Roger Grimshaw | Loughborough University (original) (raw)

Papers by Roger Grimshaw

Research paper thumbnail of Loss of energy in transition of internal solitary wave over the bottom step

ABSTRACT The significant role of the internal waves in the processes of water mixing and energy d... more ABSTRACT The significant role of the internal waves in the processes of water mixing and energy dissipation is well-known. Intense water mixing is observed the ocean shelves and underwater mountains and banks, where the intensive internal waves are generated by barotropic tide and destroyed due to propagating over the non-uniform bottom relief and horizontally variable water stratification. It is interesting to discuss the simplest situation when the internal solitary wave passes over bottom step and transforms part of its energy into eddies, turbulence and mixing. The problem is solved here using the fully nonlinear Navier-Stokes equations in the numerical tank with two-layer flow. Both polarities of incident internal solitary wave are considered. The range of parameters when asymptotic theory is applicable is found. Formation of the Kelvin-Helmholtz billows, eddies near the step and boluses is studied also. The estimations of energy losses for both internal solitary wave polarities show that they are not monotonically varied to respect ratio of step height to incident internal solitary wave amplitude.

Research paper thumbnail of On the long-term evolution of an intense localized divergent vortex on the beta-plane

Research paper thumbnail of Passage of a wave pulse through a zero-dispersion point in the nonlinear Schrödinger equation

We consider, numerically and analytically, a wave pulse passing a point where the dispersion coef... more We consider, numerically and analytically, a wave pulse passing a point where the dispersion coefficient changes its sign from focusing to defocusing. Simulations demonstrate that, in the focusing region, the pulse keeps a soliton-like shape until it is close to the zero-dispersion point, but then, after the passage of this point, the pulse decays into radiation if its energy is

Research paper thumbnail of Nonexistence of gap solitons in nonlinearly coupled systems

Research paper thumbnail of On strongly interacting internal waves in a rotating ocean and coupled Ostrovsky equations

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013

Research paper thumbnail of A note on the interaction between solitary waves in a singularly-perturbed Korteweg-de Vries equation

Research paper thumbnail of Modeling internal solitary waves on the Australian North West Shelf

Research paper thumbnail of Nonreflecting Internal Wave Beam Propagation in the Deep Ocean

Journal of Physical Oceanography, 2010

Using linear internal wave theory for an ocean stratified by both density and current, several ba... more Using linear internal wave theory for an ocean stratified by both density and current, several background profiles are identified for which internal wave beams can propagate without any internal reflection. These special profiles are favorably compared with available oceanic data.

Research paper thumbnail of Modeling Internal Solitary Waves in the Coastal Ocean

In the coastal oceans, the interaction of currents (such as the barotropic tide) with topography ... more In the coastal oceans, the interaction of currents (such as the barotropic tide) with topography can generate large-amplitude, hori- zontally propagating internal solitary waves. These waves often occur in regions where the waveguide properties vary in the direction of propagation. We consider the modeling of these waves by nonlinear evolution equations of the Korteweg-de Vries type with variable co- ecients,

Research paper thumbnail of Energy exchange in coupled sine-Gordon equations and the influence of modulational instability

A two-component system of coupled sine-Gordon equations is considered, particular solutions of wh... more A two-component system of coupled sine-Gordon equations is considered, particular solutions of which represent a continuum generalization of periodic energy exchange in a system of coupled pendulums [1]. Weakly nonlinear solutions describing periodic energy exchange between waves travelling in the two components are governed, depending on the lengthscale of variation of the amplitudes, either by two nonlocally coupled nonlinear Schrödinger

Research paper thumbnail of Nonlinear surface and internal gravity waves in a rotating ocean

Research paper thumbnail of Modulational instability in the dynamics of interacting wave packets: the extended Korteweg-de Vries equation

Research paper thumbnail of Generation of internal undular bores by transcritical flow over topography

This is a pre-print. In both the ocean and the atmosphere, the interaction of a density stratifie... more This is a pre-print. In both the ocean and the atmosphere, the interaction of a density stratified flow with topography can generate large-amplitude, horizontally propagating internal solitary waves. Often these waves appear as a wave-train, or undular bore. In this article we focus on the situation when the flow is critical, that is, the flow speed is close to that of a linear long wave mode. In the weakly nonlinear regime, this is modeled by the forced Korteweg de Vries equation. We will demonstrate how Whitham’s modulation theory may be applied to obtain an analytical description of undular bores, for flow over isolated obstacles and for flow over a step.

