Progressive internal waves on slopes

The refraction of progressive internal waves on sloping bottoms is treated for the case of constant Brunt—Väisälä frequency. In two dimensions simple, explicit expressions for the changing wavelengths and amplitudes are found. For small slopes, the solutions reduce to simple propagating waves at infinity.The singularity along a characteristic is shown to be removable, though the solutions are now inhomogeneous waves. The viscous boundary layers of the wedge geometry are briefly considered with the inviscid solutions remaining as interior solutions.A theory valid for small slopes is obtained for three-dimensional waves. The waves are refracted in the usual manner, turning parallel to the beach in shallow water.

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Internal wave excitation from stratified flow over a thin barrier

Journal of Fluid Mechanics ◽

10.1017/s0022112098003048 ◽

1998 ◽

Vol 377 ◽

pp. 223-252 ◽

Cited By ~ 37

Author(s):

BRUCE R. SUTHERLAND ◽

PAUL F. LINDEN

Keyword(s):

Internal Wave ◽

Numerical Simulations ◽

Internal Waves ◽

Wavelet Transforms ◽

Large Amplitude ◽

Internal Gravity Waves ◽

Wave Excitation ◽

Mean Flow ◽

Propagating Waves ◽

The Waves

We perform laboratory experiments in a recirculating shear flow tank of non-uniform salt-stratified water to examine the excitation of internal gravity waves (IGW) in the wake of a tall, thin vertical barrier. The purpose of this study is to characterize and quantify the coupling between coherent structures shed in the wake and internal waves that radiate from the mixing region into the deep, stationary fluid. In agreement with numerical simulations, large-amplitude internal waves are generated when the mixing region is weakly stratified and the deep fluid is sufficiently strongly stratified. If the mixing region is unstratified, weak but continuous internal wave excitation occurs. In all cases, the tilt of the phase lines of propagating waves lies within a narrow range. Assuming the waves are spanwise uniform, their amplitude in space and time is measured non-intrusively using a recently developed ‘synthetic schlieren’ technique. Using wavelet transforms to measure consistently the width and duration of the observed wavepackets, the Reynolds stress is measured and, in particular, we estimate that when large-amplitude internal wave excitation occurs, approximately 7% of the average momentum across the shear depth and over the extent of the wavepacket is lost due to transport away from the mixing region by the waves.We propose that internal waves may act back upon the mean flow modifying it so that the excitation of waves of that frequency is enhanced. A narrow frequency spectrum of large-amplitude waves is observed because the feedback is largest for waves with phase tilt in a range near 45°. Numerical simulations and analytic theories are presented to further quantify this theory.

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Oblique wave groups in deep water

Journal of Fluid Mechanics ◽

10.1017/s0022112084001749 ◽

1984 ◽

Vol 146 ◽

pp. 1-20 ◽

Cited By ~ 8

Author(s):

P. J. Bryant

Keyword(s):

Deep Water ◽

Numerical Solutions ◽

Three Dimensional ◽

Wave Pattern ◽

Periodic Wave ◽

Two Dimensions ◽

Wave Groups ◽

Oblique Wave ◽

Parallel Lines ◽

The Waves

Oblique wave groups consist of waves whose straight parallel lines of constant phase are oblique to the straight parallel lines of constant group phase. Numerical solutions for periodic oblique wave groups with envelopes of permanent shape are calculated from the equations for irrotational three-dimensional deep-water motion with nonlinear upper free-surface conditions. Two distinct families of periodic wave groups are found, one in which the waves in each group are in phase with those in all other groups, and the other in which there is a phase difference of π between the waves in consecutive groups. It is shown that some analytical solutions for oblique wave groups calculated from the nonlinear Schrödinger equation are in error because they ignore the resonant forcing of certain harmonics in two dimensions. Particular attention is given to oblique wave groups whose group-to-wave angle is in the neighbourhood of the critical angle tan−1√½, corresponding to waves on the boundary wedge of the Kelvin ship-wave pattern.

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Classical 1/3 scaling of convection holds up to Ra = 1015

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1922794117 ◽

2020 ◽

Vol 117 (14) ◽

pp. 7594-7598 ◽

Cited By ~ 9

Author(s):

Kartik P. Iyer ◽

Janet D. Scheel ◽

Jörg Schumacher ◽

Katepalli R. Sreenivasan

Keyword(s):

Heat Transport ◽

Boundary Layers ◽

Dimensionless Parameter ◽

Three Dimensional ◽

Wall Stress ◽

Global Transport ◽

Viscous Boundary ◽

Rayleigh Benard Convection ◽

Rayleigh Benard ◽

Very High

The global transport of heat and momentum in turbulent convection is constrained by thin thermal and viscous boundary layers at the heated and cooled boundaries of the system. This bottleneck is thought to be lifted once the boundary layers themselves become fully turbulent at very high values of the Rayleigh numberRa—the dimensionless parameter that describes the vigor of convective turbulence. Laboratory experiments in cylindrical cells forRa≳1012have reported different outcomes on the putative heat transport law. Here we show, by direct numerical simulations of three-dimensional turbulent Rayleigh–Bénard convection flows in a slender cylindrical cell of aspect ratio1/10, that the Nusselt number—the dimensionless measure of heat transport—follows the classical power law ofNu=(0.0525±0.006)×Ra0.331±0.002up toRa=1015. Intermittent fluctuations in the wall stress, a blueprint of turbulence in the vicinity of the boundaries, manifest at allRaconsidered here, increasing with increasingRa, and suggest that an abrupt transition of the boundary layer to turbulence does not take place.

