Basin-Scale Gravity Waves in Circular and Elliptical Containers on the Rotating Earth (original) (raw)
International Journal of Non-Linear Mechanics, 2013
The exact solutions of three-dimensional equations of motion for internal gravity waves in cylindrical coordinates in unbounded media are found by means of approximate transformation groups of equations with a small parameter. Introduction of the small parameter has been motivated by justifying the analogy of the Kelvin hypothesis on vanishing the component of the velocity u r normal to wall for the rectilinear motion. In the present case of the cylindrical domain, u r is non-zero and achieves its maximum in the interior, which also agrees with analytical predictions in [6]. However, as linear analysis shows, u r can be considered to be small in the limiting case when the aspect ratio s ¼ H=r 0 is small, in which H and r 0 are the basin's depth and radius respectively. As a particular applications to the ocean and atmospheric modeling, in terms of linear modeling, the time series of the energy density were visualized as spinning patterns that appear to be rotating in an anticlockwise sense when looking from above the North Pole. Such spinning patterns were compared with the flow around a low-pressure area that is usually being linked with a modeling of hurricanes. In terms of zeroth-order approximate transformations, the invariant solutions were visualized as funnels having something in common with the geometric structure of oceanic whirlpools.
The Role of the Earth’s Rotation: Oscillations in Semi-bounded and Bounded Basins of Constant Depth
In Chap. 7 of Volume I, the propagation of surface waves in a layer of a homogeneous fluid referred to an inertial frame was studied. It was shown that superposing the fields of two waves, with the same frequency propagating in opposite directions with the same amplitude can be combined to a standing wave. These standing waves appear as localized oscillations between fixed nodal lines of which the distance defines the semi-wave length with wave humps and wave troughs arising inbetween. Under frictionless conditions imaginary walls can be placed at any position parallel to the wave direction to confine a channel without physically violating any boundary conditions. Similarly, the locations of the nodal lines across the channel turned out to be the positions of standing waves where the longitudinal velocity component vanishes for all time so that vertical walls can equally be inserted at these positions without disturbing the solution. This then formally yields the surface wave solution for the unidirectional motion in a basin of rectangular form and constant depth, see Figs. 7.9 and 7.12 in Chap. 7 of Volume I. These standing wave solutions were subsequently generalized to two-dimensional oscillations in rectangular cells of constant depth in which non-vanishing horizontal velocity components are allowed within the cell that only persistently vanish at the four side walls, thus forming oscillations of true cellular structure (see Figs. 7.14 and 7.15 in Chap. 7 in Volume I). How does the structure of these waves change when the fluid is rotating?
An analytical and numerical study of internal gravity waves forced by isolated topography
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The present study provides a consistent and unified theory for the three types of linear waves of the shallow-water equations (SWE) in a zonal channel on the  plane: Kelvin, inertia-gravity (Poincaré), and planetary (Rossby). The new theory is formulated from the linearized SWE as an eigenvalue problem that is a variant of the classical Schrödinger equation. The results of the new theory show that Kelvin waves exist on the  plane with vanishing meridional velocity, as is the case on the f plane, without any change in the dispersion relation, while the meridional structure of their height amplitude is trivially modified from exponential on the f plane to a one-sided Gaussian on the  plane. Similarly, inertia-gravity waves are only slightly modified in the new theory in comparison with their characteristics on the f plane. For planetary waves (which exist only on the  plane) the new theory yields a similar dispersion relation to the classical theory only for large gravity wave phase speed, such as those encountered in a barotropic ocean or an equivalent barotropic atmosphere. In contrast, for low gravity wave phase speed, for example, those in an equivalent barotropic ocean where the relative density jump at the interface is 10 Ϫ3 , the phase speed of planetary waves in the new theory is 2 times those of the classical theory. The ratio between the phase speeds in the two theories increases with channel width. This faster phase propagation is consistent with recent observation of the westward propagation of crests and troughs of sea surface height made by the altimeter aboard the Ocean Topography Experiment (TOPEX)/Poseidon satellite. The new theory also admits inertial waves, that is, waves that oscillate at the local inertial frequency, as a genuine solution of the eigenvalue problem.
