Wentzel–Kramers–Brillouin approximation for atmospheric waves

2015 ◽  
Vol 777 ◽  
pp. 260-290 ◽  
Author(s):  
Oleg A. Godin
Keyword(s):  
Gravity Waves ◽  
Acoustic Waves ◽  
Ad Hoc ◽  
Berry Phase ◽  
Boussinesq Approximation ◽  
Internal Gravity Waves ◽  
Wkb Approximation ◽  
Approximation Properties ◽  
Atmospheric Waves ◽  
Acoustic Gravity Waves

Ray and Wentzel–Kramers–Brillouin (WKB) approximations have long been important tools in understanding and modelling propagation of atmospheric waves. However, contradictory claims regarding the applicability and uniqueness of the WKB approximation persist in the literature. Here, we consider linear acoustic–gravity waves (AGWs) in a layered atmosphere with horizontal winds. A self-consistent version of the WKB approximation is systematically derived from first principles and compared to ad hoc approximations proposed earlier. The parameters of the problem are identified that need to be small to ensure the validity of the WKB approximation. Properties of low-order WKB approximations are discussed in some detail. Contrary to the better-studied cases of acoustic waves and internal gravity waves in the Boussinesq approximation, the WKB solution contains the geometric, or Berry, phase. The Berry phase is generally non-negligible for AGWs in a moving atmosphere. In other words, knowledge of the AGW dispersion relation is not sufficient for calculation of the wave phase.

On the reflection of internal gravity waves

1970 ◽  
Vol 48 (15) ◽  
pp. 1830-1834
Author(s):  
J. D. Whitehead
Keyword(s):  
Phase Change ◽  
Gravity Waves ◽  
Relative Phase ◽  
Internal Gravity Waves ◽  
Acoustic Gravity Waves ◽  
Wave Parameters ◽  
Normal Propagation

Recently Einaudi and Hines discussed the fact that different wave parameters of acoustic–gravity waves may have different reflection heights. This raises the problem that there will be a range of heights over which one parameter is evanescent, whereas another parameter is in its normal propagation region. It is not obvious that for this second parameter the range of heights does not encompass many wavelengths and thus there would be a large relative phase change between the two parameters.However, it is shown that the extent of the relative phase change will often be less than 2π, and the conditions under which it may exceed 2π are derived.

Reflection properties of internal gravity waves incident upon a hyperbolic tangent shear layer

1982 ◽  
Vol 120 ◽  
pp. 505-521 ◽  
Author(s):  
Cornelis A. Van Duin ◽  
Hennie Kelder
Keyword(s):  
Shear Layer ◽  
Gravity Waves ◽  
Critical Level ◽  
Boussinesq Approximation ◽  
Internal Gravity Waves ◽  
Hyperbolic Tangent ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Heun’S Equation ◽  
Stable Flow

The properties of reflection and transmission of internal gravity waves incident upon a shear layer containing a critical level are investigated. The shear layer is modelled by a hyperbolic tangent profile. In the Boussinesq approximation, the differential equation governing the propagation of these waves can then be transformed into Heun's equation. For large Richardson numbers this equation can be approximated by an equation that has solutions in terms of hypergeometric functions. For these values of the Richardson number the reflection coefficient proves to be strongly dependent on the place of the critical level in the shear flow. If the Doppler-shifted frequency is an odd function of the height difference with respect to the critical level, the reflection and transmission coefficients can be evaluated in closed form.Over-reflection is possible for sufficiently small wavenumbers and Richardson numbers. It is pointed out that over-reflection and over-transmission cannot occur in a stable flow and that resonant over-reflection is not possible in our model.

WKB approximation in application to acoustic–gravity waves

1970 ◽  
Vol 48 (12) ◽  
pp. 1458-1471 ◽  
Author(s):  
F. Einaudi ◽  
C. O. Hines
Keyword(s):  
Wave Propagation ◽  
Gravity Waves ◽  
Wkb Approximation ◽  
Acoustic Gravity Waves ◽  
Background Temperature

The WKB approximation often provides a useful tool for the study of wave propagation through varying media. The utility of this tool in the case of acoustic–gravity waves, propagating through regions of varying background temperature, has been somewhat questionable because of an ambiguity in the assignment of an appropriate expression for the vertical wavenumber. The ambiguity is examined here, and shown to be inconsequential except in circumstances such that the WKB approximation is itself inapplicable; a unique choice of expression for the vertical wavenumber is recommended on the basis of simplicity and. utility. Conclusions reached in a number of recent papers are reexamined in the light of the insight that has been gained.

