The distortion of short internal waves produced by a long wave, with application to ocean boundary mixing

1989 ◽  
Vol 208 ◽  
pp. 395-415 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Density Gradient ◽  
Internal Waves ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Finite Amplitude ◽  
Constant Density ◽  
Boundary Mixing ◽  
Short Waves ◽  
Long Wave ◽  
Boussinesq Fluid

The propagation of a train of short, small-amplitude, internal waves through a long, finite-amplitude, two-dimensional, internal wave is studied. An exact solution of the equations of motion for a Boussinesq fluid of constant density gradient is used to describe the long wave, and its distortion of the density gradient as well as its velocity field are accounted for in determining the propagation characteristics of the short waves. To illustrate the magnitude of the effects on the short waves, particular numerical solutions are found for short waves generated by an idealized flow induced by a long wave adjacent to sloping, sinusoidal topography in the ocean, and the results are compared with a laboratory experiment. The theory predicts that the long wave produces considerably distortion of the short waves, changing their amplitudes, wavenumbers and propagation directions by large factors, and in a way which is generally consistent with, but not fully tested by, the observations. It is suggested that short internal waves generated by the interaction of relatively long waves with a rough sloping topography may contribute to the mixing observed near continental slopes.

Double diffusive instability in a tall thin slot

1998 ◽  
Vol 375 ◽  
pp. 203-233 ◽  
Author(s):  
NEIL J. BALMFORTH ◽  
JOSEPH A. BIELLO
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Finite Amplitude ◽  
Physical Conditions ◽  
Long Wave ◽  
Diffusive Convection ◽  
Wide Range ◽  
Boussinesq Fluid ◽  
Double Diffusive

The linear stability of doubly diffusive convection is considered for a two-dimensional, Boussinesq fluid in a tall thin slot. For a variety of boundary conditions on the slot walls, instability sets in through zero wavenumber over a wide range of physical conditions. Long-wave equations governing the nonlinear development of the instability are derived. The form of the long-wave equations sensitively depends on the thermal and salt boundary conditions; the possible long-wave theories are catalogued. Finite-amplitude solutions and their stability are studied. In some cases the finite-amplitude solutions are not the only possible attractors and numerical solutions presenting the alternatives are given. These reveal temporally complicated dynamics.

scholarly journals Gravity currents and internal waves in a stratified fluid

2008 ◽  
Vol 616 ◽  
pp. 327-356 ◽  
Author(s):  
BRIAN L. WHITE ◽  
KARL R. HELFRICH
Keyword(s):  
Internal Waves ◽  
Wave Speed ◽  
Numerical Solutions ◽  
Gravity Current ◽  
Gravity Currents ◽  
Stratified Fluid ◽  
Front Speed ◽  
Long Wave ◽  
Conjugate State ◽  
Energy Conserving

A steady theory is presented for gravity currents propagating with constant speed into a stratified fluid with a general density profile. Solution curves for front speed versus height have an energy-conserving upper bound (the conjugate state) and a lower bound marked by the onset of upstream influence. The conjugate state is the largest-amplitude nonlinear internal wave supported by the ambient stratification, and in the limit of weak stratification approaches Benjamin's energy-conserving gravity current solution. When the front speed becomes critical with respect to linear long waves generated above the current, steady solutions cannot be calculated, implying upstream influence. For non-uniform stratification, the critical long-wave speed exceeds the ambient long-wave speed, and the critical-Froude-number condition appropriate for uniform stratification must be generalized. The theoretical results demonstrate a clear connection between internal waves and gravity currents. The steady theory is also compared with non-hydrostatic numerical solutions of the full lock release initial-value problem. Some solutions resemble classic gravity currents with no upstream disturbance, but others show long internal waves propagating ahead of the gravity current. Wave generation generally occurs when the stratification and current speed are such that the steady gravity current theory fails. Thus the steady theory is consistent with the occurrence of either wave-generating or steady gravity solutions to the dam-break problem. When the available potential energy of the dam is large enough, the numerical simulations approach the energy-conserving conjugate state. Existing laboratory experiments for intrusions and gravity currents produced by full-depth lock exchange flows over a range of stratification profiles show excellent agreement with the conjugate state solutions.