Fluid-Dynamic Models of Geophysical Waves
2018
Geophysical waves are waves that are found naturally in the Earth’s atmosphere and oceans. Internal waves, that is waves that act as an interface between fluids of different density, are examples of geophysical waves. A fluid system with a flat bottom, flat surface and internal wave is initially considered. The system has a depth-dependent current which mimics a typical ocean set-up and, as it is based on the surface of the rotating Earth, incorporates Coriolis forces. Using well established fluid dynamic techniques, the total energy is calculated and used to determine the dynamics of the system using a procedure called the Hamiltonian approach. By tuning a variable several special cases, such as a current-free system, are easily recovered. The system is then considered with a non-flat bottom. Approximate models, including the small amplitude, long-wave, Boussinesq, Kaup-Boussinesq, Korteweg-de Vries (KdV) and Johnson models, are then generated using perturbation expansion technique...
Non-Divergent 2D Vorticity Dynamics and the Shallow Water Equations on the Rotating Earth
IUTAM Bookseries, 2008
From a physical viewpoint the assumption of flow's non-divergence, which greatly simplifies the Shallow Water Equations, is justified by the addition of a virtual "rigid lid" that overlies the surface of the fluid and which supplies the pressure gradient forces that drive the (non-divergent) velocity field. In the presence of rotation any initial vorticity field generates divergence by the Coriolis force in the same way that any initial horizontal velocity component generates the other component in finite time, which implies that an initial non-divergent flow is bound to become divergent at later times. Using a particular scaling of the Shallow Water Equations it can be shown that non-divergent flows are regular limits of the Shallow Water Equations when the layer of fluid is sufficiently thick (high) even though the required surface pressure is not determined by the height of the fluid. These analytical considerations are supported by numerical calculations of the instability of a shear flow on the f-plane that show how the nondivergent instability is the limit of the divergent instability when the mean layer thickness becomes large.
Observation of inertia-gravity wave attractors in an axisymmetric enclosed basin
Physical Review Fluids
Internal waves are ubiquitous in the ocean and play an essential role in the transport of energy and mixing. Their peculiar reflection enables the concentration of energy on a limit cycle. With wave beams viewed as rays, this reflection on an inclined slope shrinks its width and generically brings closer two initially different trajectories eventually reaching a limit cycle called an attractor. Following previous studies, a ray-tracing algorithm is used to track the convergence of wave beams onto such a structure in a 3D axisymmetric domain. This information is used to design experiments using a truncated conical shaped tank in order to form an inertia-gravity waves attractor in a 3D axisymmetric geometry. By increasing the amplitude of the forcing, an evolution of the attractor characteristics can be observed. The occurrence of waves at frequencies lower than the forcing frequency ω 0 suggests triadic resonant instability in a rotating or in a stratified case. Experiments performed in a stratification-only or a rotation-only case indicate two distinct behaviors. The existence of easily excited standing waves, resonant modes of the tank, at frequencies lower than the forcing one enables sharp triadic resonance instability for internal gravity waves, which is not possible for inertial waves. The effect of the symmetry axis is also investigated by adding a cylinder of sufficient diameter at the center of the domain for the wave to reflect on and thereby avoid the interaction on the singularity. Without it, the large amplitude of the waves on the axis triggers nonlinear effects and mixing, denying the access to the wave turbulence regime.
Barotropic and Baroclinic Basin-Scale Wave Dynamics Affected by the Rotation of the Earth
Advances in Geophysical and Environmental Mechanics and Mathematics, 2011
We have already given a detailed description of rotation affected external and internal waves in idealized containers of constant depth: straight channels, gulfs, rectangles and circular and elliptical cylinders. Pure Kelvin and Poincaré waves were shown to describe the oscillating motion in straight channels and their combination yielded the solution of the reflection of the rotation affected waves at the end wall of a rectangular gulf. The typical characterizations of Kelvin and Poincaré waves were seen to prevail (with some modification) in the fluid motion of rotating circular and elliptical cylinders with constant depth. The behaviour was termed Kelvin-type if for basin-scale dynamics the amplitudes of the surface and isopycnal displacements and velocities are shore-bound (i.e. large close to the boundaries and smaller in the interior of the basin), the motion cyclonic (that is counter-clockwise on the N.H.) and frequencies sub-or (less often) superinertial. Alternatively, for Poincaré-type behaviour, the surface and isopycnal displacements and velocities have large amplitudes in off-shore regions, their motion is anti-cyclonic and frequencies are strictly superinertial.
