On the effect of topography on surface wave propagation in the ambient noise frequency range

Due to the increasing popularity of analyzing empirical Green’s functions obtained from ambient seismic noise, more and more regional tomographical studies based on short-period surface waves are published. Results could potentially be biased in mountainous regions where topography is not small compared to the wavelength and penetration depth of the considered waves. We investigate the effect of topography on the propagation of short-period Rayleigh waves empirically by means of synthetic data using a spectral element code and a 3-D model with real topography. We show that topography along a profile through the studied area can result in an underestimation of phase velocities of up to about 0.7% at the shortest investigated period (3 s). Contrary to the expectation that this bias results from the increased surface distance along topography, we find that this error can be estimated by local topographic contrasts in the vicinity of the receiver alone. We discuss and generalize our results by considering topographic profiles through other mountain ranges and find that southern Norway is a good proxy to assess the topography effect. Nevertheless, topographic bias on phase velocity measurements is in general not large enough to significantly affect recovered velocity variations in the ambient noise frequency range.     

A local stability analysis of astronomical disks

Abstract

A fifth-order dispersion relation describing the local stability of a differentially rotating flow against small perturbations is derived. Finite viscosity and conductivity and both vertical (parallel to the rotation axis) and radial gradients in density, temperature and pressure are included. A general form is assumed for the equation of state, although this is not exploited in the paper. A number of special cases are studied: with negligible viscosity and conductivity, it is shown that modes can often be separated into two high frequency (modified acoustic), two intermediate frequency (combined inertial and internal waves) and a low frequency mode. In convectively unstable situations the intermediate frequency modes may be replaced by a damped/growing pair of instablities. Various criteria for mode excitation are given. It is shown that viscosity always inhibits instability at very short wavelengths, while non-zero conductivity may destabilize the flow. At intermediate wavelengths viscosity could also play a destabilizing role. A parameter study of the effects of fluctuations in the conductivity shows that it could cause mode excitation under certain circumstances.     

Solitary and cnoidal planetary waves

Abstract

A class of long planetary waves in a zonal channel analogous to the solitary and cnoidal waves of surface and internal gravity wave theory is discussed. On a mid-latitude β-plane, such waves exist as the result of divergence, non-uniform zonal velocity fields or bottom topography. In all cases studied the wave profile along the channel was found to satisfy the Korteweg-de Vries equation.     

三维台阶地形地震动效应研究

运用谱元法模拟了不同类型三维台阶地形对点源地震动的影响, 发现在中小地震中, 大型山系边缘的三维台阶地形对地表地震波具有一定的影响. 台阶斜坡与上水平地层交线和下水平地层交线的影响作用各不相同. 通过地表接收到的地震动描述了各种不同地形的波形快照特点, 并且通过绘制的台阶斜坡与上水平地层交线及下水平地层交线的测线图, 发现波形相对于水平地形有明显的后续波. 同时发现, 台阶地形上下测线相对于水平地形具有放大作用.下测线波形放大倍数基本在1.1—1.5, 上测线放大倍数在0.9—1之间. 通过描述山体边缘台阶结构在点源地震中产生的影响, 定量计算了三维台阶不同角度各个位置相对于水平地形的放大倍数, 为以后工程研究提供一个依据.     

Sunspots

Abstract

Some examples of research on structure and formation of sunspots are briefly recollected in historical sequence. They relate to many facets of sunspots, first: magnetic inhibition of convection, the conjecture of a fiat penumbra, the stratification beneath the umbra, the observable magnetic profile, the Evershed effect as syphon flow, the concept of a magnetopause; next: cooling by Alfven waves, evolution and stability, the “bright ring”, the observed change of umbra brightness with the phase of the sunspot cycle, the hypothetical cluster of separate flux strands underneath the umbra, the profile of the magnetopause, the structure of the penumbra and the inclination of its field and finally: the concept of a deep penumbra with volume currents, exchange convection and the concept of a second current sheet separating umbra and penumbra.

Of course, the rigorous theoretical modeling of local magnetoconvection is an essential tool for our understanding of all these processes. I do not deal with it here, but the reader has a fascinating review of magnetoconvection already in his hands (Weiss, 1991).     

