Depth-dependent Coefficient Research Articles

Snow Loss Modeling for Roof Mounted Photovoltaic Systems: Improving the Marion Snow Loss Model

Accurate photovoltaics (PV) energy yield assessments for cold climates necessitate understanding and estimation of snow loss. Estimation of snow loss for a specific system requires a snow loss model. Multiple models to estimate snow loss are suggested in the literature, but extensive validation is lacking. In this work, we describe the effect of snow on PV systems by analyzing signatures in monitoring data and we evaluate the accuracy of a modified adaption of the Marion snow loss model. Eight different systems with a total installed capacity of 1.6 MW <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> , installed on both tilted and flat roofs, are analyzed. In the modified model we use different snow clearing rate coefficients for thin and thick snow covers to model the natural snow clearing process. The snow depth dependent coefficients yield lower error in the total modeled snow loss and capture climatic variations between locations more accurately compared to the standard constant coefficient. For most of the systems the total absolute snow loss is modeled with an error of less than 11%, on average 23% points lower than with the default implementation of the Marion model. Some of the systems have larger modeling errors, which can be related to effects not taken into account in the model, such as the effect of building heat leakage and shading on snow clearing.

Data-driven dynamics of phytoplankton blooms in a reaction–diffusion NPZ model

Abstract The causes of seasonal phytoplankton blooms in the ocean are a debated topic. The disturbance-recovery hypothesis posits that blooms are initiated when seasonally changing light and mixing conditions disrupt attractive equilibrium states in the predator–prey relationship between zooplankton and phytoplankton, leading to an accumulation in phytoplankton biomass. This paper follows up on this notion with a nutrient-phytoplankton-zooplankton (NPZ) model incorporating diffusion and depth-dependent coefficients in which we explore the nature of these attractive states. The reaction–diffusion NPZ model, incorporated with seasonal solar radiation and mixed layer depth data, simulates blooms with better timing than a related ordinary differential equations model but still leaves room for further modeling and improvement. The results of the data-driven, reaction–diffusion model aid in understanding the driving forces of the simulated phytoplankton blooms. The results also reveal a significant influence on the vertical structure of plankton concentration by the attenuation of light with depth in the ocean. Models that accurately simulate blooms tend to share characteristics of the reaction–diffusion model in this paper. The dynamics described in this paper could be a key mechanism that seasonally disrupts the ecological balance between phytoplankton and zooplankton and facilitates a high-latitude marine phytoplankton bloom.

A new model approach for reactive solute transport in dual-permeability media with depth-dependent reaction coefficients

In this study, we developed a one-dimensional mobile-mobile model (MM) for contaminant transport in subsurface with depth-dependent reaction coefficients. Two local systems at micro-scale level are involved in this model: a micro-pore system and a less permeable pore system. The breakthrough curves (BTCs) obtained from the MM with depth-dependent coefficients were thoroughly analyzed, where the case with constant coefficients was used as the reference system. The effects of different parameters on BTCs were studied, and a parameter analysis was also performed to illustrate the effects of adsorption coefficient and degradation coefficient on BTCs. Results show that the BTC patterns for the depth-dependent reactions, in which the reaction rate decreases with increasing depth, vary between the two extreme cases with constant-rate reactions in the topsoil and observation depth. The mass transfer coefficient (α) plays a significant role in defining the BTC pattern. The double peak changes to a single peak when the value of α is large enough. In addition, the velocity of the main flow determines the shape of the BTCs while the velocity of the SVD defines the pattern of the dual peaks in BTCs which also tends to be represented by a single peak as the velocity decreases. Finally, we tested the applicability of the proposed model (MM-FD) in the leaching experiments of both field herbicide and conservative tracer in an agricultural area of northeastern Greece, and compared its performance with that of the advection-dispersion equation (ADE) model by Gao et al. (2013), both of which have depth-dependent coefficients. The results showed that the proposed mobile-mobile model with depth-dependent reaction coefficients was able to satisfactorily capture the evolution of metolachlor concentration both at upper and lower depths.

