Mineral nitrogen from KCl extractions of soil from the Jena Experiment (Main Experiment up to 30cm depth, year 2002)

As an estimate of plant-available N, this data set contains measurements of inorganic nitrogen (NO3-N and NH4-N, the sum of which is termed mineral N or Nmin) determined by extraction with 1 M KCl solution of soil samples from the main experiment plots of a large grassland biodiversity experiment (the Jena Experiment; see further details below). In the main experiment, 82 grassland plots of 20 x 20 m were established from a pool of 60 species belonging to four functional groups (grasses, legumes, tall and small herbs). In May 2002, varying numbers of plant species from this species pool were sown into the plots to create a gradient of plant species richness (1, 2, 4, 8, 16 and 60 species) and functional richness (1, 2, 3, 4 functional groups). Plots were maintained by bi-annual weeding and mowing. Soil sampling and analysis: Five soil cores (diameter 0.01 m) were taken at a depth of 0 to 0.15 m and 0.15 to 0.3 m of the mineral soil from each of the experimental plots in September 2002. Samples of the soil cores per plot were pooled during each sampling campaign. NO3-N and NH4-N concentrations were determined by extraction of soil samples with 1 M KCl solution and were measured in the soil extract with a Continuous Flow Analyzer (CFA, Skalar, Breda, Netherlands).

Mineral nitrogen from KCl extractions of soil from the Jena Experiment (Main Experiment up to 30cm depth, year 2004)

As an estimate of plant-available N, this data set contains measurements of inorganic nitrogen (NO3-N and NH4-N, the sum of which is termed mineral N or Nmin) determined by extraction with 1 M KCl solution of soil samples from the main experiment plots of a large grassland biodiversity experiment (the Jena Experiment; see further details below). In the main experiment, 82 grassland plots of 20 x 20 m were established from a pool of 60 species belonging to four functional groups (grasses, legumes, tall and small herbs). In May 2002, varying numbers of plant species from this species pool were sown into the plots to create a gradient of plant species richness (1, 2, 4, 8, 16 and 60 species) and functional richness (1, 2, 3, 4 functional groups). Plots were maintained by bi-annual weeding and mowing. Soil sampling and analysis: Five soil cores (diameter 0.01 m) were taken at a depth of 0 to 0.15 m and 0.15 to 0.3 m of the mineral soil from each of the experimental plots in March and October 2004. Samples of the soil cores per plot were pooled during each sampling campaign. NO3-N and NH4-N concentrations were determined by extraction of soil samples with 1 M KCl solution and were measured in the soil extract with a Continuous Flow Analyzer (CFA, Skalar, Breda, Netherlands).

Mineral nitrogen from KCl extractions of soil from the Jena Experiment (Main Experiment up to 15cm depth, year 2005)

As an estimate of plant-available N, this data set contains measurements of inorganic nitrogen (NO3-N and NH4-N, the sum of which is termed mineral N or Nmin) determined by extraction with 1 M KCl solution of soil samples from the main experiment plots of a large grassland biodiversity experiment (the Jena Experiment; see further details below). In the main experiment, 82 grassland plots of 20 x 20 m were established from a pool of 60 species belonging to four functional groups (grasses, legumes, tall and small herbs). In May 2002, varying numbers of plant species from this species pool were sown into the plots to create a gradient of plant species richness (1, 2, 4, 8, 16 and 60 species) and functional richness (1, 2, 3, 4 functional groups). Plots were maintained by bi-annual weeding and mowing. Soil sampling and analysis: Five soil cores (diameter 0.01 m) were taken at a depth of 0 to 0.15 m of the mineral soil from each of the experimental plots in April and September 2005. Samples of the soil cores per plot were pooled during each sampling campaign. NO3-N and NH4-N concentrations were determined by extraction of soil samples with 1 M KCl solution and were measured in the soil extract with a Continuous Flow Analyzer (CFA, Skalar, Breda, Netherlands).