A large class of non-zonal oceanic flows satisfying the arnold-blumen sufficient condition for stability

Abstract

Recently Andrews has discussed an example of a topographically-forced non-zonal now satisfying the Arnold-Blumen sufficient condition for stability. At large distances from the topographic centre this flow becomes purely zonal and westward. After underlining the richness of solutions of the Andrews model, the present paper goes on to show that Andrews' technique can be applied successfully to a model where the vertical profile of static stability resembles those found in the ocean. In particular we obtain a large class of hydrodynamic stable flows, forced by the bottom topography, for continuously stratified fluids (two layers each with uniform Brunt-Väisälä frequency).     

Structure of the inner core boundary

Abstract

A model of the inner-core boundary (ICB) is constructed which is consistent with current ideas of the dynamic and thermodynamic state of the core and which is capable of reflecting seismic waves with period of one second. This requires the mass fraction of solid below the ICB to grow to an appreciable fraction in roughly one kilometer. This rapid growth of solid with depth is a result of downward fluid flow from the outer core which is a part of the convective motions which sustain the geodynamo. The solid which crystallizes from this descending fluid after it crosses the ICB continually coats the dendrites which occur there. The gradual cooling of the outer core causes the ICB to advance by growth of dendrites at their tips. The balance of these two effects gives an equilibrium profile for the mass fraction of solid with depth below the ICB which is capable of yielding sharp reflection of seismic waves.     

Magneto-acoustic-gravity waves in the horizontal magnetic field

Abstract

In this paper we consider the propagation of magneto-acoustic-gravity waves in a compressible, conducting isothermal atmosphere permeated by a uniform horizontal magnetic field. The singular levels, arising in a horizontal magnetic field, are considered in their most general form. Exact analytical solutions for a number of particular cases of wave propagation are obtained. The wave transformation is analyzed for all these cases using the solutions obtained.

Based on the theory of wave propagation across a magnetic field, low-frequency wave trapping in a chromospheric resonator is explained, and some properties of running penumbral waves are discussed.     

Frictional effects on trapped shelf waves

This paper revisits models of coastal trapped waves and examines them with the inclusion of friction. In addition we use a numerical method which at least theoretically will have enhanced accuracy. The model has continuous stratification and variable depth. It is linear, quasigeostrophic and hydrostatic, and utilizes perturbation from this fundamental state. The model is applied with both idealized and realistic topography and stratification.Responsible Editior: Phil Dyke     

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     

Symmetry breaking in a nonlinear αω-dynamo

Abstract

An idealized nonlinear αω-dynamo is investigated. Emphasis is placed upon the different spatial symmetries, and the asymmetries that arise after secondary bifurcations. On varying the main control parameter D (the dynamo number), many transitions are found involving solutions without an equatorial symmetry, and solutions with quasiperiodic time dependence, but no chaos. Instead of a cascade to smaller spatial scales when D is highly supercritical it is found that additional asymmetries are introduced at tertiary bifurcations. Our complete bifurcation diagrams allow us to follow in detail how stability is passed from one solution to another as D varies. In these diagrams there are typically multiple stable solutions at any value of D, which suggests that similar stars can have different magnetic patterns.     

Barotropic waves along an eastern continental shelf

Abstract

An analysis is presented of the propagation of barotropic non-divergent oscillations along the western side of an ocean basin along which the persistent circulation in the basin is strongly intensified and laterally sheared. Because the Rossby number of a western boundary current is near unity, the properties of these waves are strongly affected by the steady circulation pattern. It is shown that for relatively long wavelengths, these waves can travel along the shelf in both directions; however, for a small range of short wavelengths they can only propagate northward and are unstable. Along the southeastern coast of North America, the unstable waves have wavelengths of order 150 km and periods of order 10 days. However, these waves can become stable oscillations in the deeper water northeast of Cape Hatteras. These oscillations are a possible explanation of the initiation of Gulf Stream meanders along the continental rise.     

