A note on the geostrophic flow past a crater in a rotating fluid

Abstract

Laboratory experiments are described on the flow past a solid obstacle in a rotating, homogeneous fluid. Specifically, the obstacle has the form of a walled crater specially constructed so that the volume of the depression is identically equal to the volume of the walls. The results show that closed streamlines occur rather more easily above such topography than above other obstacle types of the same scale but that the conditions for closure are determined essentially by the detailed geometry of the crater, the value of the Rossby number, and the depth of the fluid. The observed flow patterns are analysed and classified and attempts to quantify the most common flow type are made.     

2D finite-difference viscoelastic wave modelling including surface topography

We have pursued two-dimensional (2D) finite-difference (FD) modelling of seismic scattering from free-surface topography. Exact free-surface boundary conditions for the particle velocities have been derived for arbitrary 2D topographies. The boundary conditions are combined with a velocity–stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free-surface topography. The wave equations are numerically discretized by an eighth-order FD method on a staggered grid in space, and a leap-frog technique and the Crank–Nicholson method in time.
In order to demonstrate the capabilities of the surface topography modelling technique, we simulate incident point sources with a sinusoidal topography in seismic media of increasing complexities. We present results using parameters typical of exploration surveys with topography and heterogeneous media. Topography on homogeneous media is shown to generate significant scattering. We show additional effects of layering in the medium, with and without randomization, using a von Kármán realization of apparent anisotropy. Synthetic snapshots and seismograms indicate that prominent surface topography can cause back-scattering, wave conversions and complex wave patterns which are usually discussed in terms of inter-crust heterogeneities.     

Topographic effects on the wind‐driven ocean circulation

Abstract

This is a study of the influence of bottom topography of an ocean basin on the wind‐driven, barotropic ocean circulation. A detailed investigation is made of the role of vorticity transfer to the ocean bottom in the presence of varying topography. It is shown that the wind‐driven gyre over the topography of the North Atlantic has a transport in the western part of the basin only half of that obtained in an ocean of constant depth.     

The influence of mesoscale topography on the stability and growth rates of a two-layer model of the open ocean

Abstract

The influence of mesoscale topography on the baroclinic instability of a two-layer model of the open ocean is considered. For westward velocities in the top layer (U), and for a sinusoidal topography independent of x or longitude (a cross-stream topography), the critical value of U (U_c ) leading to instability is the same as when there is no topography. The wavelength of the unstable perturbation corresponding to U _c is shortened. For a given wavevector (k) of the perturbation the system becomes stable (as also in the absence of topography) for large values of |U|. The minimum value of the shear leading to stability is, however, significantly reduced by the topography.

For sufficiently large values of the height of the topographic features, instabilities appear which are localized within a narrow range of the shear. These instabilities are studied for a topography that depends both on x and y.

For a cross-stream topography the growth rates are somewhat smaller than those without topography and they depend only weakly on k_y . For the topographies considered here which depend both on x and y, perturbations with different values of k_y can again have roughly the same growth rate.

In the case of stable oscillations, variations in the eddy energy with very long periods are made possible by the coexistence of topographic modes with closely lying periods.     

Some effects of the air-water interface on gravity waves

Abstract

New rates of decay are presented for temporally-attenuated gravity waves in deep water, allowance being made for the energy dissipated in the Stokes interfacial boundary layer in the air. This decay-rate, involving air drag, may then be used to deduce a new “free-surface” boundary condition for the problem of the mass transport velocity due to progressive waves; for shallow-water waves, two specific velocity profiles are calculated, and indicate large differences in comparison with the corresponding profiles of Longuet-Higgins (1953) for a vacuum-water interface.     

Note on the flow of a stratified fluid over a stationary obstacle in a channel

Abstract

A new method is introduced to produce a uniform stratified flow over a stationary obstacle in an open channel. The flow is achieved by discharging the flow from the channel through a sink. The details of the sink are unimportant. The flow speed is limited only by the sink capacity. Selective withdrawal at lower densimetric Froude numbers is effectively eliminated through the use of a contraction. The standing, free-surface, long wave arising from the initiation of the flow is also eliminated by the contraction. Experiments are conducted for flow over a sphere for a range of Reynolds numbers from O(10²) to O(10³) and a range of Richardson numbers from O(10^?1) to O(10). Dye and neutrally buoyant droplets are used for quantitative analysis of the wake structure. The wake is also probed by a hot-film anemometer. The frequency of vortex shedding is obtained. Comparison with data from towed experiments is also presented.     