Mineral nitrogen from KCl extractions of soil from the Jena Experiment (Main Experiment up to 15cm depth, year 2008)

As an estimate of plant-available N, this data set contains measurements of inorganic nitrogen (NO3-N and NH4-N, the sum of which is termed mineral N or Nmin) determined by extraction with 1 M KCl solution of soil samples from the main experiment plots of a large grassland biodiversity experiment (the Jena Experiment; see further details below). In the main experiment, 82 grassland plots of 20 x 20 m were established from a pool of 60 species belonging to four functional groups (grasses, legumes, tall and small herbs). In May 2002, varying numbers of plant species from this species pool were sown into the plots to create a gradient of plant species richness (1, 2, 4, 8, 16 and 60 species) and functional richness (1, 2, 3, 4 functional groups). Plots were maintained by bi-annual weeding and mowing. Soil sampling and analysis: Five soil cores (diameter 0.01 m) were taken at a depth of 0 to 0.15 m of the mineral soil from each of the experimental plots in March and October 2008. In October 2008, also the plots of the management experiment, that altered mowing frequency and fertilized subplots (see further details below) were sampled. Samples of the soil cores per plot (subplots in case of the management experiment) were pooled during each sampling campaign. NO3-N and NH4-N concentrations were determined by extraction of soil samples with 1 M KCl solution and were measured in the soil extract with a Continuous Flow Analyzer (CFA, AutoAnalyzer, Seal, Burgess Hill, United Kingdom).

Mineral nitrogen from KCl extractions of soil from the Jena Experiment (Main Experiment up to 15cm depth, year 2006)

As an estimate of plant-available N, this data set contains measurements of inorganic nitrogen (NO3-N and NH4-N, the sum of which is termed mineral N or Nmin) determined by extraction with 1 M KCl solution of soil samples from the main experiment plots of a large grassland biodiversity experiment (the Jena Experiment; see further details below). In the main experiment, 82 grassland plots of 20 x 20 m were established from a pool of 60 species belonging to four functional groups (grasses, legumes, tall and small herbs). In May 2002, varying numbers of plant species from this species pool were sown into the plots to create a gradient of plant species richness (1, 2, 4, 8, 16 and 60 species) and functional richness (1, 2, 3, 4 functional groups). Plots were maintained by bi-annual weeding and mowing. Soil sampling and analysis: Five soil cores (diameter 0.01 m) were taken at a depth of 0 to 0.15 m of the mineral soil from each of the experimental plots in March 2006. In October 2006 also the plots of the management experiment, that altered mowing frequency and fertilized subplots (see further details below) were sampled. Measurements from the management experiment are separated into 0 to 0.08 m and 0.08 to 0.15 m. Samples of the soil cores per plot (subplots in case of the management experiment) were pooled during each sampling campaign. NO3-N and NH4-N concentrations were determined by extraction of soil samples with 1 M KCl solution and were measured in the soil extract with a Continuous Flow Analyzer (CFA, AutoAnalyzer, Seal, Burgess Hill, United Kingdom).

Mineral nitrogen from KCl extractions of soil from the Jena Experiment (Main Experiment up to 15cm depth, year 2007)

As an estimate of plant-available N, this data set contains measurements of inorganic nitrogen (NO3-N and NH4-N, the sum of which is termed mineral N or Nmin) determined by extraction with 1 M KCl solution of soil samples from the main experiment plots of a large grassland biodiversity experiment (the Jena Experiment; see further details below). In the main experiment, 82 grassland plots of 20 x 20 m were established from a pool of 60 species belonging to four functional groups (grasses, legumes, tall and small herbs). In May 2002, varying numbers of plant species from this species pool were sown into the plots to create a gradient of plant species richness (1, 2, 4, 8, 16 and 60 species) and functional richness (1, 2, 3, 4 functional groups). Plots were maintained by bi-annual weeding and mowing. Soil sampling and analysis: Five soil cores (diameter 0.01 m) were taken at a depth of 0 to 0.15 m of the mineral soil from each of the experimental plots in March and October 2007. In March and in October 2007 also the plots of the management experiment, that altered mowing frequency and fertilized subplots (see further details below) were sampled. Samples of the soil cores per plot (subplots in case of the management experiment) were pooled during each sampling campaign. NO3-N and NH4-N concentrations were determined by extraction of soil samples with 1 M KCl solution and were measured in the soil extract with a Continuous Flow Analyzer (CFA, AutoAnalyzer, Seal, Burgess Hill, United Kingdom).

