Roger Grimshaw | Loughborough University (original) (raw)
Papers by Roger Grimshaw
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.
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
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013
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.
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,
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
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.
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.
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.
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.
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
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.
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
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013
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.
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,
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
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.
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.
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.
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.
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