On the solution stability in the problem of determining the time variations of the tidal responses of a medium in the vicinity of the sources of severe earthquakes

In (Molodenskii M.S. et al., 2016), the data from horizontal pendulums recording the tilts in the closest vicinity of the Great Tohoku earthquake of March 11, 2011 in Japan were analyzed. A significantly improved method for statistical analysis of the observational data enabled the authors to reveal a slow growth in tidal tilts during a period of six years before the earthquake, which was superseded by an instantaneous drop in the amplitudes at the time of the earthquake. After this, during the subsequent four years, the tidal amplitudes have remained at a significantly lower level than their average values before the earthquake. These changes in tidal amplitudes testify to the nonlinear character of the tidal response of the medium in the presence of large tectonic stresses: as is well known, the linear relationship between stresses and strains in a real medium is only the case for stresses that are far below the yield stress. When the stresses approach the failure limit, two counteracting effects come into play: (1) the shear moduli in some areas decrease as a result of the avalanche growth of the crack formation processes, and (2) the moduli increase due to the compression in the other areas. Irrespective of which particular effect of these two is predominant, in either case the linearity of the relationship between the stresses and strains should be violated. This violation cannot but affect the amplitudes of the tidal tilts and strains characterizing this relationship in the presence of fairly low additional tidal stresses (i.e., the derivative of the off-diagonal stress tensor components with respect to the same components of the strain tensor). Since there is presently a sufficiently dense network of the horizontal pendulums recording the tilts (the global IRIS network and the particularly dense F-NET network in Japan), monitoring the changes in the amplitudes of tidal tilts can be considered as a key instrument for capturing the signs of the approach of tectonic stresses to their critical values. The increase in tidal amplitudes before the Tohoku earthquake and their drop at the moment of the earthquake, which were revealed by us, as well as the constancy of the amplitudes during four years after the event, unambiguously indicate that the accumulation of tectonic stresses caused the growth in tidal amplitudes, whereas the stress release by the earthquake caused their diminution. This does not however mean that stress accumulation is accompanied by a decrease in the elastic moduli and that the release of stresses is accompanied by the growth of elastic moduli all over the source area. As was shown in (Molodenskii M.S. et al., 2012), even in the simplest model of spatially homogeneous variations of elastic modules, the variations in tidal tilts are an odd function of the distance from the epicenter. Therefore, irrespective of whether the elastic moduli decrease or increase, the amplitudes of tidal tilts should decrease in some areas and increase in other areas. Hence, the very fact of the growth of tidal tilt amplitudes with time cannot be considered as a sign of the growth of tectonic stresses. To be positive about the latter, one should make sure that the consistent (unidirectional) changes have been observed during a sufficiently long time interval and that their magnitudes were significantly larger than the measurement errors. Hence, it is important to reliably estimate the errors of the observational data.     

A simple numerical model for inlet sedimentation at intermittently open–closed coastal lagoons

This contribution aims to model the dominant processes that control sedimentation within the ocean inlet to intermittently open–closed coastal lagoons; focussing on the role of infilling, backfilling and morphodynamic feedback. The key elements that have been included in the present model are: (1) the delivery of sediment to the mouth of the estuary by littoral processes; (2) sediment transport processes within the inlet due to non-linear tidal flow; and (3) the down-slope diffusion of sediment. The model discussed here includes a simple one-dimensional (1-D), non-linear flow parameterisation that predicts the spatial variability in the magnitude and non-linearity of the tidal flow. The predicted third and fourth velocity moments are used to drive a morphodynamic module. Down-slope diffusion of sediment is dealt with in a separate diffusion term in the bed-evolution equation. Feedback between the evolving morphology and the flow field are integral to the model. Numerical simulations are used to investigate different modes of evolution for this type of inlet system. Inlet closure due to infilling is critically controlled by the balance between sediment addition at the mouth due to littoral processes and the removal of sediment by non-linear tidal flow. Rapid widening or deepening of the inlet at its landward margin leads to the deposition of a flood shoal. Under conditions of high sedimentation (and low diffusion) the flood shoal can become sufficiently well developed to present a physical barrier to sediment entering the lagoon. Under these circumstances backfilling can become significant. The infilling and backfilling processes are ameliorated by efficient down-slope diffusion which is controlled in the present model by a diffusion parameter, D. High diffusion coefficients slow inlet closure and allow sediment to propagate further into the lagoon.     

