On steady and travelling waves over bottom topography in the model of homogeneous flow in a beta-plane channel

Abstract

A spectral low-order model is proposed in order to investigate some effects of bottom corrugation on the dynamics of forced and free Rossby waves. The analysis of the interaction between the waves and the topographic modes in the linear version of the model shows that the natural frequencies lie between the corresponding Rossby wave frequencies for a flat bottom and those applying in the “topographic limit” when the beta-effect is zero. There is a possibility of standing or eastward-travelling free waves when the integrated topograhic effect exceeds the planetary beta-effect.

The nonlinear interactions between forced waves in the presence of topography and the beta-effect give rise to a steady dynamical mode correlated to the topographic mode. The periodic solution that includes this steady wave is stable when the forcing field moves to the West with relatively large phase speed. The energy of this solution may be transferred to the steady zonal shear flow if the spatial scale of this zonal mode exceeds the scale of the directly forced large-scale dynamical mode.     

切变基本纬向流中非线性赤道Rossby长波

为了解决观测和理论研究中的一些问题以及更好地了解热带大气动力学 ,有必要进一步研究基本气流的变化对大气中赤道Rossby波动的影响 .本文研究分析基本气流对赤道Rossby长波的影响 ,利用一个简单赤道 β平面浅水模式和摄动法 ,研究纬向基本气流切变中非线性赤道Rossby波 ,推导出在切变基本纬向流中赤道Rossby长波振幅演变所满足的非线性KdV方程并得到其孤立波解 .分析表明 ,孤立波存在的必要条件是基本气流有切变 ,而且基流切变不能太强 ,否则将产生正压不稳定 .     

Weakly non-linear analysis of small-amplitude internal gravity waves forced by isolated topography

Asymptotic methods and numerical simulations are used to examine the evolution of an internal gravity wave packet comprising a continuous spectrum of horizontal wavenumbers and propagating upwards in a continuously stratified shear flow. In the multiple-scale framework for a horizontally localized wave packet generated by stratified flow over a localized mountain range with multiple peaks, there are in general two horizontal scales: the “fast” scale which is defined by the oscillations within the packet, i.e. the number of peaks, and the “slow scale” which is defined by the horizontal extent of the packet, i.e. the width of the mountain range. The focus here is on the specific case of an isolated mountain where the spectrum of horizontal wavenumbers is centred at zero and the multiple-scaling procedure is thus simplified by the absence of the fast spatial scale. The background flow is vertically sheared and critical-level interactions occur. The time frame within which non-linear critical-level effects become significant is determined by the magnitude of the non-linear terms in the governing equations. With the isolated mountain forcing this time frame is significantly longer than in the case of a multiple-peak mountain range forcing and it depends on the horizontal scale of the forcing, as well as on the amplitude. At leading-order, the non-linear asymptotic solution approaches a steady state in the outer region at late time, but the zero-wavenumber component of the solution continues to evolve with time in the vicinity of the critical level.     

Formation of one-dimensional waves for resonant flow in a channel

When a forcing moves in a shallow channel at a velocity near the phase velocity for linear long waves, energy cannot escape from the forcing at the linear group velocity and nonlinear effects become important in describing the resulting flow. This flow is termed resonant or transcritical. It has been found both experimentally and numerically that large amplitude upstream propagating waves are generated by the forcing. These waves are straight crested, even though the forcing is two-dimensional. It is shown that these upstream waves become straight crested due to geometrical effects aided by the presence of side walls. Using energy conservation, approximate values of the amplitude of the upstream waves are obtained which are compared with recent experimental and numerical results.     

On the sensitivity of the sedimentary system in the East Frisian Wadden Sea to sea-level rise and wave-induced bed shear stress

