Invariant and approximately invariant solutions of non-linear internal gravity waves forming a column of stratified fluid affected by the Earth's rotation

2013 ◽  
Vol 51 ◽  
pp. 28-44 ◽  
Author(s):  
Ranis Ibragimov ◽  
Grace Jefferson ◽  
John Carminati
Keyword(s):  
Gravity Waves ◽  
Internal Gravity Waves ◽  
Stratified Fluid ◽  
Earth’S Rotation ◽  
Invariant Solutions ◽  
Earth's Rotation ◽  
Non Linear

A non-linear investigation of critical levels for internal atmospheric gravity waves

1971 ◽  
Vol 50 (3) ◽  
pp. 545-563 ◽  
Author(s):  
R. J. Breeding
Keyword(s):  
Steady State ◽  
Linear Theory ◽  
Gravity Waves ◽  
Incident Wave ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Mean Flow ◽  
Non Linear ◽  
Energy And Momentum ◽  
Almost All

The behaviour of internal gravity waves near a critical level is investigated by means of a transient two dimensional finite difference model. All the important non-linear, viscosity and thermal conduction terms are included, but the rotational terms are omitted and the perturbations are assumed to be incompressible. For Richardson numbers greater than 2·0 the interaction of the incident wave and the mean flow is largely as predicted by the linear theory–very little of the incident wave penetrates through the critical level and almost all of the wave's energy and momentum are absorbed by changes in the original wind. However, these changes in the wind are centred above the critical level, so that the change in the wind has only a small effect on the height of the critical level. For Richardson numbers less than 2·0 and greater than 0·25 a significant fraction of the incident wave is reflected, part of which could have been predicted by the linear theory. For these stable Richardson numbers a steady state is apparently reached where the maximum wind change continues to grow slowly, but the minimum Richardson number and wave magnitudes remain constant. This condition represents a balance between the diffusion outward of the added momentum and the rate at which it is absorbed. For Richardson numbers less than 0·25, over-reflexion, predicted from the linear theory, is observed, but because the system is dynamically unstable no over-reflecting steady state is ever reached.

scholarly journals A Linear Theory of Three-Dimensional Land–Sea Breezes

2012 ◽  
Vol 69 (6) ◽  
pp. 1890-1909 ◽  
Author(s):  
Qingfang Jiang
Keyword(s):  
Gravity Waves ◽  
Three Dimensional ◽  
Vertical Motion ◽  
Early Morning ◽  
Earth’S Rotation ◽  
Earth's Rotation ◽  
Motion Patterns ◽  
Sea Breezes ◽  
Differential Heating ◽  
Downwind Side

Abstract Land–sea breezes (LSBs) induced by diurnal differential heating are examined using a three-dimensional linear model employing fast Fourier transform with emphasis on the complex coastline shape and geometry, the earth’s rotation, and background wind effects. It has been demonstrated that the low-level vertical motion associated with LSB can be significantly enhanced over a bay (peninsula) because of convergence of perturbations induced by differential heating along a seaward concave (convex) coastline. The dependence of surface winds and vertical motion patterns and their evolutions on the coastline geometries such as the width and the aspect ratio of the bay, the earth’s rotation, and the background winds are investigated. The LSB induced by an isolated tropical island is characterized by onshore flow and ascent over the island in the afternoon to early evening, with a reversal of direction from midnight to early morning. The diurnal heating–induced vertical motion is greatly enhanced over the island and weakened offshore because of the convergence and divergence of perturbations. In the presence of background flow, stronger diurnal perturbations are found at the downwind side of the island, which can extend far downstream associated with inertia–gravity waves.

Energy transfer between external and internal gravity waves

1964 ◽  
Vol 19 (3) ◽  
pp. 465-478 ◽  
Author(s):  
F. K. Ball
Keyword(s):  
Energy Transfer ◽  
Internal Wave ◽  
Surface Waves ◽  
Shallow Water ◽  
Gravity Waves ◽  
External Surface ◽  
Internal Gravity Waves ◽  
Liquid System ◽  
Resonant Interactions ◽  
Non Linear

In a two-layer liquid system non-linear resonant interactions between a pair of external (surface) waves can result in transfer of energy to an internal wave when appropriate resonance conditions are satisfied. This energy transfer is likely to be more powerful than similar transfers between external waves. The shallow water case is discussed in detail.

On the normal modes of parallel flow of inviscid stratified fluid

1976 ◽  
Vol 75 (1) ◽  
pp. 149-171 ◽  
Author(s):  
W. H. H. Banks ◽  
P. G. Drazin ◽  
M. B. Zaturska
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Richardson Number ◽  
Normal Modes ◽  
Internal Gravity Waves ◽  
Stratified Fluid ◽  
Parallel Flow ◽  
Marginal Stability ◽  
Mean Flow ◽  
Asymptotic Results

The overall pattern of normal modes of parallel flow of inviscid stratified fluid is examined. For a given flow and wavenumber the modes are divided into five classes, some of which may be empty: (i) a finite class of non-singular unstable modes; (ii) a conjugate finite class of non-singular damped stable modes; (iii) a finite class of singular stable modes, each of these having a branch point and being the limit of unstable modes; (iv) a discrete class of modified internal gravity waves, these being non-singular stable modes (if the density decreases with height everywhere); (v) a continuous class of singular stable modes. The modified internal gravity waves are described asymptotically for large values of the Richardson number. These asymptotic results are related to and extended by numerical calculations for a sinusoidal basic velocity profile and a Bickley jet. The wave speeds for small values of the Richardson number are found to depend only upon the local behaviour of the mean flow near an overall simple maximum or minimum of the velocity profile. Finally some difficulties in the use of the Howard formula for perturbation at a curve of marginal stability are elucidated.

Cyclone–anticyclone asymmetry in gravity wave radiation from a co-rotating vortex pair in rotating shallow water

2015 ◽  
Vol 772 ◽  
pp. 80-106 ◽  
Author(s):  
Norihiko Sugimoto ◽  
K. Ishioka ◽  
H. Kobayashi ◽  
Y. Shimomura
Keyword(s):  
Shallow Water ◽  
Gravity Wave ◽  
Gravity Waves ◽  
Water System ◽  
Local Maximum ◽  
Vortex Pair ◽  
Wave Radiation ◽  
Earth’S Rotation ◽  
Far Field ◽  
Earth's Rotation

Cyclone–anticyclone asymmetry in spontaneous gravity wave radiation from a co-rotating vortex pair is investigated in an $f$-plane shallow water system. The far field of gravity waves is derived analytically by analogy with the theory of aeroacoustic sound wave radiation (Lighthill theory). In the derived form, the Earth’s rotation affects not only the propagation of gravity waves but also their source. While the results correspond to the theory of vortex sound in the limit of $f\rightarrow 0$, there is an asymmetry in gravity wave radiation between cyclone pairs and anticyclone pairs for finite values of $f$. Anticyclone pairs radiate gravity waves more intensely than cyclone pairs due to the effect of the Earth’s rotation. In addition, there is a local maximum of intensity of gravity waves from anticyclone pairs at an intermediate $f$. To verify the analytical solution, a numerical simulation is also performed with a newly developed spectral method in an unbounded domain. The novelty of this method is the absence of wave reflection at the boundary due to a conformal mapping and a pseudo-hyperviscosity that acts like a sponge layer in the far field of waves. The numerical results are in excellent agreement with the analytical results even for finite values of $f$ for both cyclone pairs and anticyclone pairs.

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