The energetics of the interaction between short small-amplitude internal waves and inertial waves

1988 ◽  
Vol 196 ◽  
pp. 93-106 ◽  
Author(s):  
D. Broutman ◽  
R. Grimshaw
Keyword(s):  
Wave Field ◽  
Internal Waves ◽  
Small Amplitude ◽  
Short Wave ◽  
Finite Amplitude ◽  
Inertial Waves ◽  
Mean Flow ◽  
Total Energy Balance ◽  
Inertial Wave ◽  
Wave Induced

The interaction between a wave packet of small-amplitude short internal waves, and a finite-amplitude inertial wave field is described to second order in the short-wave amplitude. The discussion is based on the principle of wave action conservation and the equations for the wave-induced Lagrangian mean flow. It is demonstrated that as the short internal waves propagate through the inertial wave field they generate a wave-induced train of trailing inertial waves. The contribution of this wave-induced mean flow to the total energy balance is described. The results obtained here complement the finding of Broutman & Young (1986) that the short internal waves undergo a net change in energy after their encounter with the inertial wave field.

Stability of finite-amplitude interfacial waves. Part 1. Modulational instability for small-amplitude waves

1985 ◽  
Vol 160 ◽  
pp. 297-315 ◽  
Author(s):  
R. H. J. Grimshaw ◽  
D. I. Pullin
Keyword(s):  
Small Amplitude ◽  
Modulational Instability ◽  
Density Ratio ◽  
Basic Density ◽  
Finite Amplitude ◽  
Flow Equation ◽  
Mean Flow ◽  
Coupled Equations ◽  
The Stability ◽  
Wave Induced

In two previous papers (Pullin & Grimshaw 1983a, b) we studied the wave profile and other properties of finite-amplitude interfacial progressive waves in a two-layer fluid. In this and the following paper (Pullin & Grimshaw 1985) we discuss the stability of these waves to small perturbations. In this paper we obtain anatytical results for the long-wavelength modulational instability of small-amplitude waves. Using a multiscale expansion, we obtain a nonlinear Schrödinger equation coupled to a wave-induced mean-flow equation to describe slowly modulated waves. From these coupled equations we determine the stability of a plane progressive wave. Our results are expressed by determining the instability bands in the (p, q)-plane, where (p, q) is the modulation wavenumber, and are obtained for a range of values of basic density ratio and undisturbed layer depths.

scholarly journals Response of a swirl flame to inertial waves

2017 ◽  
Vol 10 (4) ◽  
pp. 277-286 ◽  
Author(s):  
Alp Albayrak ◽  
Deniz A Bezgin ◽  
Wolfgang Polifke
Keyword(s):  
Steady State ◽  
Impulse Response ◽  
Finite Impulse Response ◽  
Reactive Flow ◽  
Inertial Waves ◽  
Mean Flow ◽  
Flow Equations ◽  
Flame Response ◽  
Inertial Wave ◽  
The Impact

Acoustic waves passing through a swirler generate inertial waves in rotating flow. In the present study, the response of a premixed flame to an inertial wave is scrutinized, with emphasis on the fundamental fluid-dynamic and flame-kinematic interaction mechanism. The analysis relies on linearized reactive flow equations, with a two-part solution strategy implemented in a finite element framework: Firstly, the steady state, low-Mach number, Navier–Stokes equations with Arrhenius type one-step reaction mechanism are solved by Newton’s method. The flame impulse response is then computed by transient solution of the analytically linearized reactive flow equations in the time domain, with mean flow quantities provided by the steady-state solution. The corresponding flame transfer function is retrieved by fitting a finite impulse response model. This approach is validated against experiments for a perfectly premixed, lean, methane-air Bunsen flame, and then applied to a laminar swirling flame. This academic case serves to investigate in a generic manner the impact of an inertial wave on the flame response. The structure of the inertial wave is characterized by modal decomposition. It is shown that axial and radial velocity fluctuations related to the eigenmodes of the inertial wave dominate the flame front modulations. The dispersive nature of the eigenmodes plays an important role in the flame response.

