Internal waves around a disturbance in a fluid with arbitrary stratification and background shear flow

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

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Simulating background shear flow in local gyrokinetic simulations

Plasma Physics and Controlled Fusion ◽

10.1088/1361-6587/ab06a4 ◽

2019 ◽

Vol 61 (5) ◽

pp. 055006 ◽

Cited By ~ 1

Author(s):

B F McMillan ◽

J Ball ◽

S Brunner

Keyword(s):

Shear Flow ◽

Background Shear ◽

Gyrokinetic Simulations

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The stability of filamentary vorticity in two-dimensional geophysical vortex-dynamics models

Journal of Fluid Mechanics ◽

10.1017/s002211209100352x ◽

1991 ◽

Vol 231 ◽

pp. 575-598 ◽

Cited By ~ 26

Author(s):

D. W. Waugh ◽

D. G. Dritschel

Keyword(s):

Green Function ◽

Shear Flow ◽

Stream Function ◽

Potential Vorticity ◽

Two Dimensional ◽

Interaction Range ◽

Inversion Operator ◽

The Green Function ◽

The Stability ◽

Background Shear

The linear stability of filaments or strips of ‘potential’ vorticity in a background shear flow is investigated for a class of two-dimensional, inviscid, non-divergent models having a linear inversion relation between stream function and potential vorticity. In general, the potential vorticity is not simply the Laplacian of the stream function – the case which has received the greatest attention historically. More general inversion relationships between stream function and potential vorticity are geophysically motivated and give an impression of how certain classic results, such as the stability of strips of vorticity, hold under more general circumstances.In all models, a strip of potential vorticity is unstable in the absence of a background shear flow. Imposing a shear flow that reverses the total shear across the strip, however, brings about stability, independent of the Green-function inversion operator that links the stream function to the potential vorticity. But, if the Green-function inversion operator has a sufficiently short interaction range, the strip can also be stabilized by shear having the same sense as the shear of the strip. Such stabilization by ‘co-operative’ shear does not occur when the inversion operator is the inverse Laplacian. Nonlinear calculations presented show that there is only slight disruption to the strip for substantially less adverse shear than necessary for linear stability, while for co-operative shear, there is major disruption to the strip. It is significant that the potential vorticity of the imposed flow necessary to create shear of a given value increases dramatically as the interaction range of the inversion operator decreases, making shear stabilization increasingly less likely. This implies an increased propensity for filaments to ‘roll-up’ into small vortices as the interaction range decreases, a finding consistent with many numerical calculations performed using the quasi-geostrophic model.

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A laboratory study of stratified accelerating shear flow over a rough boundary

Journal of Fluid Mechanics ◽

10.1017/s0022112084000082 ◽

1984 ◽

Vol 138 ◽

pp. 185-196 ◽

Cited By ~ 6

Author(s):

S. A. Thorpe

Keyword(s):

Shear Flow ◽

Internal Waves ◽

Mixing Layer ◽

Internal Gravity Waves ◽

Rough Boundary ◽

Turbulent Layer ◽

Near Surface ◽

Rate Of Spread ◽

Vertical Scale ◽

The Waves

Experiments are made in which a stratified shear flow, accelerating from rest and containing a level where the direction of flow reverses, is generated over a rough floor. The roughness elements consist of parallel square bars set at regular intervals normal to the direction of flow. Radiating internal gravity waves are generated in the early stages of flow, whilst flow separation behind the bars produces turbulent mixing regions which eventually amalgamate and entirely cover the floor. This turbulent layer spreads vertically less rapidly than the internal waves. Observed features of the waves are compared with those predicted by a model in which the floor is assumed to be sinusoidal, and fair agreement is found for the amplitude, phase and vertical wavenumber of the waves, even when the latter becomes large.The rate of spread of the turbulent layer depends on the separation of the bars. Some interaction between the turbulence and the internal waves occurs near the edge of the turbulent layer. Wave-breaking is prevalent and the vertical scale of the waves is affected by turbulent eddies. The radiating internal waves are suppressed by replacing the bars by an array of square cubes, but there is continued evidence of features resembling internal waves near the boundary of the turbulent region. Structures are observed which bear some similarities to those found at the foot of the near-surface mixing layer in a lake.

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