Continental Shelf Research, 2012
officemates Basang and Pierre at the GFI, thank you. Furthermore, to the many anonymous reviewers at the various journals and conferences thank you for your constructive help and comments. I would also further like to thank the staff and students at GFI. My dear parents, Sorayya and Ahmad, I have been able to achieve this degree, indeed all academic success thus far, because you instilled within me a love of science and a yearning after understanding. Thank you. Most importantly, my dear lovely lady Maryam, thank you with all of my heart. This thesis would not have been possible without your never-ending love, friendship, encouraging support and tireless enthusiasm during our (almost) seven years of marriage. Maryam, this achievement is the product of your love and endless light of hope, thank you. And last, but not least, to those who have lent a hand during my PhD and I have failed to mention, my gratitude. you guys for ensuring I had a wonderful time outside of the academic environmen. t To my
Experimental and Numerical Analysis of the Hydrodynamics around a Vertical Cylinder in Waves
Journal of Marine Science and Engineering
The present study provides an extensive analysis on the hydrodynamics induced by a vertical slender pile under wave action. The authors carried out the study both experimentally and numerically, thus enabling a deep understanding of the flow physics. The experiments took place at a wave flume of the Università Politecnica delle Marche. Two different experimental campaigns were performed: In the former one, a mobile bed model was realized with the aims to study both the scour process and the hydrodynamics around the cylinder; in the latter one, the seabed was rigid in order to make undisturbed optical measurements, providing a deeper analysis of the hydrodynamics. The numerical investigation was made by performing a direct numerical simulation. A second order numerical discretization, both in time and in space, was used to solve the Navier–Stokes equations while a volume of fluid (VOF) approach was adopted for tracking the free surface. The comparison between experimental and numeric...
Internal gravity waves in the upper eastern equatorial Pacific: Observations and numerical solutions
Journal of Geophysical Research, 1997
On the basis of data froxn a towed thermistor chain collected near 140øW on the equator during April 1987, the zonal wavenumber and vertical structure of internal gravity waves were observed to vary significantly between wave events. Our hypothesis is that this variability is due to changes in the vertical structure of mean horizontal velocity and density. Assuming that the observed waves were the fastest growing modes for shear instability, we solve the Taylor-Goldstein equation, using different analytical basic states, including a zonal and meridional flow, to simulate the different conditions during 4 nights of intense internal wave activity. We find that while the observed waves are of finite amplitude, linear sheeu' instability is sufficient to explain the wavelength and vertical structure of vertical displacement for most of the waves. The fastest growing, unstable, mode-one solutions have e-folding growth times of less than 10 min. These solutions show wave phase speeds and vertical structures to be highly dependent upon the velocity structure of the uppermost 40 m. Near the base of the mixed layer at a flow inflection point the kinetic energy of the mean flow is extracted for wave growth. Wave vertical displacement is maximum near this inflection point. Zonal phase speeds range from-0.8 to-0.1 m/s. The propagation direction of waves with growth rates of 75% of the maximum growth rate can range from about 45 ø north to 45 • south of the zonal direction. The vertical wave-induced Reynolds stress divergence could explain a discrepancy in zonal momentum budgets of the upper 90 m of this region. Estimates of this stress divergence show that only about 100 days of intense internal wave activity is needed per year for these internal waves to explain estimated residuals of the mean zonal momentum budgets of this region at 50-to 100-m depth. eral mechanisms for this diurnal cycling have been considered, including diurnal cycles in solar heating, surface wind stress, and mean velocity shears, only internal waves can account for the cycle in turbulence extending well below the mixed layer. Indeed, the correspondence of the deeper mixing with the presence of internal waves subsequently was confirmed by two sets of independent observations. The first, a towed thermistor chain [
The Exact Seismic Response of an Ocean and a N-Layer Configuration*
Geophysical Prospecting, 1987
The space-time acoustic wave motion generated by an impulsive monopole source is calculated with the aid of the Cagniard-de Hoop technique. Two configurations with plane interfaces are discussed : an air/fluid/solid configuration with the source and the receiver located in the fluid layer; and a stack of n fluid layers between two acoustic half-spaces where the source and the receiver are located in the upper half-space. Synthetic seismograms are generated for the pressure of the reflected wavefield, using the source signature of an airgun.