Research paper thumbnail of Modulational instabilities and Fermi-Pasta-Ulam recurrence in a coupled long wave-short wave system, with a mismatch in group velocity

This is a pre-print. The resonance of two envelopes of short (capillary) waves with a common long... more This is a pre-print. The resonance of two envelopes of short (capillary) waves with a common long (gravity) wave component is considered. A mismatch in group velocity is incorporated and the resonance conditions need not be satisfied exactly. This slight detuning permits a wider choice of modes and consequently, a much richer set of dynamics. The linear instability of plane waves is studied, and the dominant unstable wave numbers are identified. The subsequent fully nonlinear evolution of these perturbed plane waves is investigated by direct numerical simulations. The Fermi – Pasta – Ulam recurrence phenomenon is observed.

Research paper thumbnail of Nonlinear Waves

Research paper thumbnail of Exact periodic steady solutions for nonlinear wave equations: A new approach

Research paper thumbnail of Solitons in nonintegrable systems

Chaos (Woodbury, N.Y.), 2005

We introduce the concept of this special focus issue on solitons in nonintegrable systems. A brie... more We introduce the concept of this special focus issue on solitons in nonintegrable systems. A brief overview of some recent developments is provided, and the various contributions are described. The topics covered in this focus issue are the modulation of solitons, bores, and shocks, the dynamical evolution of solitary waves, and existence and stability of solitary waves and embedded solitons.

Research paper thumbnail of Changing forms and sudden smooth transitions of tsunami waves

Journal of Ocean Engineering and Marine Energy, 2014

ABSTRACT In some tsunami waves travelling over the ocean, such as the one approaching the eastern... more ABSTRACT In some tsunami waves travelling over the ocean, such as the one approaching the eastern coast of Japan in 2011, the sea surface of the ocean is depressed by a small metre-scale displacement over a multi-kilometre horizontal length scale, lying in front of a positive elevation of comparable magnitude and length, which together constitute a down-up wave. Shallow water theory shows that the latter travels faster than the former, leading to an interaction, whose description is the issue addressed in this paper, using model equations of the Korteweg–de Vries type. First, we re-examine the undular bore solutions of the Korteweg–de Vries equation which describe how an initial depression wave deforms into a depression rarefaction wave followed by an undular bore of large elevation waves riding on this depression. Then we develop a new extended Korteweg–de Vries equation some of whose solutions can be used to describe the interaction of an elevation wave chasing a depression wave. These show that the two waves coincide at a given position and time producing a maximum elevation. Typically this amplitude is larger than the initial displacement magnitude by a factor which can be as large as two, which may explain anomalous elevations of tsunamis at particular positions along their trajectories. It is physically significant that for these small amplitude waves, no wave breaking occurs and there is no excess dissipation. Then, following the transition, the elevation wave moves ahead of the depression wave and the distance between them increases either linearly or logarithmically with time. The implications for how these down-up tsunami waves reach the shoreline are considered.

