Using Grain‐Size Distribution Methods for Estimation of Air Permeability

Knowledge of air permeability (k_a) at dry conditions is critical for the use of air flow models in porous media; however, it is usually difficult and time consuming to measure k_a at dry conditions. It is thus desirable to estimate k_a at dry conditions from other readily obtainable properties. In this study, the feasibility of using information derived from grain‐size distributions (GSDs) for estimating k_a at dry conditions was examined. Fourteen GSD‐based equations originally developed for estimating saturated hydraulic conductivity were tested using k_a measured at dry conditions in both undisturbed and disturbed river sediment samples. On average, the estimated k_a from all the equations, except for the method of Slichter, differed by less than ± 4 times from the measured k_a for both undisturbed and disturbed groups. In particular, for the two sediment groups, the results given by the methods of Terzaghi and Hazen‐modified were comparable to the measured k_a. In addition, two methods (e.g., Barr and Beyer) for the undisturbed samples and one method (e.g., Hazen‐original) for the undisturbed samples were also able to produce comparable k_a estimates. Moreover, after adjusting the values of the coefficient C in the GSD‐based equations, the estimation of k_a was significantly improved with the differences between the measured and estimated k_a less than ±4% on average (except for the method of Barr). As demonstrated by this study, GSD‐based equations may provide a promising and efficient way to estimate k_a at dry conditions.     

基于不同模型的大型湖泊水气界面气体传输速率估算

气体传输速率是湖泊水—气界面温室气体交换通量的重要驱动因子,但其估算具有不确定性.本研究选择3种不同的参数化方程估算大型(面积2400 km2)浅水(平均水深1.9 m)湖泊——太湖水—气界面的气体传输速率,探讨大型湖泊气体传输速率的控制因子和变化范围,为估算模型的选取提供参考.结果表明,气体传输速率的两个重要参数风应力和水体对流混合速率存在夜间高、白天低的变化特征,因此气体传输速率也存在夜间高、白天低的变化特征.总体上太湖气体传输速率主要由风力控制,可以通过风速函数估算得到.太湖水—气界面气体传输速率的年均值为1.27~1.46m/d.因气体传输速率存在空间变化,单一站点参数化的模型可能不适合其他区域湖泊水—气界面气体传输速率的估算,但湖泊的面积可能是一个有效的预测因子.     

Control strategy for variable damping element considering near‐future excitation influence

A control strategy is proposed for variable damping elements (VDEs) used together with auxiliary stiffness elements (ASEs) that compose a time‐varying non‐linear Maxwell (NMW) element, considering near‐future excitation influence. The strategy first composes a state equation for the structural dynamics and the mechanical balance in the NMW elements. Next, it establishes a cost function for estimating future responses by the weighted quadratic norms of the state vector, the controlled force and the VDEs' damping coefficients. Then, the Euler equations for the optimum values are introduced, and also approximated by the first‐order terms under the autoregressive (AR) model of excitation information. Thus, at each moment t_k, the strategy conducts the following steps: (1) identify the obtained seismic excitation information to an AR model, and convert it to a state equation; and (2) determine VDEs' damping coefficients under the initial conditions at t_k and the final state at t_k+L, using the first‐order approximation of the Euler equations. The control effects are examined by numerical experiments. Copyright © 2000 John Wiley & Sons, Ltd.     

