Residence and exposure times : when diffusion does not matter

Under constant hydrodynamic conditions and assuming horizontal homogeneity, negatively buoyant particles released at the surface of the water column have a mean residence time in the surface mixed layer of h/w, where h is the thickness of the latter and w ( > 0) is the sinking velocity Deleersnijder (Environ Fluid Mech 6(6):541–547, 2006a). The residence time does not depend on the diffusivity and equals the settling timescale. We show that this behavior is a result of the particular boundary conditions of the problem and that it is related to a similar property of the exposure time in a one-dimensional infinite domain. In 1-D advection–diffusion problem with a constant and uniform velocity, the exposure time—which is a generalization of the residence time measuring the total time spent by a particle in a control domain allowing the particle to leave and reenter the control domain—is also equal to the advection timescale at the upstream boundary of the control domain. To explain this result, the concept of point exposure is introduced; the point exposure is the time integral of the concentration at a given location. It measures the integrated influence of a point release at a given location and is related to the concept of number of visits of the theory of random walks. We show that the point exposure takes a constant value downstream the point of release, even when the diffusivity varies in space. The analysis of this result reveals also that the integrated downstream transport of a passive tracer is only effected by advection. While the diffusion flux differs from zero at all times, its integrated value is strictly zero.

     

A Note on the Estimation of Eddy Diffusivity and Dissipation Length in Low Winds over a Tropical Urban Terrain

— Urban terrain poses a challenge for modeling air pollutant diffusion. In tropics, because of the dominant low wind speed environment, the importance of understanding the turbulence diffusion is even more critical, and uncertain. The objective of this study is to estimate the vertical eddy diffusivity of an urban, tropical atmosphere in low–wind speeds. Turbulence measurements at 1 Hz were made at 4-m level over an urban terrain with a roughness length of 0.78 m during winter months. Eddy diffusivity is estimated from spectral quantities of the turbulence data involving turbulent kinetic energy (E) and its dissipation rate (?). The spectral information of the vertical velocity fluctuations is used to estimate the vertical length scale which provides information on the eddy diffusivity. In addition, the product of friction velocity and the vertical length scale has been used to non-dimensionalize the eddy diffusivity, which is shown to increase with increasing instability. Using the eddy diffusivity (K) estimates from the E – ? approach, a relation is suggested for the mixing length based eddy diffusivity models of the form: K = c _w .[2.5 ? 0.5(z/L)], where z is the measurement height, L is the Obukhov length, and c _w has an average value close to 1 for unstable and near 0.5 for stable conditions for the urban terrains.     

A stochastic model for tropical rainfall at a single location

Modelling data that correspond to rainfall accumulated over fixed periods of time presents the challenging problem of dealing with a random variable that has a point mass at zero which corresponds to dry periods that occur with positive probability. One way to overcome this difficulty is to assume that the data correspond to a normal variate w, that has been truncated and transformed. The dry periods correspond to the (unobserved) negative values and the wet periods correspond to some power of the positive ones. The serial structure that is present in rainfall can be modelled by imposing a serial structure to w. We use a dynamic linear model on w using a Fourier representation to allow for the seasonality of the data, which in the case of tropical rainfall is very marked. The model is fitted using a Markov chain Monte Carlo method that uses latent variables to handle both dry periods and missing values. We use the model to estimate and predict both the amount of rainfall and the probability of a dry period. The method is illustrated with data collected in the Venezuelan state of Guárico.     

Applicability of the Dual‐Domain Model to Nonaggregated Porous Media

More theoretical analysis is needed to investigate why a dual‐domain model often works better than the classical advection‐dispersion (AD) model in reproducing observed breakthrough curves for relatively homogeneous porous media, which do not contain distinct dual domains. Pore‐scale numerical experiments presented here reveal that hydrodynamics create preferential flow paths that occupy a small part of the domain but where most of the flow takes place. This creates a flow‐dependent configuration, where the total domain consists of a mobile and an immobile domain. Mass transfer limitations may result in nonequilibrium, or significant differences in concentration, between the apparent mobile and immobile zones. When the advection timescale is smaller than the diffusion timescale, the dual‐domain mass transfer (DDMT) model better captures the tailing in the breakthrough curve. Moreover, the model parameters (mobile porosity, mean solute velocity, dispersivity, and mass transfer coefficient) demonstrate nonlinear dependency on mean fluid velocity. The studied case also shows that when the Peclet number, Pe, is large enough, the mobile porosity approaches a constant, and the mass transfer coefficient can be approximated as proportional to mean fluid velocity. Based on detailed analysis at the pore scale, this paper provides a physical explanation why these model parameters vary in certain ways with Pe. In addition, to improve prediction in practical applications, we recommend conducting experiments for parameterization of the DDMT model at a velocity close to that of the relevant field sites, or over a range of velocities that may allow a better parameterization.     

