Mathematical Model

Abstract. Dynamics of large amplitude internal waves in two-layers of shallow water is considered. It is demonstrated that in laboratory experiments the subsurface waves of depression over a shelf may be simulated by internal symmetric solitary waves of the mode 2 ("lump-like" waves). The mathematical model describing the propagation and decaying of large internal waves in two-layer fluid is introduced. It is a variant of Choi-Camassa equations with hydrostatic pressure distribution in one of the layers. It is shown that the numerical scheme developed for the Green-Naghdi equations in open channel flows may be applied for the description of large amplitude internal waves over a shelf.

scholarly journals The effect of strong shear on internal solitary-like waves

2021 ◽  
Author(s):  
Marek Stastna ◽  
Aaron Coutino ◽  
Ryan Walter
Keyword(s):  
Solitary Waves ◽  
Large Amplitude ◽  
Initial Conditions ◽  
Theoretical Calculations ◽  
Mathematical Proof ◽  
Internal Solitary Waves ◽  
Algebraic Complexity ◽  
Background Current ◽  
Variable Environment ◽  
Strong Shear

Abstract. Large amplitude internal waves in the ocean propagate in a dynamic, highly variable environment with changes in background current, local depth, and stratification. The Dubreil-Jacotin-Long, or DJL, theory of exact internal solitary waves can account for a background shear, doing so at a cost of algebraic complexity and a lack of a mathematical proof of algorithm convergence. Waves in the presence of shear that is strong enough to preclude theoretical calculations have been reported in observations. We report on high resolution simulations of stratified adjustment in the presence of strong shear currents. We find instances of large amplitude solitary-like waves with recirculating cores in parameter regimes for which DJL theory fails, and of wave types that are completely different in shape from classical internal solitary waves. Both are spontaneously generated from general initial conditions. Some of the waves observed are associated with critical layers, but others exhibit a propagation speed that is very near the background current maximum. As such they are not freely propagating solitary waves, and a DJL theory would not apply. We thus provide a partial reconciliation between observations and theory.

Sign in / Sign up

Export Citation Format

Share Document