On the shape of progressive internal waves

1968 ◽  
Vol 263 (1145) ◽  
pp. 563-614 ◽  
Keyword(s):  
Experimental Evidence ◽  
Internal Waves ◽  
Gravity Waves ◽  
Boussinesq Approximation ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Fluid System ◽  
Expansion Technique ◽  
Theoretical Predictions ◽  
Two Fluid

An expansion technique, analogous to that of Stokes in the study of surface waves, is used to investigate the effects of finite amplitude on a progressive train of internal gravity waves. The paper is divided into two main parts, a study of interfacial waves in a two-fluid system and an examination of internal waves in a continuously stratified fluid. Experimental evidence is presented which confirms some of the theoretical predictions. The validity of the Boussinesq approximation is examined and particular examples are taken to illustrate the general results.

On the shape and breaking of finite amplitude internal gravity waves in a shear flow

1978 ◽  
Vol 85 (1) ◽  
pp. 7-31 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Shear Flow ◽  
Internal Waves ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Mixing Layer ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Depth Interval ◽  
Plane Shear

This paper is concerned with two important aspects of nonlinear internal gravity waves in a stably stratified inviscid plane shear flow, their shape and their breaking, particularly in conditions which are frequently encountered in geophysical applications when the vertical gradients of the horizontal current and the density are concentrated in a fairly narrow depth interval (e.g. the thermocline in the ocean). The present theoretical and experimental study of the wave shape extends earlier work on waves in the absence of shear and shows that the shape may be significantly altered by shear, the second-harmonic terms which describe the wave profile changing sign when the shear is increased sufficiently in an appropriate sense.In the second part of the paper we show that the slope of internal waves at which breaking occurs (the particle speeds exceeding the phase speed of the waves) may be considerably reduced by the presence of shear. Internal waves on a thermocline which encounter an increasing shear, perhaps because of wind action accelerating the upper mixing layer of the ocean, may be prone to such breaking.This work may alternatively be regarded as a study of the stability of a parallel stratified shear flow in the presence of a particular finite disturbance which corresponds to internal gravity waves propagating horizontally in the plane of the flow.

A conservation law for internal gravity waves

1976 ◽  
Vol 78 (2) ◽  
pp. 209-216 ◽  
Author(s):  
Michael Milder
Keyword(s):  
Internal Wave ◽  
Group Velocity ◽  
Internal Waves ◽  
Gravity Waves ◽  
Conservation Law ◽  
Short Wavelength ◽  
Internal Gravity Waves ◽  
Volume Integral ◽  
Weak Density ◽  
Progressive Waves

The scaled vorticity Ω/N and strain ∇ ζ associated with internal waves in a weak density gradient of arbitrary depth dependence together comprise a quantity that is conserved in the usual linearized approximation. This quantity I is the volume integral of the dimensionless density DI = ½[Ω2/N2 + (∇ ζ)2]. For progressive waves the ‘kinetic’ and ‘potential’ parts are equal, and in the short-wavelength limit the density DI and flux FI are related by the ordinary group velocity: FI = DIcg. The properties of DI suggest that it may be a useful measure of local internal-wave saturation.

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.

scholarly journals Interaction Features of Internal Wave Breathers in a Stratified Ocean

Fluids ◽  
2020 ◽  
Vol 5 (4) ◽  
pp. 205
Author(s):  
Ekaterina Didenkulova ◽  
Efim Pelinovsky
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Gravity Waves ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Vries Equation ◽  
Internal Gravity Waves ◽  
Wave Fields ◽  
Tidal Waves ◽  
Stratified Ocean

Oscillating wave packets (breathers) are a significant part of the dynamics of internal gravity waves in a stratified ocean. The formation of these waves can be provoked, in particular, by the decay of long internal tidal waves. Breather interactions can significantly change the dynamics of the wave fields. In the present study, a series of numerical experiments on the interaction of breathers in the frameworks of the etalon equation of internal waves—the modified Korteweg–de Vries equation (mKdV)—were conducted. Wave field extrema, spectra, and statistical moments up to the fourth order were calculated.

Reflection properties of internal gravity waves incident upon a hyperbolic tangent shear layer

1982 ◽  
Vol 120 ◽  
pp. 505-521 ◽  
Author(s):  
Cornelis A. Van Duin ◽  
Hennie Kelder
Keyword(s):  
Shear Layer ◽  
Gravity Waves ◽  
Critical Level ◽  
Boussinesq Approximation ◽  
Internal Gravity Waves ◽  
Hyperbolic Tangent ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Heun’S Equation ◽  
Stable Flow

The properties of reflection and transmission of internal gravity waves incident upon a shear layer containing a critical level are investigated. The shear layer is modelled by a hyperbolic tangent profile. In the Boussinesq approximation, the differential equation governing the propagation of these waves can then be transformed into Heun's equation. For large Richardson numbers this equation can be approximated by an equation that has solutions in terms of hypergeometric functions. For these values of the Richardson number the reflection coefficient proves to be strongly dependent on the place of the critical level in the shear flow. If the Doppler-shifted frequency is an odd function of the height difference with respect to the critical level, the reflection and transmission coefficients can be evaluated in closed form.Over-reflection is possible for sufficiently small wavenumbers and Richardson numbers. It is pointed out that over-reflection and over-transmission cannot occur in a stable flow and that resonant over-reflection is not possible in our model.

A Model of Rayleigh-Bénard Convection

2013 ◽  
Author(s):  
Gary A. Glatzmaier
Keyword(s):  
Gravity Waves ◽  
Thermal Convection ◽  
Fluid Flows ◽  
Boussinesq Approximation ◽  
Internal Gravity Waves ◽  
Bénard Convection ◽  
Benard Convection ◽  
Key Characteristics ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, focusing on thermal convection and internal gravity waves. Thermal convection and internal gravity waves are the two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces. The chapter then reviews the equations that govern fluid dynamics based on conservation of mass, momentum, and energy. It also examines the conditions under which the Boussinesq approximation simplifies conservation equations to a form very similar to that of an incompressible fluid. Finally, it discusses the key characteristics of the model of Rayleigh–Bénard convection.

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