Mineral nitrogen from KCl extractions of soil from the Jena Experiment (Main Experiment up to 30cm depth, year 2003)

As an estimate of plant-available N, this data set contains measurements of inorganic nitrogen (NO3-N and NH4-N, the sum of which is termed mineral N or Nmin) determined by extraction with 1 M KCl solution of soil samples from the main experiment plots of a large grassland biodiversity experiment (the Jena Experiment; see further details below). In the main experiment, 82 grassland plots of 20 x 20 m were established from a pool of 60 species belonging to four functional groups (grasses, legumes, tall and small herbs). In May 2002, varying numbers of plant species from this species pool were sown into the plots to create a gradient of plant species richness (1, 2, 4, 8, 16 and 60 species) and functional richness (1, 2, 3, 4 functional groups). Plots were maintained by bi-annual weeding and mowing. Soil sampling and analysis: Five soil cores (diameter 0.01 m) were taken at a depth of 0 to 0.15 m and 0.15 to 0.3 m of the mineral soil from each of the experimental plots in March, June, and October 2003. Samples of the soil cores per plot were pooled during each sampling campaign. NO3-N and NH4-N concentrations were determined by extraction of soil samples with 1 M KCl solution and were measured in the soil extract with a Continuous Flow Analyzer (CFA, Skalar, Breda, Netherlands).

Pollen analysis of soil samples from the A.D. 79 level. Boscoreale, Oplontis, and Pompeii

Fifty samples of Roman time soil preserved under the thick ash layer of the A.D.79 eruption of Mt Vesuvius were studied by pollen analysis: 33 samples from a former vineyard surrounding a Villa Rustica at Boscoreale (excavation site 40 x 50 m), 13 samples taken along the 60 m long swimming pool in the sculpture garden of the Villa of Poppaea at Oplontis, and four samples from the formal garden (12.4 x 17.5 m) of the House of the Gold Bracelet in Pompeii. To avoid contamination with modern pollen all samples were taken immediately after uncovering a new portion of the A.D. 79 soil. For comparison also samples of modern Italian soils were studied. Using standard methods for pollen preparation the pollen content of 15 of the archaeological samples proved to be too little to reach a pollen sum of more than 100 grains. The pollen spectra of these samples are not shown in the pollen tables. (Flotation with a sodium tungstate solution, Na2WO4, D = 2.05, following treatment with HCl and NaOH would probably have given a somewhat better result. This method was, however, not available as too expensive at that time.) Although the archaeological samples were taken a few meters apart their pollen values differ very much from one sample to the other. E.g., at Boscoreale (SW quarter). the pollen values of Pinus range from 1.5 to 54.5% resp. from 1 to 244 pine pollen grains per 1 gram of soil, the extremes even found under pine trees. Vitis pollen was present in 7 of the 11 vineyard samples from Boscoreale (NE quarter) only. Although a maximum of 21.7% is reached, the values of Vitis are mostly below 1.5%. Even the values of common weeds differ very much, not only at Boscoreale, but also at the other two sites. The pollen concentration values show similar variations: 3 to 3053 grains and spores were found in 1 g of soil. The mean value (290) is much less than the number of pollen grains, which would fall on 1 cm2 of soil surface during one year. In contrast, the pollen and spore concentrations of the recent soil samples, treated in exactly the same manner, range from 9313 to almost 80000 grains per 1 g of soil. Evidently most of the Roman time pollen has disappeared since its deposition, the reasons not being clear. Not even species which are known to have been cultivated in the garden of Oplontis, like Citrus and Nerium, plant species with easily distinguishable pollen grains, could be traced by pollen analysis. The loss of most of the pollen grains originally contained in the soil prohibits any detailed interpretation of the Pompeian pollen data. The pollen counts merely name plant species which grew in the region, but not necessarily on the excavated plots.

The relationship between soil geochemistry and the bioaccessibility of trace elements in playground soil

