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 Observations of Near-Inertial Internal Gravity Waves Radiating from a Frontal Jet

2013 ◽  
Vol 43 (6) ◽  
pp. 1225-1239 ◽  
Author(s):  
Matthew H. Alford ◽  
Andrey Y. Shcherbina ◽  
Michael C. Gregg
Keyword(s):  
Gravity Waves ◽  
Three Dimensional ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Shear Layers ◽  
Adjustment Process ◽  
Vertical Wavelength ◽  
The North ◽  
Inertial Wave ◽  
The North Pacific

Abstract Shipboard ADCP and towed CTD measurements are presented of a near-inertial internal gravity wave radiating away from a zonal jet associated with the Subtropical Front in the North Pacific. Three-dimensional spatial surveys indicate persistent alternating shear layers sloping downward and equatorward from the front. As a result, depth-integrated ageostrophic shear increases sharply equatorward of the front. The layers have a vertical wavelength of about 250 m and a slope consistent with a wave of frequency 1.01f. They extend at least 100 km south of the front. Time series confirm that the shear is associated with a downward-propagating near-inertial wave with frequency within 20% of f. A slab mixed layer model forced with shipboard and NCEP reanalysis winds suggests that wind forcing was too weak to generate the wave. Likewise, trapping of the near-inertial motions at the low-vorticity edge of the front can be ruled out because of the extension of the features well south of it. Instead, the authors suggest that the wave arises from an adjustment process of the frontal flow, which has a Rossby number about 0.2–0.3.

The evolution of a stratified turbulent cloud

2013 ◽  
Vol 739 ◽  
pp. 229-253 ◽  
Author(s):  
Andrea Maffioli ◽  
P. A. Davidson ◽  
S. B. Dalziel ◽  
N. Swaminathan
Keyword(s):  
Gravity Waves ◽  
Large Scale ◽  
Wave Packets ◽  
Three Dimensional ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Group Speed ◽  
Propagating Waves ◽  
The Subject ◽  
Energy And Momentum

AbstractLocalized regions of turbulence, or turbulent clouds, in a stratified fluid are the subject of this study, which focuses on the edge dynamics occurring between the turbulence and the surrounding quiescent region. Through laboratory experiments and numerical simulations of stratified turbulent clouds, we confirm that the edge dynamics can be subdivided into materially driven intrusions and horizontally travelling internal wave-packets. Three-dimensional visualizations show that the internal gravity wave-packets are in fact large-scale pancake structures that grow out of the turbulent cloud into the adjacent quiescent region. The wave-packets were tracked in time, and it is found that their speed obeys the group speed relation for linear internal gravity waves. The energetics of the propagating waves, which include waveforms that are inclined with respect to the horizontal, are also considered and it is found that, after a period of two eddy turnover times, the internal gravity waves carry up to 16 % of the cloud kinetic energy into the initially quiescent region. Turbulent events in nature are often in the form of decaying turbulent clouds, and it is therefore suggested that internal gravity waves radiated from an initial cloud could play a significant role in the reorganization of energy and momentum in the atmosphere and oceans.

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.

scholarly journals Solitary flexural–gravity waves in three dimensions

2018 ◽  
Vol 376 (2129) ◽  
pp. 20170345 ◽  
Author(s):  
Olga Trichtchenko ◽  
Emilian I. Părău ◽  
Jean-Marc Vanden-Broeck ◽  
Paul Milewski
Keyword(s):  
Gravity Waves ◽  
Three Dimensional ◽  
Ice Sheet ◽  
Boundary Integral ◽  
Three Dimensions ◽  
Cosserat Theory ◽  
Theme Issue ◽  
Flexural Gravity Waves ◽  
Forced Waves ◽  
Ice Phenomena

The focus of this work is on three-dimensional nonlinear flexural–gravity waves, propagating at the interface between a fluid and an ice sheet. The ice sheet is modelled using the special Cosserat theory of hyperelastic shells satisfying Kirchhoff's hypothesis, presented in (Plotnikov & Toland. 2011 Phil. Trans. R. Soc. A 369 , 2942–2956 ( doi:10.1098/rsta.2011.0104 )). The fluid is assumed inviscid and incompressible, and the flow irrotational. A numerical method based on boundary integral equation techniques is used to compute solitary waves and forced waves to Euler's equations. This article is part of the theme issue ‘Modelling of sea-ice phenomena’.

Trapped internal gravity waves in a geostrophic boundary current

1993 ◽  
Vol 247 ◽  
pp. 205-229
Author(s):  
Hong Ma
Keyword(s):  
Gravity Waves ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Combined Effects ◽  
Kelvin Wave ◽  
Boundary Current ◽  
Trapping Mechanism ◽  
Rossby Radius Of Deformation ◽  
Wave Guides ◽  
Internal Kelvin Wave

The effect of a geostrophic boundary current on internal gravity waves is studied with a reduced-gravity model. We found that the boundary current not only modifies the coastal Kelvin wave, but also forms wave guides for short internal gravity waves. The combined effects of current shear, the boundary, and the slope of the interface create the trapping mechanism. These trapped internal gravity waves appear as groups of discrete zonal modes. They have wavelengths comparable to or shorter than the internal Rossby radius of deformation. Their phase speeds are close to that of the internal Kelvin wave. However, they can propagate both in, or opposite to, the direction of the Kelvin wave. The results of the present work suggest the possibility of finding an energetic internal gravity wave phenomenon with near-inertial frequency in a broad geostrophic boundary current.