Secondary instabilities in rapidly rotating fluids: inertial wave breakdown

1999 ◽  
Vol 382 ◽  
pp. 283-306 ◽  
Author(s):  
R. R. KERSWELL
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Small Scale ◽  
Inertial Waves ◽  
Rotating Fluids ◽  
Initial Excitation ◽  
Geostrophic Flows ◽  
Inertial Wave ◽  
Secondary Instabilities ◽  
Wave Breakdown

Inertial waves are a ubiquitous feature of rapidly rotating fluids. Although much is known about their initial excitation, little is understood about their stability. Experiments indicate that they are generically unstable and in many cases catastrophically so, quickly causing the whole flow to collapse to small-scale disorder. The linear stability of two three-dimensional inertial waves observed to break down in the laboratory is considered here at experimentally small but finite Ekman numbers of [les ]10−4. Surprisingly small threshold amplitudes for instability are found. The results support the conjecture that triad resonances are the generic mechanism for secondary instability in rapidly rotating fluids but also highlight the ability of geostrophic flows to derive energy through a finite-amplitude inertial wave. This latter finding may go some way to explaining the significant mean circulations typically observed in inertial wave experiments.

Over-reflection of internal-inertial waves from the mixed layer

1984 ◽  
Vol 141 ◽  
pp. 179-196 ◽  
Author(s):  
M. Kamachi ◽  
R. Grimshaw
Keyword(s):  
Mixed Layer ◽  
Growth Rates ◽  
Vertical Profile ◽  
Body Force ◽  
Inertial Waves ◽  
Mean Flow ◽  
Energy Propagation ◽  
Frequency Space ◽  
Wave Induced ◽  
The Relationship

Near-inertial oscillations associated with downward energy propagation are commonly observed in the upper ocean. Stern (1977) has suggested that these observations may be internal-inertial waves over-reflected from the shear zone at the base of the mixed layer. In this paper we develop a criterion for over-reflection as a function of wavenumber and frequency for a class of shear flows in the mixed layer. By examining the vertical profile of the vertical wave action flux we demonstrate that the source of the over-reflection is the shear at the base of the mixed layer, which is maintained by the wind-induced turbulent Reynolds stress, here parametrized as a body force. The relationship between over-reflection and the wave-induced Lagrangian-mean flow is determined. We also determine a criterion for unstable waves, and show that these are contiguous in wavenumber-frequency space with points of resonant over-reflection. However, the growth rates of these unstable waves are quite small, and in practice unstable waves will be indistinguishable from waves generated by over-reflection.

scholarly journals Weakly nonlinear wave motions in a thermally stratified boundary layer

1996 ◽  
Vol 315 ◽  
pp. 293-316 ◽  
Author(s):  
James P. Denier ◽  
Eunice W. Mureithi
Keyword(s):  
Boundary Layer ◽  
Nonlinear Wave ◽  
Finite Amplitude ◽  
Large Reynolds Number ◽  
Neutral Stability ◽  
Interaction Theory ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Order Of Magnitude ◽  
Wave Induced

We consider weakly nonlinear wave motions in a thermally stratified boundary layer. Attention is focused on the upper branch of the neutral stability curve, corresponding to small wavelengths and large Reynolds number. In this limit the motion is governed by a first harmonic/mean flow interaction theory in which the wave-induced mean flow is of the same order of magnitude as the wave component of the flow. We show that the flow is governed by a system of three coupled partial differential equations which admit finite-amplitude periodic solutions bifurcating from the linear, neutral points.

scholarly journals Could waves mix the ocean?

