The critical layer for internal gravity waves in a shear flow

1967 ◽  
Vol 27 (3) ◽  
pp. 513-539 ◽  
Author(s):  
John R. Booker ◽  
Francis P. Bretherton
Keyword(s):  
Gravity Waves ◽  
Phase Speed ◽  
Deep Ocean ◽  
Internal Gravity Waves ◽  
Mean Flow ◽  
Lee Waves ◽  
The Mean ◽  
Horizontal Momentum ◽  
Pass Through ◽  
The Waves

Internal gravity waves of small amplitude propagate in a Boussinesq inviscid, adiabatic liquid in which the mean horizontal velocity U(z) depends on height z only. If the Richardson number R is everywhere larger than 1/4, the waves are attenuated by a factor $\exp\{-2\pi(R - \frac{1}{4})^{\frac{1}{2}}\}$ as they pass through a critical level at which U is equal to the horizontal phase speed, and momentum is transferred to the mean flow there. This effect is considered in relation to lee waves in the airflow over a mountain, and in relation to transient localized disturbances. It is significant in considering the propagation of gravity waves from the troposphere to the ionosphere, and possibly in transferring horizontal momentum into the deep ocean without substantial mixing.

On internal gravity waves in an accelerating shear flow

1978 ◽  
Vol 88 (4) ◽  
pp. 623-639 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Gravity Waves ◽  
Phase Distribution ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Initial Energy ◽  
Constant Density ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
The Waves

The investigation of the effects which a changing mean flow has on a uniform train of internal gravity waves (Thorpe 1978a) is continued by considering waves in a uniformly accelerating stratified plane Couette flow with constant density gradient. Experiments reveal a change in the mode structure and phase distribution of the waves, and their eventual breaking near the boundary where the mean flow is greatest, the phase speed of the waves being positive. A linear numerical model is devised which accurately describes the waves up to the onset of their breaking, and this is used to investigate their energetics. The working of the Reynolds stress against the mean velocity gradient results in a very rapid transfer of energy from the waves to the mean flow, so that by the time breaking occurs only a small fraction of their initial energy remains for possible transfer into potential energy of the fluid.The consequences have important applications in oceanography and meteorology, to flow stability and flow generation, and explain some earlier laboratory observations.

Nonlinear internal gravity waves in a rotating fluid

1975 ◽  
Vol 71 (3) ◽  
pp. 497-512 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Gravity Waves ◽  
Wave Structure ◽  
Internal Gravity Waves ◽  
Mean Flow ◽  
Mean Velocity ◽  
Main Effect ◽  
Local Wave ◽  
The Mean ◽  
The Waves ◽  
Circulation Theorem

The interaction between internal gravity waves in a rotating frame and the mean flow is discussed for the case when the properties of the mean flow vary slowly on a scale determined by the local wave structure. The principle of conservation of wave action is established. It is shown that the main effect of the waves on the Lagrangian mean velocity is due to an appropriate ‘radiation stress’ tensor. A circulation theorem and a potential-vorticity equation are derived for the mean velocity.

Momentum transport by gravity waves in a perfectly conducting shear flow

1972 ◽  
Vol 54 (2) ◽  
pp. 217-240 ◽  
Author(s):  
N. Rudraiah ◽  
M. Venkatachalappa
Keyword(s):  
Magnetic Field ◽  
Gravity Waves ◽  
Critical Level ◽  
Critical Levels ◽  
Mean Flow ◽  
Critical Layers ◽  
The Mean ◽  
Pass Through ◽  
The Waves ◽  
Aligned Magnetic Field

Alfvén-gravitational waves propagating in a Boussinesq, inviscid, adiabatic, perfectly conducting fluid in the presence of a uniform aligned magnetic field in which the mean horizontal velocityU(z)depends on heightzonly are considered. The governing wave equation has three singularities, at the Doppler-shifted frequencies Ωd= 0, ± ΩA, where ΩAis the Alfvén frequency. Hence the effect of the Lorentz force is to introduce two more critical levels, called hydromagnetic critical levels, in addition to the hydrodynamic critical level. To study the influence of magnetic field on the attenuation of waves two situations, one concerning waves far away from the critical levels (i.e. Ωd[Gt ] ΩA) and the other waves at moderate distances from the critical levels (i.e. Ωd> ΩA), are investigated. In the former case, if the hydrodynamic Richardson numberJHexceeds one quarter the waves are attenuated by a factor exp{−2π(JH−¼)½} as they pass through the hydromagnetic critical levels, at which Ωd= ± ΩA, and momentum is transferred to the mean flow there. Whereas in the case of waves at moderate distances from the critical levels the ratio of momentum fluxes on either side of the hydromagnetic critical levels differ by a factor exp {−2π(J−¼)½}, whereJ(> ¼) is the algebraic sum of hydrodynamic and hydromagnetic Richardson numbers. Thus the solutions to the hydromagnetic system approach asymptotically those of the hydrodynamic system sufficiently far on either side of the magnetic critical layers, though their behaviour in the vicinity of such levels is quite dissimilar. There is no attenuation and momentum transfer to the mean flow across the hydrodynamic critical level, at which Ωd= 0. The general theory is applied to a particular problem of flow over a sinusoidal corrugation. This is significant in considering the propagation of Alfvén-gravity waves, in the presence of a geomagnetic field, from troposphere to ionosphere.