Multidimensional nonlinear ion-acoustic waves in a plasma in view of relativistic effects

The structure and dynamics of ion-acoustic waves in an unmagnetized plasma, including the case of weakly relativistic collisional plasma (when it is necessary to take into account the high energy particle flows which are observed in the magnetospheric plasma), are studied analytically and numerically on the basis of a model of the Kadomtsev-Petviashvili (KP) equation. It is shown that, if the velocity of plasma particles approaches the speed of light, the relativistic effects start to strongly influence on the wave characteristics, such as its phase velocity, amplitude, and characteristic wavelength, with the propagation of the twodimensional solitary ion-acoustic wave. The results can be used in the study of nonlinear wave processes in the magnetosphere and in laser and astrophysical plasma.     

Tidal inlet response to sediment infilling of the associated bay and possible implications of human activities: the Marennes-Oléron Bay and the Maumusson Inlet,France

Tidal inlet characteristics are controlled by wave energy, tidal range, tidal prism, sediment supply and direction and rates of sand delivered to the inlet. This paper deals with the relations between inlet and lagoon evolutions, linked by the tidal prism. Our study is focused on the Maumusson Inlet and the Marennes-Oléron Bay (first oyster farming area in Europe), located on the western coast of France. The tidal range (2–6 m) and wave climate (mean height: 1.5 m) place this tidal inlet system in the mixed energy (tide, waves), tide-dominated category. The availability of high-resolution bathymetric data since 1824 permits to characterise and quantify accurately morphological changes of both the inlet and the tidal bay. Since 1824, sediment filling of the tidal bay has led to a 20% decrease in its water volume, and a 35% reduction of the inlet throat section. Furthermore, the bay is subjected to a very high anthropic pressure, mainly related to oyster farming. Thus, both natural and human-related processes seem relevant to explain high sedimentation rates. Current measurements, hydrodynamic modelling and cross-sectional area of the inlet throat are used in order to quantify tidal prism changes since 1824. Both flood and ebb tidal prism decreased by 35%. Decrease in the Marennes-Oléron Bay water volume is inferred to be responsible for a part of tidal prism decrease at the inlet. Tidal prisms decrease may also be explained by an increase in frictional resistance to tidal wave propagation, due to a general shoaling and oyster farms in the bay. A conceptual model is proposed, taking into account natural and human-related sedimentation processes, and explaining tidal inlet response to tidal bay evolutions.     

Solution of the linear diffusion equation for modelling erosion processes with a time varying diffusion coefficient

In the present paper the differential equation of the temporal development of a landform (mountain) with a time dependent diffusion coefficient is solved. It is shown that the shape and dimensions of the landform at time t are independent of the specific variation of the diffusion coefficient with time; they only depend on the mean value of the diffusion coefficient in the time interval where the erosion process takes place. Studying the behaviour of the solution of the differential equation in the wave number domain, it is concluded that Fourier analysis may help in estimating, in quantitative terms, the initial dimensions, the age or, alternatively, the value of the diffusion coefficient of the landform. The theoretical predictions are tested on a hill of the southern part of the Ural mountainous region, in order to show how the results of the mathematical analysis can be used in describing, in quantitative terms, the morphological development of landforms due to erosion processes. Copyright © 2007 John Wiley & Sons, Ltd.     

Studying Long-term Variations in the Caspian Sea with the Use of the Theory of Stochastic Differential Equations

Nonlinear mechanisms of long-term variations in the Caspian Sea level are described. It is shown that with account taken of the dependence of the evaporation depth from the Volga basin surface on soil moisture content and the dependence of the evaporation depth from the sea surface on its level, we obtain a fundamentally new (chaotic) oscillation mechanism with several attraction levels. The stochastic differential equations describing the water budget of the sea basin and the sea proper and the respective solutions of the Fokker–Planck–Kolmogorov equation are shown to have stationary bimodal density of the level probability. The random process, characterizing the sea level variations at a nonlinear dependence between the evaporation rate and the level is found to be non-Gaussian. Noise-induced transitions, caused by nonlinear evaporation processes are described. A new nonlinear stochastic theory describing the Caspian Sea level variations and based on predicted physical effects is suggested.     

