On Long's hypothesis of no upstream influence in uniformly stratified or rotating flow

1972 ◽  
Vol 52 (2) ◽  
pp. 209-243 ◽  
Author(s):  
Michael E. McIntyre
Keyword(s):  
Molecular Diffusion ◽  
Time Development ◽  
Narrow Gap ◽  
Dimensional Problem ◽  
Weakly Nonlinear ◽  
Lee Waves ◽  
Lee Wave ◽  
Wave Trains ◽  
Set Up ◽  
Boussinesq Fluid

The weakly nonlinear, two-dimensional problem for the disturbance due to a slender obstacle in a uniformly stratified, Boussinesq fluid moving past the obstacle with constant basic horizontal velocityU, is considered up to second order in the amplitude ε of the disturbance. Analogous rotating problems are also treated. Particular attention is given to calculating explicitly the columnar-disturbance strengths upstream and downstream of the obstacle, both in the stratified and in the rotating problems, with a view to discussing the truth or otherwise of Long's hypothesis (LH).Whether or not columnar disturbances are found far upstream, violating LH, depends,interalia, on whether or not the flow is externally bounded by rigid horizontal planes (or by a tube or annulus, in the rotating problem), and on whether the problem is made determinate by means of an ‘inviscid transient’ formulation, or by means of a ‘viscous’ one.The inviscid, transient, bounded problem, for time-development of lee waves from a state of no initial disturbance, always exhibits columnar disturbances oforder ε2somewhere in the fluid. They are generated, not near the obstacle, but in the ‘tails’ or transient terminal zones of the lee-wave trains. The columnar-disturbance strengths are largely independent of how the flow is set up from an initially undisturbed state. I n all but one instance the effect is non-zero far up-stream. The exception is the singly-subcritical stratified (or narrow-gap rotating) case, in which the excitation has modal structure sin(2z), the fluid region being 0 [les ]z[les ] π in this case the only columnar disturbance that can penetrate up-stream has structure sinzand so is not excited.A completely different result holds for ‘viscous’ formulations for unseparated, bounded régimes (with steady lee waves spatially attenuated by effects of small molecular diffusion). The strengths of all columnar disturbances, upstream and downstream, vanish in the limit of small diffusivity.In the inviscid, transient, unbounded problem, the upstream influence is, likewise, evanescent, beingO(ε2t−2) as timet→ ∞.The basic expansion in powers of ε will be invalid for times ∝ ε−1or greater, because of resonant-interactive instability of the lee waves.

Climatology of the Sierra Nevada Mountain-Wave Events

2008 ◽  
Vol 136 (2) ◽  
pp. 757-768 ◽  
Author(s):  
Vanda Grubišić ◽  
Brian J. Billings
Keyword(s):  
Sierra Nevada ◽  
Cloud Formation ◽  
Wave Form ◽  
Mountain Range ◽  
Mountain Wave ◽  
Single Wave ◽  
Sierra Nevada Mountain ◽  
Lee Waves ◽  
Lee Wave ◽  
Wave Trains

Abstract This note presents a satellite-based climatology of the Sierra Nevada mountain-wave events. The data presented were obtained by detailed visual inspection of visible satellite imagery to detect mountain lee-wave clouds based on their location, shape, and texture. Consequently, this climatology includes only mountain-wave events during which sufficient moisture was present in the incoming airstream and whose amplitude was large enough to lead to cloud formation atop mountain-wave crests. The climatology is based on data from two mountain-wave seasons in the 1999–2001 period. Mountain-wave events are classified in two types according to cloud type as lee-wave trains and single wave clouds. The frequency of occurrence of these two wave types is examined as a function of the month of occurrence (October–May) and region of formation (north, middle, south, or the entire Sierra Nevada range). Results indicate that the maximum number of mountain-wave events in the lee of the Sierra Nevada occurs in the month of April. For several months, including January and May, frequency of wave events displays substantial interannual variability. Overall, trapped lee waves appear to be more common, in particular in the lee of the northern sierra. A single wave cloud on the lee side of the mountain range was found to be a more common wave form in the southern Sierra Nevada. The average wavelength of the Sierra Nevada lee waves was found to lie between 10 and 15 km, with a minimum at 4 km and a maximum at 32 km.

scholarly journals Presence of Longitudinal Roll Structures during Synoptic Forced Conditions in Complex Terrain

Atmosphere ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 737
Author(s):  
Cory M. Payne ◽  
Jeffrey E. Passner ◽  
Robert E. Dumais ◽  
Abdessattar Abdelkefi ◽  
Christopher M. Hocut
Keyword(s):  
Boundary Layer ◽  
Agricultural Research ◽  
Wrf Model ◽  
Flow Characteristics ◽  
Critical Threshold ◽  
Department Of Agriculture ◽  
Synoptic Conditions ◽  
Lee Waves ◽  
Lee Wave ◽  
Usda Agricultural Research