Fermat's Variational Principle for Anisotropic Inhomogeneous Media

Fermat's variational principle states that the signal propagates from point S to R along a curve which renders Fermat's functional (l) stationary. Fermat's functional (l) depends on curves l which connect points S and R, and represents the travel times from S to R along l. In seismology, it is mostly expressed by the integral (l) = (x ^k,x ^k ')du, taken along curve l, where (x ^k,x ^k ') is the relevant Lagrangian, x ^k are coordinates, u is a parameter used to specify the position of points along l, and x ^k ' = dx ^k÷du. If Lagrangian (x ^k,x ^k ') is a homogeneous function of the first degree in x ^k ', Fermat's principle is valid for arbitrary monotonic parameter u. We than speak of the first-degree Lagrangian ⁽¹⁾(x ^k,x ^k '). It is shown that the conventional Legendre transform cannot be applied to the first-degree Lagrangian ⁽¹⁾(x ^k,x ^k ') to derive the relevant Hamiltonian ⁽¹⁾(x ^k,p _k), and Hamiltonian ray equations. The reason is that the Hessian determinant of the transform vanishes identically for first-degree Lagrangians ⁽¹⁾(x ^k,x ^k '). The Lagrangians must be modified so that the Hessian determinant is different from zero. A modification to overcome this difficulty is proposed in this article, and is based on second-degree Lagrangians ⁽²⁾. Parameter u along the curves is taken to correspond to travel time , and the second-degree Lagrangian ⁽²⁾(x ^k, ^k ) is then introduced by the relation ⁽²⁾(x ^k, ^k ) = [⁽¹⁾(x ^k, ^k )]², with ^k = dx ^k÷d. The second-degree Lagrangian ⁽²⁾(x ^k, ^k ) yields the same Euler/Lagrange equations for rays as the first-degree Lagrangian ⁽¹⁾(x ^k, ^k ). The relevant Hessian determinant, however, does not vanish identically. Consequently, the Legendre transform can then be used to compute Hamiltonian ⁽²⁾(x ^k,p _k) from Lagrangian ⁽²⁾(x ^k, ^k ), and vice versa, and the Hamiltonian canonical equations can be derived from the Euler-Lagrange equations. Both ⁽²⁾(x ^k, ^k ) and ⁽²⁾(x ^k,p _k) can be expressed in terms of the wave propagation metric tensor g _ij(x ^k, ^k ), which depends not only on position x ^k, but also on the direction of vector ^k . It is defined in a Finsler space, in which the distance is measured by the travel time. It is shown that the standard form of the Hamiltonian, derived from the elastodynamic equation and representing the eikonal equation, which has been broadly used in the seismic ray method, corresponds to the second-degree Lagrangian ⁽²⁾(x ^k, ^k ), not to the first-degree Lagrangian ⁽¹⁾(x ^k, ^k ). It is also shown that relations ⁽²⁾(x ^k, ^k ) = ; and ⁽²⁾(x ^k,p _k) = are valid at any point of the ray and that they represent the group velocity surface and the slowness surface, respectively. All procedures and derived equations are valid for general anisotropic inhomogeneous media, and for general curvilinear coordinates x ⁱ. To make certain procedures and equations more transparent and objective, the simpler cases of isotropic and ellipsoidally anisotropic media are briefly discussed as special cases.     

Probabilistic development of shear strength model for reinforced concrete squat walls

In order to reconcile the larger scatter and avoid the biased estimate from deterministic predictions for the shear strength of reinforced concrete (RC) squat structural walls, a probabilistic shear strength model is developed in this paper based on the strut‐and‐tie model and the generalized likelihood uncertainty estimation (GLUE) method. The strut‐and‐tie model is used to derive an appropriate function form for the probabilistic shear strength model, where four unknown model parameters (e.g. k₁, k₂, k₃ and k₄) are defined carefully to guarantee them having a clear physical‐based meaning so that the corresponding prior distribution ranges can be specified reasonably. Then, the GLUE method is adopted to estimate the posterior cumulative distribution of k₁, k₂, k₃ and k₄ with an available experimental database. Furthermore, to demonstrate the stability of the estimated posterior cumulative distribution, the sensitivity of three major aspects in GLUE method is investigated. Finally, based on the estimated cumulative distribution of k₁, k₂, k₃ and k₄, the developed probabilistic shear strength model is simplified as a mean prediction model and a standard deviation prediction model for facilitate using in engineering practice. Therefore, with the developed probabilistic shear strength model, not only can the squat structural walls be designed in confidence, but the accuracy of those deterministic predictions can be evaluated in a probabilistic manner. Copyright © 2016 John Wiley & Sons, Ltd.     

