On parametric instabilities of finite-amplitude internal gravity waves

1982 ◽  
Vol 119 ◽  
pp. 367-377 ◽  
Author(s):  
J. Klostermeyer
Keyword(s):  
Gravity Wave ◽  
Internal Gravity Wave ◽  
Maximum Growth ◽  
Numerical Approach ◽  
Internal Gravity Waves ◽  
Resonant Interaction ◽  
Finite Amplitude ◽  
Interaction Diagram ◽  
Parametric Instabilities ◽  
Boussinesq Fluid

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.

Nonlinear aspects of an internal gravity wave co-existing with an unstable mode associated with a Helmholtz velocity profile

1976 ◽  
Vol 76 (1) ◽  
pp. 65-84 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Wave ◽  
Unstable Mode ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Linear Stability Theory ◽  
Jump Discontinuity ◽  
Weakly Nonlinear ◽  
Boussinesq Fluid

Recently Lindzen (1974) has proposed a model of a shear-layer instability which allows unstable modes to co-exist with radiating internal gravity waves. The model is an infinite, continuously stratified, Boussinesq fluid, with a simple jump discontinuity in the velocity profile. Linear stability theory shows that the model is stable for wavenumbers k such that k2 < N2/2U2, where N is the Brunt—Väisälä frequency and 2U is the change in velocity across the discontinuity. For N2/2U2 < k2 < N2/U2 an unstable mode may co-exist with an internal gravity wave. This paper examines the weakly nonlinear aspects of this model for wavenumbers k close to the critical wavenumber N/2½U. An equation governing the evolution of the amplitude of the interfacial displacement is derived. It is shown that the interface may support a stable finite amplitude internal gravity wave.

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 Nonlinear interactions among standing surface and internal gravity waves

1974 ◽  
Vol 63 (4) ◽  
pp. 801-825 ◽  
Author(s):  
Terrence M. Joyce
Keyword(s):  
Energy Transfer ◽  
Steady State ◽  
Internal Wave ◽  
Surface Waves ◽  
Viscous Dissipation ◽  
Growth Rates ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Resonant Interaction ◽  
Initial Growth

A laboratory study has been undertaken to measure the energy transfer from two surface waves to one internal gravity wave in a nonlinear, resonant interaction. The interacting waves form triads for which \[ \sigma_{1s} - \sigma_{2s} \pm\sigma_1 = 0\quad {\rm and}\quad \kappa_{1s} - \kappa_2s} \pm \kappa_I = 0; \] σj and κj being the frequency and wavenumber of the jth wave. Unlike previously published results involving single triplets of interacting waves, all waves here considered are standing waves. For both a diffuse, two-layer density field and a linearly increasing density with depth, the growth to steady state of a resonant internal wave is observed while two deep water surface eigen-modes are simultaneously forced by a paddle. Internal-wave amplitudes, phases and initial growth rates are compared with theoretical results derived assuming an arbitrary Boussinesq stratification, viscous dissipation and slight detuning of the internal wave. Inclusion of viscous dissipation and slight detuning permit predictions of steady-state amplitudes and phases as well as initial growth rates. Satisfactory agreement is found between predicted and measured amplitudes and phases. Results also suggest that the internal wave in a resonant triad can act as a catalyst, permitting appreciable energy transfer among surface waves.

scholarly journals Analysis of internal gravity waves with GPS RO density profiles

2014 ◽  
Vol 7 (12) ◽  
pp. 4123-4132 ◽  
Author(s):  
P. Šácha ◽  
U. Foelsche ◽  
P. Pišoft
Keyword(s):  
Gravity Wave ◽  
Gravity Waves ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Atmospheric Density ◽  
Retrieval Process ◽  
Density Profiles ◽  
Height Range ◽  
Upper Troposphere ◽  
The Past

Abstract. GPS radio occultation (RO) data have proved to be a great tool for atmospheric monitoring and studies. In the past decade, they were frequently used for analyses of the internal gravity waves in the upper troposphere and lower stratosphere region. Atmospheric density is the first quantity of state gained in the retrieval process and is not burdened by additional assumptions. However, there are no studies elaborating in detail the utilization of GPS RO density profiles for gravity wave analyses. In this paper, we introduce a method for density background separation and a methodology for internal gravity wave analysis using the density profiles. Various background choices are discussed and the correspondence between analytical forms of the density and temperature background profiles is examined. In the stratosphere, a comparison between the power spectrum of normalized density and normalized dry temperature fluctuations confirms the suitability of the density profiles' utilization. In the height range of 8–40 km, results of the continuous wavelet transform are presented and discussed. Finally, the limits of our approach are discussed and the advantages of the density usage are listed.

