Laplacian Diffusion Research Articles

Global qualitative analysis of coupled three-level food chains

A coupled model for three-level food chain is considered, the coupling rule is discretized Laplacian diffusion type. The focus of this paper is on the global behavior of solutions. We will show that the differences of the same level population density approach zero as t → ∞ , provided that the coupling strength is sufficiently large.

Image analysis using p ‐Laplacian and geometrical PDEs

AbstractMinimizing the integral ∫Ω1/p |∇L |p d Ω for an image L under suitable boundary conditions gives PDEs that are well‐known for p = 1, 2, namely Total Variation evolution and Laplacian diffusion (also known as Gaussian scale space and heat equation), respectively. Without fixing p, one obtains a framework related to the p ‐Laplace equation. The partial differential equation describing the evolution can be simplified using gauge coordinates (directional derivatives), yielding an expression in the two second order gauge derivatives and the norm of the gradient. Ignoring the latter, one obtains a series of PDEs that form a weighted average of the second order derivatives, with Mean Curvature Motion as a specific case. Both methods have the Gaussian scale space in common. Using singularity theory, one can use properties of the heat equation (namely. the role of scale) in the full L (x, t) space and obtain a framework for topological image segmentation. In order to be able to extend image analysis aspects of Gaussian scale space in future work, relations between these methods are investigated, and general numerical schemes are developed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Assessment of water exchange between a discharge region and the open sea – A comparison of different methodological concepts

Two different methods of estimating the water exchange through the Baltic coastal region of Laxemar have been used, consisting of particle trajectories and passive tracers. Water is traced from and to a small discharge region near the coast. The discharge material in this region is treated as zero-dimensional particles or tracers with neutral buoyancy. The real discharge material could be a leakage of radio-nuclides through the sea floor from an underground repository of nuclear waste. Water exchange rates between the discharge region and the model domain are estimated using both forward and backward trajectories as well as passive tracers. The Lagrangian trajectories can account for the time evolution of the water exchange while the tracers give one average age per model grid box. Water exchange times such as residence time, age and transient times have been calculated with trajectories but only the average age (AvA) for tracers. The trajectory calculations provide a more detailed time evolution than the tracers. On the other hand the tracers are integrated “on-line” simultaneously in the sea circulation model with the same time step while the Lagrangian trajectories are integrated “off-line” from the stored model velocities with its inherent temporal resolution, presently 1 h. The sub-grid turbulence is parameterised as the Laplacian diffusion for the passive tracers and with an extra stochastic velocity for trajectories. The importance of the parameterised sub-grid turbulence for the trajectories is estimated to give an extra diffusion of the same order as the Laplacian diffusion by comparing the Lagrangian dispersions with and without parameterisation. The results of the different methods are similar but depend on the chosen diffusivity coefficient with a slightly higher correlation between trajectories and tracers when integrated with a lower diffusivity coefficient.

Overshootings and spurious oscillations caused by biharmonic mixing

Biharmonic mixing is often used in large scale numerical models of the ocean because of its scale selectivity; it effectively damps small scale noise and leaves the large scale dynamics nearly unaffected. The biharmonic operator lacks however positiveness and monotonicity and can therefore produce unphysical results exhibiting spurious overshootings and oscillations. This problematic behaviour cannot be avoided by the addition of an ordinary Laplacian diffusion term. It appears in both continuous and discrete approaches/solutions in both unbounded and bounded domains. The overshootings and oscillations are induced by the strong damping of the smaller scale modes and are therefore comparable to the Gibbs’ phenomenon. With appropriate boundary conditions, the variance of the field decreases monotonically and the oscillations are expected to remain small. The lack of positiveness is however a severe drawback for (dynamic) tracer studies.

Reducing metal artifacts in cone-beam CT images by preprocessing projection data

Computed tomography (CT) streak artifacts caused by metallic implants remain a challenge for the automatic processing of image data. The impact of metal artifacts in the soft-tissue region is magnified in cone-beam CT (CBCT), because the soft-tissue contrast is usually lower in CBCT images. The goal of this study was to develop an effective offline processing technique to minimize the effect. The geometry calibration cue of the CBCT system was used to track the position of the metal object in projection views. The three-dimensional (3D) representation of the object can be established from only two user-selected viewing angles. The position of the shadowed region in other views can be tracked by projecting the 3D coordinates of the object. Automatic image segmentation was used followed by a Laplacian diffusion method to replace the pixels inside the metal object with the boundary pixels. The modified projection data were then used to reconstruct a new CBCT image. The procedure was tested in phantoms, prostate cancer patients with implanted gold markers and metal prosthesis, and a head-and-neck patient with dental amalgam in the teeth. Both phantom and patient studies demonstrated that the procedure was able to minimize the metal artifacts. Soft-tissue visibility was improved near or away from the metal object. The processing time was 1-2 s per projection. We have implemented an effective metal artifact-suppressing algorithm to improve the quality of CBCT images.

