An asymptotic solution of the tidal equations

1967 ◽  
Vol 30 (3) ◽  
pp. 417-438 ◽  
Author(s):  
Stanley J. Jacobs
Keyword(s):  
Gravity Waves ◽  
Delta Function ◽  
Boundary Layer Theory ◽  
Geostrophic Adjustment ◽  
Initial Development ◽  
Local Boundary ◽  
Geostrophic Balance ◽  
The Earth ◽  
The Cauchy Problem ◽  
Plane Form

The Cauchy problem for the β-plane form of the tidal equations is solved for both oscillatory and delta function initial data. The radius of deformation is assumed to be much less than the radius of the earth, and in accord with this assumption a ray approximation is employed.It is shown that, owing to the rapid rate of propagation of inertio-gravity waves, the motion in its initial development tends towards geostrophic balance. However, the solution given by the ray approximation is singular on certain surfaces in space and time, the envelopes of the rays. A local boundary-layer theory is employed to correct this deficiency. The existence of these caustics implies that the process of geostrophic adjustment is more complicated than hitherto imagined.

scholarly journals Solitons in the Close Binary Systems

1998 ◽  
Vol 11 (1) ◽  
pp. 398-398
Author(s):  
Kenji Tanabe
Keyword(s):  
Gravity Waves ◽  
Binary Systems ◽  
Kdv Equation ◽  
Frame Of Reference ◽  
Close Binary ◽  
Flare Activity ◽  
Lagrangian Point ◽  
Nls Equation ◽  
Close Binary Systems ◽  
The Earth

Propagation of the surface waves of the lobe-filing components of close binary systems is investigated theoretically. Such waves are considered to be analogous to the gravity waves of water on the earth. As a result, the equations of the surface wave in the rotating frame of reference are reduced to the so-called Kortewegde Vries (KdV) equation and non-linear Schroedinger (NLS) equation according to its ”depth”. Each of these equations is known to have the solution of soliton. When this soliton is sent to the other component of the binary system through the Lagrangian point, it can give rise to the flare activity observed in some kinds of close binary systems.

Extreme vertical drafts in the polar summer mesosphere: A super mesospheric bore?

2021 ◽  
Author(s):  
Jorge Luis Chau ◽  
Raffaele Marino ◽  
Fabio Feraco ◽  
Juan M. Urco ◽  
Gerd Baumgarten ◽  
...  
Keyword(s):  
Gravity Waves ◽  
Extreme Event ◽  
Ice Clouds ◽  
Wind Variability ◽  
Near Space ◽  
The Earth ◽  
Radar Echoes ◽  
Gaussian Statistics ◽  
Non Gaussian ◽  
Impact Planning

<p>The polar summer mesosphere is the Earth’s coldest region, allowing the formation of mesospheric ice clouds, potentially linked to climate change. These clouds produce strong radar echoes that are used as tracers of mesospheric dynamics. Here we report the first observations of extreme vertical drafts in the mesosphere, characterized by velocities larger than 40 m/s, i.e., more than five standard deviations larger than the observed wind variability. The morphology seems to resemble mesospheric bores, however the scales observed are much larger. Powerful vertical drafts, intermittent in space and time, emerge also in direct numerical simulations of stratified flows, predicting non-Gaussian statistics of vertical velocities. This evidence suggests that mesospheric bores might result from the interplay of gravity waves and turbulent motions. Our extreme event is interpreted as a mesospheric "super-bore", impacting mesospheric mixing and ice-formation, and would potentially impact planning of sub-orbital flights, and the investigation of biological material in the near space.</p>

scholarly journals Blow-Up Criterion of Weak Solutions for the 3D Boussinesq Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Zhaohui Dai ◽  
Xiaosong Wang ◽  
Lingrui Zhang ◽  
Wei Hou
Keyword(s):  
Weak Solutions ◽  
Blow Up ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Interpolation Inequality ◽  
Embedding Theorem ◽  
The Earth ◽  
The Cauchy Problem ◽  
Homogeneous Besov Space ◽  
Blow Up Criterion

The Boussinesq equations describe the three-dimensional incompressible fluid moving under the gravity and the earth rotation which come from atmospheric or oceanographic turbulence where rotation and stratification play an important role. In this paper, we investigate the Cauchy problem of the three-dimensional incompressible Boussinesq equations. By commutator estimate, some interpolation inequality, and embedding theorem, we establish a blow-up criterion of weak solutions in terms of the pressurepin the homogeneous Besov spaceḂ∞,∞0.