A total of 32 samples of surficial soil were collected from 16 playground areas in Madrid (Spain), in order to investigate the importance of the geochemistry of the soil on subsequent bioaccessibility of trace elements. The in vitro bioaccessibility of As, Co, Cr, Cu, Ni, Pb and Zn was evaluated by means of two extraction processes that simulate the gastric environment and one that reproduces a gastric + intestinal digestion sequence. The results of the in vitro bioaccessibility were compared against aqua regia extractions (“total” concentration), and it was found that total concentrations of As, Cu, Pb and Zn were double those of bioaccessible values, whilst that of Cr was ten times higher. Whereas the results of the gastric + intestinal extraction were affected by a high uncertainty, both gastric methods offered very similar and consistent results, with bioaccessibilities following the order: As = Cu = Pb = Zn > Co > Ni > Cr, and ranging from 63 to 7 %. Selected soil properties including pH, organic matter, Fe and CaCO3 content were determined to assess their influence on trace element bioaccessibility, and it was found that Cu, Pb and Zn were predominantly bound to organic matter and, to a lesser extent, Fe oxides. The former fraction was readily accessible in the gastric solution, whereas Fe oxides seemed to recapture negatively charged chloride complexes of these elements in the gastric solution, lowering their bioaccessibility. The homogeneous pH of the playground soils included in the study does not influence trace element bioaccessibility to any significant extent except for Cr, where the very low gastric accessibility seems to be related to the strongly pH-dependent formation of complexes with organic matter. The results for As, which have been previously described and discussed in detail in Mingot et al. (Chemosphere 84: 1386–1391, 2011), indicate a high gastric bioaccessibility for this element as a consequence of its strong association with calcium carbonate and the ease with which these bonds are broken in the gastric solution. The calculation of risk assessments are therefore dependant on the methodology used and the specific environment they address. This has impacts on management strategies formulated to ensure that the most vulnerable of society, children, can live and play without adverse consequences to their health.

Seasonal evolution of soil respiration in a mixed forestry plantation of walnuts (Juglans × intermedia Carr.) and alders (Alnus cordata (Loisel.) Duby) in Domaine de Restinclières (France)

The ecological intensification of crops is proposed as a solution to the growing demand of agricultural and forest resources, in opposition to intensive monocultures. The introduction of mixed cultures as mixtures between nitrogen fixing species and non nitrogen fixing species intended to increase crop yield as a result of an improvement of the available nitrogen and phosphorus in soil. Relationship between crops have received little attention despite the wide range of advantages that confers species diversity to these systems, such as increased productivity, resilience to disruption and ecological sustainability. Forests and forestry plantations can develop an important role in storing carbon in their tissues, especially in wood which become into durable product. A simplifying parameter to analyze the amount allocated carbon by plantation is the TBCA (total belowground carbon allocation), whereby, for short periods and mature plantations, is admitted as the subtraction between soil carbon efflux and litterfall. Soil respiration depends on a wide range of factors, such as soil temperature and soil water content, soil fertility, presence and type of vegetation, among others. The studied orchard is a mixed forestry plantation of hybrid walnuts(Juglans × intermedia Carr.) for wood and alders (Alnus cordata (Loisel.) Duby.), a nitrogen fixing specie through the actinomycete Frankia alni ((Woronin, 1866) Von Tubeuf 1895). The study area is sited at Restinclières, a green area near Montpellier (South of France). In the present work, soil respiration varied greatly throughout the year, mainly influenced by soil temperature. Soil water content did not significantly influence the response of soil respiration as it was constant during the measurement period and under no water stress conditions. Distance between nearest walnut and measurement was also a highly influential factor in soil respiration. Generally there was a decreasing trend in soil respiration when the distance to the nearest tree increased. It was also analyzed the response of soil respiration according to alder presence and fertilizer management (50 kg N·ha-1·año-1 from 1999 to 2010). None of these treatments significantly influenced soil respiration, although previous studies noticed an inhibition in rates of soil respiration under fertilized conditions and high rates of available nitrogen. However, treatments without fertilization and without alder presence obtained higher respiration rates in those cases with significant differences. The lack of significant differences between treatments may be due to the high coefficient of variation experienced by soil respiration measurements. Finally an asynchronous fluctuation was observed between soil respiration and litterfall during senescence period. This is possibly due to the slowdown in the emission of exudates by roots during senescence period, which are largely related to microbial activity.

On soil-structure interaction in large non-slender partially buried structures

This paper addresses the seismic analysis of a deeply embedded non-slender structure hosting the pumping unit of a reservoir. The dynamic response in this type of problems is usually studied under the assumption of a perfectly rigid structure using a sub-structuring procedure (three-step solution) proposed specifically for this hypothesis. Such an approach enables a relatively simple assessment of the importance of some key factors influencing the structural response. In this work, the problem is also solved in a single step using a direct approach in which the structure and surrounding soil are modelled as a coupled system with its actual geometry and flexibility. Results indicate that, quite surprisingly, there are significant differences among prediction using both methods. Furthermore, neglecting the flexibility of the structure leads to a significant underestimation of the spectral accelerations at certain points of the structure.