2009 ◽  
Vol 638 ◽  
pp. 1-4 ◽  
Author(s):  
G. FALKOVICH
Keyword(s):  
Shallow Water ◽  
Internal Waves ◽  
Particle Velocity ◽  
Small Amplitude ◽  
Water System ◽  
Effective Diffusivity ◽  
Finite Amplitude ◽  
Random Set ◽  
Shallow Water System ◽  
Rotating Shallow Water

A finite-amplitude propagating wave induces a drift in fluids. Understanding how drifts produced by many waves disperse pollutants has broad implications for geophysics and engineering. Previously, the effective diffusivity was calculated for a random set of small-amplitude surface and internal waves. Now, this is extended by Bühler & Holmes-Cerfon (J. Fluid Mech., 2009, this issue, vol. 638, pp. 5–26) to waves in a rotating shallow-water system in which the Coriolis force is accounted for, a necessary step towards oceanographic applications. It is shown that interactions of finite-amplitude waves affect particle velocity in subtle ways. An expression describing the particle diffusivity as a function of scale is derived, showing that the diffusivity can be substantially reduced by rotation.

Near-inertial wave dispersion by geostrophic flows

2017 ◽  
Vol 817 ◽  
pp. 406-438 ◽  
Author(s):  
Jim Thomas ◽  
K. Shafer Smith ◽  
Oliver Bühler
Keyword(s):  
Shallow Water ◽  
Wave Field ◽  
Wave Energy ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Deep Ocean ◽  
Boussinesq Equations ◽  
Amplitude Equation ◽  
Mean Flow ◽  
Inertial Wave

We investigate theoretically and numerically the modulation of near-inertial waves by a larger-amplitude geostrophically balanced mean flow. Because the excited wave is initially trapped in the mixed layer, it projects onto a broad spectrum of vertical modes, each mode $n$ being characterized by a Burger number, $Bu_{n}$, proportional to the square of the vertical scale of the mode. Using numerical simulations of the hydrostatic Boussinesq equations linearized about a prescribed balanced background flow, we show that the evolution of the wave field depends strongly on the spectrum of $Bu_{n}$ relative to the Rossby number of the balanced flow, $\unicode[STIX]{x1D716}$, with smaller relative $Bu_{n}$ leading to smaller horizontal scales in the wave field, faster accumulation of wave amplitude in anticyclones and faster propagation of wave energy into the deep ocean. This varied behaviour of the wave may be understood by considering the dynamics in each mode separately; projecting the linearized hydrostatic Boussinesq equations onto modes yields a set of linear shallow water equations, with $Bu_{n}$ playing the role of the reduced gravity. The wave modes fall into two asymptotic regimes, defined by the scalings $Bu_{n}\sim O(1)$ for low modes and $Bu_{n}\sim O(\unicode[STIX]{x1D716})$ for high modes. An amplitude equation derived for the former regime shows that vertical propagation is weak for low modes. The high-mode regime is the basis of the Young & Ben Jelloul (J. Mar. Res., vol. 55, 1997, pp. 735–766) theory. This theory is here extended to $O(\unicode[STIX]{x1D716}^{2})$, from which amplitude equations for the subregimes $Bu_{n}\sim O(\unicode[STIX]{x1D716}^{1/2})$ and $Bu_{n}\sim O(\unicode[STIX]{x1D716}^{2})$ are derived. The accuracy of each approximation is demonstrated by comparing numerical solutions of the respective amplitude equation to simulations of the linearized shallow water equations in the same regime. We emphasize that since inertial wave energy and shear are distributed across vertical modes, their overall modulation is due to the collective behaviour of the wave field in each regime. A unified treatment of these regimes is a novel feature of this work.

An exact Lagrangian-mean wave activity for finite-amplitude disturbances to barotropic flow on a sphere

2012 ◽  
Vol 693 ◽  
pp. 69-92 ◽  
Author(s):  
Abraham Solomon ◽  
N. Nakamura
Keyword(s):  
Small Amplitude ◽  
Correction Term ◽  
Momentum Density ◽  
Finite Amplitude ◽  
Wave Activity ◽  
New Wave ◽  
Mean Flow ◽  
Flow Modification ◽  
Barotropic Flow ◽  
Generalized Lagrangian