The paper addresses the individual and collective contribution of different forcing factors (tides, wind waves, and sea-level rise) to the dynamics of sediment in coastal areas. The results are obtained from simulations with the General Estuarine Transport Model coupled with a sediment transport model. The wave-induced bed shear stress is formulated using a simple model based on the concept that the turbulent kinetic energy (TKE) associated with wind waves is a function of orbital velocity, the latter depending on the wave height and water depth. A theory is presented explaining the controls of sediment dynamics by the TKE produced by tides and wind waves. Several scenarios were developed aiming at revealing possible trends resulting from realistic (observed or expected) changes in sea level and wave magnitude. The simulations demonstrate that these changes not only influence the concentration of sediment, which is very sensitive to the magnitude of the external forcing, but also the temporal variability patterns. The joint effect of tides and wave-induced bed shear stress revealed by the comparison between theoretical results and simulations is well pronounced. The intercomparison between different scenarios demonstrates that the spatial patterns of erosion and deposition are very sensitive to the magnitude of wind waves and sea-level rise. Under a changing climate, forcing the horizontal distribution of sediments adjusts mainly through a change in the balance of export and import of sediment from the intertidal basins. The strongest signal associated with this adjustment is simulated North of the barrier islands where the evolution of sedimentation gives an integrated picture of the processes in tidal basins.     

Acoustic gravity wave growth and damping in convecting plasma

The propagation of acoustic gravity waves through steadily convecting plasma in the thermosphere has been analysed theoretically. The growth and damping rates of internal gravity waves due to the feedback effects of wave-modulated Joule heating and Laplace forcing have been calculated. It is found that large convection flow velocities lead to the growth of large-scale internal gravity waves, whilst small- and medium-scale waves are heavily damped, under similar conditions. It has also been shown that wave growth is favoured for waves travelling against the plasma flow direction. The effects of critical coupling when wave phase speeds match the plasma flow speed have also been investigated. The results of these calculations are discussed in the context of the atmospheric energy budget and thermosphere-ionosphere coupling.     

Modelling offshore sand wave evolution

We present a two-dimensional vertical (2DV) flow and morphological numerical model describing the behaviour of offshore sand waves. The model contains the 2DV shallow water equations, with a free water surface and a general bed load formula. The water movement is coupled to the sediment transport equation by a seabed evolution equation. Using this model, we investigate the evolution of sand waves in a marine environment. As a result, we find sand wave saturation for heights of 10–30% of the average water depth on a timescale of decades. The stabilization mechanism, causing sand waves to saturate, is found to be based on the balance between the shear stress at the seabed and the principle that sediment is transported more easily downhill than uphill. The migration rate of the sand waves decreases slightly during their evolution. For a unidirectional steady flow the sand waves become asymmetrical in the horizontal direction and for a unidirectional block current asymmetrical in the vertical. A sensitivity analysis showed the slope effect of the sediment transport plays an important role herein. Furthermore, the magnitude of the resistance at the seabed and the eddy viscosity influence both the timescale and height of sand waves. The order of magnitudes found of the time and spatial scales coincide with observations made in the southern bight of the North Sea, Japan and Spain.     

The influence of tides, wind and waves on the net sand transport in the North Sea

Large-scale redistribution of sand by hydrodynamical processes in shelf seas is important for basin and coastal evolution on time scales of a thousand to tens of thousands of years. The influence of tides on the large-scale net sand-transport patterns in the North Sea has received much attention, but the influence of wind-driven flow and wind waves has hardly been investigated. Here, to establish the present-day situation and to develop a method that can also be used for palaeo-situations and forecasts for different sea levels, this influence is assessed for the present southern North Sea using a numerical flow model, a parametric wave model and a wave-averaged sand-transport formulation. Various forcing combinations are used to identify the dominant transport mechanisms: tides only, tides and wind, tides and waves, and combined tides, wind and waves. Wind forcing is applied in two ways to find an efficient, but still representative, method of incorporating this stochastic process: a statistical wind climatology and an observed time series. The results show that (i) the wind climatology yields a good approximation of the sand transport computed using the time series; (ii) wind-driven flow and waves only contribute significantly to the net sand transport by tides when acting together where tidal currents are small; and (iii) various combinations of forcings dominate the net sand transport in different regions of the southern North Sea: (a) tides dominate in the southern, middle and northwestern parts of the Southern Bight and in the region of The Wash; (b) tides, wind-driven flow and waves all are important in the northeastern part of the Southern Bight; and (c) wind-driven flow and waves dominate north of the Friesian Islands, in the German Bight and on the Dogger Bank. Qualitative comparison with observations shows good agreement.     