Ion microprobe studies of water in silicate melts. Concentration-dependent water diffusion in obsidian

The ion microprobe at Johnson Space Center has been calibrated for in situ water determinations on a 10-μm scale over the range 0.2 wt.% H₂O to 1.8, 6.8, and 3.7 wt.%, for basaltic, albitic, and rhyolitic glasses, respectively. The basalt glass calibration curve differs substantially from those of albite and rhyolite glasses, indicating a need to carefully match composition and/or melt structure between H₂O standards and unknowns.A value for the diffusivity of water as a function of concentration and time has been calculated from water diffusion profiles measured in rhyolite glasses prepared at 850°C and 700 barsP_t(H₂O) [1]. Transient diffusion into a semi-infinite medium is described by the equation:?(φ/2)?¸/?φ=?(D_w?¸/?φ)/?φ #x003B8;=1, φ=0, θ→ 0, θ→∞, wherex =distance from the cylinder edge,t =time,C₀ =initial concentration,C_s =concentration at the edge,C =concentration at x,θ = C ? C₀/C_s ? C₀,φ = x/t^1/2, andD_w =diffusivity of water. An iterative technique has been used to calculate solutions to the diffusion equation as a function ofD_w [2]. Comparison of these solutions with the ion probe data indicate that, for0.2wt.% ≤ C ≤ 3.7wt.%H₂O,D_w can be described by an exponential function of θ, of the formD_w = D₀exp(bθ), withD₀ (i.e.,D_w at 0.2%) = (0.8?2.2) × 10^?8 cm²/s and2 ≤ b ≤ 4.     

Temporal and spatial variability of radium in the coastal ocean and its impact on computation of nearshore cross-shelf mixing rates

Constraining the exchange of water from the shoreline to the mid-shelf is necessary for the development of accurate and predictive models of nearshore circulation. Ra isotopes, which emanate from sediments and have a variety of half-lives, may be useful in measuring cross-shelf mixing rates. The distributions of Ra isotopes were measured in transects extending perpendicular from the shoreline at Sunset Beach and Huntington Beach, CA. The average inventory at Sunset Beach was four times greater than at Huntington Beach. Building on previous research on Ra inputs and circulation in San Pedro Bay, a two-dimensional model for surface water Ra was developed to identify the importance of onshore flow and cross-shelf mixing near Huntington Beach. For the mean summertime conditions, the eddy diffusivity (K_h) was 1.4±0.4 m² s⁻¹, with 8% of the water from Sunset Beach moving down the coast. The remaining water must be low-Ra water that has moved onshore. At time scales greater than a week, the short-lived Ra inventory at Huntington Beach varied by 50%, which reflects changes in the fractions of water moving down-coast and/or in the longshore advection rate. The shoreline Ra concentration varied on time scales of hours, which may be generated by tidal changes in the Ra input at the shoreline and short-period fluctuations in the mixing rate. The low K_h observed in this study in comparison to higher values measured further offshore is evidence that K_h increases with distance offshore. When scale-dependent mixing beyond 455 m offshore is incorporated into the model, the results are consistent with the observed data for ²²³Ra, ²²⁴Ra, and ²²⁸Ra. Using the model, the ²²⁸Ra input flux to the summertime mixed layer was between 3.4×10⁶ and 4.0×10⁶ atoms s⁻¹ (m shoreline)⁻¹.     