TH‐E‐330A‐05: Reducing Metal Artifacts in Cone‐Beam CT by Tracking and Eliminating Metal Shadows in Raw Projection Data

Purpose: CT streaky artifacts, caused by metallic implants such as fiducial markers or dental fillings, remain a challenge for automatic processing of image data. The effect of these metal artifacts is magnified in cone‐beam CT (CBCT) images due to the fact that the soft tissue contrast is usually lower in these images and therefore is more sensitive to the artifacts. The goal of this study is to develop an effective offline processing technique to minimize the effect. Method and Materials: The geometry calibration cue of the CBCT system was used for tracking the position of the metal implant in the raw projection data. The 3D representation of the metallic object can be established from only two user‐selected viewing angles. The position of the shadowed region in any view can be accurately tracked by re‐projecting the 3D coordinates of the metal object. Then automatic image segmentation was performed to obtain a binary mask of the shadow at each projection angle. Finally, a Laplacian diffusion filter was used to replace the pixels in the masked region with the boundary pixels. The modified projection data were then sent back to the CBCT reconstruction engine to create a new CBCT image. Varian's Trilogy system was used in this study. The procedure was tested phantoms and patient cases. Results: It was demonstrated that this procedure can significantly minimize the metal artifacts and at the same time restore soft tissue contrast near the metallic object, even for the more difficult head and neck case with irregularly shaped dental fillings. Soft tissue visibility was improved drastically. Although not designed for on‐line applications, the processing time is approximately 1–2 second per projection on an Intel Pentium processor at 2.6GHz. Conclusion: We have implemented an effective metal artifact suppressing algorithm to improve the quality of CBCT images.

Convergence of Laplacian diffusion versus resolution of an ocean model

This paper presents a convergence study for second order finite difference Laplacian diffusion used in ocean models. For demonstration, ocean model simulations are performed over a rectangular domain, based on the North Pacific subtropical gyre region with grid resolution between 1/2° and 1/32° and with horizontal eddy viscosity coefficient (AH) ranging from 8000 to 30 m2 s−1. A range of AH which is appropriate for useful model simulations of an oceanic domain is found to exist. This range is determined by examining the spatial patterns of Eddy kinetic energy and mean sea surface height. The results fall into three broad categories: (a) converged, (b) converging, and (c) numerical problems. Solutions in the “converged” category do not change with increased grid resolution, and solutions in the “numerical problems” category exhibit distinct differences to the converged result at the same AH.

Extremal solutions of [formula omitted]-Laplacian–diffusion scalar problems with nonlinear functional boundary conditions in a unified way

The aim of this paper is to ensure the existence of extremal solutions lying between a pair of lower and upper solutions for a second-order differential equation that includes both the φ -Laplacian and some classes diffusion equations. The results hold from a suitable change of variables. An example is given to expose the applicability of the obtained results.

Parameter dependent quasi-linear parabolic equations

In this work we are concerned with non-linear reaction diffusion equations with principal part given by p -laplacian and positive diffusion coefficient a ɛ ∈ L ∞ , 0 < δ < a ɛ , which has global attractors A ɛ . We obtain some properties about the convergence of the solutions of these equations and show the upper semicontinuity of the attractors.

Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach.

The use of reaction-diffusion models rests on the key assumption that the diffusive process is Gaussian. However, a growing number of studies have pointed out the presence of anomalous diffusion, and there is a need to understand reactive systems in the presence of this type of non-Gaussian diffusion. Here we study front dynamics in reaction-diffusion systems where anomalous diffusion is due to asymmetric Levy flights. Our approach consists of replacing the Laplacian diffusion operator by a fractional diffusion operator of order alpha, whose fundamental solutions are Levy alpha-stable distributions that exhibit power law decay, x(-(1+alpha)). Numerical simulations of the fractional Fisher-Kolmogorov equation and analytical arguments show that anomalous diffusion leads to the exponential acceleration of the front and a universal power law decay, x(-alpha), of the front's tail.

Laplacian growth and diffusion limited aggregation: different universality classes.