Statistical dynamics of internal gravity waves-turbulence

Abstract

Numerical simulations of internal gravity waves-turbulence are carried out for the inviscid, viscous and forced-dissipative two-dimensional primitive equations using the spectral method. Some of the results are compared with the predictions of the eddy damped quasi-normal Markovian (EDQNM) closure for internal waves of Carnevale and Frederiksen, generalized for periodic boundary conditions and possible random forcing and dissipation. The EDQNM reduces to the Boltzman equation of resonant interaction theory in the continuum space limit and as the forcing and dissipation vanish. However, the limit is singular in the sense that as well as conserving total energy, E, and total cross-correlation between the vorticity and buoyancy fields, C, an additional conservation law, viz. z-momentum, P_z , occurs in the limit. This means that the resonant interaction equilibrium (RIE) solution of the Boltzmann equation differs from the statistical mechanical equilibrium (SME) solution of the EDQNM closure.

The statistical stability of the SME and RIE spectra for the primitive equations is tested by integrating the inviscid equations using initial realizations of these spectra with random phases. It is found that E and C are accurately conserved while P_z undergoes large amplitude variations. The approach to equilibrium of initial disequilibrium spectra is monitored by examining the evolution of the entropy. The increase and asymptotic approach to a constant value corresponding to complete chaos is consistent with the behaviour predicted by the EDQNM closure.

For the viscous decay and forced-dissipative experiments, the behaviour of the entropy is also consistent with that predicted by the EDQNM closure. There is approximate equipartition of potential and total kinetic energies throughout the integrations from initial conditions having equal potential and total kinetic energies and as well equal vertical and horizontal energies, but as expected, the ratio of horizontal to vertical kinetic energy increases with time to a value greater than unity.

With Laplacian viscous dissipation and thermal diffusivity, the statistical steady states produced in the forced-dissipative experiments have k^?3 power laws for k≧7. A comparison with the power laws for kinetic energy and passive scalar variance produced in a numerical simulation of the two-dimensional passive scalar problem is also presented.     

Flood routing based on diffusion wave equation using mixing cell method

A one-dimensional non-linear diffusion wave equation is derived from the Saint Venant equations with neglect of the inertia terms. This non-linear equation has no general analytical solution. Numerical schemes are therefore employed to discretize the space and time axes and convert the differential equation to difference form. In this study, the mixing cell method is used to convert the diffusion wave equation to difference form, in which the difference term can be eliminated by selecting an optimal space step size Δx when time step size Δt is given. When the time step size Δt→0, the space step size Δx=Q/(2S₀BC]_k) where Q is discharge, S₀ is bed slope, B is channel width and C_k is kinematic wave celerity, which is the same as the characteristic length proposed by Kalinin and Milyukov. The results of application to two cases show that the mixing cell and linear channel flow routing methods produce hydrographs that are in agreement with the observed flood hydrographs. © 1997 John Wiley & Sons, Ltd.     

The null distribution of sample serial correlation coefficient

 The null distribution of the lag-k sample serial correlation coefficient (r _k , k=1,2,3) was investigated by Monte Carlo simulation. For a time series with normal, exponential, Pearson 3, EV1 (Gumbel), or generalized Pareto (GP) distribution type, the null distribution of its r _k can be approximated by the normal distribution with mean −1/(n−k) and variance 1/(n−1). But for a time series with the lognormal, EV2 or EV3 (Weibull) distribution type, the null distribution of r _k is skewed distributed. In such cases, a simulation technique is suggested to construct percentile confidence intervals at a given significance level.     

Fourier amplitude spectra of strong motion acceleration: Extension to high and low frequencies

It is shown how the empirical equations for scaling the Fourier amplitude spectra in the frequency band from ～0.1 to 25 Hz can be extended to describe the strong motion amplitudes in a much broader frequency range. At long periods, the proposed equations are in excellent agreement with (1) the seismological and field estimates of permanent ground displacement (near field) and (2) the independent estimates of seismic moment (far field). At high frequencies, f ≥ 25 Hz, the spectral amplitudes can be described by exp (? πkf), where k ranges from 0·02 (near source) to about 0·06 at an epicentral distance of about 200 km. It is also shown how amplification by local soil and geological site conditions can be defined to apply in the same broad frequency range.     

