4 results
On radiating solitons in a model of the internal wave–shear flow resonance
- VYACHESLAV V. VORONOVICH, IGOR A. SAZONOV, VICTOR I. SHRIRA
-
- Journal:
- Journal of Fluid Mechanics / Volume 568 / 10 December 2006
- Published online by Cambridge University Press:
- 10 November 2006, pp. 273-301
-
- Article
- Export citation
-
The work concerns the nonlinear dynamics of oceanic internal waves in resonance with a surface shear current. The resonance occurs when the celerity of the wave matches the mean flow speed at the surface. The evolution of weakly nonlinear waves long compared to the thickness of the upper mixed layer is found to be described by two linearly coupled equations (a linearized intermediate long wave equation and the Riemann wave equation). The presence of a pseudodifferential operator leads to qualitatively new features of the wave dynamics compared to the previously studied case of shallow water. The system is investigated primarily by means of numerical analysis. It possesses a variety of both periodic and solitary wave stationary solutions, including ‘delocalized solitons’ with a localized core and very small non-decaying oscillatory tails (throughout the paper we use the term ‘soliton’ as synonymous with ‘solitary wave’ and do not imply any integrability of the system). These ‘solitons’ are in linear resonance with infinitesimal waves, which in the evolutionary problem normally results in radiative damping. However, the rate of the energy losses proves to be so small, that these delocalized radiating solitons can be treated as quasi-stationary, that is, effectively, as true solitons at the characteristic time scales of the system. Moreover, they represent a very important class of intermediate asymptotics in the evolution of initial localized pulses. A typical pulse evolves into a sequence of solitary waves of all kinds, including the ‘delocalized’ ones, plus a decaying train of periodic waves. The remarkable feature of this evolution is that of all the products of the pulse fission (in a wide range of parameters of the initial pulse) the radiating solitons have by far the largest amplitudes. We argue that the radiating solitons acting as intermediate asymptotics of initial-value problems are a generic phenomenon not confined to the particular model under consideration.
Nonlinear dynamics of vorticity waves in the coastal zone
- Victor I. Shrira, Vyacheslav V. Voronovich
-
- Journal:
- Journal of Fluid Mechanics / Volume 326 / 10 November 1996
- Published online by Cambridge University Press:
- 26 April 2006, pp. 181-203
-
- Article
- Export citation
-
Vorticity waves are wave-like motions occurring in various types of shear flows. We study the dynamics of these motions in alongshore shear currents in situations where it can be described within weakly nonlinear asymptotic theory. The principal mechanism of vorticity waves can be interpreted as potential vorticity conservation with the background vorticity gradient provided both by the mean current shear and the variation of depth. Under the assumption that the mean potential vorticity distibution is monotonic in the cross-shore direction, the nonlinear stage of the dynamics of weakly nonlinear vorticity waves, long in comparison with the current cross-shore scale, is found to be governed by an evolution equation of the generalized Benjamin–Ono type. The dispersive terms are given by an integro-differential operator with the kernel determined by the large-scale cross-shore depth and current dependence. The derived equations form a wide new class of nonlinear evolution equations. They all tend to the Benjamin–Ono equation in the short-wave limit, while in the long-wave limit their asymptotics depend on the specific form of the depth and current profiles. For a particular family of model bottom profiles the equations are ‘intermediate’ between Benjamin–Ono and Korteweg–de Vries equations, but are distinct from the Joseph intermediate equation. Solitary-wave solutions to the equations for these depth profiles are found to decay exponentially. Taking into account coastline inhomogeneity or/and alongshore depth variations adds a linear forcing term to the evolution equation, thus providing an effective generation mechanism for vorticity waves.
Wave breaking due to internal wave–shear flow resonance over a sloping bottom
- VICTOR I. SHRIRA, VYACHESLAV V. VORONOVICH, IGOR A. SAZONOV
-
- Journal:
- Journal of Fluid Mechanics / Volume 425 / 25 December 2000
- Published online by Cambridge University Press:
- 01 December 2000, pp. 187-211
-
- Article
- Export citation
-
A new mechanism of internal wave breaking in the subsurface ocean layer is considered. The breaking is due to the ‘resonant’ interaction of shoaling long internal gravity waves with the subsurface shear current occurring in a resonance zone. Provided the wind-induced shear current is oriented onshore, there exists a wide resonance zone, where internal wave celerity is close to the current velocity at the water surface and a particularly strong resonant interaction of shoaling internal waves with the current takes place. A model to describe the coupled dynamics of the current perturbations treated as ‘vorticity waves’ and internal waves propagating over a sloping bottom is derived by asymptotic methods. The model generalizes the earlier one by Voronovich, Pelinovsky & Shrira (1998) by taking into account the mild bottom slope typical of the oceanic shelf. The focus of the work is upon the effects on wave evolution due to the presence of the bottom slope. If the bottom is flat, the model admits a set of stationary solutions, both periodic and of solitary wave type, their amplitude being limited from above. The limiting waves are sharp crested. Space–time evolution of the waves propagating over a sloping bottom is studied both by the adiabatic Whitham method for comparatively mild slopes and numerically for an arbitrary one. The principal result is that all onshore propagating waves, however small their initial amplitudes are, inevitably reach the limiting amplitude within the resonance zone and break. From the mathematical viewpoint the unique peculiarity of the problem lies in the fact that the wave evolution remains weakly nonlinear up to breaking. To address the situations when the subsurface current becomes strongly turbulent due to particularly intense wind-wave breaking, the effect of turbulent viscosity on the wave evolution is also investigated. The damping due to the turbulence results in a threshold in the initial amplitudes of perturbations: the ‘subcritical’ perturbations are damped, the ‘supercritical’ ones inevitably break. As the breaking events occur mainly in the subsurface layer, they may contribute significantly to the mixing and exchange processes at the air/sea interface and in creating significant surface signatures.
On internal wave–shear flow resonance in shallow water
- VYACHESLAV V. VORONOVICH, DMITRY E. PELINOVSKY, VICTOR I. SHRIRA
-
- Journal:
- Journal of Fluid Mechanics / Volume 354 / 10 January 1998
- Published online by Cambridge University Press:
- 10 January 1998, pp. 209-237
-
- Article
- Export citation
-
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.