Mean motions and impulse of a guided internal gravity wave packet

1973 ◽  
Vol 60 (4) ◽  
pp. 801-811 ◽  
Author(s):  
Michael E. Mcintyre
Keyword(s):  
Wave Packet ◽  
Gravity Wave ◽  
Electromagnetic Waves ◽  
Internal Gravity Wave ◽  
Single Frequency ◽  
Buoyancy Frequency ◽  
Mean Fields ◽  
The Mean ◽  
Boussinesq Fluid ◽  
The Waves

Second-order mean fields of motion and density are calculated for the two-dimensional problem of an internal gravity wave packet (the waves are predominantly of a single frequency o and wavenumberk) propagating as a wave-guide mode in an inviscid, diffusionless Boussinesq fluid of constant buoyancy frequencyN, confined between horizontal boundaries. (The same mathematical analysis applies to the formally identical problem for inertia waves in a homogeneous rotating fluid.)To leading order the mean motions turn out to be zero outside the wave packet, which consequently possesses a well-defined fluid impulse [Iscr ]. This is directed horizontally, and is given in magnitude and sense by\[ {\cal I} = \alpha{\cal M};\quad\alpha = \frac{2c_{\rm g}(c-c_{\rm g})(c+2c_{\rm g})}{c^3-4c^3_{\rm g}}. \]Here [Mscr ] is the so-called ‘wave momentum’, defined as wave energy divided by horizontal phase velocity c ≡ ω/k, andcg=c(N2–ω2)/N2, the group velocity.If the wave packet is supposed generated by a horizontally towed obstacle, [Mscr ] appears as the total fluid impulse, but of this a portion [Mscr ]-[Iscr ] in general propagates independently away from the wave packet in the form of long waves. When the wave packet itself is totally reflected by a vertical barrier immersed in the fluid, the time-integrated horizontal force on the barrier equals 2 [Iscr ] (and not 2 [Mscr ] as might have been expected from a naive analogy with the radiation pressure of electromagnetic waves.)

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.

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.

Topographic Form Drag on Tides and Low-Frequency Flow: Observations of Nonlinear Lee Waves over a Tall Submarine Ridge near Palau

2020 ◽  
Vol 50 (5) ◽  
pp. 1489-1507 ◽  
Author(s):  
Gunnar Voet ◽  
Matthew H. Alford ◽  
Jennifer A. MacKinnon ◽  
Jonathan D. Nash
Keyword(s):  
Ocean Circulation ◽  
Neap Tide ◽  
Spring Tide ◽  
Tidal Flow ◽  
Low Frequency ◽  
Buoyancy Frequency ◽  
Mean Flow ◽  
Submarine Ridge ◽  
The Mean ◽  
The Waves

AbstractTowed shipboard and moored observations show internal gravity waves over a tall, supercritical submarine ridge that reaches to 1000 m below the ocean surface in the tropical western Pacific north of Palau. The lee-wave or topographic Froude number, Nh0/U0 (where N is the buoyancy frequency, h0 the ridge height, and U0 the farfield velocity), ranged between 25 and 140. The waves were generated by a superposition of tidal and low-frequency flows and thus had two distinct energy sources with combined amplitudes of up to 0.2 m s−1. Local breaking of the waves led to enhanced rates of dissipation of turbulent kinetic energy reaching above 10−6 W kg−1 in the lee of the ridge near topography. Turbulence observations showed a stark contrast between conditions at spring and neap tide. During spring tide, when the tidal flow dominated, turbulence was approximately equally distributed around both sides of the ridge. During neap tide, when the mean flow dominated over tidal oscillations, turbulence was mostly observed on the downstream side of the ridge relative to the mean flow. The drag exerted by the ridge on the flow, estimated to for individual ridge crossings, and the associated power loss, thus provide an energy sink both for the low-frequency ocean circulation and the tidal flow.

scholarly journals Enhanced internal gravity wave activity and breaking over the northeastern Pacific–eastern Asian region

2015 ◽  
Vol 15 (22) ◽  
pp. 13097-13112 ◽  
Author(s):  
P. Šácha ◽  
A. Kuchař ◽  
C. Jacobi ◽  
P. Pišoft
Keyword(s):  
Gravity Wave ◽  
Coastal Region ◽  
Internal Gravity Wave ◽  
Wave Activity ◽  
Satellite Mission ◽  
Density Profiles ◽  
Cycle Amplitude ◽  
Northeastern Pacific ◽  
Middle Atmospheric Dynamics ◽  
The Mean

Abstract. We have found a stratospheric area of anomalously low annual cycle amplitude and specific dynamics in the stratosphere over the northeastern Pacific–eastern Asia coastal region. Using GPS radio occultation density profiles from the Formosat Satellite Mission 3/Constellation Observing System for Meteorology, Ionosphere, and Climate (FORMOSAT-3/COSMIC), we have discovered an internal gravity wave (IGW) activity and breaking hotspot in this region. Conditions supporting orographic wave sourcing and propagation were found. Other possible sources of wave activity in this region are listed. The reasons why this particular IGW activity hotspot was not discovered before as well as why the specific dynamics of this region have not been pointed out are discussed together with the weaknesses of using the mean potential energy as a wave activity proxy. Possible consequences of the specific dynamics in this region on the middle atmospheric dynamics and transport are outlined.

