Effect of background mean flow on PSI of internal wave beams

2019 ◽  
Vol 869 ◽  
Author(s):  
Boyu Fan ◽  
T. R. Akylas
Keyword(s):  
Internal Wave ◽  
Evolution Equations ◽  
Internal Gravity Wave ◽  
Necessary Condition ◽  
Finite Width ◽  
Asymptotic Model ◽  
Mean Flow ◽  
Short Scale ◽  
The Mean ◽  
Parametric Subharmonic Instability

An asymptotic model is developed for the parametric subharmonic instability (PSI) of finite-width nearly monochromatic internal gravity wave beams in the presence of a background constant horizontal mean flow. The subharmonic perturbations are taken to be short-scale wavepackets that may extract energy via resonant triad interactions while in contact with the underlying beam, and the mean flow is assumed to be small so that its advection effect on the perturbations is as important as dispersion, triad nonlinearity and viscous dissipation. In this ‘distinguished limit’, the perturbation dynamics are governed by the same evolution equations as those derived in Karimi & Akylas (J. Fluid Mech., vol. 757, 2014, pp. 381–402), except for a mean flow term that affects the group velocity of the perturbations and imposes an additional necessary condition for PSI, which stabilizes very short-scale perturbations. As a result, it is possible for a small amount of mean flow to weaken PSI dramatically.

scholarly journals On strongly nonlinear gravity waves in a vertically sheared atmosphere

2020 ◽  
Vol 6 (1) ◽  
pp. 63-74
Author(s):  
Mark Schlutow ◽  
Georg S. Voelker
Keyword(s):  
Gravity Waves ◽  
Lower Layer ◽  
Vertical Shear ◽  
Horizontal Wind ◽  
Necessary Condition ◽  
Mean Flow ◽  
Individual Layer ◽  
Strongly Nonlinear ◽  
The Mean ◽  
The Stability

Abstract We investigate strongly nonlinear stationary gravity waves which experience refraction due to a thin vertical shear layer of horizontal background wind. The velocity amplitude of the waves is of the same order of magnitude as the background flow and hence the self-induced mean flow alters the modulation properties to leading order. In this theoretical study, we show that the stability of such a refracted wave depends on the classical modulation stability criterion for each individual layer, above and below the shearing. Additionally, the stability is conditioned by novel instability criteria providing bounds on the mean-flow horizontal wind and the amplitude of the wave. A necessary condition for instability is that the mean-flow horizontal wind in the upper layer is stronger than the wind in the lower layer.

On strongly nonlinear gravity waves in a vertically sheared atmosphere

2021 ◽  
Author(s):  
Georg Sebastian Voelker ◽  
Mark Schlutow
Keyword(s):  
Gravity Wave ◽  
Gravity Waves ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Wave Drag ◽  
Mean Flow ◽  
Flow Strength ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
The Mean

<p>Internal gravity waves are a well-known mechanism of energy redistribution in stratified fluids such as the atmosphere. They may propagate from their generation region, typically in the Troposphere, up to high altitudes. During their lifetime internal waves couple to the atmospheric background through various processes. Among the most important interactions are the exertion of wave drag on the horizontal mean-flow, the heat generation upon wave breaking, or the mixing of atmospheric tracers such as aerosols or greenhouse gases.</p><p>Many of the known internal gravity wave properties and interactions are covered by linear or weakly nonlinear theories. However, for the consideration of some of the crucial effects, like a reciprocal wave-mean-flow interaction including the exertion of wave drag on the mean-flow, strongly nonlinear systems are required. That is, there is no assumption on the wave amplitude relative to the mean-flow strength such that they may be of the same order.</p><p>Here, we exploit a strongly nonlinear Boussinesq theory to analyze the stability of a stationary internal gravity wave which is refracted at the vertical edge of a horizontal jet. Thereby we assume that the incident wave is horizontally periodic, non-hydrostatic, and vertically modulated. Performing a linear stability analysis in the vicinity of the jet edge we find necessary and sufficient criteria for instabilities to grow. In particular, the refracted wave becomes unstable if its incident amplitude is large enough and both mean-flow horizontal winds, below and above the edge of the jet, do not exceed particular upper bounds.</p>

The fully nonlinear development of Görtler vortices in growing boundary layers

1988 ◽  
Vol 415 (1849) ◽  
pp. 421-444 ◽  
Keyword(s):  
Boundary Layer ◽  
Evolution Equations ◽  
Mean Flow ◽  
Initial Development ◽  
Fully Nonlinear ◽  
Nonlinear Development ◽  
Görtler Vortices ◽  
Starting Point ◽  
The Mean ◽  
Gortler Vortices