Flow structure over square bars at intermediate submergence: Large Eddy Simulation study of bar spacing effect

Turbulent open-channel flow over 2D roughness elements is investigated numerically by Large Eddy Simulation (LES). The flow over square bars for two roughness regimes (k-type roughness and transitional roughness between d-type and k-type) at a relative submergence of H/k = 6.5 is considered, where H is the maximum water depth and k is the roughness height. The selected roughness configurations are based on laboratory experiments, which are used for validating numerical simulations. Results from the LES, in turn, complement the experiments in order to investigate the time-averaged flow properties at much higher spatial resolution. The concept of the double-averaging (DA) of the governing equations is utilized to quantify roughness effects at a range of flow properties. Double-averaged velocity profiles are analysed and the applicability of the logarithmic law for rough-wall flows of intermediate submergence is evaluated. Momentum flux components are quantified and roughness effect on their vertical distribution is assessed using an integral form of the DA-equations. The relative contributions of pressure drag and viscous friction to the overall bed shear stress are also reported.     

ω-k FILTER DESIGN*

Two-dimensional band-pass filters can be constructed by a simple extension of the theory of one-dimensional band-pass filters. Similarly to the one-dimensional analogue the shape of the two-dimensional filter is important in determining its effectiveness. The band-pass filter formulation can be further refined so that the filter will concentrate its rejection energies in certain areas of the ω, k plane. Such band-pass, band-reject filters are found by solving a set of simultaneous equations.     

Vortical and wave modes in 3D rotating stratified flows: random large-scale forcing

Utilizing an eigenfunction decomposition, we study the growth and spectra of energy in the vortical (geostrophic) and wave (ageostrophic) modes of a three-dimensional (3D) rotating stratified fluid as a function of ε = f/N, where f is the Coriolis parameter and N is the Brunt–Vaisala frequency. Throughout, we employ a random large-scale forcing in a unit aspect ratio domain and set these parameters such that the Froude and Rossby numbers are roughly comparable and much less than unity. Working in regimes characterized by moderate Burger numbers, i.e. Bu = 1/ε² < 1 or Bu ≥ 1, our results indicate profound change in the character of vortical and wave mode interactions with respect to Bu = 1. Indeed, previous analytical work concerning the qualitatively different nature of these interactions has been in limiting conditions of rotation or stratification domination (i.e. when Bu ? 1 or Bu ? 1, respectively). As with the reference state of ε = 1, for ε < 1 the wave mode energy saturates quite quickly and the ensuing forward cascade continues to act as an efficient means of dissipating ageostrophic energy. Further, these saturated spectra steepen as ε decreases: we see a shift from k ^?1 to k ^?5/3 scaling for k _f < k < k _d (where k _f and k _d are the forcing and dissipation scales, respectively). On the other hand, when ε > 1 the wave mode energy never saturates and comes to dominate the total energy in the system. In fact, in a sense the wave modes behave in an asymmetric manner about ε = 1. With regard to the vortical modes, for ε ≤ 1, the signatures of 3D quasigeostrophy are clearly evident. Specifically, we see a k ^?3 scaling for k _f < k < k _d and, in accord with an inverse transfer of energy, the vortical mode energy never saturates but rather increases for all k < k _f . In contrast, for ε > 1 and increasing, the vortical modes contain a progressively smaller fraction of the total energy indicating that the 3D quasigeostrophic subsystem, though always present, plays an energetically smaller role in the overall dynamics. Combining the vortical and wave modes, the total energy for k > k _f and ε ≤ 1 shows a transition as k increases wherein the vortical modes contain a large portion of the energy at large scales, while the wave modes dominate at smaller scales. There is no such transition when ε > 1 and the wave modes dominate the total energy for all k > k _f .     