The effect of wave-induced turbulence on intertidal mudflats: Impact of boat traffic and wind

Semi-diurnal and fortnightly surveys were carried out to quantify the effects of wind- and navigation-induced high-energy events on bed sediments above intertidal mudflats. The mudflats are located in the upper fluvial part (Oissel mudflat) and at the mouth (Vasière Nord mudflat) of the macrotidal Seine estuary. Instantaneous flow velocities and mudflat bed elevation were measured at a high frequency and high resolution with an acoustic doppler velocimeter (ADV) and an ALTUS altimeter, respectively. Suspended particulate matter concentrations were estimated by calibrating the ADV acoustic backscattered intensity with bed sediments collected at the study sites. Turbulent bed shear stress values were estimated by the turbulent kinetic energy method, using velocity variances filtered from the wave contribution. Wave shear stress and maximum wave–current shear stress values were calculated with the wave–current interaction (WCI) model, which is based on the bed roughness length, wave orbital velocities and the wave period (T_S). In the fluvial part of the estuary, boat passages occurred unevenly during the surveys and were characterized by long waves (T_S>50 s) induced by the drawdown effect and by short boat-waves (T_S<10 s). Boat waves generated large bottom shear stress values of 0.5 N m⁻² for 2–5 min periods and, in burst of several seconds, larger bottom shear stress values up to 1 N m⁻². At the mouth of the estuary, west south-west wind events generated short waves (T_S<10 s) of H_S values ranging from 0.1 to 0.3 m. In shallow-water environment (water depth <1.5 m), these waves produced bottom shear stress values between 1 and 2 N m⁻². Wave–current shear stress values are one order of magnitude larger than the current-induced shear stress and indicate that navigation and wind are the dominant hydrodynamic forcing parameters above the two mudflats. Bed elevation and SPM concentration time series showed that these high energy events induced erosion processes of up to several centimetres. Critical erosion shear stress (τ_ce) values were determined from the SPM concentration and bed elevation measurements. Rough τ_ce values were found above 0.2 N m⁻² for the Oissel mudflat and about 1 N m⁻² for the Vasière Nord mudflat.     

Wave-induced topographic formstress in baroclinic channel flow

Large-scale zonal flow driven across submarine topography establishes standing Rossby waves. In the presence of stratification, the wave pattern can be represented by barotropic and baroclinic Rossby waves of mixed planetary topographic nature, which are locked to the topography. In the balance of momentum, the wave pattern manifests itself as topographic formstress. This wave-induced formstress has the net effect of braking the flow and reducing the zonal transport. Locally, it may lead to acceleration, and the parts induced by the barotropic and baroclinic waves may have opposing effects. This flow regime occurs in the circumpolar flow around Antarctica. The different roles that the wave-induced formstress plays in homogeneous and stratified flows through a zonal channel are analyzed with the BARBI (BARotropic-Baroclinic-Interaction ocean model, Olbers and Eden, J Phys Oceanogr 33:2719–2737, 2003) model. It is used in complete form and in a low-order version to clarify the different regimes. It is shown that the barotropic formstress arises by topographic locking due to viscous friction and the baroclinic one due to eddy-induced density advection. For the sinusoidal topography used in this study, the transport obeys a law in which friction and wave-induced formstress act as additive resistances, and windstress, the effect of Ekman pumping on the density stratification, and the buoyancy forcing (diapycnal mixing of the stratified water column) of the potential energy stored in the stratification act as additive forcing functions. The dependence of the resistance on the system parameters (lateral viscosity ε, lateral diffusivity κ of eddy density advection, Rossby radius λ, and topography height δ) as well as the dependence of transport on the forcing functions are determined. While the current intensity in a channel with homogeneous density decreases from the viscous flat bottom case in an inverse quadratic law ~δ ^–2 with increasing topography height and always depends on ε, a stratified system runs into a saturated state in which the transport becomes independent of δ and ε and is determined by the density diffusivity κ rather than the viscosity: κ/λ ² acts as a vertical eddy viscosity, and the transport is λ ²/κ times the applied forcing. Critical values for the topographic heights in these regimes are identified.     

Resonance induced by Rossby Lee waves

Abstract

The mutual interaction of fields induced by spatially separated potential vorticity sources in a quasi-geostrophic barotropic flow is investigated using the weakly nonlinear approach. It is found that a powerful nonlinear response can be triggered by Rossby lee waves. This resonance phenomenon which dominates all other nonlinear corrections depends on certain global resonance conditions and on the change in the phase of the Rossby lee wave across the distance separating the sources. The response is particularly strong for topographic forcing possessing δ-function characterisitics.     