scholarly journals Time-Dependent Propagation of Tsunami-Generated Acoustic–Gravity Waves in the Atmosphere

2020 ◽  
Vol 77 (4) ◽  
pp. 1233-1244 ◽  
Author(s):  
Yue Wu ◽  
Stefan G. Llewellyn Smith ◽  
James W. Rottman ◽  
Dave Broutman ◽  
Jean-Bernard H. Minster
Keyword(s):  
Internal Waves ◽  
Gravity Waves ◽  
Acoustic Waves ◽  
Early Stage ◽  
Vertical Displacement ◽  
Inverse Laplace Transform ◽  
Acoustic Gravity Waves ◽  
Numerical Inverse Laplace Transform ◽  
Wave Development

Abstract Tsunami-generated linear acoustic–gravity waves in the atmosphere with altitude-dependent vertical stratification and horizontal background winds are studied with the long-term goal of real-time tsunami warning. The initial-value problem is examined using Fourier–Laplace transforms to investigate the time dependence and to compare the cases of anelastic and compressible atmospheres. The approach includes formulating the linear propagation of acoustic–gravity waves in the vertical, solving the vertical displacement of waves and pressure perturbations numerically as a set of coupled ODEs in the Fourier–Laplace domain, and employing den Iseger’s algorithm to carry out a fast and accurate numerical inverse Laplace transform. Results are presented for three cases with different atmospheric and tsunami profiles. Horizontal background winds enhance wave advection in the horizontal but hinder the vertical transmission of internal waves through the whole atmosphere. The effect of compressibility is significant. The rescaled vertical displacement of internal waves at 100-km altitude shows an arrival at the early stage of wave development due to the acoustic branch that is not present in the anelastic case. The long-term displacement also shows an O(1) difference between the compressible and anelastic results for the cases with uniform and realistic stratification. Compressibility hence affects both the speed and amplitude of energy transmitted to the upper atmosphere because of fast acoustic waves.

A Model of Rayleigh-Bénard Convection

2013 ◽  
Author(s):  
Gary A. Glatzmaier
Keyword(s):  
Gravity Waves ◽  
Thermal Convection ◽  
Fluid Flows ◽  
Boussinesq Approximation ◽  
Internal Gravity Waves ◽  
Bénard Convection ◽  
Benard Convection ◽  
Key Characteristics ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, focusing on thermal convection and internal gravity waves. Thermal convection and internal gravity waves are the two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces. The chapter then reviews the equations that govern fluid dynamics based on conservation of mass, momentum, and energy. It also examines the conditions under which the Boussinesq approximation simplifies conservation equations to a form very similar to that of an incompressible fluid. Finally, it discusses the key characteristics of the model of Rayleigh–Bénard convection.

Gravity waves as a mechanism of coupling oceanic and atmospheric acoustic waveguides to seismic sources

2020 ◽  
Author(s):  
Oleg Godin
Keyword(s):  
Gravity Waves ◽  
Acoustic Waves ◽  
Seismic Waves ◽  
Wind Waves ◽  
Normal Modes ◽  
Internal Gravity Waves ◽  
Body Waves ◽  
Surface Scattering ◽  
Seismic Sources ◽  
T Waves

<p>Direct excitation of acoustic normal modes in horizontally stratified oceanic waveguides is negligible even for shallow earthquakes because of the disparity between velocities of seismic waves and the sound speed in the water column. T-phases, which propagate at the speed of sound in water, are often reported to originate in the open ocean in the vicinity of the epicenter of an underwater earthquake, even in the absence of prominent bathymetric features or significant seafloor roughness. This paper aims to evaluate the contribution of scattering by hydrodynamic waves into generation of abyssal T-waves. Ocean is modeled as a range-independent waveguide with superimposed volume inhomogeneities due to internal gravity waves and surface roughness due to wind waves and sea swell. Guided acoustic waves are excited by volume and surface scattering of ballistic body waves. The surface scattering mechanism is shown to explain key observational features of abyssal T-waves, including their ubiquity, low-frequency cutoff, presence on seafloor sensors, and weak dependence on the earthquake focus depth. On the other hand, volume scattering due to internal gravity waves proves to be ineffective in coupling the seismic sources to T-waves. The theory is extended to explore a possible role that scattering by gravity waves may play in excitation of infrasonic normal modes of tropospheric and stratospheric waveguides by underwater earthquakes. Model predictions are compared to observations [L. G. Evers, D. Brown, K. D. Heaney, J. D. Assink, P. S. M. Smets, and M. Snellen (2014), Geophys. Res. Lett., <strong>41</strong>, 1644–1650] of infrasonic signals generated by the 2004 Macquarie Ridge earthquake.</p>