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The energy flux spectrum of internal waves generated by turbulent convection

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.669 ◽

2018 ◽

Vol 854 ◽

Cited By ~ 18

Author(s):

Louis-Alexandre Couston ◽

Daniel Lecoanet ◽

Benjamin Favier ◽

Michael Le Bars

Keyword(s):

Internal Waves ◽

Energy Flux ◽

Wave Energy ◽

Three Dimensional ◽

Turbulent Convection ◽

Reynolds Stresses ◽

Wave Energy Flux ◽

Flux Spectrum ◽

The Waves ◽

Good Agreement

We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of mixed convective and stably stratified fluids. We demonstrate that in the limit of large Rayleigh number ($Ra\in [4\times 10^{7},10^{9}]$) and large stratification (Brunt–Väisälä frequencies$f_{N}\gg f_{c}$, where$f_{c}$is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region as described in Lecoanet & Quataert (Mon. Not. R. Astron. Soc., vol. 430 (3), 2013, pp. 2363–2376). Specifically, we demonstrate that the wave energy flux spectrum scales like$k_{\bot }^{4}\,f^{-13/2}$for weakly damped waves (with$k_{\bot }$and$f$the waves’ horizontal wavenumbers and frequencies, respectively), and that the total wave energy flux decays with$z$, the distance from the convective region, like$z^{-13/8}$.

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Boundary streaming by internal waves

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.786 ◽

2018 ◽

Vol 858 ◽

pp. 71-90 ◽

Cited By ~ 4

Author(s):

A. Renaud ◽

A. Venaille

Keyword(s):

Boundary Layers ◽

Internal Waves ◽

Acoustic Streaming ◽

Early Time ◽

Mean Flow ◽

Viscous Boundary Layer ◽

Quasi Biennial Oscillation ◽

Flow Interactions ◽

Viscous Boundary ◽

Boussinesq Fluid

Damped internal wave beams in stratified fluids have long been known to generate strong mean flows through a mechanism analogous to acoustic streaming. While the role of viscous boundary layers in acoustic streaming has been thoroughly addressed, it remains largely unexplored in the case of internal waves. Here we compute the mean flow generated close to an undulating wall that emits internal waves in a viscous, linearly stratified two-dimensional Boussinesq fluid. Using a quasi-linear approach, we demonstrate that the form of the boundary conditions dramatically impacts the generated boundary streaming. In the no-slip scenario, the early-time Reynolds stress divergence within the viscous boundary layer is much stronger than within the bulk while also driving flow in the opposite direction. Whatever the boundary condition, boundary streaming is however dominated by bulk streaming at larger time. Using a Wentzel–Kramers–Brillouin approach, we investigate the consequences of adding boundary streaming effects to an idealised model of wave–mean flow interactions known to reproduce the salient features of the quasi-biennial oscillation. The presence of wave boundary layers has a quantitative impact on the flow reversals.

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The generation of three-dimensional internal waves and attendant boundary layers in a viscous continuously stratified fluid. Construction of an analytical solution

Fluid Dynamics ◽

10.1007/s10697-006-0109-9 ◽

2006 ◽

Vol 41 (6) ◽

pp. 949-956 ◽

Cited By ~ 1

Author(s):

A. Yu. Vasil’ev ◽

Yu. D. Chashechkin

Keyword(s):

Analytical Solution ◽

Boundary Layers ◽

Internal Waves ◽

Three Dimensional ◽

Stratified Fluid

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Laser Scanning Confocal Microscopy and Three-Dimesnsional Reconstruction of the Mitotic Spindles of Unequally Dividing Embryonic Cells

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100158376 ◽

1990 ◽

Vol 48 (3) ◽

pp. 166-167

Author(s):

J. Holy ◽

G. Schatten

Keyword(s):

Confocal Microscopy ◽

Mitotic Spindle ◽

Laser Scanning ◽

Three Dimensional ◽

Laser Scanning Confocal Microscopy ◽

Two Dimensions ◽

Embryonic Cells ◽

Mitotic Spindles ◽

Cleavage Division ◽

Scanning Confocal Microscopy

One of the classic limitations of light microscopy has been the fact that three dimensional biological events could only be visualized in two dimensions. Recently, this shortcoming has been overcome by combining the technologies of laser scanning confocal microscopy (LSCM) and computer processing of microscopical data by volume rendering methods. We have employed these techniques to examine morphogenetic events characterizing early development of sea urchin embryos. Specifically, the fourth cleavage division was examined because it is at this point that the first morphological signs of cell differentiation appear, manifested in the production of macromeres and micromeres by unequally dividing vegetal blastomeres.The mitotic spindle within vegetal blastomeres undergoing unequal cleavage are highly polarized and develop specialized, flattened asters toward the micromere pole. In order to reconstruct the three-dimensional features of these spindles, both isolated spindles and intact, extracted embryos were fluorescently labeled with antibodies directed against either centrosomes or tubulin.

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