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The development of any atmosphere or ocean model warrants a suite of test cases (TCs) to verify its spatial and temporal discretizations, order of accuracy, stability, reproducibility, portability, scalability, etc. In this paper, we present a suite of shallow water TCs designed to verify the barotropic solver of atmosphere and ocean models. These include the non‐dispersive coastal Kelvin wave; the dispersive inertia‐gravity wave; the dispersive planetary and topographic Rossby waves; the barotropic tide; and a non‐linear manufactured solution. These TCs check the implementation of the linear pressure gradient term; the linear constant or variable‐coefficient Coriolis and bathymetry terms; and the non‐linear advection terms. Simulation results are presented for a variety of time‐stepping methods as well as two spatial discretizations: a mimetic finite volume method based on the TRiSK scheme, and a high‐order discontinuous Galerkin spectral element method. The experimental procedure for conducting these numerical experiments is detailed. It underscores several key considerations that vary depending on the chosen spatial discretization method. Finally, convergence studies of every TC are conducted with refinement in both space and time, only in space, and only in time. The convergence slopes match the expected theoretical predictions. Plain Language Summary: Before running an atmosphere, ocean, or a coupled climate simulation, every model developer should ensure the correct implementation of each term in the governing equations that drive the models forward in time. This motivates the development of idealized test cases (TCs), each of which verifies a subset of terms in the governing equations with different initial and boundary conditions. Here we present a suite of six TCs for the momentum equation and sea surface height equation for ocean models in a single‐layer configuration. The computed results from the ocean model can be compared to exact solutions. The computed solution always has a small error, but is said to converge to the exact solution with reduction in grid cell size and time step. If the model converges at the expected rate, then we know that it is solving the governing equations correctly. We show results of convergence tests from two models, and share the full specifications of these TCs so that other ocean modelers may reproduce them. Key Points: A suite of test cases is presented for the verification of barotropic dynamics of ocean models, with exact and manufactured solutionsSpecifications are provided for coastal Kelvin wave, inertia‐gravity wave, planetary and topographic Rossby waves, barotropic tide, and non‐linear casesResults are presented for a variety of time‐stepping methods and two types of spatial discretizations: TRiSK and discontinuous Galerkin spectral element method [ABSTRACT FROM AUTHOR]

We consider the problem of determining the arrival statistics of unbiased planar random walkers to complex target configurations. In contrast to problems posed in finite domains, simple moments of the first passage time distribution, such as its mean and variance, are not defined. Therefore, it is necessary to obtain the full arrival statistics. We describe several methods to obtain these distributions and other associated quantities such as splitting probabilities. One approach combines a Laplace transform of the underlying parabolic equation with matched asymptotic analysis followed by numerical transform inversion. The second approach is similar but uses a boundary integral equation method to solve for the transformed variable. To validate the results of this theory, and to obtain the arrival time statistics in very general configurations of absorbers, we introduce an efficient kinetic Monte Carlo (KMC) method that describes trajectories as a combination of large but exactly solvable projection steps. The effectiveness of these methodologies is demonstrated on a variety of challenging examples highlighting the applicability of these methods to a variety of practical scenarios, such as source inference. A particularly useful finding arising from these results is that homogenization theories, in which complex configurations are replaced by equivalent simple ones, are remarkably effective at describing arrival time statistics. [ABSTRACT FROM AUTHOR]

Most wildland and prescribed fire spread occurs through ground fuels, and the rate of spread (RoS) in such environments is often summarized with empirical models that assume uniform environmental conditions and produce a unique RoS. On the other hand, representing the effects of local, small-scale variations of fuel and wind experienced in the field is challenging and, for landscape-scale models, impractical. Moreover, the level of uncertainty associated with characterizing RoS and flame dynamics in the presence of turbulent flow demonstrates the need for further understanding of fire dynamics at small scales in realistic settings. This work describes adapted computer vision techniques used to form fine-scale measurements of the spatially and temporally varying RoS in a natural setting. These algorithms are applied to infrared and visible images of a small-scale prescribed burn of a quasi-homogeneous pine needle bed under stationary wind conditions. A large number of distinct fire front displacements are then used statistically to analyze the fire spread. We find that the fine-scale forward RoS is characterized by an exponential distribution, suggesting a model for fire spread as a random process at this scale. [ABSTRACT FROM AUTHOR]

Transport of viscous fluid through porous media is a direct consequence of the pore structure. Here we investigate transport through a specific class of two-dimensional porous geometries, namely those formed by fluid-mechanical erosion. We investigate the tortuosity and dispersion by analyzing the first two statistical moments of tracer trajectories. For most initial configurations, tortuosity decreases in time as a result of erosion increasing the porosity. However, we find that tortuosity can also increase transiently in certain cases. The porosity-tortuosity relationships that result from our simulations are compared with models available in the literature. Asymptotic dispersion rates are also strongly affected by the erosion process, as well as by the number and distribution of the eroding bodies. Finally, we analyze the pore size distribution of an eroding geometry. The simulations are performed by combining a boundary integral equation solver for the fluid equations, a second-order stable time-stepping method to simulate erosion, and high-order numerical methods to stably and accurately resolve nearly touching eroded bodies and particle trajectories near the eroding bodies. [ABSTRACT FROM AUTHOR]