A probabilistic model of seismicity: Kamchatka earthquakes

The catalog of Kamchatka earthquakes is represented as a probability space of three objects {Ω, $ \tilde F $ \tilde F P}. Each earthquake is treated as an outcome ω _i in the space of elementary events Ω whose cardinality for the period under consideration is given by the number of events. In turn, ω _i is characterized by a system of random variables, viz., energy class k_i, latitude φ _i , longitude λ _i , and depth h _i . The time of an outcome has been eliminated from this system in this study. The random variables make up subsets in the set $ \tilde F $ \tilde F and are defined by multivariate distributions, either by the distribution function $ \tilde F $ \tilde F (φ, λ, h, k) or by the probability density f(φ, λ, h, k) based on the earthquake catalog in hand. The probabilities P are treated in the frequency interpretation. Taking the example of a recurrence relation (RR) written down in the form of a power law for probability density f(k), where the initial value of the distribution function f(k ₀) is the basic data [Bogdanov, 2006] rather than the seismic activity A ₀, we proceed to show that for different intervals of coordinates and time the distribution f _elim(k) of an earthquake catalog with the aftershocks eliminated is identical to the distribution f _full(k), which corresponds to the full catalog. It follows from our calculations that f ₀(k) takes on nearly identical numeral values for different initial values of energy class k ₀ (8 ≤ k ₀ ≤ 12) f(k ₀). The difference decreases with an increasing number of events. We put forward the hypothesis that the values of f(k ₀) tend to cluster around the value 2/3 as the number of events increases. The Kolmogorov test is used to test the hypothesis that statistical recurrence laws are consistent with the analytical form of the probabilistic RR based on a distribution function with the initial value f(k ₀) = 2/3. We discuss statistical distributions of earthquake hypocenters over depth and the epicenters over various areas for several periods     

An error analysis of truncated starting conditions in step-by-step time integration: Consequences for structural dynamics

Most of the step-by-step time integration algorithms for structural dynamics require an initial acceleration vector to be specified, in addition to displacement and velocity vectors. A consistent initial acceleration vector may be calculated by solving the equations of motion at the initial time, while a truncated initial acceleration vector is obtained by setting the acceleration values to zero. Although the truncated starting procedure decreases computational effort, it is shown to affect accuracy adversely. For the structural dynamics algorithms considered herein, the rate at which the numerical solution converges to the exact solution is O(Δt) when the truncated starting procedure is used, compared to O(Δt²) when consistent initial acceleration values are used.     

ω-k FILTER DESIGN*

Two-dimensional band-pass filters can be constructed by a simple extension of the theory of one-dimensional band-pass filters. Similarly to the one-dimensional analogue the shape of the two-dimensional filter is important in determining its effectiveness. The band-pass filter formulation can be further refined so that the filter will concentrate its rejection energies in certain areas of the ω, k plane. Such band-pass, band-reject filters are found by solving a set of simultaneous equations.     

Turbulence in vegetated flows: Volume-average analysis and modelling aspects

Two approaches for the modelling of turbulence in vegetated flows have been developed in the past. The “microscopic” approach which is straightforward but limited to simple cases and the “macroscopic” approach which is based on Volume Average Theory (VAT). In this study, aspects of Volume-Average (VA) analysis and modelling are investigated for turbulent vegetated flow using computed three-dimensional results from the solution of the Reynolds-Averaged Navier-Stokes (RANS) equations around a representative vegetal element. In particular (a) the VA transport equations for k and ε, based on VAT, are properly derived, (b) the Boussinesq hypothesis for the VA quantities, employed in 〈k〉-〈ε〉 turbulence models is tested, and (c) the values of the coefficients used in such turbulence models are assessed in comparison with those used in the classical turbulence models.     