Excitation of long internal waves by groups of short surface waves incident on a barrier

1988 ◽  
Vol 192 ◽  
pp. 17-31 ◽  
Author(s):  
Yehuda Agnon ◽  
Chiang C. Mei
Keyword(s):  
Surface Waves ◽  
Group Velocity ◽  
Internal Waves ◽  
Long Waves ◽  
Baroclinic Wave ◽  
Short Waves ◽  
Phase Velocities ◽  
Long Wave ◽  
Incident Waves ◽  
Model Ocean

The effects of diffraction by a long barrier on second-order long waves forced by sinusoidally modulated short incident waves are examined for a two-layered model ocean. When the group velocity of the short waves lies between the phase velocities of the longest baroclinic and barotropic modes, long internal waves of the frequency equal to twice the modulational frequency of the short waves are found to radiate away from the edge ray which divides the geometrical shadow and the illuminated region. In particular the baroclinic wave can penetrate the shadow. This penetration occurs when the internal long wave is not resonated by short surface waves.

Modelling the morning glory of the Gulf of Carpentaria

2002 ◽  
Vol 454 ◽  
pp. 1-20 ◽  
Author(s):  
ANNE PORTER ◽  
NOEL F. SMYTH
Keyword(s):  
Numerical Solutions ◽  
Equations Of Motion ◽  
Simple Wave ◽  
Morning Glory ◽  
Pressure Jump ◽  
Large Wave ◽  
Long Wave ◽  
Westerly Direction ◽  
Cape York ◽  
Good Agreement

The morning glory is a meteorological phenomenon which occurs in northern Australia and takes the form of a series of roll clouds. The morning glory is generated by the interaction of nocturnal seabreezes over Cape York Peninsula and propagates in a south-westerly direction over the Gulf of Carpentaria. In the present work, it is shown that the morning glory can be modelled by the resonant flow of a two-layer fluid over topography, the topography being the mountains of Cape York Peninsula. In the limit of a deep upper layer, the equations of motion reduce to a forced Benjamin–Ono equation. In this context, resonant means that the underlying flow velocity of the seabreezes is near a linear long-wave velocity for one of the long-wave modes. The morning glory is then modelled by the undular bore (simple wave) solution of the modulation equations for the Benjamin–Ono equation. This modulation solution is compared with full numerical solutions of the forced Benjamin–Ono equation and good agreement is found when the wave amplitudes are not too large. The reason for the difference between the numerical and modulation solutions for large wave amplitude is also discussed. Finally, the predictions of the modulation solution are compared with observational data on the morning glory and good agreement is found for the pressure jump due to the lead wave of the morning glory, but not for the speed and half-width of this lead wave. The reasons for this are discussed.

On standing internal gravity waves of finite amplitude

1968 ◽  
Vol 32 (3) ◽  
pp. 489-528 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Density Gradient ◽  
Gravity Waves ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Constant Density ◽  
Interfacial Wave ◽  
Large Wave ◽  
Fractional Density ◽  
Rectangular Container ◽  
Good Agreement

Two-dimensional internal gravity waves in a rectangular container are examined theoretically and experimentally in (a) fluids which contain a single density discontinuity and (b) fluids in which the density gradient is everywhere continuous. The fractional density difference between the top and bottom of the fluid is small.Good agreement is found between the observed and calculated wave profiles in case (a). Unlike surface standing waves, which tend to sharpen at their crests as the wave amplitude increases, and which eventually break at the crests when fluid accelerations become equal to that of gravity, internal wave crests are found to be flat and exhibit no instability. In the case (a) breaking is found to occur at the nodes of the interfacial wave, where the current shear, generated by the wave itself, is greatest. For sufficiently large wave amplitudes, a disturbance with the form of a vortex but with direction of rotation reversing twice every cycle, grows at the wave node and causes mixing. This instability is found to be followed by the generation of cross-waves, of which two different forms are observed.Several modes of oscillation can be generated and are observed in a fluid with constant density gradient. The wave frequencies and shape are well predicted by theory. The experiments failed to establish any limitation of the possible wave amplitudes.

Nonlinear Internal Waves in Stratified Fluid With Homogeneous Layer

2005 ◽  
Author(s):  
Nikolai I. Makarenko ◽  
Janna L. Maltseva
Keyword(s):  
Internal Waves ◽  
Lower Layer ◽  
Nonlinear Ordinary Differential Equation ◽  
Homogeneous Layer ◽  
Constant Density ◽  
Long Wave ◽  
Scaling Procedure ◽  
Internal Bores ◽  
Long Wave Approximation ◽  
Exponential Stratification

The problem of steady internal waves in a weakly stratified two-layered fluid is studied analytically. We discuss the model with a constant density in lower layer and exponential stratification in the other one. The long-wave approximation using a scaling procedure with small Boussinesq parameter is constructed. The nonlinear ordinary differential equation describing large amplitude solitary waves and internal bores is obtained.

