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.     

Instability of planetary waves in a high vertical resolution model

Abstract

A high vertical resolution model is used to examine the instability of a baroclinic zonal flow and a finite amplitude topographically forced wave. Two families of unstable modes are found, consisting of zonally propagating most unstable modes, and stationary unstable modes. The former have time scale and spatial structure similar to baroclinic synoptic disturbances, but are localized in space due to interaction with the zonally asymmetric forcing. These modes transport heat efficiently in both the zonal and meridional directions. The second family of stationary unstable modes has characteristics of modes of low frequency variability of the atmosphere. They have time scales of 10 days and longer, and are of planetary scale with an equivalent barotropic vertical structure. The horizontal structure resembles blocking flows. They are maintained by available potential energy of the basic wave, and have large zonal heat fluxes. The results for both families of modes are interpreted in terms of an interaction between forcing and baroclinic instability to create favoured regions for eddy development. Applications to baroclinic planetary waves are also considered.     

Propagation of oblique waves over small bottom undulation in an ice-covered two-layer fluid

Scattering of oblique incident waves by small bottom undulation in a two-layer fluid, where the upper layer has a thin ice-cover while the lower one has the undulation, is investigated within the framework of linearized water wave theory. The ice-cover is being modeled as an elastic plate of very small thickness. There exist two modes of time-harmonic waves–one with lower wave number propagating along the ice-cover (ice-cover mode) and the other with higher wave number along the interface (interfacial mode). A perturbation analysis is employed to solve the corresponding boundary value problem governed by modified Helmholtz equation and thereby evaluating the reflection and transmission coefficients approximately up to first order for both modes. A patch of sinusoidal ripples, having two different wave numbers over two consecutive stretches, is considered as an example and the related coefficients are determined. It is observed that when the wave is incident on the ice-cover surface we always find energy transfer to the interface, but for interfacial incident waves there are parameter ranges for which no energy transfer to the ice-cover surface is possible. Also it is observed that for small angles of incidence, the reflected energy is more as compared to the other angles of incidence. These results are demonstrated in graphical form. From the derived results, the solutions for problems with free surface can be obtained as particular cases.     

Lagrangian motion of air parcels in the stratosphere in the presence of planetary waves

Development of thoughts on tracer transport mechanisms in the stratosphere which lead to new approaches to two-dimensional modeling of the tracer problem is reviewed.Three-dimensional motions of individual air parcels affected by a planetary wave are investigated theoretically, treating a steady, upward propagating wave in a uniform flow. It is shown that trajectories of air parcels are of elliptical form when projected onto the meridional plane and that they have no mean meridional or vertical motion, even though the usual zonal Eulerian-mean vertical motion exists. The origin of the difference between the mean air parcel motion and the Eulerian-mean motion is discussed.On the basis of the knowledge of air parcel motion, two approaches to two-dimensional modeling are considered. The generalized Lagrangian mean motion (quasi-zonal weighted mean taken over a meandering material tube), recently introduced by Andrews and McIntyre, is identical with the mean motion of an air parcel in a steady state. Such a mean meridional circulation may be used for advecting a tracer in the meridional plane in a two-dimensional model. The transport effect is represented solely by the advection and an eddy transport does not appear in this scheme, to a first approximation.The finding that trajectories of air parcels are elliptical necessitates a reexamination of the Reed-German eddy diffusivity currently used in two-dimensional chemical-dynamical models. By applying a mixing length type hypothesis, we derive an eddy diffusivity formula for use in Eulerian-mean calculations, which, in the case of a conservative tracer is dominated by an anti-symmetric tensor. The eddy transport due to this anti-symmetric tensor diffusivity is of advective type (not diffusive) and has the effect of taking the Stoke drift effect into account, when used in the usual Eulerian-mean formulation.     

具有非线性地形的正压流体中孤立Rossby波的mKdV方程

正压流体中,采用摄动方法将准地转位涡方程推导出地形效应的mKdV方程,得到Rossby波振幅的演变满足地形效应的mKdV方程的结论,说明地形效应是诱导Rossby孤立波的重要因素.     