A study of the Joule and Lorentz inputs in the production of atmospheric gravity waves in the upper thermosphere

First results of a modelling study of atmospheric gravity waves (AGWs) are presented. A fully-coupled global thermosphere-ionosphere-plasmasphere model is used to examine the relative importance of Lorentz forcing and Joule heating in the generation of AGWs. It is found that Joule heating is the dominant component above 110km. The effects of the direction of the Lorentz forcing component on the subsequent propagation of the AGW are also addressed. It is found that enhancement of zonal E × B forcing results in AGWs at F-region altitudes of similar magnitudes travelling from the region of forcing in both poleward and equatorward directions, whilst enhancement of equatorward meridional E × B forcing results in AGWs travelling both poleward and equatorward, but with the magnitude of the poleward wave severely attenuated compared with the equatorward wave.     

Simplified longitudinal seismic response of tunnels linings subjected to surface waves

This paper presents an engineering approach for analysing the longitudinal behaviour of tunnels subjected to earthquakes. The tunnel is modelled as a Timoshenko beam connected to the far soil by means of continuous elastic support (Winkler model). Seismic free-field inputs, such as those caused by surface waves travelling parallel to the tunnel axis, were imposed at the base of the springs of the Winkler model, generating bending moments and shear forces on the cross-sections of the tunnel. Closed-form expressions of the tunnel displacements, shear forces, and bending moments were determined at any tunnel section in terms of the seismic excitation, tunnel geometry and material properties, and subgrade reaction modulus of the soil. A dimensional analysis was carried out to ascertain directly the maximum tunnel displacement, bending moment, and shear force.     

EFFECT OF SHOCK WAVES ON RILL FORMATION

Hydraulic mechanism of fill formation was studied theoretically and experimentally. It was assumed that the impact of varied boundary on overland flow results in fluctuating of water surface, and shock waves that may contribute to the formation of fills. Both theoretical derivation and laboratory experiments were used to compare the hydraulic characteristics of flows with and without shock waves. Results showed that shock waves can lead to an increase in flow depth, flow velocity, and turbulence intensity. Consequently, flow shear stress or stream energy increase dramatically and fill headcuts may occur where shock waves converge.     

Numerical experiments on the generation of long ocean waves in coastal waters of the Buenos Aires province,Argentina

A numerical model (two horizontal dimensions, vertically integrated) is used to investigate the generation of long ocean waves, ranging from 20 min to almost 2 h, at Buenos Aires continental shelf. The domain includes the Río de la Plata estuary and the continental shelf together and extends from 33.5° to 40.5°S latitude, and from 51° to 63°W longitude. Sea-level oscillations are modeled by forcing with passage of atmospheric cold fronts and atmospheric gravity waves. Both forcing mechanisms, which have been present during high activity lapses of long ocean waves, are mathematically implemented. After several numerical simulations, it is concluded that the pressure and wind fields associated to cold fronts do not generate long ocean waves in the area, though they do produce disturbances with periods longer than the tidal ones. On the other hand, it is so concluded that atmospheric gravity waves are an effective mechanism to force long ocean waves. Results obtained show that generation of long ocean waves is highly sensitive depending on the propagation direction and the phase speed of the atmospheric gravity waves. The long ocean wave event detected during the large-amplitude gravity-wave event of 13 October 1985 is successfully simulated. Finally, all our results suggest that atmospheric gravity waves are a highly effective mechanism forcing for the generation of long ocean waves in Buenos Aires coastal waters.     

A practical method for analysis of river waves and for kinematic wave routing in natural channel networks

The behaviour of river waves is described using a simplified dimensionless form of the momentum equation in conjunction with the continuity equation. Three dimensionless parameters were derived based on a quantitative linear analysis. These parameters, which depend on the Froude number of the steady uniform flow and the geometric characteristics of the river, permit quantification of the influence of inertia and pressure in the momentum equation. It was found that dynamic and diffusion waves occur mainly on gentle channel slopes and the transition between them is characterized by the Froude number. On the other hand, the kinematic wave has a wide range of applications. If the channel slope is greater than 1%, the kinematic wave is particularly suitable for describing the hydraulics of flow. Since slopes in natural channel networks are often greater than 1%, an analytical solution of the linearized kinematic wave equation with lateral inflow uniformly distributed along the channel is desirable and was therefore derived. The analytical solution was then implemented in a channel routing module of an existing simple rainfall–runoff model. The results obtained using the analytical solution compared well with those obtained from a non‐linear kinematic wave model. Copyright © 2000 John Wiley & Sons, Ltd.     