Efficient PGD-based dynamic calculation of non-linear soil behavior

Non-linear behavior of soils during a seismic event has a predominant role in current site response analysis. Soil response analysis consistently indicates that the stress-strain relationship of soils is nonlinear and shows hysteresis. When focusing in forced response simulations, time integrations based on modal analysis are widely considered, however parametric analysis, non-linear behavior and complex damping functions make difficult the online use of standard discretization strategies, e.g. those based on the use of finite element. In this paper we propose a new harmonic analysis formulation, able to address forced response simulation of soils exhibiting their characteristic nonlinear behavior. The solution can be evaluated in real-time from the offline construction of a parametric solution of the associated linearized problem within the Proper Generalized Decomposition framework.

Real-Time Dynamic PGD Calculation of Non-Linear Soil Behavior

El estudio sísmico en los últimos 50 años y el análisis del comportamiento dinámico del suelo revelan que el comportamiento del suelo es altamente no lineal e histéretico incluso para pequeñas deformaciones. El comportamiento no lineal del suelo durante un evento sísmico tiene un papel predominante en el análisis de la respuesta de sitio. Los análisis unidimensionales de la respuesta sísmica del suelo son a menudo realizados utilizando procedimientos lineales equivalentes, que requieren generalmente pocos parámetros conocidos. Los análisis de respuesta de sitio no lineal tienen el potencial para simular con mayor precisión el comportamiento del suelo, pero su aplicación en la práctica se ha visto limitada debido a la selección de parámetros poco documentadas y poco claras, así como una inadecuada documentación de los beneficios del modelado no lineal en relación al modelado lineal equivalente. En el análisis del suelo, el comportamiento del suelo es aproximado como un sólido Kelvin-Voigt con un módulo de corte elástico y amortiguamiento viscoso. En el análisis lineal y no lineal del suelo se están considerando geometrías y modelos reológicos más complejos. El primero está siendo dirigido por considerar parametrizaciones más ricas del comportamiento linealizado y el segundo mediante el uso de multi-modo de los elementos de resorte-amortiguador con un eventual amortiguador fraccional. El uso del cálculo fraccional está motivado en gran parte por el hecho de que se requieren menos parámetros para lograr la aproximación exacta a los datos experimentales. Basándose en el modelo de Kelvin-Voigt, la viscoelasticidad es revisada desde su formulación más estándar a algunas descripciones más avanzada que implica la amortiguación dependiente de la frecuencia (o viscosidad), analizando los efectos de considerar derivados fraccionarios para representar esas contribuciones viscosas. Vamos a demostrar que tal elección se traduce en modelos más ricos que pueden adaptarse a diferentes limitaciones relacionadas con la potencia disipada, amplitud de la respuesta y el ángulo de fase. Por otra parte, el uso de derivados fraccionarios permite acomodar en paralelo, dentro de un análogo de Kelvin-Voigt generalizado, muchos amortiguadores que contribuyen a aumentar la flexibilidad del modelado para la descripción de los resultados experimentales. Obviamente estos modelos ricos implican muchos parámetros, los asociados con el comportamiento y los relacionados con los derivados fraccionarios. El análisis paramétrico de estos modelos requiere técnicas numéricas eficientemente capaces de simular comportamientos complejos. El método de la Descomposición Propia Generalizada (PGD) es el candidato perfecto para la construcción de este tipo de soluciones paramétricas. Podemos calcular off-line la solución paramétrica para el depósito de suelo, para todos los parámetros del modelo, tan pronto como tales soluciones paramétricas están disponibles, el problema puede ser resuelto en tiempo real, porque no se necesita ningún nuevo cálculo, el solucionador sólo necesita particularizar on-line la solución paramétrica calculada off-line, que aliviará significativamente el procedimiento de solución. En el marco de la PGD, parámetros de los materiales y los diferentes poderes de derivación podrían introducirse como extra-coordenadas en el procedimiento de solución. El cálculo fraccional y el nuevo método de reducción modelo llamado Descomposición Propia Generalizada han sido aplicado en esta tesis tanto al análisis lineal como al análisis no lineal de la respuesta del suelo utilizando un método lineal equivalente. ABSTRACT Studies of earthquakes over the last 50 years and the examination of dynamic soil behavior reveal that soil behavior is highly nonlinear and hysteretic even at small strains. Nonlinear behavior of soils during a seismic event has a predominant role in current site response analysis. One-dimensional seismic ground response analysis are often performed using equivalent-linear procedures, which require few, generally well-known parameters. Nonlinear analyses have the potential to more accurately simulate soil behavior, but their implementation in practice has been limited because of poorly documented and unclear parameter selection, as well as inadequate documentation of the benefits of nonlinear modeling relative to equivalent linear modeling. In soil analysis, soil behaviour is approximated as a Kelvin-Voigt solid with a elastic shear modulus and viscous damping. In linear and nonlinear analysis more complex geometries and more complex rheological models are being considered. The first is being addressed by considering richer parametrizations of the linearized behavior and the second by using multi-mode spring-dashpot elements with eventual fractional damping. The use of fractional calculus is motivated in large part by the fact that fewer parameters are required to achieve accurate approximation of experimental data. Based in Kelvin-Voigt model the viscoelastodynamics is revisited from its most standard formulation to some more advanced description involving frequency-dependent damping (or viscosity), analyzing the effects of considering fractional derivatives for representing such viscous contributions. We will prove that such a choice results in richer models that can accommodate different constraints related to the dissipated power, response amplitude and phase angle. Moreover, the use of fractional derivatives allows to accommodate in parallel, within a generalized Kelvin-Voigt analog, many dashpots that contribute to increase the modeling flexibility for describing experimental findings. Obviously these rich models involve many parameters, the ones associated with the behavior and the ones related to the fractional derivatives. The parametric analysis of all these models require efficient numerical techniques able to simulate complex behaviors. The Proper Generalized Decomposition (PGD) is the perfect candidate for producing such kind of parametric solutions. We can compute off-line the parametric solution for the soil deposit, for all parameter of the model, as soon as such parametric solutions are available, the problem can be solved in real time because no new calculation is needed, the solver only needs particularize on-line the parametric solution calculated off-line, which will alleviate significantly the solution procedure. Within the PGD framework material parameters and the different derivation powers could be introduced as extra-coordinates in the solution procedure. Fractional calculus and the new model reduction method called Proper Generalized Decomposition has been applied in this thesis to the linear analysis and nonlinear soil response analysis using a equivalent linear method.

