Database Analysis to Support Nutrient Criteria Development (Phase I)

The intent of this publication of the Arkansas Water Resources Center is to provide a location whereby a final report on water research to a funding agency can be archived. The Texas Commission on Environmental Quality (TCEQ) contracted with University of Arkansas researchers for a multiple year project titled “Database Analysis to Support Nutrient Criteria Development”. This publication covers the first of three phases of that project and has maintained the original format of the report as submitted to TCEQ. This report can be cited either as an AWRC publication (see below) or directly as the final report to TCEQ

A key-formula to compute the gravitational potential of inhomogeneous discs in cylindrical coordinates

We have established the exact expression for the gravitational potential of a homogeneous polar cell - an elementary pattern used in hydrodynamical simulations of gravitating discs. This formula, which is a closed-form, works for any opening angle and radial extension of the cell. It is valid at any point in space, i.e. in the plane of the distribution (inside and outside) as well as off-plane, thereby generalizing the results reported by Durand (1953) for the circular disc. The three components of the gravitational acceleration are given. The mathematical demonstration proceeds from the "incomplete version of Durand's formula" for the potential (based on complete elliptic integrals). We determine first the potential due to the circular sector (i.e. a pie-slice sheet), and then deduce that of the polar cell (from convenient radial scaling and subtraction). As a by-product, we generate an integral theorem stating that "the angular average of the potential of any circular sector along its tangent circle is 2/PI times the value at the corner". A few examples are presented. For numerical resolutions and cell shapes commonly used in disc simulations, we quantify the importance of curvature effects by performing a direct comparison between the potential of the polar cell and that of the Cartesian (i.e. rectangular) cell having the same mass. Edge values are found to deviate roughly like 2E-3 x N/256 in relative (N is the number of grid points in the radial direction), while the agreement is typically four orders of magnitude better for values at the cell's center. We also produce a reliable approximation for the potential, valid in the cell's plane, inside and close to the cell. Its remarkable accuracy, about 5E-4 x N/256 in relative, is sufficient to estimate the cell's self-acceleration.Comment: Accepted for publication in Celestial Mechanics and Dynamical Astronom

On the fluctuations of jamming coverage upon random sequential adsorption on homogeneous and heterogeneous media

The fluctuations of the jamming coverage upon Random Sequential Adsorption (RSA) are studied using both analytical and numerical techniques. Our main result shows that these fluctuations (characterized by $\sigma_{\theta_J}$) decay with the lattice size according to the power-law $\sigma_{\theta_J} \propto L^{-1/ \nu}$. The exponent $\nu$ depends on the dimensionality $D$ of the substrate and the fractal dimension of the set where the RSA process actually takes place ($d_f$) according to $\nu = 2 / (2D - d_f)$.This theoretical result is confirmed by means of extensive numerical simulations applied to the RSA of dimers on homogeneous and stochastic fractal substrates. Furthermore, our predictions are in excellent agreement with different previous numerical results. It is also shown that, studying correlated stochastic processes, one can define various fluctuating quantities designed to capture either the underlying physics of individual processes or that of the whole system. So, subtle differences in the definitions may lead to dramatically different physical interpretations of the results. Here, this statement is demonstrated for the case of RSA of dimers on binary alloys.Comment: 20 pages, 8 figure

Surface Polymer Network Model and Effective Membrane Curvature Elasticity

A microscopic model of a surface polymer network - membrane system is introduced, with contact polymer surface interactions that can be either repulsive or attractive and sliplinks of functionality four randomly distributed over the supporting membrane surface anchoring the polymers to it. For the supporting surface perturbed from a planar configuration and a small relative number of surface sliplinks, we investigate an expansion of the free energy in terms of the local curvatures of the surface and the surface density of sliplinks, obtained through the application of the Balian - Bloch - Duplantier multiple surface scattering method. As a result, the dependence of the curvature elastic modulus, the Gaussian modulus as well as of the spontaneous curvature of the "dressed" membrane, ~{\sl i.e.} polymer network plus membrane matrix, is obtained on the mean polymer bulk end to end separation and the surface density of sliplinks.Comment: 15 pages with one included compressed uuencoded figure

Micro-computed tomographic analysis of the radial geometry of intrarenal artery-vein pairs in rats and rabbits: Comparison with light microscopy

We assessed the utility of synchrotron-radiation micro-computed tomography (micro-CT) for quantification of the radial geometry of the renal cortical vasculature. The kidneys of nine rats and six rabbits were perfusion fixed and the renal circulation filled with Microfil. In order to assess shrinkage of Microfil, rat kidneys were imaged at the Australian Synchrotron immediately upon tissue preparation and then post fixed in paraformaldehyde and reimaged 24 hours later. The Microfil shrank only 2-5% over the 24 hour period. All subsequent micro-CT imaging was completed within 24 hours of sample preparation. After micro-CT imaging, the kidneys were processed for histological analysis. In both rat and rabbit kidneys, vascular structures identified in histological sections could be identified in two-dimensional (2D) micro-CT images from the original kidney. Vascular morphology was similar in the two sets of images. Radial geometry quantified by manual analysis of 2D images from micro-CT was consistent with corresponding data generated by light microscopy. However, due to limited spatial resolution when imaging a whole organ using contrast-enhanced micro-CT, only arteries ≥100 and ≥60 μm in diameter, for the rat and rabbit respectively, could be assessed. We conclude that it is feasible and valid to use micro-CT to quantify vascular geometry of the renal cortical circulation in both the rat and rabbit. However, a combination of light microscopic and micro-CT approaches are required to evaluate the spatial relationships between intrarenal arteries and veins over an extensive range of vessel size

Water wave propagation and scattering over topographical bottoms

Here I present a general formulation of water wave propagation and scattering over topographical bottoms. A simple equation is found and is compared with existing theories. As an application, the theory is extended to the case of water waves in a column with many cylindrical steps

Persistent global power fluctuations near a dynamic transition in electroconvection

This is a study of the global fluctuations in power dissipation and light transmission through a liquid crystal just above the onset of electroconvection. The source of the fluctuations is found to be the creation and annihilation of defects. They are spatially uncorrelated and yet temporally correlated. The temporal correlation is seen to persist for extremely long times. There seems to be an especially close relation between defect creation/annihilat ion in electroconvection and thermal plumes in Rayleigh-B\'enard convection

Fractional moment bounds and disorder relevance for pinning models

We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(.) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n)=n^{-\alpha-1}L(n), with L(.) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For \alpha<1/2 it is known that disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents. The same has been proven also for \alpha=1/2, but under the assumption that L(.) diverges sufficiently fast at infinity, an hypothesis that is not satisfied in the (1+1)-dimensional wetting model considered by Forgacs et al. (1986) and Derrida et al. (1992), where L(.) is asymptotically constant. Here we prove that, if 1/21, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case \alpha=1/2, under the assumption that L(.) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is known to be smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered by Forgacs et al. (1986) and Derrida et al. (1992) is out of our analysis and remains open.Comment: 20 pages, 1 figure; v2: few typos corrected, references revised. To appear on Commun. Math. Phy