Outer Approximation of the Spectrum of a Fractal Laplacian

We present a new method to approximate the Neumann spectrum of a Laplacian on a fractal K in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains that approximate K from the outside. The method allows a numerical approximation of eigenvalues and eigenfunctions for lower portions of the spectrum. We present experimental evidence that the method works by looking at examples where the spectrum of the fractal Laplacian is known (the unit interval and the Sierpinski Gasket (SG)). We also present a speculative description of the spectrum on the standard Sierpinski carpet (SC), where existence of a self-similar Laplacian is known, and also on nonsymmetric and random carpets and the octagasket, where existence of a self-similar Laplacian is not known. At present we have no explanation as to why the method should work. Nevertheless, we are able to prove some new results about the structure of the spectrum involving "miniaturization" of eigenfunctions that we discovered by examining the experimental results obtained using our method.Comment: 61 pages, 24 figures, to appear, Exp. Mat

The Nashua agronomic, water quality, and economic dataset

This paper describes a dataset relating management to nitrogen (N) loading and crop yields from 1990 to 2003 on 36, 0.4 ha (1 ac) individually tile-drained plots on the Northeast Research and Demonstration Farm near Nashua, Iowa, United States. The field-measured data were used to calibrate the Root Zone Water Quality Model (RZWQM), and the results were summarized in a special issue ofGeoderma (Ahuja and Hatfield 2007). With a comprehensive, long-term measured dataset and a model that simulates many of the components of the agricultural system, one can begin to understand the effects of management practices on N loading, crop yields, and net income to the farmers. Other researchers can use this dataset to assess the effects of management on similar tile-drained systems occurring some distance from Nashua, under alternative climates and soils, with other management systems, or with simulation models using different process representations. By integrating the understanding developed at Nashua with datasets from other highly monitored sites and other sources, progress can be made in addressing problems related to excessive N fluxes in the Mississippi Basin. An example 30-year RZWQM simulation of 18 management systems implies that significant management changes are needed to meet the goal of reducing N loads to the Gulf of Mexico by 45%. This paper and the associated datasets are intended to be used in conjunction with the analyses and process descriptions presented in the Geoderma special issue. The datasets and additional explanatory materials are available for download at http://apps.tucson.ars.ag.gov/nashua

RZWQM simulated effects of crop rotation, tillage, and controlled drainage on crop yield and nitrate-N loss in drain flow

Accurate simulation of agricultural management effects on N loss in tile drainage is vitally important for understanding hypoxia in the Gulf of Mexico. An experimental study was initiated in 1978 at Nashua, Iowa of the USA to study long-term effects of tillage, crop rotation, and N management practices on subsurface drainage flow and associated N losses. The Root Zone Water Quality Model (RZWQM) was applied to evaluate various management effects in several previous studies. In this study, the simulation results were further analyzed for management effects (tillage, crop rotation, and controlled drainage) on crop production and N loss in drain flow. RZWQM simulated the observed increase in N concentration in drain flow with increasing tillage intensity from NT (no-till) to RT (ridge till) to CP (chisel plow) and to MP (moldboard plow). It also adequately simulated tillage effects on yearly drain flow and yearly N loss in drain flow. However, the model failed to simulate lower corn and soybean yields under NT than under MP, CP, and RT. On the other hand, RZWQM adequately simulated lower yearly drain flow and lower flow-weighted N concentration in drain flow under CS (corn–soybean) and SC (soybean–corn) than under CC (continuous corn). The model adequately simulated higher corn yield under CS and SC than under CC. Applying the newly suggested N management practice for the Midwest of controlled drainage, the model simulated a 30% reduction in drain flow and a 29% decrease in N losses in drain flow under controlled drainage (CD) compared to free drainage (FD). With most of the simulations in reasonably close agreement with observations, we concluded that RZWQM is a promising tool for quantifying the relative effects of tillage, crop rotation, and controlled drainage on N loss in drainage flow. Further improvements on simulated management effects on crop yield and N mineralization are needed, however

Sensitivity of tile drainage flow and crop yield on measured and calibrated soil hydraulic properties

Process-based agricultural system models require detailed description of soil hydraulic properties that are usually not available. The objectives of this study were to evaluate the sensitivity of model simulation results to variability in measured soil hydraulic properties and to compare simulation results using measured and default soil parameters. To do so, we measured soil water retention curves and saturated soil hydraulic conductivity (Ksat) from intact soil cores taken from a long-term experimental field near Nashua, Iowa for the Kenyon–Clyde–Floyd–Readlyn soil association. The soil water retention curves could be well described using the pore size distribution index (λ). Measured λ values from undisturbed soil cores ranged from 0.04 to 0.12 and the measured Ksat values ranged from 1.8 to 14.5 cm/h. These hydraulic properties were then used to calibrate the Root Zone Water Quality Model (RZWQM) for simulating soil water content, water table, tile drain flow, and crop yield (corn and soybean) by optimizing the lateral Ksat(LKsat) and hydraulic gradient (HG) for subsurface lateral flow. The measured soil parameters provided better simulations of soil water storage, water table, and N loss in tile flow than using the default soil parameters based on soil texture classes in RZWQM. Sensitivity analyses were conducted for λ, Ksat, saturated soil water content (θs) or drainable porosity, LKsat, and HG using the Latin Hypercubic Sampling (LHS) and for LKsat and HG also using a single variable analysis. Results of sensitivity analyses showed that RZWQM-simulated yield and biomass were not sensitive to soil hydraulic properties. Simulated tile flow and N losses in tile flow were not sensitive to λ and Ksat either, but they were sensitive to LKsat and HG. Further sensitivity analyses using a single variable showed that LKsat in the tile layer was a more sensitive parameter compared to LKsat in other soil layers, and HG was the most sensitive parameter for tile flow under the experimental soil and weather conditions