The internal gravity wave field emitted by a stably stratified turbulent wake

2013 ◽  
Vol 720 ◽  
pp. 104-139 ◽  
Author(s):  
Ammar M. Abdilghanie ◽  
Peter J. Diamessis
Keyword(s):  
Gravity Wave ◽  
Wavelet Transforms ◽  
Internal Gravity Wave ◽  
Early Time ◽  
Reynolds Numbers ◽  
Turbulent Wake ◽  
Buoyancy Frequency ◽  
Continuous Wavelet Transforms ◽  
Wide Range ◽  
Body Velocity

AbstractThe internal gravity wave (IGW) field emitted by a stably stratified, initially turbulent, wake of a towed sphere in a linearly stratified fluid is studied using fully nonlinear numerical simulations. A wide range of Reynolds numbers, $\mathit{Re}= UD/ \nu \in [5\times 1{0}^{3} , 1{0}^{5} ] $ and internal Froude numbers, $\mathit{Fr}= 2U/ (ND)\in [4, 16, 64] $ ($U$, $D$ are characteristic body velocity and length scales, and $N$ is the buoyancy frequency) is examined. At the higher $\mathit{Re}$ examined, secondary Kelvin–Helmholtz instabilities and the resulting turbulent events, directly linked to a prolonged non-equilibrium (NEQ) regime in wake evolution, are responsible for IGW emission that persists up to $Nt\approx 100$. In contrast, IGW emission at the lower $\mathit{Re}$ investigated does not continue beyond $Nt\approx 50$ for the three $\mathit{Fr}$ values considered. The horizontal wavelengths of the most energetic IGWs, obtained by continuous wavelet transforms, increase with $\mathit{Fr}$ and appear to be smaller at the higher $\mathit{Re}$, especially at late times. The initial value of these wavelengths is set by the wake height at the beginning of the NEQ regime. At the lower $\mathit{Re}$, consistent with a recently proposed model, the waves propagate over a narrow range of angles that minimize viscous decay along their path. At the higher $\mathit{Re}$, wave motion is much less affected by viscosity, at least initially, and early-time wave propagation angles extend over a broader range of values which are linked to increased efficiency in momentum extraction from the turbulent wake source.

On strongly nonlinear gravity waves in a vertically sheared atmosphere

2021 ◽  
Author(s):  
Georg Sebastian Voelker ◽  
Mark Schlutow
Keyword(s):  
Gravity Wave ◽  
Gravity Waves ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Wave Drag ◽  
Mean Flow ◽  
Flow Strength ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
The Mean

<p>Internal gravity waves are a well-known mechanism of energy redistribution in stratified fluids such as the atmosphere. They may propagate from their generation region, typically in the Troposphere, up to high altitudes. During their lifetime internal waves couple to the atmospheric background through various processes. Among the most important interactions are the exertion of wave drag on the horizontal mean-flow, the heat generation upon wave breaking, or the mixing of atmospheric tracers such as aerosols or greenhouse gases.</p><p>Many of the known internal gravity wave properties and interactions are covered by linear or weakly nonlinear theories. However, for the consideration of some of the crucial effects, like a reciprocal wave-mean-flow interaction including the exertion of wave drag on the mean-flow, strongly nonlinear systems are required. That is, there is no assumption on the wave amplitude relative to the mean-flow strength such that they may be of the same order.</p><p>Here, we exploit a strongly nonlinear Boussinesq theory to analyze the stability of a stationary internal gravity wave which is refracted at the vertical edge of a horizontal jet. Thereby we assume that the incident wave is horizontally periodic, non-hydrostatic, and vertically modulated. Performing a linear stability analysis in the vicinity of the jet edge we find necessary and sufficient criteria for instabilities to grow. In particular, the refracted wave becomes unstable if its incident amplitude is large enough and both mean-flow horizontal winds, below and above the edge of the jet, do not exceed particular upper bounds.</p>

Mean flows induced by internal gravity wave packets propagating in a shear flow

1979 ◽  
Vol 292 (1393) ◽  
pp. 391-417 ◽  
Keyword(s):  
Shear Flow ◽  
Gravity Wave ◽  
Internal Gravity Wave ◽  
Basic Flow ◽  
Channel Depth ◽  
Amplitude Wave ◽  
Generalized Lagrangian ◽  
Basic Shear ◽  
Wave Induced ◽  
The Waves

An inviscid, incompressible, stably stratified fluid occupies a horizontal channel, along which an internal gravity wave packet is propagating in the presence of a basic shear flow. By using a generalized Lagrangian mean formulation, the equation for wave action conservation is derived to describe the manner in which the basic flow affects the waves. Equations describing the second-order (in amplitude) wave-induced Lagrangian mean flows are obtained. Two kinds of applications are discussed: (i) steady mean flows, due to waves encountering an inhomogeneity in their environment, such as a varying channel depth; (ii) mean flows induced by modulations in the wave amplitude.

On the instability of an internal gravity wave

1977 ◽  
Vol 356 (1686) ◽  
pp. 411-432 ◽  
Keyword(s):  
Gravity Wave ◽  
Parametric Instability ◽  
Normal Modes ◽  
Internal Gravity Wave ◽  
Marginal Stability ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Physical Conclusion ◽  
Arbitrary Amplitude ◽  
Boussinesq Fluid

The linear stability of an internal gravity wave of arbitrary amplitude in an unbounded stratified inviscid Boussinesq fluid is considered mathematically. The instability is shown to be governed by a Floquet system and treated by a generalization of the method of normal modes. Some properties of the Floquet system, and in particular those of its parametric instability, are analysed. The parametric instability is related to the theory of resonant wave interactions; and the surface of marginal stability in the control space of the amplitude and wavenumbers is shown to be describable by the catastrophe theory of Thom. Finally some results of numerical calculations of the marginal surface are shown. The main physical conclusion is to confirm that the internal gravity wave is unstable always, even when its amplitude is small and so its local Richardson number is large everywhere for all time. It is suggested, by various illustrations and arguments, that the methods developed in this paper are applicable to the instability of many symmetric nonlinear waves.

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