Summary: We consider suspensions of rigid bodies in a two‐dimensional viscous fluid. Even with high‐fidelity numerical methods, unphysical contact between particles occurs because of spatial and temporal discretization errors. We extend a time stepping method that avoids overlap by imposing a minimum separation distance between all pairs of bodies. In its original form, the method discretizes interactions between different particles explicitly. Therefore, to avoid stiffness, a large minimum separation distance is used. In this paper, we introduce a new implicit time stepping method that is able to simulate dense suspensions with large time step sizes and a small minimum separation distance. The method is tested on various unbounded and bounded flows, and rheological properties of the resulting suspensions are computed. [ABSTRACT FROM AUTHOR]

We consider an efficient preconditioner for a boundary integral equation (BIE) formulation of the two-dimensional Stokes equations in porous media. While BIEs are well-suited for resolving the complex porous geometry, they lead to a dense linear system of equations that is computationally expensive to solve for large problems. This expense is further amplified when a significant number of iterations is required in an iterative Krylov solver such as generalized minimial residual method (GMRES). In this paper, we apply a fast inexact direct solver, the inverse fast multipole method, as an efficient preconditioner for GMRES. This solver is based on the framework of [ABSTRACT FROM AUTHOR]

SUMMARY The discretization of the double-layer potential integral equation for the interior Dirichlet-Laplace problem in a domain with smooth boundary results in a linear system that has a bounded condition number. Thus, the number of iterations required for the convergence of a Krylov method is, asymptotically, independent of the discretization size N. Using the fast multipole method to accelerate the matrix-vector products, we obtain an optimal [ABSTRACT FROM AUTHOR]

A model is developed to explore fire–atmosphere interactions due to the convective sink and vorticity sources in a highly simplified and idealized form, in order to examine their effect on spread and the stability of various fire front geometries. The model is constructed in a cellular automata framework, is linear, and represents a background flow, convective sink, and vortices induced by the fire plume at every burning cell. We use standard techniques to solve the resulting Poisson equations with careful attention to the boundary conditions. A modified Bresenham algorithm is developed to represent convection. The three basic flow types—large-scale background flow, sink flow, and vortex circulation—interact in a complex fashion as the geometry of the fire evolves. Fire-generated vortex–sink interactions produce a range of fire behavior, including unsteady spread rate, lateral spreading, and dynamic fingering. In this simplified framework, pulsation is found associated with evolving fire-line width, a fire-front acceleration in junction fires, and the breakup of longer initial fire lines into multiple head fires. Fuel is very simply represented by a single burn time parameter. The model fuel is uniform yet patchiness occurs due to a dynamic interaction of diffusive and convective effects. The interplay of fire-induced wind and the geometry of the fire front depends also on the fuel burn time. [ABSTRACT FROM AUTHOR]

Wind profile observations are used to estimate turbulent mixing in the atmospheric boundary layer from 1 m up to 300 m height in two locations of pine forests characteristic of the southeast US region, and to 30 m height at one location in the northeast. Basic turbulence characteristics of the boundary layers above and within the canopy were measured near prescribed fires for time periods spanning the burns. Together with theoretical models for the mean horizontal velocity and empirical relations between mean flow and variance, we derive the lateral diffusivity using Taylor's frozen turbulence hypothesis in the thin surface-fuel layer. This parameter is used in a simple 1D model to predict the spread of surface fires in different wind conditions. Initial assessments of sensitivity of the fire spread rates to the lateral diffusivity are made. The lateral diffusivity with and without fire-induced wind is estimated and associated fire spread rates are explored. Our results support the conceptual framework that eddy dynamics in the fuel layer is set by larger eddies developed in the canopy layer aloft. The presence of fire modifies the wind, hence spread rate, depending on the fire intensity. [ABSTRACT FROM AUTHOR]

Bacteria are able to coordinate their movement, growth and biochemical activities through cell-cell communication. While the biophysical mechanism of bacterial chemotaxis has been well understood in individual cells, the role of communication in the chemotaxis of bacterial populations is not clear. Here we report experimental evidence for cell-cell communication that significantly enhances the chemotactic migration of bacterial populations, a finding that we further substantiate using numerical simulations. Using a microfluidic approach, we find that E. coli cells respond to the gradient of chemoattractant not only by biasing their own random-walk swimming pattern through the well-understood intracellular chemotaxis signaling, but also by actively secreting a chemical signal into the extracellular medium, possibly through a hitherto unknown communication signal transduction pathway. This extracellular signaling molecule is a strong chemoattractant that attracts distant cells to the food source. The observed behavior may represent a common evolved solution to accelerate the function of biochemical networks of interacting cells.