It had been conjectured that diffusion limited aggregates and Laplacian growth patterns (with small surface tension) are in the same universality class. Using iterated conformal maps we construct a one-parameter family of fractal growth patterns with a continuously varying fractal dimension. This family can be used to bound the dimension of Laplacian growth patterns from below. The bound value is higher than the dimension of diffusion limited aggregates, showing that the two problems belong to two different universality classes.

Remarks on the planetary geostrophic model of gyre scale ocean circulation

We study in this article the mathematical formulation of the planetary geostrophic (PG) equations of large-scale ocean circulation, in the case where small-scale processes are parameterized by the traditional Laplacian eddy diffusion and eddy viscosity. We prove the existence and uniqueness of global in time strong solutions of these equations with either $L^\infty$ or $H^2$ initial data. Due essentially to the high nonlinearity (comparable to a squared gradient) of the equations, two problems remain open. First, the existence of more regular solutions with $L^\infty \cap H^1$ initial data is still unknown, although more regular solutions are obtained with $H^2$ initial data. Second, the existence of global attractor and its dimension estimates are open, and related to that are the time uniform boundedness of the norm in $H^2$ and higher order Sobolev spaces of the solutions.

The Deep Interior Downwelling, the Veronis Effect, and Mesoscale Tracer Transport Parameterizations in an OGCM

Numerous numerical simulations of basin-scale ocean circulation display a vast interior downwelling and a companion intense western boundary layer upwelling at midlatitude below the thermocline. These features, related to the so-called Veronis effect, are poorly rationalized and depart strongly from the classical vision of the deep circulation where upwelling is considered to occur in the interior. Furthermore, they significantly alter results of ocean general circulation models (OGCMs) using horizontal Laplacian diffusion. Recently, some studies showed that the parameterization for mesoscale eddy effects formulated by Gent and McWilliams allows integral quantities like the streamfunction and meridional heat transport to be free of these undesired effects. In this paper, an idealized OGCM is used to validate an analytical rationalization of the processes at work and help understand the physics. The results show that the features associated with the Veronis effect can be related quantitatively to three different width scales that characterize the baroclinic structure of the deep western boundary current. In addition, since one of these scales may be smaller than the Munk barotropic layer, usually considered to determine the minimum resolution and horizontal viscosity for numerical models, the authors recommend that it be taken into account. Regarding the introduction of the new parameterization, diagnostics in terms of heat balances underline some interesting similarities between local heat fluxes by eddy-induced velocities and horizontal diffusion at low and midlatitudes when a common large diffusivity (here 2000 m2 s−1) is used. The near-quasigeostrophic character of the flow explains these results. As a consequence, the response of the Eulerian-mean circulation is locally similar for runs using either of the two parameterizations. However, it is shown that the advective nature of the eddy-induced heat fluxes results in a very different effective circulation, which is the one felt by tracers.

The Silicate Garden Reaction in Microgravity: A Fluid Interfacial Instability

In the “silicate garden” reaction, crystals of a metal salt are placed in sodium silicate solution. The crystals become coated with a semipermeable membrane of metal silicate reaction product, from which hollow tubes of metal silicate rise convectively upward, against gravity. In the absence of gravity, and free of convective influences, the reaction might be expected to reveal more fundamental organizing principles. Accordingly, we have grown silicate gardens in microgravity, from salts of calcium, cobalt, and magnesium with added nickel. Even in these isotropic conditions, complex structures developed. They included tubes and hollow spheroids, and also novel dendritic “fingers” which grew by continuous plastic deformation. This new mode of growth is favored by the slow, diffusion-limited rate of reaction in microgravity, which greatly reduces the rate of hardening of the reaction products. The magnesium-nickel garden grew almost entirely as a fluid interfacial instability between the metal salt solution inside and the silicate solution outside, by deformation of the semipermeable fluid membrane between them. The resulting shape had similarities to that of a solid front advancing through a supercooled melt. The morphology of such a solid is determined by the diffusion of released latent heat away from it, according to the Laplacian diffusion equation. We suggest that Laplacian-growth morphology arises in a microgravity silicate garden when its development is controlled by the analogous diffusion of dissolved ions away from it.