The fully nonlinear development of small-wavelength Görtler vortices in a growing boundary layer is investigated by a combination of asymptotic and numerical methods. The starting point for the analysis is the weakly nonlinear theory of Hall ( J. Inst. Math. Applies 29, 173 (1982)) who discussed the initial development of large-wavenumber small-amplitude vortices in a neighbourhood of the location where they first become linearly unstable. That development is unusual in the context of nonlinear stability theory in that it is not described by the Stuart-Watson approach. In fact, the development is governed by a pair of coupled nonlinear partial differential evolution equations for the vortex flow and the mean flow correction. Here the further development of this interaction is considered for vortices so large that the mean flow correction driven by them is as large as the basic state. Surprisingly it is found that such a nonlinear interaction can still be described by asymptotic means. It is shown that the vortices spread out across the boundary layer and effectively drive the boundary layer. In fact, the system obtained by the equations for the fundamental component of the vortex generates a differential equation for the basic state. Thus the mean flow adjusts so as to make these large amplitude vortices locally neutral. Moreover in the region where the vortices exist the mean flow has a ‘square-root’ profile and the vortex velocity field can be written down in closed form. The upper and lower boundaries of the region of vortex activity are determined by a free-boundary problem involving the boundary-layer equations. In general it is found that this region ultimately includes almost all of the original boundary layer and much of the free stream. In this situation the mean flow has essentially no relation to the flow that exists in the absence of the vortices.

scholarly journals Energetics of Seamount Wakes. Part I: Energy Exchange

2020 ◽  
Vol 50 (5) ◽  
pp. 1365-1382 ◽  
Author(s):  
B. Perfect ◽  
N. Kumar ◽  
J. J. Riley
Keyword(s):  
Kinetic Energy ◽  
Energy Exchange ◽  
Evolution Equations ◽  
Mean Flow ◽  
Wake Turbulence ◽  
Barotropic Flow ◽  
Unsteady Wake ◽  
Basin Scale ◽  
Wake Effects ◽  
The Mean

AbstractSeamounts have been theorized to act as “stirring rods,” converting barotropic flow into an unsteady wake, turbulence, and diapycnal mixing. The energetics of these processes are not well understood, but they may have implications for basin-scale mixing calculations. This study presents the results of a series of simulations for idealized seamounts in steady barotropic flow, with varying degrees of stratification and rotation. The kinetic energy within each simulation domain is decomposed into the mean kinetic energy, unsteady eddy energy, and turbulent kinetic energy; evolution equations are derived for each. Within the evolution equations, energy exchange terms arise, which relate the various forms of kinetic energy and potential energy. Key exchange terms, such as the rate at which the mean flow is converted into eddy energy, are compared across the Froude–Rossby parameter space. It is shown that the conversion terms associated with mesoscale motions are a function of the Burger number, which is consistent with a quasigeostrophic flow regime. Conversely, conversion terms associated with turbulent processes scale with the product of the Froude and Rossby number. The amount of energy extracted from the mean flows suggests that wake effects may be significant for the parameter range and model assumptions studied. These results suggest that some seamounts may indeed act as oceanic stirring rods.

Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains

2014 ◽  
Vol 757 ◽  
pp. 381-402 ◽  
Author(s):  
Hussain H. Karimi ◽  
T. R. Akylas
Keyword(s):  
Small Amplitude ◽  
Evolution Equations ◽  
Plane Waves ◽  
Inviscid Limit ◽  
Short Scale ◽  
Time Harmonic ◽  
Minimum Number ◽  
Boussinesq Fluid ◽  
Parametric Subharmonic Instability ◽  
Triad Interactions

AbstractInternal gravity wavetrains in continuously stratified fluids are generally unstable as a result of resonant triad interactions which, in the inviscid limit, amplify short-scale perturbations with frequency equal to one half of that of the underlying wave. This so-called parametric subharmonic instability (PSI) has been studied extensively for spatially and temporally monochromatic waves. Here, an asymptotic analysis of PSI for time-harmonic plane waves with locally confined spatial profile is made, in an effort to understand how such wave beams differ, in regard to PSI, from monochromatic plane waves. The discussion centres upon a system of coupled evolution equations that govern the interaction of a small-amplitude wave beam with short-scale subharmonic wavepackets in a nearly inviscid uniformly stratified Boussinesq fluid. For beams with general localized profile, it is found that triad interactions are not strong enough to bring about instability in the limited time that subharmonic perturbations overlap with the beam. On the other hand, for quasi-monochromatic wave beams whose profile comprises a sinusoidal carrier modulated by a locally confined envelope, PSI is possible if the beam is wide enough. In this instance, a stability criterion is proposed which, under given flow conditions, provides the minimum number of carrier wavelengths a beam of small amplitude must comprise for instability to arise.

scholarly journals A Closure for Internal Wave–Mean Flow Interaction. Part I: Energy Conversion

2017 ◽  
Vol 47 (6) ◽  
pp. 1389-1401 ◽  
Author(s):  
Dirk Olbers ◽  
Carsten Eden
Keyword(s):  
Radiative Transfer ◽  
Internal Wave ◽  
Ocean Circulation ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Exchange Energy ◽  
Mean Flow ◽  
Flow Interaction ◽  
The Mean ◽  
Internal Inertia