Estimating Reaction Rate Coefficients Within a Travel‐Time Modeling Framework

A generalized, efficient, and practical approach based on the travel‐time modeling framework is developed to estimate in situ reaction rate coefficients for groundwater remediation in heterogeneous aquifers. The required information for this approach can be obtained by conducting tracer tests with injection of a mixture of conservative and reactive tracers and measurements of both breakthrough curves (BTCs). The conservative BTC is used to infer the travel‐time distribution from the injection point to the observation point. For advection‐dominant reactive transport with well‐mixed reactive species and a constant travel‐time distribution, the reactive BTC is obtained by integrating the solutions to advective‐reactive transport over the entire travel‐time distribution, and then is used in optimization to determine the in situ reaction rate coefficients. By directly working on the conservative and reactive BTCs, this approach avoids costly aquifer characterization and improves the estimation for transport in heterogeneous aquifers which may not be sufficiently described by traditional mechanistic transport models with constant transport parameters. Simplified schemes are proposed for reactive transport with zero‐, first‐, nth‐order, and Michaelis‐Menten reactions. The proposed approach is validated by a reactive transport case in a two‐dimensional synthetic heterogeneous aquifer and a field‐scale bioremediation experiment conducted at Oak Ridge, Tennessee. The field application indicates that ethanol degradation for U(VI)‐bioremediation is better approximated by zero‐order reaction kinetics than first‐order reaction kinetics.     

The boundary layer of the residence time field

The residence time of a tracer in a control domain is usually computed by releasing tracer parcels and registering the time when each of these tracer parcels cross the boundary of the control domain. In this Lagrangian procedure, the particles are discarded or omitted as soon as they leave the control domain. In a Eulerian approach, the same approach can be implemented by integrating forward in time the advection–diffusion equation for a tracer. So far, the conditions to be applied at the boundary of the control domain were uncertain. We show here that it is necessary to prescribe that the tracer concentration vanishes at the boundary of the control domain to ensure the compatibility between the Lagrangian and Eulerian approaches. When we use the Constituent oriented Age and Residence time Theory (CART), this amounts to solving the differential equation for the residence time with boundary conditions forcing the residence time to vanish at the open boundaries of the control domain. Such boundary conditions are likely to induce the development of boundary layers (at outflow boundaries for the tracer concentration and at inflow boundaries for the residence time). The thickness of these boundary layers is of the order of the ratio of the diffusivity to the velocity. They can however be partly smoothed by tidal and other oscillating flows.     

Space–Time Earthquake Prediction: The Error Diagrams

The quality of earthquake prediction is usually characterized by a two-dimensional diagram n versus τ, where n is the rate of failures-to-predict and τ is a characteristic of space–time alarm. Unlike the time prediction case, the quantity τ is not defined uniquely. We start from the case in which τ is a vector with components related to the local alarm times and find a simple structure of the space–time diagram in terms of local time diagrams. This key result is used to analyze the usual 2-d error sets {n, τ _w } in which τ _w is a weighted mean of the τ components and w is the weight vector. We suggest a simple algorithm to find the (n, τ _w ) representation of all random guess strategies, the set D, and prove that there exists the unique case of w when D degenerates to the diagonal n + τ _w  = 1. We find also a confidence zone of D on the (n, τ _w ) plane when the local target rates are known roughly. These facts are important for correct interpretation of (n, τ _w ) diagrams when we discuss the prediction capability of the data or prediction methods.     

Roundhouse (RND) Mountain Top Research Site: Measurements and Uncertainties for Winter Alpine Weather Conditions

The objective of this work is to better understand and summarize the mountain meteorological observations collected during the Science of Nowcasting Winter Weather for the Vancouver 2010 Olympics and Paralympics (SNOW-V10) project that was supported by the Fog Remote Sensing and Modeling (FRAM) project. The Roundhouse (RND) meteorological station was located 1,856 m above sea level that is subject to the winter extreme weather conditions. Below this site, there were three additional observation sites at 1,640, 1,320, and 774 m. These four stations provided some or all the following measurements at 1 min resolution: precipitation rate (PR) and amount, cloud/fog microphysics, 3D wind speed (horizontal wind speed, U _h; vertical air velocity, w _a), visibility (Vis), infrared (IR) and shortwave (SW) radiative fluxes, temperature (T) and relative humidity with respect to water (RH_w), and aerosol observations. In this work, comparisons are made to assess the uncertainties and variability for the measurements of Vis, RH_w, T, PR, and wind for various winter weather conditions. The ground-based cloud imaging probe (GCIP) measurements of snow particles using a profiling microwave radiometer (PMWR) data have also been shown to assess the icing conditions. Overall, the conclusions suggest that uncertainties in the measurements of Vis, PR, T, and RH can be as large as 50, >60, 50, and >20 %, respectively, and these numbers may increase depending on U _h, T, Vis, and PR magnitude. Variability of observations along the Whistler Mountain slope (~500 m) suggested that to verify the models, model space resolution should be better than 100 m and time scales better than 1 min. It is also concluded that differences between observed and model based parameters are strongly related to a model’s capability of accurate prediction of liquid water content (LWC), PR, and RH_w over complex topography.     