Algorithm for calculating synthetic seismograms in a layered half-space with application of matrix impedance

A new algorithm is proposed for calculating the complete synthetic seismograms from a point source in the form of the sum of a single force and a dipole with an arbitrary seismic moment tensor in a plane layered medium composed of homogenous elastic isotropic layers. Following the idea of (Alekseev and Mikhailenko, 1978), an artificial cylindrical boundary is introduced, on which the boundary conditions are specified. For this modified problem, the exact solution (in terms of the displacements and stresses on the horizontal plane areal element) in the frequency domain is derived and substantiated. The unknown depth-dependent coefficients form the motion-stress vector, whose components satisfy the known system of ordinary differential equations. This system is solved by the method that involves the matrix impedance and propagator for the vector of motion, as previously suggested by the author in (Pavlov, 2009). In relation to the initial problem, the reflections from the artificial boundary are noise, which, to a certain degree, can be suppressed by selecting a long enough distance to this boundary and owing to the presence of a purely imaginary addition to the frequency. The algorithm is not constrained by the thickness of the layers, is applicable for any frequency range, and is suitable for computing the static offset.

Influences of lower-mantle properties on the formation of asthenosphere in oceanic upper mantle

Asthenosphere is a venerable concept based on geological intuition of Reginald Daly nearly 100 years ago. There have been various explanations for the existence of the asthenosphere. The concept of a plume-fed asthenosphere has been around for a few years due to the ideas put forth by Yamamoto et al.. Using a two-dimensional Cartesian code based on finite-volume method, we have investigated the influences of lower-mantle physical properties on the formation of a low-viscosity zone in the oceanic upper mantle in regions close to a large mantle upwelling. The rheological law is Newtonian and depends on both temperature and depth. An extended-Boussinesq model is assumed for the energetics and the olivine to spinel, the spinel to perovskite and perovskite to post-perovskite (ppv) phase transitions are considered. We have compared the differences in the behavior of hot upwellings passing through the transition zone in the mid-mantle for a variety of models, starting with constant physical properties in the lower-mantle and culminating with complex models which have the post-perovskite phase transition and depth-dependent coefficient of thermal expansion and thermal conductivity. We found that the formation of the asthenosphere in the upper mantle in the vicinity of large upwellings is facilitated in models where both depth-dependent thermal expansivity and conductivity are included. Models with constant thermal expansivity and thermal conductivity do not produce a hot low-viscosity zone, resembling the asthenosphere. We have also studied the influences of a cylindrical model and found similar results as the Cartesian model with the important difference that upper-mantle temperatures were much cooler than the Cartesian model by about 600 to 700 K. Our findings argue for the potentially important role played by lower-mantle material properties on the development of a plume-fed asthenosphere in the oceanic upper mantle.

Correction to “Constraints on the stress state of the San Andreas Fault with analysis based on core and cuttings from San Andreas Fault Observatory at Depth (SAFOD) drilling phases 1 and 2”

[1] Analysis of field data has led different investigators to conclude that the San Andreas Fault (SAF) has either anomalously low frictional sliding strength (μ 0.6). Arguments for the apparent weakness of the SAF generally hinge on conceptual models involving intrinsically weak gouge or elevated pore pressure within the fault zone. Some models assert that weak gouge and/or high pore pressure exist under static conditions while others consider strength loss or fluid pressure increase due to rapid coseismic fault slip. The present paper is composed of three parts. First, we develop generalized equations, based on and consistent with the Rice (1992) fault zone model to relate stress orientation and magnitude to depth-dependent coefficient of friction and pore pressure. Second, we present temperature- and pressure-dependent friction measurements from wet illite-rich fault gouge extracted from San Andreas Fault Observatory at Depth (SAFOD) phase 1 core samples and from weak minerals associated with the San Andreas Fault. Third, we reevaluate the state of stress on the San Andreas Fault in light of new constraints imposed by SAFOD borehole data. Pure talc (μ≈0.1) had the lowest strength considered and was sufficiently weak to satisfy weak fault heat flow and stress orientation constraints with hydrostatic pore pressure. Other fault gouges showed a systematic increase in strength with increasing temperature and pressure. In this case, heat flow and stress orientation constraints would require elevated pore pressure and, in some cases, fault zone pore pressure in excess of vertical stress.