Flow of grounded abyssal ocean currents along zonally-varying topography on a rotating sphere

A steady nonlinear planetary–geostrophic model in spherical coordinates is presented describing the hemispheric-scale meridional flow of grounded abyssal currents on a zonally-sloping bottom. The model, which corresponds mathematically to a quasi-linear hyperbolic partial differential equation, can be solved explicitly for a cross-slope isopycnal field that is grounded (i.e. intersects the bottom on the up slope and down slope sides). As a consequence of the conservation of potential vorticity, the abyssal currents possess decreasing thickness in the equatorward direction while maintaining constant meridional volume transport. There is a small westward zonal transport in the interior of these currents that results in westward intensification as they flow toward the equator. Conditions for the possible formation of a shock to develop on the up slope flank of the current are derived.     

Hydromagnetic waves in a thin rotating spherical shell

We consider an electrically conducting fluid confined to a thin rotating spherical shell in which the Elsasser and magnetic Reynolds numbers are assumed to be large while the Rossby number is assumed to vanish in an appropriate limit. This may be taken as a simple model for a possible stable layer at the top of the Earth's outer core. It may also be a model for the thin shells which are thought to be a source of the magnetic fields of some planets such as Mercury or Uranus. Linear hydromagnetic waves are studied using a multiple scale asymptotic scheme in which boundary layers and the associated boundary conditions determine the structure of the waves. These waves are assumed to be of the form of an asymptotic series expanded about an ambient magnetic field which vanishes on the equatorial plane and velocity and pressure fields which do not. They take the form of short wave, slowly varying wave trains. The results are compared to the author's previous work on such waves in cylindrical geometry in which the boundary conditions play no role. The approximation obtained is significantly different from that obtained in the previous work in that an essential singularity appears at the equator and nonequatorial wave regions appear.     

Spectral analysis of planetary waves seen in ionospheric total electron content (TEC): First results using GPS differential TEC and stratospheric reanalyses

The coupling of the neutral atmosphere and the ionosphere through planetary waves (PW) (zonal wavenumber 0–5) is investigated by spherical harmonic analyses of the ionospheric total electron content (TEC). These analyses detect mean variations, standing and travelling waves which are assumed to be signatures of PW. Database used for TEC analyses are 3 years of hourly TEC maps covering the higher middle and polar latitudes. They are regularly produced by DLR Neustrelitz. The obtained results are compared with PW analyses using NCEP/NCAR and Met Office stratospheric reanalyses. Case studies show that signatures of PW occur simultaneously in the middle atmosphere and ionosphere.     

Trapped internal waves over undular topography in a partially mixed estuary

The flow of a stratified fluid over small-scale topographic features in an estuary may generate significant internal wave activity. Lee waves and upstream influence generated at isolated topographic features have received considerable attention during the past few decades. Field surveys of a partially mixed estuary, the Rotterdam Waterway, in 1987, also showed a plethora of internal wave activity generated by isolated topography, banks and groynes. Additionally it revealed a spectacular series of resonant internal waves trapped above low-amplitude bed waves. The internal waves reached amplitudes of 3–4 m in an estuary with a mean depth of 16 m. The waves were observed during the decreasing flood tide and are thought to make a significant contribution to turbulence production and mixing. However, while stationary linear and finite amplitude theories can be used to explain the presence of these waves, it is important to further investigate their time-dependent and non-linear behaviour. With the development of advanced non-hydrostatic models it now becomes possible to further investigate these waves through numerical experimentation. This is the focus of the work presented here. The non-hydrostatic finite element numerical model FINEL3D developed by Labeur was used in the experiments presented here. The model has been shown to work well in a number of stratified flow investigations. Here, we first show that the model reproduces the field data and for idealised stationary flow scenarios that the results are in agreement with the resonant response predicted by linear theory. Then we explore the effects of non-linearity and time dependence and consider the importance of resonant internal waves for turbulence production in stratified coastal environments.Responsible Editior: Hans Burchard     

Scattering of internal waves in a two-layer fluid flowing through a channel with small undulations