The Influence of Wind Effects upon Internal Tides in Shelf Edge Regions

The interaction of the internal tide with wind-induced currents in the shelf edge region off the west coast of Scotland is studied using a baroclinic shelf edge model. The model is used in cross-shelf form with a horizontal finite-difference grid of the order of 0.6 km and 50 sigma levels in the vertical to study the modification of the internal tide produced by upwelling and downwelling winds. Horizontal mixing in the model is parameterized using either the Laplacian form of the horizontal diffusion or the biharmonic form and the sensitivity of the solution to both forms is examined. Coefficients for the vertical diffusion of momentum and density are determined using either an algebraic expression involving the Richardson number or from a two-equation turbulence energy submodel. Calculations show that in the case of an upwelling-favorable wind the density gradient in the near-bed region is increased leading to a slight modification (compared to the tide only solution) of the internal tide at the fundamental frequency with significant increases in amplitude of the higher harmonics due to the increase in the nonlinear terms produced by the increase in the density gradient. With a downwelling-favorable wind the amplitude of the current and internal displacement of the internal tide at the fundamental frequency are significantly reduced due to the change in the density field in the region of internal tidal production. This also leads to a reduction in the amplitude of the higher tidal harmonics. By using a fine grid in the horizontal, the coefficients in the horizontal diffusion terms were set at a minimum and no significant difference in solutions computed with the Laplacian or biharmonic diffusion terms was found. Similarly there are no differences in the major features of the flow field computed with the various parameterizations of the vertical diffusion, although there are some differences in the magnitude of the diffusion coefficients.

Scaling of surface roughness in a heterogeneous film growth system: GexSi1-x on Si.

We examine how the roughness of compositionally graded, relaxed GeSi/Si(001) films can be understood in terms of theoretical ideas based on continuum descriptions of surface growth. The roughness of all samples shows a universal behavior; it increases linearly with length at small length scales and becomes constant at large length scales. Remarkably, all data can be collapsed onto one universal curve. These results can be described by a linear model composed of Laplacian and surface diffusion terms. We also examine the nonuniversal amplitudes and the up-down symmetry to investigate rigorously the linearity of the model.

Upwind‐weighted advection schemes for ocean tracer transport: An evaluation in a passive tracer context

Centered‐in‐space, centered‐in‐time integration has generally been used for the advection of scalars in ocean models. An assessment is made of the implications of centered leapfrog integration in the context of two‐dimensional passive tracer advection within a Stommel (1948) gyre. Nonphysical ripples in the tracer field grow to alarming levels in purely advective integrations. Diffusive parameterizations of eddy mixing moderate these ripples, but it is found that Laplacian diffusion greatly reduces the peak amplitude of the tracer field, while biharmonic or weaker Laplacian diffusion allows ripples of large amplitude. Several forward‐in‐time, upwind‐weighted schemes are found to provide better solutions. Smolarkiewicz's (1984) Multi‐Dimensional Positive‐Definite Advection and Transport Algorithm (MPDATA) scheme is slightly superior for an integration at moderate resolution within which the western boundary current is poorly resolved in typical fashion. Third‐order, upwind‐based schemes exhibit little sensitivity to the details of multidimensional treatment for this problem of passive tracer advection, with results nearly as good as for MPDATA.

The growth kinetics of individual ledges during solid-solid phase transformations

The problem of step motion during lateral growth in solid-solid phase transformations is re-examined. Results are obtained for the steady motion of an individual ledge when volume diffusion in the parent phase is the predominant contribution to the growth rate. A comparison is made between our results and the earlier work of Jones &amp; Trivedi (1971). There are significant differences between the two sets of results particularly in the limit of small perturbations to the Laplacian diffusion field. To confirm the accuracy of the results presented here the calculations have been made by two different methods.

Generalized finite-difference approximations f or the parallel solution of initial value problems

Methods used in analog computation for the parallel- finite-differences solution of partial differential equations have been almost universally based on the "classical" derivation of finite-difference approximations. That is, the space-dependence of the approximate solution is lo cally assumed to be appropriately represented by a Tay lor (or polynomial) truncated series of low1, or higher7, order. Although known to introduce higher truncation errors than the use of other functional series2,3, this re striction has been mostly accepted for reasons of con venience because no simple method for obtaining more general approximations was available in practice. We show in this paper that completely general finite- difference approximations of linear partial differential operators in space, based on functional (rather than, but including, polynomial) approximations, can be easily ob tained in a direct method involving only the inversion and multiplication of constant matrices. One of the drawbacks of the analog computer imple mentation of equations obtained by application of the method presented here is that it requires more attenu ators than the classical low-order method. However, as suggested by Deiters and Nomura3, this is alleviated in a hybrid computer by introducing at discrete time inter vals the higher-order terms as digital corrections super imposed upon an analog low-order implementation. Examples of the derivation of generalized finite-differ ence equations are shown for the representation of the Laplacian (diffusion) operator in circular (r, θ) and linear (x) coordinates.