The Role of Heterogeneities as a Tuning Parameter of Earthquake Dynamics

We investigate the influence of spatial heterogeneities on various aspects of brittle failure and seismicity in a model of a large strike-slip fault. The model dynamics is governed by realistic boundary conditions consisting of constant velocity motion of regions around the fault, static/kinetic friction laws, creep with depth-dependent coefficients, and 3-D elastic stress transfer. The dynamic rupture is approximated on a continuous time scale using a finite stress propagation velocity (“quasidynamic model”). The model produces a “brittle-ductile” transition at a depth of about 12.5 km, realistic hypocenter distributions, and other features of seismicity compatible with observations. Previous work suggested that the range of size scales in the distribution of strength-stress heterogeneities acts as a tuning parameter of the dynamics. Here we test this hypothesis by performing a systematic parameter-space study with different forms of heterogeneities. In particular, we analyze spatial heterogeneities that can be tuned by a single parameter in two distributions: (1) high stress drop barriers in near-vertical directions and (2) spatial heterogeneities with fractal properties and variable fractal dimension. The results indicate that the first form of heterogeneities provides an effective means of tuning the behavior while the second does not. In relatively homogeneous cases, the fault self-organizes to large-scale patches and big events are associated with inward failure of individual patches and sequential failures of different patches. The frequency-size event statistics in such cases are compatible with the characteristic earthquake distribution and large events are quasi-periodic in time. In strongly heterogeneous or near-critical cases, the rupture histories are highly discontinuous and consist of complex migration patterns of slip on the fault. In such cases, the frequency-size and temporal statistics follow approximately power-law relations.

Three-Dimensional Fully Spectral Numerical Method for Mantle Convection with Depth-Dependent Properties

A semi-implicit fully spectral collocation method for the simulation of three-dimensional mantle convection with depth-dependent thermodynamic and transport properties is presented. The variable property Navier-Stokes equation expressed in terms of the primitive variable velocity and pressure is solved with the mass continuity and temperature equations. The periodic horizontal boundary conditions allow a Fourier expansion for the two horizontal directions. The stress-free, impermeable isothermal boundary conditions along with the depth dependent coefficients are handled with a Chebyshev expansion in the vertical direction. In the limit of an infinite Prandtl number appropriate to mantle convection, the inertial terms in the momentum equation are unimportant. In this case an explicit solution of a Poisson equation for pressure can be avoided; instead a fourth-order equation for vertical velocity can be solved. Simultaneous imposition of both impermeable and continuity boundary conditions during the vertical velocity evaluation is discussed. The pressure distributions on the top and bottom bounding planes were determined by means of an influence matrix technique. The numerical method employed here avoids time-splitting errors and enforces velocity boundary conditions and continuity over the entire domain, including the boundaries, to machine accuracy. Strongy time-dependent three-dimensional solutions up to a surface Rayleigh number of 1 × 107 have been obtained. Strong upwellings, pulsating chaotically, are formed by the collective merging of cylindrical plumes.

Effective irrigation uniformity as related to root zone depth

In models used for relating the yield to irrigation uniformity it has been assumed that the spatial distribution of irrigation water, as measured at the soil surface, is indeed the water distribution at any depth throughout the root zone. In the present paper the distribution of infiltrated water within the soil bulk, as determined by an analytic solution of the two-dimensional unsaturated flow equation, did not conform to this assumption. A new alternative definition of irrigation uniformity is proposed under the assumption that water uptake by roots does not affect the flux distribution within the soil profile. In this analysis the spatial distribution of irrigation water flux at the soil surface, which is the upper boundary condition of the flow equation, is assumed to be a sine function. The solution to this problem indicates that there is a “damping” effect, which increases with soil depth, on the surface flux fluctuations. Furthermore, the actual irrigation uniformity at a given depth below the soil surface depends upon the initial uniformity at the surface and the distance between adjacent water sources. The closer the water sources are to each other, the shallower is the depth needed to damp the oscillations down to a certain level. This may explain why the actual uniformity of drip irrigation is high while the detailed distribution is very nonuniform and on the other hand, why the actual uniformity of sprinkler guns is low while the detailed actual distribution is close to uniform. Two uniformity coefficients are derived in this study: 1. A depth dependent coefficient which is made up of the damping factor that multiplies the flux fluctuations at the soil surface; 2. An effective uniformity coefficient, which is an average of the depth dependent coefficient over a part or the entire root zone. Different degrees of uniformity are expected when water is applied by different irrigation systems having similar uniformity coefficients at the soil surface, but dissimilar distances between the emitters. Assuming that crop yield depends to some extent on the uniformity of water depth actually available to the roots, the yields associated with such irrigation systems will probably also vary.

Wave reflection from a sediment layer with depth-dependent properties

The prediction of reflection coefficients for the water-sediment interface is strongly influenced by the physical nature of the sediment (a viscoelastic porous material saturated by a liquid) and the depth dependence of the mechanical properties [E. L. Hamilton, J. Acoust. Soc. Am. 68, 1313–1340 (1980)]. An efficient computational scheme for calculating reflection coefficients has been developed for a sediment layer using the Biot equations with depth-dependent coefficients. The qualitative effects of the inhomogeneous sediment properties (in comparison with uniform properties) are studied numerically. [Work supported by the Office of Naval Research.]