Theory of azimuthally small-scale hydromagnetic waves in the axisymmetric magnetosphere with finite plasma pressure

The structure of monochromatic MHD-waves with large azimuthal wave number m 1 in a two-dimensional model of the magnetosphere has been investigated. A joint action of the field line curvature, finite plasma pressure, and transversal equilibrium current leads to the phenomenon that waves, standing along the field lines, are travelling across the magnetic shells. The wave propagation region, the transparency region, is bounded by the poloidal magnetic surface on one side and by the resonance surface on the other. In their meaning these surfaces correspond to the usual and singular turning points in the WKB-approximation, respectively. The wave is excited near the poloidal surface and propagates toward the resonance surface where it is totally absorbed due to the ionospheric dissipation. There are two transparency regions in a finite-beta magnetosphere, one of them corresponds to the Alfvén mode and the other to the slow magneto-sound mode.     

Resonant forcing of coastally trapped waves in a continuously stratified ocean

In a previous paper (Grimshaw, 1987) the resonant forcing of coastally trapped waves was discussed in the barotropic case. In order to extend that theory to more realistic situations, we have considered the analogous theory whereby a longshore current interacts with a longshore topographic feature, or the forcing is due to longshore wind stress, for the case of the continuously stratified ocean. As in the previous theory, near resonance, when a long-wave phase speed is close to zero (in the reference frame of the forcing), the wave motion is governed by a forced evolution equation of the KdV-type. The behaviour of the wave field is characterized by three parameters representing the bandwidth for resonance, the forcing amplitude and the dissipation. We have evaluated these parameters in various practical cases, and found that the bandwidths, which scale with ^1/2 when the forcing has dimensionless amplitude , can often be quite broad. Typically the second, third, or higher, modes may be resonant. Concurrently, the dissipation is also usually significant, leading to a steady state balance between the forcing, dissipation and nonlinear terms.     

电离层Alfven谐振器对地面观测到的地磁信号的影响初步研究

本文研究了0.1~10 Hz频率范围内的ULF波从磁层到地面的传播,得到了解析解,分析了电离层Alfven谐振器、磁倾角、电离层电导率、以及波频率对地面观测到的地磁信号的影响.数值结果表明:在磁层中剪切波在竖直方向有明显的谐振结构;地面观测到的信号在IAR谐振频率出现极大值,其谐振频率随磁倾角的增大而增大;电离层电导率的变化可以改变IAR的谐振频率,并能改变波的透射,从而影响地面地磁信号的频谱.     

Can zonally symmetric inertial waves drive an oscillating zonal mean flow?

In the present paper zonal mean flow excitation by inertial waves is studied in analogy to mean flow excitation by gravity waves that plays an important role for the quasi-biennial oscillation in the equatorial atmosphere. In geophysical flows that are stratified and rotating, pure gravity and inertial waves correspond to the two limiting cases: gravity waves neglect rotation, inertial waves neglect stratification. The former are more relevant for fluids like the atmosphere, where stratification is dominant, the latter for the deep oceans or planet cores, where rotation dominates. In the present study a hierarchy of simple analytical and numerical models of zonally symmetric inertial wave-mean flow interactions is considered and the results are compared with data from a laboratory experiment. The main findings can be summarised as follows: (i) when the waves are decoupled from the mean flow they just drive a retrograde (eastward) zonal mean flow, independent of the sign of the meridional phase speed; (ii) when coupling is present and the zonal mean flow is assumed to be steady, the waves can drive vertically alternating jets, but still, in contrast to the gravity wave case, the structure is independent of the sign of the meridional phase speed; (iii) when coupling is present and time-dependent zonal mean flows are considered the waves can drive vertically and temporarily oscillating mean flows. The comparison with laboratory data from a rotating annulus experiment shows a qualitative agreement. It appears that the experiment captures the basic elements of the inertial wave mean flow coupling. The results might be relevant to understand how the Equatorial Deep Jets can be maintained against dissipation, a process currently discussed controversially.