AbstractWhen internal (inertia-)gravity waves propagate in a vertically sheared geostrophic (eddying or mean) flow, they exchange energy with the flow. A novel concept parameterizing internal wave–mean flow interaction in ocean circulation models is demonstrated, based on the description of the entire wave field by the wave-energy density in physical and wavenumber space and its prognostic computation by the radiative transfer equation. The concept enables a simplification of the radiative transfer equation with a small number of reasonable assumptions and a derivation of simple but consistent parameterizations in terms of spectrally integrated energy compartments that are used as prognostic model variables. The effect of the waves on the mean flow in this paradigm is in accordance with the nonacceleration theorem: only in the presence of dissipation do waves globally exchange energy with the mean flow in the time mean. The exchange can have either direction. These basic features of wave–mean flow interaction are theoretically derived in a Wentzel–Kramers–Brillouin (WKB) approximation of the wave dynamics and confirmed in a suite of numerical experiments with unidirectional shear flow.

On three-dimensional internal gravity wave beams and induced large-scale mean flows

2015 ◽  
Vol 769 ◽  
pp. 621-634 ◽  
Author(s):  
T. Kataoka ◽  
T. R. Akylas
Keyword(s):  
Gravity Wave ◽  
Large Scale ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Internal Gravity Wave ◽  
Nonlinear Effects ◽  
Beam Width ◽  
Finite Amplitude ◽  
Horizontal Line ◽  
Mean Flow

The three-dimensional propagation of internal gravity wave beams in a uniformly stratified Boussinesq fluid is discussed, assuming that variations in the along-beam and transverse directions are of long length scale relative to the beam width. This situation applies, for instance, to the far-field behaviour of a wave beam generated by a horizontal line source with weak transverse dependence. In contrast to the two-dimensional case of purely along-beam variations, where nonlinear effects are minor even for beams of finite amplitude, three-dimensional nonlinear interactions trigger the transfer of energy to a circulating horizontal time-mean flow. This resonant beam–mean-flow coupling is analysed, and a system of two evolution equations is derived for the propagation of a small-amplitude beam along with the induced mean flow. This model explains the salient features of the experimental observations of Bordes et al. (Phys. Fluids, vol. 24, 2012, 086602).

Anelastic Internal Wave Packet Evolution and Stability

2011 ◽  
Vol 68 (12) ◽  
pp. 2844-2859 ◽  
Author(s):  
Hayley V. Dosser ◽  
Bruce R. Sutherland
Keyword(s):  
Nonlinear Dynamics ◽  
Growth Rate ◽  
Wave Packet ◽  
Internal Wave ◽  
Internal Gravity Wave ◽  
Nonlinear Effects ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Stability And Instability ◽  
Anelastic Equations

Abstract As upward-propagating anelastic internal gravity wave packets grow in amplitude, nonlinear effects develop as a result of interactions with the horizontal mean flow that they induce. This qualitatively alters the structure of the wave packet. The weakly nonlinear dynamics are well captured by the nonlinear Schrödinger equation, which is derived here for anelastic waves. In particular, this predicts that strongly nonhydrostatic waves are modulationally unstable and so the wave packet narrows and grows more rapidly in amplitude than the exponential anelastic growth rate. More hydrostatic waves are modulationally stable and so their amplitude grows less rapidly. The marginal case between stability and instability occurs for waves propagating at the fastest vertical group velocity. Extrapolating these results to waves propagating to higher altitudes (hence attaining larger amplitudes), it is anticipated that modulationally unstable waves should break at lower altitudes and modulationally stable waves should break at higher altitudes than predicted by linear theory. This prediction is borne out by fully nonlinear numerical simulations of the anelastic equations. A range of simulations is performed to quantify where overturning actually occurs.

Numerical study of internal wave–caustic and internal wave–shear interactions in a stratified fluid

2000 ◽  
Vol 415 ◽  
pp. 89-116 ◽  
Author(s):  
A. JAVAM ◽  
J. IMBERGER ◽  
S. W. ARMFIELD
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Turning Point ◽  
Critical Level ◽  
Numerical Study ◽  
Shear Instability ◽  
Level Energy ◽  
Mean Flow ◽  
Reflected Waves ◽  
The Mean

The behaviour of internal waves at a caustic level, turning point and critical layer have been investigated numerically. At a caustic reflection, a triad interaction was formed within the reflection region and the internal wave energy was transferred to lower frequencies (subharmonics). This resulted in a local subharmonic instability. One of the excited internal waves penetrated the caustic level and propagated downwards. This downward propagating wave then produced a second caustic where further reflection could take place. At a turning point, nonlinear interaction between the incident and reflected waves transferred energy to higher frequencies (evanescent trapped waves) which resulted in a superharmonic instability. At the critical level, energy was transferred to the mean flow. As the degree of nonlinearity increased, more energy was found to be transferred and overturning resulted due to a shear instability.

Sign in / Sign up

Export Citation Format

Share Document