Bioturbation and erosion rates along the soil-hillslope conveyor belt,part 2: Quantification using an analytical solution of the diffusion–advection equation

Particles on soil-mantled hillslopes are subject to downslope transport by erosion processes and vertical mixing by bioturbation. Both are key processes for understanding landscape evolution and soil formation, and affect the functioning of the critical zone. We show here how the depth–age information, derived from feldspar-based single grain post-infrared infrared stimulated luminescence (pIRIR), can be used to simultaneously quantify erosion and bioturbation processes along a hillslope. In this study, we propose, for the first time, an analytical solution for the diffusion–advection equation to calculate the diffusivity constant and erosion–deposition rates. We have fitted this model to age–depth data derived from 15 soil samples from four soil profiles along a catena located under natural grassland in the Santa Clotilde Critical Zone Observatory, in the south of Spain. A global sensitivity analysis was used to assess the relative importance of each model parameter in the output. Finally, the posterior probability density functions were calculated to evaluate the uncertainty in the model parameter estimates. The results show that the diffusivity constant at the surface varies from 11.4 to 81.9 mm² a^-1 for the hilltop and hill-base profile, respectively, and between 7.4 and 64.8 mm² a^-1 at 50 cm depth. The uncertainty in the estimation of the erosion–deposition rates was found to be too high to make a reliable estimate, probably because erosion–deposition processes are much slower than bioturbation processes in this environment. This is confirmed by a global sensitivity analysis that shows how the most important parameters controlling the age–depth structure in this environment are the diffusivity constant and regolith depth. Finally, we have found a good agreement between the soil reworking rates proposed by earlier studies, considering only particle age and depth, and the estimated diffusivity constants. The soil reworking rates are effective rates, corrected for the proportion of particles actually participating in the process. © 2019 John Wiley & Sons, Ltd.     

An exact solution for a constant-strength line-sink satisfying the modified Helmholtz equation for groundwater flow

We obtained an exact solution in terms of the discharge potential for a constant-strength line-sink that satisfies the modified Helmholtz equation for groundwater flow, for example for semi-confined flow and transient flow. The solution is obtained by integrating the potential for a point sink (well) along a straight line element. The potential for the point-sink is the modified Bessel function of the second kind and zero order K₀. Since K₀ cannot be integrated directly (in closed form) along a line-element, earlier solutions for a line-sink have been obtained by integrating polynomial approximations to K₀. These approximations, however, are only valid up to a certain distance from the well and consequently impose a limit on the length of the line-sink. In this paper we integrate an exact series representation for K₀ that is valid at any distance from the well, thus allowing integration along line-elements of any length, at least in theory. Numerical difficulties arise when evaluating our expressions at large distances from the line-sink, but these are shown to be of little consequence in practice. We made use of Wirtinger calculus to facilitate integration and also to allow us to arrive at exact expressions for the integrated flux over a poly-line and the total leakage over a domain. These properties are essential when using the solution in the context of the Analytic Element Method (AEM). We demonstrate our solution for the case of semi-confined flow (with leakage) and for the case of transient flow in the context of the Laplace Transform Analytic Element Method (LT-AEM).     

Rapid formation of taylor columns: Obstacles against sidewalls

Abstract

The problem of topographic forcing by an obstacle against the boundary of a rotating flow is considered in various parameter regimes. The timescale for the motion is the topographic vortex-stretching time, which is inversely proportional to the background rotation rate and the fractional height of the obstacle. For slow flows this time is short compared with the advection time and the governing equation of conservation of potential vorticity is linear. The final state satisfies the non-linear equation for the advection of potential vorticity, however, and so time dependence has given a specific solution to a non-linear problem. The presence of the sidewall causes a stagnant Taylor column to be set up far more rapidly than cases with no sidewall. It is shown that viscosity and mixing arrests the inviscid evolution at some stage, thus some fluid still crosses the obstacle in the steady state. These solutions suggest that experimental results on separation obtained by Griffiths and Linden (1983) can tentatively be ascribed to entrainment and expulsion of fluid through vertical shear layers at the edge of the topography.     