Modeling the propagation of atmospheric waves from various tropospheric disturbances and studying their influence on the upper atmosphere

2020 ◽  
Author(s):  
Yuliya Kurdyaeva ◽  
Olga Borchevkina ◽  
Sergey Kshevetskii
Keyword(s):  
Gravity Waves ◽  
Coastal Region ◽  
Internal Gravity Waves ◽  
Travelling Wave ◽  
The State ◽  
Upper Atmosphere ◽  
Small Scale ◽  
Research Project ◽  
Atmospheric Waves ◽  
The Earth

<p>The atmosphere and ionosphere are a complex dynamic system, which is affected by sources, caused both by internal processes and external ones. It is known that atmospheric waves propagating from the troposphere to the upper atmosphere make a significant contribution to the state of this system. One of the regular sources of such waves are various tropospheric disturbances caused, for example, by meteorological processes. Numerical modeling is an effective tool for studying these processes and the effects they cause. However, a number of problems arise, while setting up numerical experiments. The first is that most atmospheric models use hydrostatic approximation (which does not allow the resolution of small-scale perturbations) and work for a limited range of heights (which does not allow studying the relationship between the lower and upper atmosphere). This demands an accurate selection of the model in accordance with the stated research goals. The second problem is the difficulty of direct definition of the wave tropospheric sources, that was mentioned before, due to the lack of experimental information for their detailed description. The authors proposed, researched and tested a way to solve this problem. It was shown that the solution of the problem of waves propagation from a certain tropospheric source is completely determined by the pressure field at the surface of the Earth. This work is devoted to solving various problems using this approach.</p><p>This study presents the results of calculations of the propagation of infrasound and internal gravity waves from tropospheric disturbances given by pressure variations at the surface of the Earth. The experimental data associated with various meteorological events and the passage of the solar terminator were obtained both directly - by a network of microbarographs in the studied region, and indirectly - based on the data from the LIDAR signal intensity and temperature changes in the coastal region. The calculations were done using the non-hydrostatic numerical model “AtmoSym”. The characteristics of atmospheric waves generated by such sources are estimated. The effect from a tropospheric sources on the state of the upper atmosphere and ionosphere is investigated. The physical processes that determine the change in atmospheric parameters are discussed.  It is shown that the main contribution from wave disturbances generated by meteorological sources belongs to infrasound. Infrasound and internal gravity waves can be sources of travelling wave packets and can also cause a sporadic E-layer.</p><p>The study was funded by RFBR and Kaliningrad region according to the research project  19-45-390005 (Y. Kurdyaeva) and  RFBR to the research project  18-05-00184 (O. Borchevkina).</p>

On over-reflexion

1976 ◽  
Vol 77 (3) ◽  
pp. 433-472 ◽  
Author(s):  
D. J. Acheson
Keyword(s):  
Gravity Waves ◽  
Incident Wave ◽  
Compressible Fluid ◽  
Acoustic Waves ◽  
Vortex Sheet ◽  
Internal Gravity Waves ◽  
Negative Energy ◽  
Helmholtz Instability ◽  
The Mean ◽  
Parameter Values

Reflexion coefficients greater than unity have now been predicted for a variety of different systems involving waves propagating towards a shear layer, but almost invariably only in regions of parameter space for which the layer exhibits Kelvin-Helmholtz instability. This paper contains a study of two examples in which, for appropriate parameter values, there are no such instabilities to obscure (or even prevent) the ‘over-reflexion’ of an incident wave, namely(a)hydro-magnetic internal gravity waves meeting a vortex-current sheet in a stratified fluid and(b)magneto-acoustic waves meeting a vortex sheet in a compressible fluid. In the former case the energetic aspects of over-reflexion are examined in detail, thus displaying the way in which the excess reflected energy is extracted from the mean motion and the sense in which the transmitted wave may be viewed, by analogy with certain concepts employed in plasma physics, as a carrier of so-called ‘negative energy’.

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