Double reflection of capillary/gravity waves by a non-uniform current: a boundary-layer theory

1993 ◽  
Vol 251 ◽  
pp. 239-271 ◽  
Author(s):  
Karsten Trulsen ◽  
Chiang C. Mei
Keyword(s):  
Boundary Layer ◽  
Gravity Waves ◽  
Boundary Layer Theory ◽  
Boundary Layer Approximation ◽  
Uniform Current ◽  
The Waves

When a train of gravity waves encounters an opposing current, the wavelength is shortened and the waves may be reflected. If capillarity is included, the shortened waves may be reflected for a second time and experience further shortening. By this process the initially long gravity waves can be damped by viscosity quickly without breaking. In this paper a boundary-layer approximation is obtained close to the reflection points, and is matched to the ray approximations outside. This is then applied to the propagation of a wavepacket. Damping is accounted for in the ray solution and the result is compared to the undamped solution. The case where the two reflection points coalesce is also considered. It is found that as the separation between the reflection points decreases, the wavepacket appears to remain longer in the region of reflections relative to the width of this region.

scholarly journals Convergence of eigenfunction expansions for flexural gravity waves in infinite water depth

2021 ◽  
Vol 2070 (1) ◽  
pp. 012006
Author(s):  
Santanu Koley ◽  
Kottala Panduranga
Keyword(s):  
Gravity Waves ◽  
Water Depth ◽  
Water Waves ◽  
Eigenfunction Expansion ◽  
Velocity Potential ◽  
Wave Theory ◽  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Flexural Gravity Waves

Abstract In the present paper, point-wise convergence of the eigenfunction expansion to the velocity potential associated with the flexural gravity waves problem in water wave theory is established for infinite water depth case. To take into account the hydroelastic boundary condition at the free surface, a flexible membrane is assumed to float in water waves. In this context, firstly the eigenfunction expansion for the velocity potentials is obtained. Thereafter, an appropriate Green’s function is constructed for the associated boundary value problem. Using suitable properties of the Green’s functions, the vertical components of the eigenfunction expansion is written in terms of the Dirac delta function. Finally, using the property of the Dirac delta function, the convergence of the eigenfunction expansion to the velocity potential is shown.

scholarly journals Identification of the sources of inertia-gravity waves in the Andes Cordillera region

2008 ◽  
Vol 26 (9) ◽  
pp. 2551-2568 ◽  
Author(s):  
A. Spiga ◽  
H. Teitelbaum ◽  
V. Zeitlin
Keyword(s):  
Gravity Waves ◽  
Mesoscale Model ◽  
Jet Stream ◽  
Earth Atmosphere ◽  
Jet Streams ◽  
The Andes ◽  
Wave Parameters ◽  
The Earth ◽  
Andes Cordillera ◽  
Inertia Gravity Waves

Abstract. Four major sources of inertia-gravity waves are known in the Earth atmosphere: upper-tropospheric jet-streams, lower-tropospheric fronts, convection and topography. The Andes Cordillera region is an area where all of these major sources are potentially present. By combining ECMWF and NCEP-NCAR reanalysis, satellite and radiosoundings data and mesoscale WRF simulations in the Andes Cordillera region, we were able to identify the cases where, respectively, the jet-stream source, the convective source and the topography source are predominantly in action. We retrieve emitted wave parameters for each case, compare them, and analyse possible emission mechanisms. The WRF mesoscale model shows very good performance in reproducing the inertia-gravity waves identified in the data analysis, and assessing their likely sources.

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>

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

2018 ◽  
Vol 26 (6) ◽  
pp. 835-857 ◽  
Author(s):  
Dinh Nho Hào ◽  
Le Thi Thu Giang ◽  
Sergey Kabanikhin ◽  
Maxim Shishlenin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Weak Sense ◽  
Very Weak Solution ◽  
Local Boundary ◽  
The Cauchy Problem ◽  
Non Local

Abstract We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

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