On a possible origin of the anomalous effects that were observed during the May 24, 2013 earthquake in the Sea of Okhotsk

The deep-focus Sea of Okhotsk earthquake that occurred on May 24, 2013 (h = 630 km, M _w = 8.3) was accompanied by anomalous effects that were unknown previously. A combined analysis of published data concerning the source rupture evolution and some features of the deep structure provided an explanation of some anomalous effects, such as the large number of aftershocks and the low level of ground shaking in the epicentral area. However, GPS observations revealed high coseismic vertical displacements in the area. The seafloor uplift in the Sea of Okhotsk and the adjacent coasts was 3–12 mm, peaking at the approximate center of the sea, while Kamchatka and the North Kuril Islands subsided by 3–18 mm, peaking at the Apacha station 190 km east of the earthquake epicenter. These maximum estimates are 1.2–1.8 times the analogous values (10 mm) for the Chile mega-earthquake of May 20, 1960 (M _w ~ 9.5). It is known that the large distances at which ground shaking is felt during deep-focus earthquakes are due to the fact that the body waves travel through the high-Q lower mantle. However, this does not explain the paradox of the present earthquake in the Sea of Okhotsk, viz., a constant intensity of shaking (two grades) in the range of epicentral distances between 1300 and 9500 km. The explanation requires consideration of the earth’s free oscillations excited by the earthquake.     

Screening model for volatile pollutants in dual porosity soils

This paper develops mass fraction models for transport and fate of agricultural pollutants in structured two-region soils. Mass fraction index models, based on a semi-infinite domain solution, are derived that describe leaching at depth, vapor losses through soil surface, absorption, and degradation in the dynamic- and stagnant-water soil regions. The models predict that leaching is the result of the combined effect of the upward vapor-phase transport relative to downward advection, residence time relative to half-life, dispersion, and lateral diffusive mass transfer. Simulations show that leached fraction of volatile compounds does not always decrease monotonically with increased residence time relative to the pollutant half-life, as a result of complex interactions among the different physical and biochemical processes. The results show that leaching, volatilization, and degradation losses can be affected significantly by lateral diffusive mass transfer into immobile-water regions and advection relative to dispersion (i.e. Peclet number) in the mobile-water regions. It is shown that solute diffusion into the immobile phase and subsequent biochemical decay reduces leaching and vapor losses through soil surface. Potential use of the modified leaching index for the screening of selected pesticides is illustrated for different soil textures and infiltration rates. The analysis may be useful to the management of pesticides and the design of landfills.     

Radiation from a Finite Reverse Fault in a Half Space

— Using a set of well-known results for the seismic field radiated by a simple dip-slip dislocation in a half space, we study interesting details of the motion at the surface of the half space. The static solution for a dislocation in a half space was found by Freund and Barnett in 1976. The corresponding elastodynamic solution was solved exactly in the Fourier and Laplace domain by several authors about 20 years ago, however its properties remained unexplored because of analytical difficulties. We remove these difficulties and show that the solution contains three important phenomena: Seismic wave fronts of P, S and SP type; the near-field pulse associated with the propagation of the dislocation front; and the long-time elastic response that converges toward the static solution of Freund and Barnett. Based on these results we show that solutions to all these problems are self-similar and homogeneous in x/h and αt/h so that when the fault depth h approaches 0, the solutions become concentrated near the origin and around the P, S and surface wave travel times. This explains several paradoxes in the radiation from dip-slip faults; among these the most notable are the presence of a point force singularity at the tip of a surface breaking fault and the reduction in high frequency radiation near the surface.     