The generation of internal waves by flow over the rough topography of a continental slope

1992 ◽  
Vol 439 (1905) ◽  
pp. 115-130 ◽  
Keyword(s):  
Internal Gravity Waves ◽  
Sea Surface ◽  
Small Scale ◽  
Buoyancy Frequency ◽  
Mean Flow ◽  
Sea Bed ◽  
Lee Waves ◽  
Inclined Slope ◽  
The Mean ◽  
The Waves

Internal gravity waves generated as standing lee waves by the flow, V , of a uniformly stratified fluid over small-scale topography on an inclined slope are examined in the particular case in which the flow is parallel to the mean slope isobaths. Attention is given to the magnitude and direction of the energy flux and to its dependence on Vl / N , α and β , where l is the wavenumber of the topography, N is the buoyancy frequency of the fluid, α is the mean slope and β defines the orientation of the two-dimensional topography on the slope. In general there is a greater probability of energy transfer towards shallow water, but in particular regions the direction of the flux depends on the orientation of the topography on the slope. For a given scale of topography and for fixed longslope current, V , and stratification, N , the drag associated with the lee waves is greatest when β = 0 (when the topography is oriented normal to the mean slope isobaths) and it can be as large as the turbulent stress on the sea bed. The lee waves may induce large variations in the currents on the sea surface. It is found that although stationary lee waves may be formed over sloping topography, the waves reflected at the sea surface may subsequently be scattered from the topography to produce waves that propagate in the mean flow.

Application of the IDEMIX Concept for Internal Gravity Waves in the Atmosphere

2020 ◽  
Vol 77 (10) ◽  
pp. 3601-3618
Author(s):  
B. Quinn ◽  
C. Eden ◽  
D. Olbers
Keyword(s):  
Energy Balance ◽  
Wave Field ◽  
Gravity Wave ◽  
Gravity Waves ◽  
Momentum Flux ◽  
Middle Atmosphere ◽  
Internal Gravity Waves ◽  
Wave Drag ◽  
Mean Flow ◽  
The Mean

AbstractThe model Internal Wave Dissipation, Energy and Mixing (IDEMIX) presents a novel way of parameterizing internal gravity waves in the atmosphere. IDEMIX is based on the spectral energy balance of the wave field and has previously been successfully developed as a model for diapycnal diffusivity, induced by internal gravity wave breaking in oceans. Applied here for the first time to atmospheric gravity waves, integration of the energy balance equation for a continuous wave field of a given spectrum, results in prognostic equations for the energy density of eastward and westward gravity waves. It includes their interaction with the mean flow, allowing for an evolving and local description of momentum flux and gravity wave drag. A saturation mechanism maintains the wave field within convective stability limits, and a closure for critical-layer effects controls how much wave flux propagates from the troposphere into the middle atmosphere. Offline comparisons to a traditional parameterization reveal increases in the wave momentum flux in the middle atmosphere due to the mean-flow interaction, resulting in a greater gravity wave drag at lower altitudes. Preliminary validation against observational data show good agreement with momentum fluxes.

The effect of viscosity and heat conduction on internal gravity waves at a critical level

1967 ◽  
Vol 30 (4) ◽  
pp. 775-783 ◽  
Author(s):  
Philip Hazel
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Gravity Waves ◽  
Order Differential Equation ◽  
Fluid Velocity ◽  
Attenuation Factor ◽  
Internal Gravity Waves ◽  
Constant Factor ◽  
The Mean ◽  
The Waves

The differential equation for the vertical velocity of a gravity wave in an inviscid shear flow is singular at a level where the mean fluid velocity is equal to the horizontal phase velocity of the waves. It has been shown that a wave travelling through such a layer has its amplitude attenuated by a constant factor dependent on the local Richardson number. In this paper the results obtained by solving numerically the full sixth order differential equation, which is derived by including viscosity and heat conduction in the problem, (and is not singular) are discussed, and the same attenuation factor is found. Some experiments which confirm certain aspects of the theory are described in an appendix.

scholarly journals Parametrization of momentum and energy depositions from gravity waves generated by tropospheric hydrodynamic sources