Using two-dimensional linear water wave theory, we consider the problem of normal water wave (internal wave) propagation over small undulations in a channel flow consisting of a two-layer fluid in which the upper layer is bounded by a fixed wall, an approximation to the free surface, and the lower one is bounded by a bottom surface that has small undulations. The effects of surface tension at the surface of separation is neglected. Assuming irrotational motion, a perturbation analysis is employed to calculate the first-order corrections to the velocity potentials in the two-layer fluid by using Green’s integral theorem in a suitable manner and the reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom undulation. Two special forms of the shape function are considered for which explicit expressions for reflection and transmission coefficients are evaluated. For the specific case of a patch of sinusoidal ripples having the same wave number throughout, the reflection coefficient up to the first order is an oscillatory function in the quotient of twice the interface wave number and the ripple wave number. When this quotient approaches one, the theory predicts a resonant interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of the incident wave energy occurs if this number is large. Again, when a patch of sinusoidal ripples having two different wave numbers for two consecutive stretches is considered, the interaction between the bed and the interface near resonance attains in the neighborhood of two (singular) points along the x-axis (when the ripple wave number of the bottom undulation become approximately twice as large as the interface wave number). The theoretical observations are presented in graphical form.     

The 11-year solar cycle, the Sun's rotation, and the middle stratosphere in winter. Part II: Response of planetary waves

The effect of the 11-year solar cycle on the response of planetary wavenumbers 1 and 2 at 10 and 30 hPa in winter to solar activity oscillations on the time scale of the Sun's rotation (27.2 day) is discussed in terms of statistical spectral analysis. The three oscillations studied are the 27.2 d (period of the Sun's rotation), 25.3 d (periodicity caused by modulation of the 27.2 d stratospheric response by annual atmospheric variation), and 54.4 d (doubled period of the solar rotation). A significant effect of the 11-year solar cycle is found for the 54.4 d periodicity in planetary wavenumber 1, and for the 27.2 and 25.3 d periodicities in planetary wavenumber 2. The effect of the 11-year solar cycle is expressed in the evident differences between the amplitudes of responses of planetary waves at maximum and minimum of the solar cycle: the amplitudes are much larger at high than at low solar activity. The 11-year modulation of planetary wave activity is most pronounced at mid-latitudes, mainly at 40–60°N, where the observed variability of planetary waves is large. The results obtained are in good agreement with results of the recent modeling study by Shindell et al. (Science 284 (1999) 305).     

正压大气模式下具有β效应与地形效应的Rossby孤立波

正压大气模式下,采用摄动方法和时空伸长变换推导了具有β效应、地形效应和耗散的mKdV-Burgers方程,得到Rossby孤立波振幅的演变满足带有β效应,地形与耗散的mKdV-Burgersm方程的结论.说明β效应、地形效应是诱导Rossby孤立波的重要因素.     

Oblique wave scattering by undulating porous bottom in a two-layer fluid: Fourier transform approach

The problem involving scattering of oblique waves by small undulation on the porous ocean bed in a two-layer fluid is investigated within the framework of linearised theory of water waves where the upper layer is free to the atmosphere. In such a two-layer fluid, there exist waves with two different wave numbers (modes): wave with lower wave number propagates along the free surface whilst that with higher wave number propagates along the interface. When an oblique incident wave of a particular mode encounters the undulating bottom, it gets reflected and transmitted into waves of both modes so that some of the wave energy transferred from one mode to another mode. Perturbation analysis in conjunction with Fourier transform technique is used to derive the first-order corrections of velocity potentials, reflection and transmission coefficients at both modes due to oblique incident waves of both modes. One special type of undulating bottom topography is considered as an example to evaluate the related coefficients in detail. These coefficients are shown in graphical forms to demonstrate the transformation of water wave energy between the two modes. Comparisons between the present results with those in the literature are made for particular cases and the agreements are found to be satisfactory. In addition, energy identity, an important relation in the study of water wave theory, is derived with the help of the Green’s integral theorem.     

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.     

Variation of overtides by wave enhanced bottom drag in a North Florida tidal inlet

Month-long observations of waves and tidal currents at Ponce de Leon Inlet, North Florida are used to investigate the importance of wave-induced bottom drag as a mechanism for overtide generation in estuaries. While bottom drag can in theory lead to overtide generation, in practice, resolving unambiguously this effect is difficult as it tends to be overshadowed by the stronger effect of diurnal–semidiurnal tidal variance. Bottom boundary layer numerical simulations based on observational data suggest that waves can cause the bottom drag experienced by currents to increase by a factor of 1.7, compared with relatively calm conditions. Despite the relatively short duration and limited scope of the experiment, the analysis suggests that overtide modulations (East–West velocity components of the 5th and 6th diurnal constituents) are correlated with wave-enhanced drag trends. Therefore, wave-enhanced bottom drags may be enhancing generation of overtides. Further work is necessary to understand the scope and the strength of this mechanism, in relation to the characteristics (e.g., flow direction) of individual overtides.     