3D free‐boundary conditions for coordinate‐transform finite‐difference seismic modelling

New alternative formulations of exact boundary conditions for arbitrary three-dimensional (3D) free-surface topographies on seismic media have been derived. They are shown to be equivalent to previously published formulations, thereby verifying the validity of each set of formulations. The top of a curved grid represents the free-surface topography while the interior of the grid represents the physical medium. We assume the velocity–stress version of the viscoelastic wave equations to be valid in this grid before transforming the equations to a rectangular grid. In order to perform the numerical discretization we apply the latter version of the equations for seismic wave propagation simulation in the medium. The numerical discretization of the free-surface topography boundary conditions by second-order finite differences (FDs) is shown, as well as the spatially unconditional stability of the resulting system of equations. The FD order is increased by two for each point away from the free surface up to eight, which is the order used in the interior. We use staggered grids in both space and time and the second-order leap-frog and Crank– Nicholson methods for wavefield time propagation. An application using parameters typical of teleseismic earthquakes and explosions is presented using a 200 × 100 km² area of real topography from southwestern Norway over a homogeneous medium. A dipping plane wave simulates a teleseismic P-wave incident on the surface topography. Results show conversion from P- to Rg- (short period fundamental mode Rayleigh) waves in the steepest and/or roughest topography, as well as attenuated waves in valleys and fjords. The codes are parallelized for simulation on fast supercomputers and PC-clusters to model high frequencies and/or large areas.     

Jet instability and the stabilizing effect of topography on jets in two-layer rotating systems

Abstract

Laboratory experiments concerning azimuthal jets in two-layer rotating systems in the absence and presence of bottom topography aligned along the jets have been conducted. The jets were forced by the selective withdrawal of fluid from the upper layer of a two-fluid system contained in a circular dishpan geometry. The principal parameters measured in the experiments were the jet Rossby number, Ro, and a stratification parameter F = r ₁/(λ₁λ₂)^1/2 where r ₁ is the radius of the circular disc used for the selective withdrawal (i.e., r ₁ is the approximate radius of curvature of the jet) and λ₁,λ₂ are the internal Rossby radii of deformation in the upper and lower fluids, respectively.

The no-topography experiments show that for a sufficiently small F, the particular value depending on Ro, the jet is stable for the duration of the experiment. For sufficiently large F, again as a function of Ro, the jet becomes unstable, exhibiting horizontal wave disturbances from modes three to seven. An Ro against F flow regime diagram is presented.

Experiments are then conducted in the presence of a bottom topography having constant cross-section and extending around a mid-radius of the dishpan. The axis of the topography is in the vicinity of the jet axis forced in the no-topography experiments and the crest of the topography is in the vicinity of the interface between the two fluids (i.e., the front associated with the jet). The experiments show that in all cases investigated the jet tends to be stabilized by the bottom topography. Experiments with the topography in place, but with the interface between the fluids being above the topography crest, are shown to be unstable but more irregular than their no-topography counterparts.

Various quantitative measurements of the jet are presented. It is shown, for example, that the jet Rossby number defined in terms of the fluid withdrawal rate from the tank. Q, can be well correlated with a dimensionless vorticity gradient, V_G , across the upper layer jet. This allows for an assessment of the stability characteristics of a jet based on a knowledge of V_G (which can be estimated given a jet profile) and F.     

Hydro-elastic wave proliferation over an impermeable seabed with bottom deformation

ABSTRACT

A hydro-elastic frame has been considered to investigate the proliferation of waves over small base deformation on an infinitely extended flexible seabed. The flexible base surface is assumed as a thin elastic plate of very small thickness and it depends on the Euler–Bernoulli beam equation. For any particular frequency, there are two different modes of time-harmonic propagating wave exists rather than one mode of propagating wave along the positive horizontal direction. The waves with smaller wavenumber spread along the free-surface of the sea (say, free-surface mode) and the waves with higher wavenumber spread along the flexible base surface (say, flexural mode). A simplified perturbation approach is utilised to bring down the entire equations which govern the original boundary value problem (bvp) to a less complex bvp for the first-order velocity potential function. The first-order potential function along with the first-order reflection and transmission coefficients for both modes are calculated by a procedure based upon Fourier transform approach. A shape of sinusoidal swells flexible base surface is taken as an example to approve the scientific results. It is observed that when the train of normal incident propagating wave spreads over base distortion because of either the free-surface unsettling influence or the flexural wave movement in the sea, the reflected and transmitted energy are always feasible to be exchanged from one particular wave mode to another wave mode. Furthermore, we notice that the realistic changes in the flexural rigidity behaviour on the flexible base surface of the sea have a significant effect on the problem of water wave proliferation over small base deformation. Moreover, the energy conservation equation is derived with the help of the Green's integral theorem. The results for the values of reflection and transmission coefficients obtained for both the free-surface unsettling influence as well as flexural wave movement in the fluid are found to satisfy the energy conservation equation almost accurately.     