An explicit potential-vorticity-conserving approach to modelling nonlinear internal gravity waves

2002 ◽  
Vol 458 ◽  
pp. 75-101 ◽  
Author(s):  
ÁLVARO VIÚDEZ ◽  
DAVID G. DRITSCHEL
Keyword(s):  
Gravity Waves ◽  
Potential Vorticity ◽  
Three Dimensional ◽  
Internal Gravity Wave ◽  
Numerical Approach ◽  
Internal Gravity Waves ◽  
Static Stability ◽  
Three Dimensions ◽  
Fluid Particles ◽  
Monge Ampere

This paper discusses a potential-vorticity-conserving approach to modelling nonlinear internal gravity waves in a rotating Boussinesq fluid. The focus of the work is on the pseudo-plane motion (motion in the x, z-plane), for which we present a broad range of numerical results. In this case there are two material coordinates, the density and the y-component of the velocity in the inertial frame of reference, which are related to the x and z displacements of fluid particles relative to a reference configuration. The amount of potential vorticity within a fluid region bounded by isosurfaces of these material coordinates is proportional to the area within this region, and is therefore conserved as well. Two new potentials, defined in terms of the displacements and combining the vorticity and density fields, are introduced as new dependent variables. These potentials entirely govern the dynamics of internal gravity waves for the linearized system when the basic state has uniform potential vorticity. The final system of equations consists of three prognostic equations (for the potential vorticity and the Laplacians of the two potentials) and one diagnostic equation, of Monge–Ampère type, for a third potential. This diagnostic equation arises from the nonlinear definition of potential vorticity. The ellipticity of the Monge–Ampère equation implies both inertial and static stability. In three dimensions, the three potentials form a vector, whose (three-dimensional) Laplacian is equal to the vorticity plus the gradient of the perturbation density.Numerical simulations are carried out using a novel algorithm which directly evolves the potential vorticity, in a Lagrangian manner (following fluid particles), without diffusion. We present results which emphasize the way in which potential vorticity anomalies modify the characteristics of internal gravity waves, e.g. the propagation of internal wave packets, including reflection, refraction, and amplification. We also show how potential vorticity anomalies may generate internal gravity waves, along with the subsequent ‘geostrophic adjustment’ of the flow to a ‘balanced’ wave-less state. These examples, and the straightforward extension of the theoretical and numerical approach to three dimensions, point to a direct and accurate means to elucidate the role of potential vorticity in internal gravity wave interactions. As such, this approach may help a better understanding of the observed characteristics of internal gravity waves in the oceans.

scholarly journals Optimal perturbation growth on a breaking internal gravity wave

2021 ◽  
Vol 925 ◽  
Author(s):  
J.P. Parker ◽  
C.J. Howland ◽  
C.P. Caulfield ◽  
R.R. Kerswell
Keyword(s):  
Shear Flow ◽  
Gravity Wave ◽  
Three Dimensional ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Mixing Efficiency ◽  
Two Dimensional ◽  
Systematic Analysis ◽  
Optimal Perturbation ◽  
Energy Growth

The breaking of internal gravity waves in the abyssal ocean is thought to be responsible for much of the mixing necessary to close oceanic buoyancy budgets. The exact mechanism by which these waves break down into turbulence remains an active area of research and can have significant implications on the mixing efficiency. Recent evidence has suggested that both shear instabilities and convective instabilities play a significant role in the breaking of an internal gravity wave in a high Richardson number mean shear flow. We perform a systematic analysis of the stability of a configuration of an internal gravity wave superimposed on a background shear flow first considered by Howland et al. (J. Fluid Mech., vol. 921, 2021, A24), using direct–adjoint looping to find the perturbation giving maximal energy growth on this evolving flow. We find that three-dimensional, convective mechanisms produce greater energy growth than their two-dimensional counterparts. In particular, we find close agreement with the direct numerical simulations of Howland et al. (J. Fluid Mech., 2021, in press), which demonstrated a clear three-dimensional mechanism causing breakdown to turbulence. The results are shown to hold at realistic Prandtl numbers. At low mean Richardson numbers, two-dimensional, shear-driven mechanisms produce greater energy growth.

Spatial and temporal analysis of high-frequency internal gravity wave signatures in brightness temperature satellite images

2021 ◽  
Author(s):  
Robert Vicari
Keyword(s):  
Water Vapor ◽  
Brightness Temperature ◽  
Gravity Wave ◽  
Gravity Waves ◽  
High Frequency ◽  
Satellite Images ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Small Scale ◽  
Wave Patterns

&lt;p&gt;Highly idealized model studies suggest that convectively generated internal gravity waves in the troposphere with horizontal wavelengths on the order of a few kilometers may affect the lifetime, spacing, and depth of clouds and convection. To answer whether such a convection-wave coupling occurs in the real atmosphere, one needs to find corresponding events in observations. In general, the study of high-frequency internal gravity wave-related phenomena in the troposphere is a challenging task because they are usually small-scale and intermittent. To overcome case-by-case studies, it is desirable to have an automatic method to analyze as much data as possible and provide enough independent and diverse evidence.&lt;br&gt;Here, we focus on brightness temperature satellite images, in particular so-called satellite water vapor channels. These channels measure the radiation at wavelengths corresponding to the energy emitted by water vapor and provide cloud-independent observations of internal gravity waves, in contrast to visible and other infrared satellite channels where one relies on the wave impacts on clouds. In addition, since these water vapor channels are sensitive to certain vertical layers in the troposphere, combining the images also reveals some vertical structure of the observed waves.&lt;br&gt;We propose an algorithm based on local Fourier analyses to extract information about high-frequency wave patterns in given brightness temperature images. This method allows automatic detection and analysis of many wave patterns in a given domain at once, resulting in a climatology that provides an initial observational basis for further research. Using data from the instrument ABI on board the satellite GOES-16 during the field campaign EUREC&lt;sup&gt;4&lt;/sup&gt;A, we demonstrate the capabilities and limitations of the method. Furthermore, we present the respective climatology of the detected waves and discuss approaches based on this to address the initial question.&lt;/p&gt;

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

&lt;p&gt;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.&lt;/p&gt;&lt;p&gt;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.&lt;/p&gt;&lt;p&gt;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.&lt;/p&gt;

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