To investigate synoptic interactions with the San Andres Mountains in southern New Mexico, the Weather Research and Forecasting (WRF) model was used to simulate several days in the period 2018–2020. The study domain was centered on the U.S. Department of Agriculture (USDA) Agricultural Research Service’s Jornada Experimental Range (JER) and the emphasis was on synoptic conditions that favor strong to moderate winds aloft from the southwest, boundary layer shear, a lack of moisture (cloud coverage), and modest warming of the surface. The WRF simulations on these synoptic days revealed two distinct regimes: lee waves aloft and SW-to-NE oriented Longitudinal Roll Structures (LRS) that have typical length scales of the width of the mountain basin in the horizontal and the height of the boundary layer (BL) in the vertical. Analysis of the transitional periods indicate that the shift from the lee wave to LRS regime occurs when the surface heating and upwind flow characteristics reach a critical threshold. The existence of LRS is confirmed by satellite observations and the longitudinal streak patterns in the soil of the JER that indicate this is a climatologically present BL phenomenon.

Three-dimensional tidal sand waves

2009 ◽  
Vol 618 ◽  
pp. 1-11 ◽  
Author(s):  
PAOLO BLONDEAUX ◽  
GIOVANNA VITTORI
Keyword(s):  
Stability Analysis ◽  
Nonlinear Interaction ◽  
Small Amplitude ◽  
Three Dimensional ◽  
Time Development ◽  
Resonance Mechanism ◽  
Weakly Nonlinear ◽  
Sand Waves ◽  
Tidal Sand ◽  
Harmonic Components

The process which leads to the formation of three-dimensional sand waves is investigated by means of a stability analysis which considers the time development of a small-amplitude bottom perturbation of a shallow tidal sea. The weakly nonlinear interaction of a triad of resonant harmonic components of the bottom perturbation is considered. The results show that the investigated resonance mechanism can trigger the formation of a three-dimensional bottom pattern similar to that observed in the field.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

Dissipating and reflecting internal waves

2021 ◽  
Author(s):  
Callum J. Shakespeare ◽  
Brian K. Arbic ◽  
Andrew McC. Hogg
Keyword(s):  
Numerical Simulations ◽  
Linear Theory ◽  
Internal Waves ◽  
Energy Flux ◽  
Internal Tide ◽  
Internal Tides ◽  
Linear Wave ◽  
Lee Waves ◽  
Lee Wave ◽  
Upper Boundary

AbstractInternal waves generated at the seafloor propagate through the interior of the ocean, driving mixing where they break and dissipate. However, existing theories only describe these waves in two limiting cases. In one limit, the presence of an upper boundary permits bottom-generated waves to reflect from the ocean surface back to the seafloor, and all the energy flux is at discrete wavenumbers corresponding to resonant modes. In the other limit, waves are strongly dissipated such that they do not interact with the upper boundary and the energy flux is continuous over wavenumber. Here, a novel linear theory is developed for internal tides and lee waves that spans the parameter space in between these two limits. The linear theory is compared with a set of numerical simulations of internal tide and lee wave generation at realistic abyssal hill topography. The linear theory is able to replicate the spatially-averaged kinetic energy and dissipation of even highly non-linear wave fields in the numerical simulations via an appropriate choice of the linear dissipation operator, which represents turbulent wave breaking processes.

scholarly journals Comparison Between Experiments and a Multibody Weakly Nonlinear Potential Flow Approach for Modeling of Marine Operations

2018 ◽  
Author(s):  
Pierre-Yves Wuillaume ◽  
Pierre Ferrant ◽  
Aurélien Babarit ◽  
François Rongère ◽  
Mattias Lynch ◽  
...  
Keyword(s):  
Potential Flow ◽  
Wave Structure ◽  
Body Motion ◽  
Dynamics Modeling ◽  
Flow Solver ◽  
Weakly Nonlinear ◽  
Validation Test ◽  
Nonlinear Potential ◽  
Set Up ◽  
Validation Tests

This paper presents validation tests for a new numerical tool for the numerical simulation of marine operations. It involves multibody dynamics modeling, wave-structure interactions with large amplitude body motion and cable’s dynamic modeling. Hydrodynamic loads are computed using the WS_CN weakly nonlinear potential flow solver, based on the weak-scatterer hypothesis. Large deformation of the wetted body surfaces can be taken into account. Firstly the ECN’s WS_CN solver capabilities are extended to multibody simulations. A first validation test is performed by comparing numerical results to the experimental data of [1]. Then, a second validation test is proposed. It consists in the ballasting operation of a spar. The experimental set-up is described.