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.     

Theoretical analyses and numerical experiments of variational assimilation for one-dimensional ocean temperature model with techniques in inverse problems

There is an international focus on the develop-ments of data assimilation systems for meteorology and physical oceanography models. Data assimilation and inverse methods are normally used for optimal control of poorly known initial boundary conditions and model parameters by taking into account both the information about dynamics of a model and the infor-mation about the true state which is constrained by a set of measurements. The research methodology of parameter estimation in meteorology ca…     

Residual vorticity induced by the action of tidal currents in combination with bottom topography in the Southern Bight of the North Sea

Abstract

Results are presented of calculations on the generation of residual vorticity by tidal currents over the bottom topography of the Southern Bight of the North Sea. A typical order of magnitude is 10^?6 to 10^?7 s ^?1. This is compared with current measurements on calm days, when similar magnitudes are found. At windspeeds less than about 5 m/s tidal generation of residual vorticity is important; at higher windspeeds wind effects begin to dominate. Our results are relevant in understanding the spatial variability of residual currents, because a non-zero vorticity implies the existence of horizontal gradients in the residual current field.     

含起伏地形地层中点源激发的震电响应研究

地震诱导电磁现象是国内外地学领域十分关注的前沿问题,前人对地震波和电磁场耦合波场的认识主要是基于规则模型获得的.为研究含起伏地表和地下界面的地层中震电波场激发、传播特性,本文采用有限元软件COMSOL Multiphysics模拟点震源激发的电磁场.首先给出频率域二维SHTE模式震电耦合方程组,然后利用COMSOL软件建立计算模型,并求解出点力源激发震电波场的频率域响应,最后利用FFT变换得到地震波场和电磁场的时间域波形.模拟结果表明,震电波场中存在三种类型的电磁信号,第一种是震源直接激发的电磁波;第二种是地震波在分界面处激发的电磁波（包括自由表面、地下不同介质分界面）;第三种是伴随地震波的同震信号,前两种电磁波比地震波更早到达远处观测台站,对地震预警有重要意义.此外,研究还发现：当地震波传播至地表并沿着地表传播时,在地表附近空气层中同样记录到了伴随地震波传播的电磁扰动信号,该信号与相同水平源距条件下、地下观测点接收到的电磁信号相同,这与前人的一些观测结果相符.本文研究结果为今后地震电磁信号的解释提供了理论证据.

     

Spatio‐temporal changes in bog pool bottom topography – temperature effect and its influence on pool development: an example from a raised bog in Estonia

Increases in pool water and peat temperature in summer accelerate peat decomposition and production of biogenic gases, which can be trapped in peat pores and cause oscillation of peatland surfaces and the rise of peat from the bottom of bog pools. Associated changes in peat water conductivity, holding capacity and transpiration also affect bog hydrology. Our multi‐year study is the first to show in detail the extent and dynamics of changes in bog pool depth and bottom topography associated with changes in temperature, peat type and other factors. The true seasonal rise of peat from the pool bottom begins once the water temperature at the pool bottom exceeds 13–14 °C, although the speed and extent of the rise depends on peat properties, making the rise more erratic than its subsequent descent. The more rapid descent occurs after the first large drop in the temperature of the pool's surface water at the end of summer, resulting from the combination of reduced methane production and increased gas solubility with less influence by peat properties. Much higher dissolved organic carbon concentrations (216 ± 26 mg l^?1) in the pore water of peat risen from the bottom to the pool surface compared with that in the same type of peat at the pool bottom (62 ± 20 mg l^?1) indicate an acceleration of peat decomposition at the warmer pool surface. We show the extent and character of changes in pool depth and bottom topography and how annual differences relate to temperature. Only a few degrees' increase in pool water temperature could induce the pool bottom to rise faster and more extensively for a longer period and enhance decomposition in the peat at the pool surface. This should be evaluated in greater detail to assess the effects of temperature increase on the carbon budget and hydrology of peatlands. Copyright © 2012 John Wiley & Sons, Ltd.