An experimental study of dispersion of an interactive tracer in chemically heterogeneous medium at a column scale

This paper aims at examining the increase of phenol adsorption breakthrough curves spreading caused by the chemical heterogeneity of granular activated carbon fixed beds. The local and the thermodynamic equilibrium assumption, as well as the nonlinear adsorption obeying to Langmuir isotherm, are considered. This study particularly tempts to link the reduced variance of phenol breakthrough curves to a measurable quantifying parameter of the chemical heterogeneity. The investigated artificial heterogeneous media are prepared by alternating layers of two types of granular activated carbon, active and non-active ones, that have similar physical properties. On the one hand, the chemical heterogeneity is quantified by the active layer relative thickness of the column length, l₁/L. On the other hand, it is quantified by the mean value of the probability distribution γ. The latter also represents the mean active grains mass ratio of the total medium mass, hence the medium mean capacity. The obtained results show an increase in the reduced variance and thus the effective global dispersion with the heterogeneity; the increase is as important as the medium capacity decreases. However, the dispersion increase achieves a limit value, even when the heterogeneity increases. The results are statistically modelled using a regression equation function of the capacity variation in terms of γ and the chemical heterogeneity in terms of l₁/L. The relationship combining the medium capacity and the chemical heterogeneity is obtained. The relationship implicitly takes into account the effect of the column length.     

On the using cumulant expansion method and van Kampen’s lemma for stochastic differential equations with forcing

Second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is obtained by using the cumulant expansion ensemble averaging method and by taking the time dependent sure part of the multiplicative operator into account. It is shown that the satisfaction of the commutativity and the reversibility requirements proposed earlier for linear stochastic differential equations without forcing are necessary for the linear stochastic differential equations with forcing when the cumulant expansion ensemble averaging method is used. It is shown that the applicability of the operator equality, which is used for the separation of operators in the literature, is also subjected to the satisfaction of the commutativity and the reversibility requirements. The van Kampen’s lemma, which is proposed for the analysis of nonlinear stochastic differential equations, is modified in order to make the probability density function obtained through the lemma depend on the forcing terms too. The second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is also obtained by using the modified van Kampen’s lemma in order to validate the correctness of the modified lemma. Second-order exact ensemble averaged equation for one dimensional convection diffusion equation with reaction and source is obtained by using the cumulant expansion ensemble averaging method. It is shown that the van Kampen’s lemma can yield the cumulant expansion ensemble averaging result for linear stochastic differential equations when the lemma is applied to the interaction representation of the governing differential equation. It is found that the ensemble averaged equations given for one the dimensional convection diffusion equation with reaction and source in the literature obtained by applying the lemma to the original differential equation are restricted with small sure part of multiplicative operator. Second-order exact differential equations for the evolution of the probability density function for the one dimensional convection diffusion equation with reaction and source and one dimensional nonlinear overland flow equation with source are obtained by using the modified van Kampen’s lemma. The equation for the evolution of the probability density function for one dimensional nonlinear overland flow equation with source given in the literature is found to be not second-order exact. It is found that the differential equations for the evolution of the probability density functions for various hydrological processes given in the literature are not second-order exact. The significance of the new terms found due to the second-order exact ensemble averaging performed on the one dimensional convection diffusion equation with reaction and source and during the application of the van Kampen’s lemma to the one dimensional nonlinear overland flow equation with source is investigated.     

Simultaneous measurements of tidal straining and advection at two parallel transects far downstream in the Rhine ROFI

This study identifies and unravels the processes that lead to stratification and destratification in the far field of a Region of Freshwater Influence (ROFI). We present measurements that are novel for two reasons: (1) measurements were carried out with two vessels that sailed simultaneously over two cross-shore transects; (2) the measurements were carried out in the far field of the Rhine ROFI, 80 km downstream from the river mouth. This unique four dimensional dataset allows the application of the 3D potential energy anomaly equation for one of the first times on field data. With this equation, the relative importance of the depth mean advection, straining and nonlinear processes over one tidal cycle is assessed. The data shows that the Rhine ROFI extends 80 km downstream and periodic stratification is observed. The analysis not only shows the important role of cross-shore tidal straining but also the significance of along-shore straining and depth mean advection. In addition, the nonlinear terms seem to be small. The presence of all the terms influences the timing of maximum stratification. The analysis also shows that the importance of each term varies in the cross-shore direction. One of the most interesting findings is that the data are not inline with several hypotheses on the functioning of straining and advection in ROFIs. This highlights the dynamic behaviour of the Rhine ROFI, which is valuable for understanding the distribution of fine sediments, contaminants and the protection of coasts.     