Research paper thumbnail of Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

Research paper thumbnail of The extended Gardner equation for internal waves in a stratified ocean

The Korteweg - de Vries equation (KdV) is a well-known model for description of nonlinear long in... more The Korteweg - de Vries equation (KdV) is a well-known model for description of nonlinear long internal waves in the ocean. Its coefficients are defined by vertical density and currents stratification. In a shelf zone the coefficient of quadratic nonlinearity may tend to zero, and the nonlinear model should be modified. The Gardner equation, which differs from the KdV by

Research paper thumbnail of Loss of energy in transition of internal solitary wave over the bottom step

ABSTRACT The significant role of the internal waves in the processes of water mixing and energy d... more ABSTRACT The significant role of the internal waves in the processes of water mixing and energy dissipation is well-known. Intense water mixing is observed the ocean shelves and underwater mountains and banks, where the intensive internal waves are generated by barotropic tide and destroyed due to propagating over the non-uniform bottom relief and horizontally variable water stratification. It is interesting to discuss the simplest situation when the internal solitary wave passes over bottom step and transforms part of its energy into eddies, turbulence and mixing. The problem is solved here using the fully nonlinear Navier-Stokes equations in the numerical tank with two-layer flow. Both polarities of incident internal solitary wave are considered. The range of parameters when asymptotic theory is applicable is found. Formation of the Kelvin-Helmholtz billows, eddies near the step and boluses is studied also. The estimations of energy losses for both internal solitary wave polarities show that they are not monotonically varied to respect ratio of step height to incident internal solitary wave amplitude.

Research paper thumbnail of On the long-term evolution of an intense localized divergent vortex on the beta-plane

Research paper thumbnail of Passage of a wave pulse through a zero-dispersion point in the nonlinear Schrödinger equation

We consider, numerically and analytically, a wave pulse passing a point where the dispersion coef... more We consider, numerically and analytically, a wave pulse passing a point where the dispersion coefficient changes its sign from focusing to defocusing. Simulations demonstrate that, in the focusing region, the pulse keeps a soliton-like shape until it is close to the zero-dispersion point, but then, after the passage of this point, the pulse decays into radiation if its energy is

Research paper thumbnail of Nonexistence of gap solitons in nonlinearly coupled systems

Research paper thumbnail of On strongly interacting internal waves in a rotating ocean and coupled Ostrovsky equations

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013

Research paper thumbnail of A note on the interaction between solitary waves in a singularly-perturbed Korteweg-de Vries equation

Research paper thumbnail of Modeling internal solitary waves on the Australian North West Shelf

Research paper thumbnail of Nonreflecting Internal Wave Beam Propagation in the Deep Ocean

Journal of Physical Oceanography, 2010

Using linear internal wave theory for an ocean stratified by both density and current, several ba... more Using linear internal wave theory for an ocean stratified by both density and current, several background profiles are identified for which internal wave beams can propagate without any internal reflection. These special profiles are favorably compared with available oceanic data.

Research paper thumbnail of Modeling Internal Solitary Waves in the Coastal Ocean

In the coastal oceans, the interaction of currents (such as the barotropic tide) with topography ... more In the coastal oceans, the interaction of currents (such as the barotropic tide) with topography can generate large-amplitude, hori- zontally propagating internal solitary waves. These waves often occur in regions where the waveguide properties vary in the direction of propagation. We consider the modeling of these waves by nonlinear evolution equations of the Korteweg-de Vries type with variable co- ecients,

Research paper thumbnail of Energy exchange in coupled sine-Gordon equations and the influence of modulational instability

A two-component system of coupled sine-Gordon equations is considered, particular solutions of wh... more A two-component system of coupled sine-Gordon equations is considered, particular solutions of which represent a continuum generalization of periodic energy exchange in a system of coupled pendulums [1]. Weakly nonlinear solutions describing periodic energy exchange between waves travelling in the two components are governed, depending on the lengthscale of variation of the amplitudes, either by two nonlocally coupled nonlinear Schrödinger

Research paper thumbnail of Nonlinear surface and internal gravity waves in a rotating ocean

Research paper thumbnail of Modulational instability in the dynamics of interacting wave packets: the extended Korteweg-de Vries equation

Research paper thumbnail of Generation of internal undular bores by transcritical flow over topography

This is a pre-print. In both the ocean and the atmosphere, the interaction of a density stratifie... more This is a pre-print. In both the ocean and the atmosphere, the interaction of a density stratified flow with topography can generate large-amplitude, horizontally propagating internal solitary waves. Often these waves appear as a wave-train, or undular bore. In this article we focus on the situation when the flow is critical, that is, the flow speed is close to that of a linear long wave mode. In the weakly nonlinear regime, this is modeled by the forced Korteweg de Vries equation. We will demonstrate how Whitham’s modulation theory may be applied to obtain an analytical description of undular bores, for flow over isolated obstacles and for flow over a step.