Global assessment of soil phosphorus retention potential

Limited availability of P in soils to crops may be due to deficiency and/or severe P retention. Earlier studies that drew on large soil profile databases have indicated that it is not (yet) feasible to present meaningful values for "plant-available" soil P, obtained according to comparable analytical methods, that may be linked to soil geographical databases derived from 1:5 million scale FAO Digital Soil Map of the World, such as the 5 x 5 arc-minute version of the ISRIC-WISE database. Therefore, an alternative solution for studying possible crop responses to fertilizer-P applied to soils, at a broad scale, was sought. The approach described in this report considers the inherent capacity of soils to retain phosphorus (P retention), in various forms. Main controlling factors of P retention processes, at the broad scale under consideration, are considered to be pH, soil mineralogy, and clay content. First, derived values for these properties were used to rate the inferred capacity for P retention of the component soil units of each map unit (or grid cell) using four classes (i.e., Low, Moderate, High, and Very High). Subsequently, the overall soil phosphorus retention potential was assessed for each mapping unit, taking into account the P-ratings and relative proportion of each component soil unit. Each P retention class has been assigned to a likely fertilizer P recovery fraction, derived from the literature, thereby permitting spatially more detailed, integrated model-based studies of environmental sustainability and agricultural production at the global and continental level (< 1:5 million). Nonetheless, uncertainties remain high; the present analysis provides an approximation of world soil phosphorus retention potential.

Similarity solution of axisymmetric flow in porous media

Applications of the axisymmetric Boussinesq equation to groundwater hydrology and reservoir engineering have long been recognised. An archetypal example is invasion by drilling fluid into a permeable bed where there is initially no such fluid present, a circumstance of some importance in the oil industry. It is well known that the governing Boussinesq model can be reduced to a nonlinear ordinary differential equation using a similarity variable, a transformation that is valid for a certain time-dependent flux at the origin. Here, a new analytical approximation is obtained for this case. The new solution,, which has a simple form, is demonstrated to be highly accurate. (c) 2005 Elsevier Ltd. All rights reserved.