The occurrence of parametric instabilities in finite-amplitude internal gravity waves

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.

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

Journal of Fluid Mechanics ◽

10.1017/s0022112079002123 ◽

1979 ◽

Vol 90 (1) ◽

pp. 161-178 ◽

Cited By ~ 11

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

Journal of Fluid Mechanics ◽

10.1017/s0022112078000518 ◽

1978 ◽

Vol 85 (1) ◽

pp. 7-31 ◽

Cited By ~ 58

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.

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On the shape of progressive internal waves

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1968.0033 ◽

1968 ◽

Vol 263 (1145) ◽

pp. 563-614 ◽

Cited By ~ 106

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.

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On standing internal gravity waves of finite amplitude

Journal of Fluid Mechanics ◽

10.1017/s002211206800087x ◽

1968 ◽

Vol 32 (3) ◽

pp. 489-528 ◽

Cited By ~ 82

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.

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Resonant over-reflection of internal gravity waves from a thin shear layer

Journal of Fluid Mechanics ◽

10.1017/s0022112081001110 ◽

1981 ◽

Vol 109 ◽

pp. 349-365 ◽

Cited By ~ 8

Author(s):

R. H. J. Grimshaw

Keyword(s):

Growth Rate ◽

Shear Layer ◽

Gravity Waves ◽

Finite Time ◽

Linear Growth ◽

Vortex Sheet ◽

Internal Gravity Waves ◽

Finite Amplitude ◽

Linear Growth Rate ◽

Weakly Nonlinear

In a previous paper (Grimshaw 1979) the resonant over-reflection of internal gravity waves from a vortex sheet was considered in the weakly nonlinear regime. It was shown there that the time evolution of the amplitude of the vortex sheet displacement was balanced by a cubic nonlinearity. For one vortex sheet mode, symmetrical with respect to the interface, it was shown that a steady finite-amplitude wave was possible. For the other, asymmetric modes, a singularity develops in a finite time. In the present paper, that analysis is extended by replacing the vortex sheet with a thin shear layer of thickness α2, where α is the amplitude of the shear layer displacement. The effect of this extension is to introduce a linear growth rate term in the amplitude equation, which is otherwise unaltered. The linear growth rate can be computed from a formula due to Drazin & Howard (1966, p. 67). The effect on the modes is that the symmetric mode is linearly damped and requires sustained forcing to be observed, while the asymmetric modes are slightly destabilized by the linear term and, as in the vortex-sheet model, develop a singularity in finite time.

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