Small-scale magnetic buoyancy and magnetic pumping effects in a turbulent convection

We determine the nonlinear drift velocities of the mean magnetic field and nonlinear turbulent magnetic diffusion in a turbulent convection. We show that the nonlinear drift velocities are caused by three kinds of the inhomogeneities; i.e., inhomogeneous turbulence, the nonuniform fluid density and the nonuniform turbulent heat flux. The inhomogeneous turbulence results in the well-known turbulent diamagnetic and paramagnetic velocities. The nonlinear drift velocities of the mean magnetic field cause the small-scale magnetic buoyancy and magnetic pumping effects in the turbulent convection. These phenomena are different from the large-scale magnetic buoyancy and magnetic pumping effects which are due to the effect of the mean magnetic field on the large-scale density stratified fluid flow. The small-scale magnetic buoyancy and magnetic pumping can be stronger than these large-scale effects when the mean magnetic field is smaller than the equipartition field. We discuss the small-scale magnetic buoyancy and magnetic pumping effects in the context of the solar and stellar turbulent convection. We demonstrate also that the nonlinear turbulent magnetic diffusion in the turbulent convection is anisotropic even for a weak mean magnetic field. In particular, it is enhanced in the radial direction. The magnetic fluctuations due to the small-scale dynamo increase the turbulent magnetic diffusion of the toroidal component of the mean magnetic field, while they do not affect the turbulent magnetic diffusion of the poloidal field.     

Analytical approximations for flow in compressible,saturated, one-dimensional porous media

A nonlinear model for single-phase fluid flow in slightly compressible porous media is presented and solved approximately. The model assumes state equations for density, porosity, viscosity and permeability that are exponential functions of the fluid (either gas or liquid) pressure. The governing equation is transformed into a nonlinear diffusion equation. It is solved for a semi-infinite domain for either constant pressure or constant flux boundary conditions at the surface. The solutions obtained, although approximate, are extremely accurate as demonstrated by comparisons with numerical results. Predictions for the surface pressure resulting from a constant flux into a porous medium are compared with published experimental data.     

Mass transport in rotatory tidal currents

Summary A perturbation method is used to calculate steady non-linear effects in rotatory tidal currents. The inclusion of the Coriolis acceleration leads to a uniquely determined second order current system in the tidal stream. This is shown to be absent in high frequency gravity waves. It is found that particle drift, or mass transport, is critically dependent on this mean current, and consequently on the value off/,f being the Coriolis parameter, and the angular frequency of the oscillation.     

Deformation autowaves in fault zones

Empirical data on the spatiotemporal migration of recent deformation processes in fault zones are presented. Direct geodesic measurements reveal two types of waves, the interfault and intrafault waves. The velocity of the interfault wave is two-threefold higher than that of the intrafault wave. The phenomenological model describing the formation of autowave deformation processes is developed. The obtained nonlinear diffusion equation for the displacements of the Earth’s surface is shown to be structurally similar to the well-known Kolmogorov-Petrovsky-Piskunov equation derived in the population wave theory. Satisfactory agreement between the suggested model and the geodetic measurements is demonstrated. The concept of a pseudowave is discussed, and a way to expand the developed approaches to the spatiotemporal migration of earthquakes is discussed.     

Hydrodynamic effects at the entry of tidal waves into estuaries

The phenomenon of an increase in tidal wave height in cone-shaped estuaries is studied. The effect of estuary narrowing in the direction of tidal wave propagation (the hydrodynamic effect of confusor) is among the factors amplifying the tide. An opposite effect of turbulent friction, whose manifestation increases with decreasing bay’s depth, conversely, reduces tide amplitude because of the dissipation of tidal wave energy. Stokes diffusion layer also plays a significant role in the formation of wave transformation regime. In an estuary with a median depth, which is much greater than the Stokes layer thickness, the confusor effect is stronger and tide amplitude increases at estuary head. At depths lesser than Stokes layer thickness, the turbulent friction dominates over the confusor effect and the amplitude of tidal wave decreases at the head of the estuary. The depths of the order of Stokes layer thickness cause an interesting intermediate phenomenon—at the entrance into the estuary, first the effect of friction manifests itself, resulting in a decrease in the amplitude of tidal wave, but later, the effect of confusor starts dominating, and the amplitude of tidal wave again increases toward estuary head. When the period of tidal wave coincides with seiche period, a resonance enhancement of seiche oscillations takes place in the estuary.     