1997 ◽  
Vol 15 (12) ◽  
pp. 1570-1580 ◽  
Author(s):  
N. M. Gavrilov
Keyword(s):  
Gravity Waves ◽  
Wave Energy ◽  
Internal Gravity Waves ◽  
Mean Flow ◽  
Energy Fluxes ◽  
The Mean ◽  
Turbulence Production

Abstract. The mechanism of generation of internal gravity waves (IGW) by mesoscale turbulence in the troposphere is considered. The equations that describe the generation of waves by hydrodynamic sources of momentum, heat and mass are derived. Calculations of amplitudes, wave energy fluxes, turbulent viscosities, and accelerations of the mean flow caused by IGWs generated in the troposphere are made. A comparison of different mechanisms of turbulence production in the atmosphere by IGWs shows that the nonlinear destruction of a primary IGW into a spectrum of secondary waves may provide additional dissipation of nonsaturated stable waves. The mean wind increases both the effectiveness of generation and dissipation of IGWs propagating in the direction of the wind. Competition of both effects may lead to the dominance of IGWs propagating upstream at long distances from tropospheric wave sources, and to the formation of eastward wave accelerations in summer and westward accelerations in winter near the mesopause.

On the mean motion induced by internal gravity waves

1969 ◽  
Vol 36 (4) ◽  
pp. 785-803 ◽  
Author(s):  
Francis P. Bretherton
Keyword(s):  
Gravity Waves ◽  
Wave Energy ◽  
Wave Train ◽  
Three Dimensional ◽  
Internal Gravity Waves ◽  
Wave Number ◽  
Mean Flow ◽  
Mean Velocity ◽  
Mean Motion ◽  
The Mean

A train of internal gravity waves in a stratified liquid exerts a stress on the liquid and induces changes in the mean motion of second order in the wave amplitude. In those circumstances in which the concept of a slowly varying quasi-sinusoidal wave train is consistent, the mean velocity is almost horizontal and is determined to a first approximation irrespective of the vertical forces exerted by the waves. The sum of the mean flow kinetic energy and the wave energy is then conserved. The circulation around a horizontal circuit moving with the mean velocity is increased in the presence of waves according to a simple formula. The flow pattern is obtained around two- and three-dimensional wave packets propagating into a liquid at rest and the results are generalized for any basic state of motion in which the internal Froude number is small. Momentum can be associated with a wave packet equal to the horizontal wave-number times the wave energy divided by the intrinsic frequency.

scholarly journals Incidence and reflection of internal waves and wave-induced currents at a jump in buoyancy frequency

2014 ◽  
Vol 1 (1) ◽  
pp. 269-315
Author(s):  
J. P. McHugh
Keyword(s):  
Wave Packet ◽  
Internal Gravity Waves ◽  
Buoyancy Frequency ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Transmitted Waves ◽  
The Mean ◽  
Induced Currents ◽  
Wave Induced ◽  
The Waves

Abstract. Weakly nonlinear internal gravity waves are treated in a two-layer fluid with a set of nonlinear Schrodinger equations. The layers have a sharp interface with a jump in buoyance frequency approximately modelling the tropopause. The waves are periodic in the horizontal but modulated in the vertical and Boussinesq flow is assumed. The equation governing the incident wave packet is directly coupled to the equation for the reflected packet, while the equation governing transmitted waves is only coupled at the interface. Solutions are obtained numerically. The results indicate that the waves create a mean flow that is strong near and underneath the interface, and discontinuous at the interface. Furthermore, the mean flow has an oscillatory component with a vertical wavelength that decreases as the wave packet interacts with the interface.

scholarly journals Incidence and reflection of internal waves and wave-induced currents at a jump in buoyancy frequency

2015 ◽  
Vol 22 (3) ◽  
pp. 259-274 ◽  
Author(s):  
J. P. McHugh
Keyword(s):  
Wave Packet ◽  
Internal Gravity Waves ◽  
Buoyancy Frequency ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Transmitted Waves ◽  
The Mean ◽  
Induced Currents ◽  
Wave Induced ◽  
The Waves

Abstract. Weakly nonlinear internal gravity waves are treated in a two-layer fluid with a set of nonlinear Schrodinger equations. The layers have a sharp interface with a jump in buoyancy frequency approximately modeling the tropopause. The waves are periodic in the horizontal but modulated in the vertical and Boussinesq flow is assumed. The equation governing the incident wave packet is directly coupled to the equation for the reflected packet, while the equation governing transmitted waves is only coupled at the interface. Solutions are obtained numerically. The results indicate that the waves create a mean flow that is strong near and underneath the interface, and discontinuous at the interface. Furthermore, the mean flow has an oscillatory component that can contaminate the wave envelope and has a vertical wavelength that decreases as the wave packet interacts with the interface.

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