Broken ergodicity in magnetohydrodynamic turbulence

Turbulent magnetofluids appear in various geophysical and astrophysical contexts, in phenomena associated with planets, stars, galaxies and the universe itself. In many cases, large-scale magnetic fields are observed, though a better knowledge of magnetofluid turbulence is needed to more fully understand the dynamo processes that produce them. One approach is to develop the statistical mechanics of ideal (i.e. non-dissipative), incompressible, homogeneous magnetohydrodynamic (MHD) turbulence, known as “absolute equilibrium ensemble” theory, as far as possible by studying model systems with the goal of finding those aspects that survive the introduction of viscosity and resistivity. Here, we review the progress that has been made in this direction. We examine both three-dimensional (3-D) and two-dimensional (2-D) model systems based on discrete Fourier representations. The basic equations are those of incompressible MHD and may include the effects of rotation and/or a mean magnetic field B _o. Statistical predictions are that Fourier coefficients of the velocity and magnetic field are zero-mean random variables. However, this is not the case, in general, for we observe non-ergodic behavior in very long time computer simulations of ideal turbulence: low wavenumber Fourier modes that have relatively large means and small standard deviations, i.e. coherent structure. In particular, ergodicity appears strongly broken when B _o?=?0 and weakly broken when B _o?≠?0. Broken ergodicity in MHD turbulence is explained by an eigenanalysis of modal covariance matrices. This produces a set of modal eigenvalues inversely proportional to the expected energy of their associated eigenvariables. A large disparity in eigenvalues within the same mode (identified by wavevector k ) can occur at low values of wavenumber k?=?| k |, especially when B _o?=?0. This disparity breaks the ergodicity of eigenvariables with smallest eigenvalues (largest energies). This leads to coherent structure in models of ideal homogeneous MHD turbulence, which can occur at lowest values of wavenumber k for 3-D cases, and at either lowest or highest k for ideal 2-D magnetofluids. These ideal results appear relevant for unforced, decaying MHD turbulence, so that broken ergodicity effects in MHD turbulence survive dissipation. In comparison, we will also examine ideal hydrodynamic (HD) turbulence, which, in the 3-D case, will be seen to differ fundamentally from ideal MHD turbulence in that coherent structure due to broken ergodicity can only occur at maximum k in numerical simulations. However, a nonzero viscosity eliminates this ideal 3-D HD structure, so that unforced, decaying 3-D HD turbulence is expected to be ergodic. In summary, broken ergodicity in MHD turbulence leads to energetic, large-scale, quasistationary magnetic fields (coherent structures) in numerical models of bounded, turbulent magnetofluids. Thus, broken ergodicity provides a large-scale dynamo mechanism within computer models of homogeneous MHD turbulence. These results may help us to better understand the origin of global magnetic fields in astrophysical and geophysical objects.     

Runoff modelling at two field slopes: use of in situ measurements of air permeability to characterize spatial variability of saturated hydraulic conductivity

The time required at a field site to obtain a few measurements of saturated hydraulic conductivity (K_s) will allow for many measurements of soil air permeability (k_a). This study investigates if k_a measured in situ (k_{a, in situ}) can be a substitute for measurement of K_s in relation to infiltration and surface runoff modelling. Measurements of k_{a, in situ} were carried out in two small agricultural catchments. A spatial correlation of the log‐transformed values existed having a range of approximately 100 m. A predictive relationship between K_s and k_a measured on 100‐cm³ soil samples in the laboratory was derived for one of the field slopes and showed good agreement with an earlier suggested predictive K_s–k_a relationship. In situ measurements of K_s and k_a suggested that the predictive relationships also could be used at larger scale. The K_s–k_a relationships together with the k_{a, in situ} data were applied in a distributed surface runoff (DSR) model, simulating a high‐intensity rainfall event. The DSR simulation results were highly dependent on whether the geometric average of k_{a, in situ} or kriged values of k_{a, in situ} was used as model input. When increasing the resolution of K_s in the DSR model, a limit of 30–40 m was found for both field slopes. Below this limit, the simulated runoff and hydrograph peaks were independent of resolution scale. If only a few randomly chosen values of K_s were used to represent the spatial variation within the field slope, very large deviations in repeated DSR simulation results were obtained, both with respect to peak height and hydrograph shape. In contrast, when using many predicted K_s values based on a K_s–k_a relationship and measured k_{a, in situ} data, the DSR model generally captured the correct hydrograph shape although simulations were sensitive to the chosen K_s–k_a relationship. As massive measurement efforts normally will be required to obtain a satisfactory representation of the spatial variability in K_s, the use of k_{a, in situ} to assess spatial variability in K_s appears a promising alternative. Copyright © 2004 John Wiley & Sons, Ltd.