Observations of inertial waves in a rectangular basin with one sloping boundary
Journal of Fluid Mechanics, 2003
Inertial waves in a homogeneous rotating fluid travel along rays that are inclined with respect to the rotation axis. The angle of inclination depends solely on the ratio of the wave frequency and twice the angular frequency. Because of this fixed angle, the waves can become focused when reflected at a sloping wall. In an infinitely long channel with a sloping wall, the repeated action of focusing may lead to the approach to a limit cycle, the so-called wave attractor, where the energy is concentrated. This effect is studied in the laboratory in a rectangular tank with one sloping wall, placed excentrically on a rotating table. The waves are excited by modulation of the background rotation. Several frequency ratios are used to study different wave attractors and one standing wave. The observations consist mainly of particle image velocimetry data in horizontal and vertical cross-sections in one half of the basin. The attractors are observed in the vertical cross-sections. They show continuous phase propagation, which distinguishes them from the standing wave where the phase changes at the same time over the whole cross-section. However, particle motion of inertial waves is three-dimensional and the actual basin is not an ideal twodimensional channel but is of finite length. This implies that the waves must adapt to the vertical endwalls, although a prediction of the nature of these adaptations and the structure of the three-dimensional wave field is at present lacking. For critical waves, whose rays are parallel to the slope, clear three-dimensional behaviour is observed. The location of most intense motion along this critical slope attractor changes in the horizontal direction and horizontal phase propagation is observed, with a wavelength between 1/5 and 1/4 of the basin length. For the other attractors there is little evidence of phase propagation in the horizontal direction. The motion along the attractor is however stronger near the vertical endwalls for attractors with wave rays of slopes close to 1 or larger. The standing wave and the other attractors are more clearly visible near 1/5 of the tank length.
Geophysical Journal International, 2009
Within the normal mode relaxation theory, we thoroughly analyze and compare the exact compressible and incompressible solution for a viscoelastic stratified Earth model with an approximated analytical one, where the ratio between gravity and radial distance from the Earth's centre is considered constant in each layer of the model. We implement an algorithm, based on the Runge–Kutta scheme, to integrate in the r variable the spheroidal part of the linearized momentum and Poisson equations. This numerical scheme allows us to unravel the impact of such an approximation. We disclose new aspects of the physics underneath the terms entering the system of differential equations, such as the advection and the buoyancy force. These issues are relevant for a wide range of geophysical applications and timescales, from the 1–102 yr related to postseismic deformation, to the 103 yr of postglacial rebound, to the 106 yr of True Polar Wander. We show that such an approximation affects the buoyancy force term, due to the sensitivity of the tangential displacement component in the compressible deformation, and the advection term, for that part containing an effective unstable radial force, dependent on the radial displacement. Relative errors between exact and approximated Love numbers are sizable for low harmonic degrees (13, 62, 17 per cent and 4, 39, 4 per cent for the elastic and viscous radial, tangential and gravitational Love numbers), but can become lower than 1 per cent at high harmonic degrees, for appropriate choices of the constant gravity term entering advection and buoyancy force. Our findings shed light on the role of the latter in the readjustment of the interfaces where the planet is subject to variations in its physical properties, in quasi-static deformation case. We show that the contribution of the denumerably infinite compressional D-modes can be dealt with accurately by normal mode approach and that the D-mode cluster point does not contribute to the deformation.
Evolution and decay of gravity wavefields in weak-rotating environments: a laboratory study
Environmental Fluid Mechanics, 2018
Gravity waves are prominent physical features that play a fundamental role in transport processes of stratified aquatic ecosystems. In a two-layer stratified basin, the equations of motion for the first vertical mode are equivalent to the linearised shallow water equations for a homogeneous fluid. We adopted this framework to examine the spatiotemporal structure of gravity wavefields weakly affected by the background rotation of a single-layer system of equivalent thickness h , via laboratory experiments performed in a cylindrical basin mounted on a turntable. The wavefield was generated by the release of a diametral linear tilt of the air-water interface, , inducing a basin-scale perturbation that evolved in response to the horizontal pressure gradient and the rotation-induced acceleration. The basin-scale wave response was controlled by an initial perturbation parameter, * = 0 ∕h , where 0 was the initial displacement of the air-water interface, and by the strength of the background rotation controlled by the Burger number, . We set the experiments to explore a transitional regime from moderate-to weak-rotational environments, 0.65 ≤ ≤ 2 , for a wide range of initial perturbations, 0.05 ≤ * ≤ 1.0. The evolution of was registered over a diametral plane by recording a laser-induced optical fluorescence sheet and using a capacitive sensor located near the lateral boundary. The evolution of the gravity wavefields showed substantial variability as a function of the rotational regimes and the radial position. The results demonstrate that the strength of rotation and nonlinearities control the bulk decay rate of the basin-scale gravity waves. The ratio between the experimentally estimated damping timescale, T d , and the seiche period of the basin, T g , has a median value of T d ∕T g ≈ 11 , a maximum value of T d ∕T g ≈ 10 3 and a minimum value of T d ∕T g ≈ 5. The results of this study are significant for the understanding the dynamics of gravity waves in waterbodies weakly affected by Coriolis acceleration, such as mid-to small-size lakes.