Finite-Amplitude thermal convection within a self-gravitating fluid sphere

Abstract

Finite-difference calculations have been carried out to determine the structure of finite-amplitude thermal convection within a self-gravitating fluid sphere with uniform heat release. For a fixed-surface boundary condition single-cell convection breaks up into double-cell convection at a Rayleigh number of 3 × 10⁴, at a Rayleigh number of 5 × 10⁵ four-cell convection is observed. With a free-surface boundary condition only single cell convection is obtained up to a Rayleigh number of 5 × 10⁶.     

Baroclinic and topographic influences on the transport in western boundary currents

Abstract

One of the central unsolved theoretical problems of the large scale ocean circulation is concerned with explaining the very large transports measured in western boundary currents such as the Gulf Stream and the Kuroshio. The only theory up to now that can explain the size of these transports is that of non-linear recirculation in which the advective terms in the momentum equations became important near the western boundary. In this paper an alternative explanation is suggested. When bottom topography and baroclinic effects are included in a wind-driven ocean model it is shown that the western boundary current can have a transport larger than that predicted from the wind stress distribution even when the nonlinear advective terms are ignored. The explanation lies in the presence of pressure torques associated with bottom topography which can contribute to the vorticity balance in the same sense as the wind stress curl.

Three numerical experiments have been carried out to explore the nature of this process using a three dimensional numerical model. The first calculation is done for a baroclinic ocean of constant depth, the second for a homogeneous ocean with an idealized continental slope topography, and the third for a baroclinic ocean with the same continental slope topography. The nature of the vorticity balance and of the circulation around closed paths is examined in each case, and it is shown that bottom pressure torques lead to enhanced transport in the western boundary current only for the baroclinic case with variable depth.     

2D surface topography boundary conditions in seismic wave modelling

New formulations of boundary conditions at an arbitrary two-dimensional (2D) free-surface topography are derived. The top of a curved grid represents the free-surface topography while the grid's interior represents the physical medium. The velocity–stress version of the viscoelastic wave equations is assumed to be valid in this grid. However, the rectangular grid version attained by grid transformation is used to model wave propagation in this work in order to achieve the numerical discretization. We show the detailed solution of the particle velocities at the free surface resulting from discretizing the boundary conditions by second-order finite-differences (FDs). The resulting system of equations is spatially unconditionally stable. The FD order is gradually increased with depth up to eighth order inside the medium. Staggered grids are used in both space and time, and the second-order leap-frog and Crank–Nicholson methods are used for time-stepping. We simulate point sources at the surface of a homogeneous medium with a plane free surface containing a hill and a trench. Applying parameters representing exploration surveys, we present examples with a randomly realized surface topography generated by a 1D von Kármán function of order 1. Viscoelastic simulations are presented using this surface with a homogeneous medium and with a layered, randomized medium realization, all generating significant scattering.     

Low frequency edge waves over a trench-ridge topography adjoining a straight coastline

Abstract

The possible interaction of trapped midoceanic boundary waves with a nearby coastline is examined by considering a step trench-ridge topography adjoining a semi-infinite straight coastline. The full dispersion equation, including the effect of the earth's rotation, is derived for long waves over this topography. It is shown that the presence of the coastline begins to have a significant effect on the behaviour of quasigeostrophic ridge waves whenever the wave length is greater than three times the ridge coastline separation.

As an example, the dispersion curves are presented for the topography of the Heceta Bank off the coast of Oregon and it is conjectured that the presence of this off-shore ridge may provide an explanation for the anomalous direction of propagation of the 0.1 c.p.d. shelf wave reported by Mooers and Smith (1968).     

Topographic effects on nonlinear edge waves

Abstract

Edge waves are known to give rise to beach cusps. This paper investigates the topographic feed-back upon the waves. For edge waves generated by subharmonic resonance with incident waves, the topography acts to decrease the edge wave response. As well as causing frequency detuning (Guza and Bowen, 1981) the topography can cause the scattering of edge wave energy. For synchronous waves the topographic irregularities have the opposite effect, and there can be a feed of energy into the edge waves by scattering from the incident waves.     

On the exact shape of the horizontal profile of a topographically rectified tidal flow

Abstract

Starting from the nonlinear shallow water equations of a homogeneous rotating fluid we derive the equation describing the evolution of vorticity by a fluctuating bottom topography of small amplitude, using a multiple scale expansion in a small parameter, which is the topographic length scale relative to the tidal wave length. The exact response functions of residual vorticity for a sinusoidal bottom topography are compared with those obtained by a primitive perturbation series and by harmonic truncation, showing the former to be invalid for small topographic length scales and the latter to be only a fair approximation for vorticity produced by planetary vortex stretching. In deriving the exact shape of the horizontal residual velocity profile at a step-like break in the bottom topography, it is shown that the Lagrangian profile only exists in a strip having the width of the amplitude of the tidal excursion at both sides of the break, and that it vanishes outside that interval. Moreover, in the limit of small amplitude topography at least, it vanishes altogether for the generation mechanism by means of planetary vortex stretching. The Eulerian profile is shown to extend over twice the interval of the Lagrangian profile both for production by vortex stretching and by differential bottom friction. These finite intervals over which the residual velocity profiles exist for a step-like topography are not reproduced by harmonic truncation of the basic equation. This method gives exponentially decaying profiles, indicating spurious horizontal diffusion of vorticity. In terms of orders of magnitude, the method of harmonic truncation is reliable for residual velocity produced by vortex stretching but it overestimates the residual velocity produced by differential bottom friction by a factor 2.     