Modelling the influence of spatially varying hydrodynamics on the cross-sectional stability of double inlet systems

The cross-sectional stability of double inlet systems is investigated using an exploratory model that combines Escoffier’s stability concept for the evolution of the inlet’s cross-sectional area with a two-dimensional, depth-averaged (2DH) hydrodynamic model for tidal flow. The model geometry consists of four rectangular compartments, each with a uniform depth, associated with the ocean, tidal inlets and basin. The water motion, forced by an incoming Kelvin wave at the ocean’s open boundary and satisfying the linear shallow water equations on the f -plane with linearised bottom friction, is in each compartment written as a superposition of eigenmodes, i.e. Kelvin and Poincaré waves. A collocation method is employed to satisfy boundary and matching conditions. The analysis of resulting equilibrium configurations is done using flow diagrams. Model results show that internally generated spatial variations in the water motion are essential for the existence of stable equilibria with two inlets open. In the hydrodynamic model used in the paper, both radiation damping into the ocean and basin depth effects result in these necessary spatial variations. Coriolis effects trigger an asymmetry in the stable equilibrium cross-sectional areas of the inlets. Furthermore, square basin geometries generally correspond to significantly larger equilibrium values of the inlet cross-sections. These model outcomes result from a competition between a destabilising (caused by inlet bottom friction) and a stabilising mechanism (caused by spatially varying local pressure gradients over the inlets).     

A model for grain-size sorting over tidal sand ridges

A model was developed and analyzed to quantify the effect of graded sediment on the formation of tidal sand ridges. Field data reveal coarse (fine) sediment at the crests (in the troughs), but often phase shifts between the mean grain-size distribution and the bottom topography occur. Following earlier work, this study is based on a linear stability analysis of a basic state with respect to small bottom perturbations. The basic state describes an alongshore tidal current on a coastal shelf. Sediment is transported as bed load and dynamic hiding effects are accounted for. A one-layer model for the bed evolution is used and two grain size classes (fine and coarse sand) are considered. Results indicate an increase in growth and migration rates of tidal sand ridges for a bimodal mixture, whilst the wavelength of the ridges remains unchanged. A symmetrical externally forced tidal current results in a grain-size distribution which is in phase with the ridges. Incorporation of an additional external M₄ tidal constituent or a steady current results in a phase shift between the grain-size distribution and ridge topography. These results show a general agreement with observations. The physical mechanism responsible for the observed grain-size distribution over the ridges is also discussed.Responsible Editor: Jens Kappenberg     

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.     

The generation of baroclinic Rossby waves by stationary and and translating currents part 1. Stationary current forcing

Abstract

This paper investigates the generation of linear, baroclinic Rossby waves by an imposed current distribution, in a reduced gravity ocean, both with and without an eastern coast. A zonal current is impulsively applied and maintained along the northern edge of the domain of solution. Using Green's function techniques, analytical solutions are found, and these are evaluated for small times. Numerical solutions are obtained for larger times. The upper layer depth field consists of a transient response, due to the sudden application of the current. Maintenance of the current causes a response which is singular along the line of imposed non-zero h _y. The interior field decays with time (this is shown asymptotically). The parameters used are appropriate for the mid-latitude North Pacific, and the results are relevant to sudden transport changes in the North Pacific Current.     

Modelling the influence of spatially varying hydrodynamics on the cross-sectional stability of double inlet systems

The cross-sectional stability of double inlet systems is investigated using an exploratory model that combines Escoffier’s stability concept for the evolution of the inlet’s cross-sectional area with a two-dimensional, depth-averaged (2DH) hydrodynamic model for tidal flow. The model geometry consists of four rectangular compartments, each with a uniform depth, associated with the ocean, tidal inlets and basin. The water motion, forced by an incoming Kelvin wave at the ocean’s open boundary and satisfying the linear shallow water equations on the f -plane with linearised bottom friction, is in each compartment written as a superposition of eigenmodes, i.e. Kelvin and Poincaré waves. A collocation method is employed to satisfy boundary and matching conditions. The analysis of resulting equilibrium configurations is done using flow diagrams.

Model results show that internally generated spatial variations in the water motion are essential for the existence of stable equilibria with two inlets open. In the hydrodynamic model used in the paper, both radiation damping into the ocean and basin depth effects result in these necessary spatial variations. Coriolis effects trigger an asymmetry in the stable equilibrium cross-sectional areas of the inlets. Furthermore, square basin geometries generally correspond to significantly larger equilibrium values of the inlet cross-sections. These model outcomes result from a competition between a destabilising (caused by inlet bottom friction) and a stabilising mechanism (caused by spatially varying local pressure gradients over the inlets).