Experiments with a self-propelled body submerged in a fluid with a vertical density gradient

1963 ◽  
Vol 15 (1) ◽  
pp. 83-96 ◽  
Author(s):  
Allen H. Schooley ◽  
R. W. Stewart
Keyword(s):  
Density Gradient ◽  
Internal Waves ◽  
Vertical Direction ◽  
Turbulent Wake ◽  
The Body ◽  
Uniform Density ◽  
Constant Density ◽  
Density Level ◽  
Vertical Density ◽  
Mixed Fluid

It is shown that the turbulent wake of a self-propelled body moving in a fluid with a vertical density gradient is considerably different than of the same body moving in a fluid having no density gradient. In the uniform density case, the turbulent mixed fluid behind the body expands into an irregular conical shape. In the case of a density gradient, the initial expansion of the mixed fluid is quickly followed by a collapse in the vertical direction which is accompanied by a further spreading in the horizontal direction. This phenomenon is caused by the force of gravity. The volume of fluid behind the self-propelled body has a more or less constant density due to mixing. Thus, it is forced to seek its own density level in the undisturbed fluid.The collapsing vertical wake is shown to be an efficent generator of internal waves, many of high order. These manifest themselves in surface movements.The assumption that the internal waves are damped only by viscosity, not by turbulence, leads to results in general accord with the observations.

A multi-layer model for nonlinear internal wave propagation in shallow water

2012 ◽  
Vol 695 ◽  
pp. 341-365 ◽  
Author(s):  
Philip L.-F. Liu ◽  
Xiaoming Wang
Keyword(s):  
Wave Propagation ◽  
Internal Wave ◽  
Shallow Water ◽  
Internal Waves ◽  
Bottom Topography ◽  
Laboratory Data ◽  
Constant Density ◽  
Layer Model ◽  
Fluid System ◽  
Nonlinear Internal Wave

AbstractIn this paper, a multi-layer model is developed for the purpose of studying nonlinear internal wave propagation in shallow water. The methodology employed in constructing the multi-layer model is similar to that used in deriving Boussinesq-type equations for surface gravity waves. It can also be viewed as an extension of the two-layer model developed by Choi & Camassa. The multi-layer model approximates the continuous density stratification by an $N$-layer fluid system in which a constant density is assumed in each layer. This allows the model to investigate higher-mode internal waves. Furthermore, the model is capable of simulating large-amplitude internal waves up to the breaking point. However, the model is limited by the assumption that the total water depth is shallow in comparison with the wavelength of interest. Furthermore, the vertical vorticity must vanish, while the horizontal vorticity components are weak. Numerical examples for strongly nonlinear waves are compared with laboratory data and other numerical studies in a two-layer fluid system. Good agreement is observed. The generation and propagation of mode-1 and mode-2 internal waves and their interactions with bottom topography are also investigated.

On parametric instabilities of finite-amplitude internal gravity waves

1982 ◽  
Vol 119 ◽  
pp. 367-377 ◽  
Author(s):  
J. Klostermeyer
Keyword(s):  
Gravity Wave ◽  
Internal Gravity Wave ◽  
Maximum Growth ◽  
Numerical Approach ◽  
Internal Gravity Waves ◽  
Resonant Interaction ◽  
Finite Amplitude ◽  
Interaction Diagram ◽  
Parametric Instabilities ◽  
Boussinesq Fluid

The equations describing parametric instabilities of a finite-amplitude internal gravity wave in an inviscid Boussinesq fluid are studied numerically. By improving the numerical approach, discarding the concept of spurious roots and considering the whole range of directions of the Floquet vector, Mied's work is generalized to its full complexity. In the limit of large disturbance wavenumbers, the unstable disturbances propagate in the directions of the two infinite curve segments of the related resonant-interaction diagram. They can therefore be classified into two families which are characterized by special propagation directions. At high wavenumbers the maximum growth rates converge to limits which do not depend on the direction of the Floquet vector. The limits are different for both families; the disturbance waves propagating at the smaller angle to the basic gravity wave grow at the larger rate.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

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