Research paper thumbnail of Modulational instabilities and Fermi-Pasta-Ulam recurrence in a coupled long wave-short wave system, with a mismatch in group velocity

This is a pre-print. The resonance of two envelopes of short (capillary) waves with a common long... more This is a pre-print. The resonance of two envelopes of short (capillary) waves with a common long (gravity) wave component is considered. A mismatch in group velocity is incorporated and the resonance conditions need not be satisfied exactly. This slight detuning permits a wider choice of modes and consequently, a much richer set of dynamics. The linear instability of plane waves is studied, and the dominant unstable wave numbers are identified. The subsequent fully nonlinear evolution of these perturbed plane waves is investigated by direct numerical simulations. The Fermi – Pasta – Ulam recurrence phenomenon is observed.

Research paper thumbnail of Nonlinear Waves

Research paper thumbnail of Exact periodic steady solutions for nonlinear wave equations: A new approach

Research paper thumbnail of Solitons in nonintegrable systems

Chaos (Woodbury, N.Y.), 2005

We introduce the concept of this special focus issue on solitons in nonintegrable systems. A brie... more We introduce the concept of this special focus issue on solitons in nonintegrable systems. A brief overview of some recent developments is provided, and the various contributions are described. The topics covered in this focus issue are the modulation of solitons, bores, and shocks, the dynamical evolution of solitary waves, and existence and stability of solitary waves and embedded solitons.

Research paper thumbnail of Changing forms and sudden smooth transitions of tsunami waves

Journal of Ocean Engineering and Marine Energy, 2014

ABSTRACT In some tsunami waves travelling over the ocean, such as the one approaching the eastern... more ABSTRACT In some tsunami waves travelling over the ocean, such as the one approaching the eastern coast of Japan in 2011, the sea surface of the ocean is depressed by a small metre-scale displacement over a multi-kilometre horizontal length scale, lying in front of a positive elevation of comparable magnitude and length, which together constitute a down-up wave. Shallow water theory shows that the latter travels faster than the former, leading to an interaction, whose description is the issue addressed in this paper, using model equations of the Korteweg–de Vries type. First, we re-examine the undular bore solutions of the Korteweg–de Vries equation which describe how an initial depression wave deforms into a depression rarefaction wave followed by an undular bore of large elevation waves riding on this depression. Then we develop a new extended Korteweg–de Vries equation some of whose solutions can be used to describe the interaction of an elevation wave chasing a depression wave. These show that the two waves coincide at a given position and time producing a maximum elevation. Typically this amplitude is larger than the initial displacement magnitude by a factor which can be as large as two, which may explain anomalous elevations of tsunamis at particular positions along their trajectories. It is physically significant that for these small amplitude waves, no wave breaking occurs and there is no excess dissipation. Then, following the transition, the elevation wave moves ahead of the depression wave and the distance between them increases either linearly or logarithmically with time. The implications for how these down-up tsunami waves reach the shoreline are considered.

Research paper thumbnail of Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

Research paper thumbnail of The extended Gardner equation for internal waves in a stratified ocean

The Korteweg - de Vries equation (KdV) is a well-known model for description of nonlinear long in... more The Korteweg - de Vries equation (KdV) is a well-known model for description of nonlinear long internal waves in the ocean. Its coefficients are defined by vertical density and currents stratification. In a shelf zone the coefficient of quadratic nonlinearity may tend to zero, and the nonlinear model should be modified. The Gardner equation, which differs from the KdV by