Nonlinear advection,Coriolis force,and frictional influence in the South Channel of the Yangtze Estuary,China

Observation data of along-estuary and lateral current velocities over a transect located at the South Channel of the Yangtze estuary was obtained during a spring tide in August 2011.Harmonic analysis was done on the current velocities to get a mean component and a semi-diurnal component.Based on these two components,the driving mechanisms of mean lateral flow and M2 lateral tidal flow are shown and analyzed respectively.The dominant driving force of mean lateral flow is nonlinear advection and that of lateral M2 tidal flow is Coriolis force.The friction plays an important role near the bottom and surface for both lateral mean flow and M2 tidal flow.     

Numerical computation of the tidally induced Lagrangian residual current in a model bay

With a depth-averaged numerical model, the tidally induced Lagrangian residual current in a model bay was studied. To correctly reflect the long-term mass transport, it is appropriate to use the Lagrangian residual velocity (LRV) rather than the Eulerian residual velocity (ERV) or the Eulerian residual transport velocity (ETV) to describe the residual current. The parameter κ, which is defined as the ratio of the typical tidal amplitude at the open boundary to the mean water depth, is considered to be the indicator of the nonlinear effect in the system. It is found that the feasibility of making the mass transport velocity (MTV) approximate the LRV is strongly dependent on κ. The error between the MTV and the LRV tends to increase with a growing κ. An additional error will come from the various initial tidal phases due to the Lagrangian drift velocity (LDV) when κ is no longer small. According to the residual vorticity equation based on the MTV, the Coriolis effect is found to influence the residual vorticity mainly through the curl of the tidal stress. A significant difference in the flow pattern indicates that the LRV is sensitive to the bottom friction in different forms.     

鲁14井数字化水位潮汐响应特性初析

以井—含水层系统潮汐理论为基础,利用Baytap-G潮汐分析程序计算鲁14井数字化水位的潮汐响应特征参数,提取井水位潮汐因子和相位差,分析远场大震前后井水位潮汐响应特征参数的变化情况,同时对固体潮和气压影响量进行分离,以期深入了解井水位微动态变化特征,为该井水位观测资料的动态评价和效能评估提供参考。结果表明：鲁14井潮汐地下水流类型以径向流为主,潮汐因子和相位差变化相对稳定,但受远场大震影响显著,其含水层渗透性随潮汐参数的增大（减小）而增大（减小）;地震波作用可使含水层渗透性发生变化,但并非导致含水层渗透性变化的主要因素。     

Richardson number profiles in laboratory experiments applied to shallow seas

Abstract

A unified analysis is given of the critical conditions for the onset of stratification due to either a vertical or a horizontal buoyancy flux, with tidal or wind stirring.

The critical conditions for the onset of stratification with a horizontal buoyancy flux are found to be of the form of ratios of the tidal slope, or wind setup, to the equivalent surface slope due to the lateral density gradient. These ratios, which are easily determined from sea data, indicate that the profiles of critical flux Richardson Number, averaged over the stirring cycle, are similar to those inferred from the laboratory experiments of Hopfinger and Linden (1982) in which there is zero mean shear turbulence with a stabilising buoyancy flux, and also that the efficiency for the conversion of kinetic energy to potential energy for tidal stirring is similar to that for wind stirring.

The observed much greater efficiency for wind stirring, compared with tidal stirring with a vertical buoyancy flux, is also consistent with the existence of flux Richardson Number profiles in the sea similar to those occurring in the corresponding laboratory experiments. Using the solution of the turbulent kinetic energy equation for the water column, the relative importance of the production of turbulent kinetic energy, and its diffusion by turbulence are assessed, and the critical conditions for the onset of stratification with a vertical buoyancy flux are shown to reduce the classical Simpson—Hunter form.     

A master equation for reactive solute transport in porous media

The mean value of a density of a cloud of points described by a generalized Liouville equation associated with a convection dispersion equation governing adsorbing solute transport yields a joint concentration probability density. The general technique can be applied for either linear or nonlinear adsorption; here the application is restricted to linear adsorption in one-dimensional transport. The equation generated for the joint concentration probability density is in the general form of a Fokker-Planck equation, but with a suitable coordinate transformation, it is possible to represent it as a diffusion equation with variable coefficients.