AbstractThe finite-amplitude Rossby wave activity introduced recently by Nakamura and co-workers measures disturbances in terms of the areal displacement of potential vorticity (PV) from zonal symmetry and possesses exact Eliassen–Palm and non-acceleration theorems. This article investigates both theoretically and numerically how this wave activity, denoted ${A}^{\ensuremath{\ast} } $, relates to previously defined quantities such as the generalized Lagrangian-mean (GLM) pseudomomentum density and the impulse-Casimir (IC) wave activity in the context of barotropic flow on a sphere. It is shown that under the barotropic constraint both the new and GLM formalisms derive the non-acceleration theorem from the conservation of Kelvin’s circulation, but the two differ in the way the circulation is partitioned into a mean flow and wave activity/pseudomomentum density. The new wave activity differs from the (negative of) GLM pseudomomentum density by the Stokes correction to angular momentum density, which is not negligible even in the small-amplitude limit. In contrast, ${A}^{\ensuremath{\ast} } $ converges to the IC wave activity and the familiar linear pseudomomentum density in the conservative small-amplitude limit, provided that their reference states are identical. Both the GLM pseudomomentum density and the zonal-mean IC wave activity may be cast in a flux conservation form in equivalent latitude, which may then be related to an exact Eliassen–Palm theorem through a gauge transformation. However, of the three wave activity forms, only ${A}^{\ensuremath{\ast} } $ satisfies an exact non-acceleration theorem for the zonal-mean zonal wind $\bar {u} $. A simple jet forcing experiment is used to examine the quantitative differences among these diagnostics. In this experiment, ${A}^{\ensuremath{\ast} } $ and the IC wave activity behave similarly in the domain average; however, they differ substantially in the local profiles, the former being more closely related to the flow modification. Despite their close conceptual relationship, the GLM pseudomomentum fails to capture the meridional structure of ${A}^{\ensuremath{\ast} } $ because the Stokes correction term dominates the former. This demonstrates various advantages of ${A}^{\ensuremath{\ast} } $ as a diagnostic of eddy–mean flow interaction.

Tilting at wave beams: a new perspective on the St. Andrew’s Cross

2017 ◽  
Vol 830 ◽  
pp. 660-680 ◽  
Author(s):  
T. Kataoka ◽  
S. J. Ghaemsaidi ◽  
N. Holzenberger ◽  
T. Peacock ◽  
T. R. Akylas
Keyword(s):  
Wave Field ◽  
Finite Length ◽  
Line Source ◽  
Cutoff Frequency ◽  
Three Dimensional ◽  
Resonance Phenomenon ◽  
Buoyancy Frequency ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Viscous Effects

The generation of internal gravity waves by a vertically oscillating cylinder that is tilted to the horizontal in a stratified Boussinesq fluid of constant buoyancy frequency, $N$, is investigated. This variant of the widely studied horizontal configuration – where a cylinder aligned with a plane of constant gravitational potential induces four wave beams that emanate from the cylinder, forming a cross pattern known as the ‘St. Andrew’s Cross’ – brings out certain unique features of radiated internal waves from a line source tilted to the horizontal. Specifically, simple kinematic considerations reveal that for a cylinder inclined by a given angle $\unicode[STIX]{x1D719}$ to the horizontal, there is a cutoff frequency, $N\sin \unicode[STIX]{x1D719}$, below which there is no longer a radiated wave field. Furthermore, three-dimensional effects due to the finite length of the cylinder, which are minor in the horizontal configuration, become a significant factor and eventually dominate the wave field as the cutoff frequency is approached; these results are confirmed by supporting laboratory experiments. The kinematic analysis, moreover, suggests a resonance phenomenon near the cutoff frequency as the group-velocity component perpendicular to the cylinder direction vanishes at cutoff; as a result, energy cannot be easily radiated away from the source, and nonlinear and viscous effects are likely to come into play. This scenario is examined by adapting the model for three-dimensional wave beams developed in Kataoka & Akylas (J. Fluid Mech., vol. 769, 2015, pp. 621–634) to the near-resonant wave field due to a tilted line source of large but finite length. According to this model, the combination of three-dimensional, nonlinear and viscous effects near cutoff triggers transfer of energy, through the action of Reynolds stresses, to a circulating horizontal mean flow. Experimental evidence of such an induced mean flow near cutoff is also presented.

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