Impact of anthropogenic drivers on subaqueous topographical change in the Datong to Xuliujing reach of the Yangtze River

Changes of subaqueous topography in shallow offshore water pose safety risks for embankments,navigation,and ports.This study conducted measurements of subaqueous topography between Datong and Xuliujing in the Yangtze River using a Sea Bat 7125 multi-beam echo sounder,and the channel change from 1998 to 2013 was calculated using historical bathymetry data.The study revealed several important results:(1)the overall pattern of changes through the studied stretch of the river was erosion–deposition–erosion.Erosion with a volume 700×10~6m~3occurred in the upper reach,deposition of about 204×10~6m~3occurred in the middle reach,and erosion of about 602×10~6m~3occurred in the lower reach.(2)Dunes are the most common microtopographic feature,accounting for 64.3%of the Datong to Xuliujing reach,followed by erosional topography and flat river topography,accounting for 27.6%and 6.6%,respectively.(3)Human activities have a direct impact on the development of the microtopography.For instance,the mining of sand formed holes on the surface of dunes with lengths of 20–35 m and depths of 3–5 m.We concluded that the overall trend of erosion(net erosion volume of 468×10~6m~3)occurred in the study area mainly because of the decreased sediment discharge following the closure of the Three Gorges Dam.However,other human activities were also impact factors of topographic change.Use of embankments and channel management reduced channel width,restricted river meandering,and exacerbated the erosion phenomenon.     

VTI介质起伏地表地震波场模拟

起伏地表下地震波场模拟有助于解释主动源和被动源地震探测中穿过山脉和盆地的测线所获得的资料.然而传统的有限差分法处理起伏的自由边界比较困难,为了克服这一困难,我们将笛卡尔坐标系的各向异性介质弹性波方程和自由边界条件变换到曲线坐标系中,采用一种稳定的、显式的二阶精度有限差分方法离散(曲线坐标系)VTI介质中的弹性波方程;对...     

Propagation of barotropic modons over topography

Abstract

This is a broad survey of the interaction of modons with topography in a one-layer, quasigeostrophic model. Numerical simulations of modons interacting with ridges, hills, random topography and other obstacles are presented. The behavior of the modon is compared to numerical simulations of a two-point-vortex model, which proves a useful guide to the basic trajectory deflection mechanism. Under sufficiently strong but quasigeostrophically valid topographic perturbations, the modon is shown to fission into two essentially independent, oppositely-signed vortices. In the breakup of a modon near a hill it is found that the positive vortex migrates to the top of the hill. The resulting correlation between the positive vorticity trapped over the hill and the topography is in sharp contrast with the theories of turbulent flow over topography and generation of flow over topography by large scale forcing, both of which describe the development of vorticity anticorrelated with topography. A heuristic explanation of this new behavior is provided in terms of the dynamics of β bT-plane vortices. Further, it is found that a modon travelling over rough topography homogenizes the field of potential vorticity in its vicinity. This behavior is explained in terms of the induced eddy activity near the modon.     

Comparison of infiltration models with regard to design of rectangular infiltration trenches

ABSTRACT

Guidelines for the design of infiltration trenches of rectangular cross-section are compared. Four concepts of infiltration into a native soil from linear soakaways of rectangular cross-section are discussed. The results of calculations using analytical equations are compared with numerical two-dimensional (2D) simulations using HYDRUS computer code and field measurements. Satisfactory agreement is achieved for Kozeny’s concept of infiltration and the free-surface approach, applied to a trench in a highly saturated sandy loam. Theoretical solutions, neglecting matric potential, suggest that the mean infiltration rate along the wetted perimeter of a rectangular trench varies in the range 1.0 to 1.5 of the value of saturated hydraulic conductivity.     

On the mean atmospheric circulation over Antarctica

Abstract

The south-easterly surface flow down the slopes of Antarctica induces a transfer of westerly angular momentum to the atmosphere, which must be removed from the Antarctic domain by atmospheric transports. It is suggested that synoptic eddies protruding from the northern baroclinic zone into the polar regions are modified by the topography such that they are able to perform these meridional transports. A simple linear two-layer model of the axisymmetric circulation of Antarctica is presented where the eddy effects are incorporated via a K-ansatz. It is shown that qualitatively realistic mean flow patterns can be obtained with this model. The limitations of this approach are exposed.