Numerical Investigation of Mechanisms Underlying Oceanic Internal Gravity Wave Power-Law Spectra

2020 ◽  
Vol 50 (9) ◽  
pp. 2713-2733
Author(s):  
Yulin Pan ◽  
Brian K. Arbic ◽  
Arin D. Nelson ◽  
Dimitris Menemenlis ◽  
W. R. Peltier ◽  
...  
Keyword(s):  
Power Law ◽  
Gravity Waves ◽  
High Frequency ◽  
Internal Gravity Wave ◽  
Numerical Data ◽  
Field Measurements ◽  
Internal Gravity Waves ◽  
Collision Integral ◽  
Turbulence Theory ◽  
Nonlinear Interactions

AbstractWe consider the power-law spectra of internal gravity waves in a rotating and stratified ocean. Field measurements have shown considerable variability of spectral slopes compared to the high-wavenumber, high-frequency portion of the Garrett–Munk (GM) spectrum. Theoretical explanations have been developed through wave turbulence theory (WTT), where different power-law solutions of the kinetic equation can be found depending on the mechanisms underlying the nonlinear interactions. Mathematically, these are reflected by the convergence properties of the so-called collision integral (CL) at low- and high-frequency limits. In this work, we study the mechanisms in the formation of the power-law spectra of internal gravity waves, utilizing numerical data from the high-resolution modeling of internal waves (HRMIW) in a region northwest of Hawaii. The model captures the power-law spectra in broad ranges of space and time scales, with scalings ω−2.05±0.2 in frequency and m−2.58±0.4 in vertical wavenumber. The latter clearly deviates from the GM76 spectrum but is closer to a family of induced-diffusion-dominated solutions predicted by WTT. Our analysis of nonlinear interactions is performed directly on these model outputs, which is fundamentally different from previous work assuming a GM76 spectrum. By applying a bicoherence analysis and evaluations of modal energy transfer, we show that the CL is dominated by nonlocal interactions between modes in the power-law range and low-frequency inertial motions. We further identify induced diffusion and the near-resonances at its spectral vicinity as dominating the formation of power-law spectrum.

The piecewise constant symmetric potential vorticity vortex in geophysical flows

2008 ◽  
Vol 614 ◽  
pp. 145-172 ◽  
Author(s):  
ÁLVARO VIÚDEZ
Keyword(s):  
Potential Vorticity ◽  
Three Dimensional ◽  
Static Stability ◽  
Geophysical Flows ◽  
Radial Dependence ◽  
Radial Solutions ◽  
Finite Radius ◽  
Piecewise Constant ◽  
Order Of Magnitude ◽  
Horizontal Cylinders

The concept of piecewise constant symmetric vortex in the context of three-dimensional baroclinic balanced geophysical flows is explored. The pressure gradients generated by horizontal cylinders and spherical balls of uniform potential vorticity (PV), or uniform material invariants, are obtained either analytically or numerically, in the general case of Boussinesq and f-plane dynamics as well as under the quasi-geostrophic and semigeostrophic dynamical approximations. Based on the order of magnitude of the different terms in the PV inversion equation, approximated PV equations are deduced. In some of these cases, radial solutions are possible and the interior and exterior solutions are found analytically. In the case of non-radial dependence, exterior solutions can be found numerically. Linear, and upper and lower bound approximations to the full PV inversion equations, and their respective solutions, are also included. However, the general solution for the pressure gradient in the vortex exterior does not have spherical symmetry and remains as an important theoretical challenge. It is suggested that, in order to maintain everywhere the inertial and static stability of the balanced geophysical flows, small balls of finite radius, rather than PV singularities, could become, specially in numerical applications, useful mathematical objects.

scholarly journals Momentum Flux Spectrum of Convectively Forced Internal Gravity Waves and Its Application to Gravity Wave Drag Parameterization. Part I: Theory

2005 ◽  
Vol 62 (1) ◽  
pp. 107-124 ◽  
Author(s):  
In-Sun Song ◽  
Hye-Yeong Chun
Keyword(s):  
Gravity Waves ◽  
Momentum Flux ◽  
Three Dimensional ◽  
Internal Gravity Waves ◽  
Buoyancy Frequency ◽  
Spectral Structure ◽  
Two Dimensional ◽  
Resonance Factor ◽  
Flux Spectrum ◽  
Wave Momentum

Abstract The phase-speed spectrum of momentum flux by convectively forced internal gravity waves is analytically formulated in two- and three-dimensional frameworks. For this, a three-layer atmosphere that has a constant vertical wind shear in the lowest layer, a uniform wind above, and piecewise constant buoyancy frequency in a forcing region and above is considered. The wave momentum flux at cloud top is determined by the spectral combination of a wave-filtering and resonance factor and diabatic forcing. The wave-filtering and resonance factor that is determined by the basic-state wind and stability and the vertical configuration of forcing restricts the effectiveness of the forcing, and thus only a part of the forcing spectrum can be used for generating gravity waves that propagate above cumulus clouds. The spectral distribution of the wave momentum flux is largely determined by the wave-filtering and resonance factor, but the magnitude of the momentum flux varies significantly according to spatial and time scales and moving speed of the forcing. The wave momentum flux formulation in the two-dimensional framework is extended to the three-dimensional framework. The three-dimensional momentum flux formulation is similar to the two-dimensional one except that the wave propagation in various horizontal directions and the three-dimensionality of forcing are allowed. The wave momentum flux spectrum formulated in this study is validated using mesoscale numerical model results and can reproduce the overall spectral structure and magnitude of the wave momentum flux spectra induced by numerically simulated mesoscale convective systems reasonably well.

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