Forcing of oceanic mean flows by dissipating internal tides

2012 ◽  
Vol 708 ◽  
pp. 250-278 ◽  
Author(s):  
Nicolas Grisouard ◽  
Oliver Bühler
Keyword(s):  
Numerical Study ◽  
Three Dimensional ◽  
Wave Drag ◽  
Perturbation Series ◽  
Internal Tides ◽  
Time Averaging ◽  
Mean Flow ◽  
Wave Source ◽  
Lee Waves ◽  
Lee Wave

AbstractWe present a theoretical and numerical study of the effective mean force exerted on an oceanic mean flow due to the presence of small-amplitude internal waves that are forced by the oscillatory flow of a barotropic tide over undulating topography and are also subject to dissipation. This extends the classic lee-wave drag problem of atmospheric wave–mean interaction theory to a more complicated oceanographic setting, because now the steady lee waves are replaced by oscillatory internal tides and, most importantly, because now the three-dimensional oceanic mean flow is defined by time averaging over the fast tidal cycles rather than by the zonal averaging familiar from atmospheric theory. Although the details of our computation are quite different, we recover the main action-at-a-distance result from the atmospheric setting, namely that the effective mean force that is felt by the mean flow is located in regions of wave dissipation, and not necessarily near the topographic wave source. Specifically, we derive an explicit expression for the effective mean force at leading order using a perturbation series in small wave amplitude within the framework of generalized Lagrangian-mean theory, discuss in detail the range of situations in which a strong, secularly growing mean-flow response can be expected, and then compute the effective mean force numerically in a number of idealized examples with simple topographies.

Weakly nonlinear dispersive waves: group velocity

1982 ◽  
Vol 27 (3) ◽  
pp. 507-514
Author(s):  
Bhimsen K. Shivamoggi
Keyword(s):  
Group Velocity ◽  
Acoustic Waves ◽  
Specific Problem ◽  
Electron Plasma ◽  
Hot Electron ◽  
Ion Acoustic Waves ◽  
Dispersive Waves ◽  
Weakly Nonlinear ◽  
Longitudinal Waves ◽  
Wave Trains

For slowly varying wave trains in a linear system, it is known that a quantity proportional to the square of the amplitude propagates with the group velocity. It is shown here, by considering a specific problem of longitudinal waves in a hot electron-plasma and using an asymptotic analysis, that this result continues to be valid even when weak nonlinearities are introduced into the system provided they produce slowly varying wave trains. The method of analysis fails, however, for weakly nonlinear ion-acoustic waves.

Diapycnal diffusivity induced by the breaking of lee waves

2020 ◽  
Author(s):  
Thomas Eriksen ◽  
Carsten Eden ◽  
Dirk Olbers
Keyword(s):  
Internal Wave ◽  
Internal Waves ◽  
Energy Flux ◽  
Wave Energy ◽  
Diapycnal Mixing ◽  
Overturning Circulation ◽  
Lee Waves ◽  
Lee Wave ◽  
Wave Energy Flux ◽  
Diapycnal Diffusivity

<p>A key component in setting the large scale ocean circulation is the process of diapycnal mixing, since this can drive the meridional overturning circulation. Diapycnal mixing in the interior ocean is predominantly associated with the breaking of internal waves. Traditionally, diapycnal mixing has been represented in ocean models by a diapycnal diffusivity either constant or exponentially decreasing with depth. This approach, however, does not take into account the actual physics behind the breaking of internal waves. The energetically consistent internal wave model IDEMIX (Internal wave Dissipation, Energetics and MIXing), on the other hand, computes diffusivities directly on the basis of internal wave energetics. One such type of internal waves are lee waves. These are generated and subsequently dissipated when geostrophic currents interact with bottom topography and are therefore believed to be a source of energy for deep ocean mixing. In this study IDEMIX is coupled to a 1/12<sup>th</sup> degree regional model of the Atlantic. The lee wave energy flux is calculated and used as a bottom flux at each time step effectively allowing lee waves to propagate, interact with mean flow and waves, and subsequently dissipate. This setup enables not only an estimate of the lee wave energy flux but also a direct investigation of the influence of lee waves on dissipation, stratification and horizontal and overturning circulation.</p>

Upstream influence and Long's model in stratified flows

1977 ◽  
Vol 82 (1) ◽  
pp. 147-159 ◽  
Author(s):  
P. G. Baines
Keyword(s):  
Vertical Structure ◽  
Gravitational Instability ◽  
Fluid Velocity ◽  
Finite Depth ◽  
First Order ◽  
Fluid Velocities ◽  
Lee Waves ◽  
Theoretical Predictions ◽  
Lee Wave ◽  
And Diffusion

This paper describes an experimental study of a stratified fluid which is flowing over a smooth two-dimensional obstacle which induces no flow separation and in which effects of viscosity and diffusion are not important. The results are restricted to fluid of finite depth. Various properties of the flow field, in particular the criterion for the onset of gravitational instability in the lee-wave field, are measured and compared with the theoretical predictions of Long's model. The agreement is found to be generally poor, and the consequent inapplicability of Long's model is explained by the failure of Long's hypothesis of no upstream influence, which is demonstrably invalid when stationary lee waves are possible. The obstacle generates upstream motions with fluid velocities which appear to be of first order in the obstacle height. These motions have some of the character of shear fronts or columnar disturbance modes and have the same vertical structure as the corresponding lee-wave modes generated downstream. They result in a reduced fluid velocity upstream below the level of the top of the obstacle, together with a jet of increased fluid velocity above this level which pours down the lee side of the obstacle. This phenomenon becomes more pronounced as the number of modes is increased.

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