Theoretical and Observed Depth Correction for M s

—?Modal summation technique is used to generate 5000, three-component theoretical seismograms of Love and Rayleigh waves, assuming modified PREM (PREM-C) and AK135F global earth models. The focal depth h and the geometrical fault parameters are randomly chosen so as to uniformly cover possible source mechanisms and obtain uniform distribution of log h in the interval 1?h?h?M _s of the form:¶ΔM _s (h)=0 forh< 20km, ΔM _s (h)=0.314log(h)-0.409 for 20≠h< 60km, ΔM _s (h)=1.351log(h)-2.253 for 60≠h< 100km, ΔM _s (h)=0.400log(h)-0.350 for 100≠h< 600km .¶After applying the above correction, the relationship between the surface wave magnitude and the scalar seismic moment for the observational data set significantly improves, and becomes independent of the source depth. In relation to CTBT, no depth correction is needed for M _S when the m _b ???M _S discriminant is computed, because the proposed correction is zero for earthquakes with foci above 20?km.     

Analysis of interfacial error in saturated–unsaturated flow models

Interfacial error results from estimation of interblock conductivities related to the saturated–unsaturated interface. Both interfacial conductivity error (IE_K) and interfacial pressure error (IE_h) were analyzed under the arithmetic mean scheme while IE_K was numerically investigated under the averaging schemes arithmetic, geometric and harmonic. IE_K, dependent on the media pore size, is regularly less than zero while IE_h, associated with the height of capillary fringe, may be greater than zero. An interfacial discretization technique was developed to add two complementary equations into the saturated–unsaturated model with respect to the interface. The proposed interfacial approach may eliminate interfacial error from the approximations of interblock conductivities. Underestimation of the water-table response to infiltration is related to the negative IE_K. The water-table response error reaches −5.13% in our investigation, which is an accumulated result from IE_K.     

Continuous-cyclic variations in the b-value of the earthquake frequency-magnitude distribution

Seismicity of the Earth (M ≥ 4.5) was compiled from NEIC, IRIS and ISC catalogues and used to compute b-value based on various time windows. It is found that continuous cyclic b-variations occur on both long and short time scales, the latter being of much higher value and sometimes in excess of 0.7 of the absolute b-value. These variations occur not only yearly or monthly, but also daily. Before the occurrence of large earthquakes, b-values start increasing with variable gradients that are affected by foreshocks. In some cases, the gradient is reduced to zero or to a negative value a few days before the earthquake occurrence. In general, calculated b-values attain maxima 1 day before large earthquakes and minima soon after their occurrence. Both linear regression and maximum likelihood methods give correlatable, but variable results. It is found that an expanding time window technique from a fixed starting point is more effective in the study of b-variations. The calculated b-variations for the whole Earth, its hemispheres, quadrants and the epicentral regions of some large earthquakes are of both local and regional character, which may indicate that in such cases, the geodynamic processes acting within a certain region have a much regional effect within the Earth. The b-variations have long been known to vary with a number of local and regional factors including tectonic stresses. The results reported here indicate that geotectonic stress remains the most significant factor that controls b-variations. It is found that for earthquakes with M _w ≥ 7, an increase of about 0.20 in the b-value implies a stress increase that will result in an earthquake with a magnitude one unit higher.     

An alternative approach for the Istanbul earthquake early warning system

Two recent catastrophic earthquakes that struck the Marmara Region on 17 August 1999 (M_w=7.4) and 12 November 1999 (M_w=7.2) caused major concern about future earthquake occurrences in Istanbul and the Marmara Region. As a result of the preparations for an expected earthquake may occur around Istanbul region, an earthquake early warning system has been established in 2002 with a simple and robust algorithm, based on the exceedance of specified thresholds of time domain amplitudes and the cumulative absolute velocity (CAV) levels (Erdik et al., 2003 [1]). In order to improve the capability of Istanbul earthquake early warning system (IEEWS) for giving early warning of a damaging earthquake in the Marmara Region, we explored an alternative approach with the use of a period parameter (τ_c) and a high-pass filtered vertical displacement amplitude parameter (Pd) from the initial 3 s of the P waveforms as proposed by Kanamori (2005) [2] and Wu and Kanamori (2005) 3 and 4. The empirical relationships both between τ_c and moment magnitude (M_w), and between Pd and peak ground velocity (PGV) for the Marmara Region are presented. These relationships can be used to detect a damaging earthquake within seconds after the arrival of P waves, and can provide on-site warning in the Marmara Region.