Eulerian Advection Schemes Research Articles

Smoothed Particle Hydrodynamics for anisotropic dispersion in heterogeneous porous media

In the context of modeling solute transport through heterogeneous porous media, particle methods possess inherent advantages with respect to mesh-based (Eulerian) methods. In Smoothed Particle Hydrodynamics (SPH), particles represent fluid volumes exchanging concentrations with their neighbors to emulate hydrodynamic dispersion, and advection is simulated by the particles’ displacement. This crucially prevents problems that are otherwise typically associated with Eulerian advection schemes (especially at high grid-Péclet numbers), such as numerical diffusion. Despite the advantages of SPH, modeling dispersion with anisotropic coefficients remains a challenge for the approach, with studies reporting unphysical negative concentrations in conservative problems. This has likely hindered its practical use because dispersion is intrinsically anisotropic in porous media. This article provides a review and numerical evaluation of SPH for simulating dispersion, focusing on three formulations compatible with anisotropic dispersion coefficients. The analysis includes a scheme for which negative concentrations have been formerly reported, plus two more recently developed methods which are applied here for the first time to the problem of anisotropic dispersion in heterogeneous porous media. The SPH schemes are tested under different degrees of dispersion anisotropy for both homogeneous and heterogeneous velocity fields. The results indicate that the newer SPH schemes can produce accurate results without negative concentrations while considering anisotropic dispersion, providing a valid alternative to simulate solute transport through heterogeneous domains.

Modeling Performance of SCALE‐AMPS: Simulations of Arctic Mixed‐Phase Clouds Observed During SHEBA

AbstractThe Advanced Microphysics Prediction System (AMPS), which adopts a two‐moment hybrid‐bin habit‐predicting scheme, was previously developed to study cloud microphysical processes that depend on ice habit; however, only one particular atmospheric model, the University of Wisconsin‐Nonhydrostatic Modeling System, has been used to test the AMPS. In this study, AMPS is implemented into the Scalable Computing for Advanced Library and Environment (SCALE) large‐eddy simulation model. The AMPS Eulerian advection scheme for non‐mass characteristic variables of ice particles, such as axis lengths, is refined to minimize numerical artifacts. The resulting SCALE‐AMPS model successfully reproduces features of mixed‐phase clouds observed during the Surface Heat Budget of the Arctic campaign, including liquid water path (LWP), ice particle size distributions, and ice habits, when ice particle number concentrations (Nice) are reproduced. Sensitivity studies show that increases in Nice result in reductions of LWP that are generally consistent with previous results. Interestingly, LWP reductions lead to changes in ice habits through increases in cloud temperature due to weaker cloud top radiative cooling. Furthermore, aspect ratios of precipitating particles also change following LWP reductions, because in Bigg's immersion freezing scheme, adopted in this study, the aspect ratios depend on the initial size of the ice particles and freezing rates depend on both temperature and droplet size. Because habits of ice particles affect their growth rates, fall speeds, and collision rates, the results obtained in this study reveal possible feedback processes of Arctic mixed‐phase clouds operating through ice habits.

A Computationally Efficient Linear Semi-Lagrangian Scheme for the Advection of Microphysical Variables in Cloud-Resolving Models

AbstractAn efficient semi-Lagrangian (SL) numerical scheme for the advection of positive-definite microphysical variables in cloud-resolving models is proposed. The scheme is a linear combination of SL schemes of the first and second order. The proposed scheme is compared with two high-order, monotonic, nonlinear Eulerian schemes, in two 2D tests. In the first test, advection of a positive scalar field with large gradients is performed within complicated idealized steering flows. In the second test, a hail storm is simulated using the 2D Hebrew University Cloud Model, describing the interactions between 15 size distributions of different microphysical quantities (each distribution is described by 43 mass bins). The proposed scheme is computationally efficient, has a low numerical diffusion and a high level of mass conservation accuracy, and preserves the sum of multiple advected variables as well as the linear relationships between the advected scalar variables. Although the proposed SL scheme is not exactly positive definite, the negative values and their number, which may appear only in the case of very strong gradients, are negligible. The scheme produces results similar to those of nonlinear Eulerian advection schemes while reducing computation time by approximately a factor of 10.

A consistent balanced force refined moment-of-fluid method for surface tension dominant two-phase flows

Inconsistent implementation of surface tension in an interfacial solver for incompressible two-phase flow results in spurious currents near the interface. The balanced force algorithm (BFA) enforces consistent implementation of surface tension force and pressure gradient at the cell faces thereby generating conservative face center velocity field, even though the cell center velocity may not be divergence free and spurious current free. The implication of this inconsistency is directly reflected in interface smoothness while using a Lagrangian–Eulerian interface evolution in contrary to Eulerian advection where divergence free face velocities are used. This paper describes a consistent balanced force algorithm (CBFA) in collocated grid for consistent implementation of surface tension force. The interface evolution is tracked by refined moment-of-fluid (RMOF) method using Lagrangian–Eulerian advection scheme. In CBFA, additional Poisson equation is solved to obtain cell face corrections for conservative cell center velocity computation. The effectiveness of the proposed method is demonstrated in this article for several test cases.

Ocean surface waves in an ice-free Arctic Ocean

The retreat of the Arctic ice edge implies that global ocean surface wave models have to be extended at high latitudes or even to cover the North Pole in the future. The obstacles for conventional latitude-longitude grid wave models to cover the whole Arctic are the polar problems associated with their Eulerian advection schemes, including the Courant-Friedrichs-Lewy (CFL) restriction on diminishing grid length towards the Pole, the singularity at the Pole and the invalid scalar assumption for vector components defined relative to the local east direction. A spherical multiple-cell (SMC) grid is designed to solve these problems. It relaxes the CFL restriction by merging the longitudinal cells towards the Poles. A round polar cell is used to remove the singularity of the differential equation at the Pole. A fixed reference direction is introduced to define vector components within a limited Arctic part in mitigation of the scalar assumption errors at high latitudes. The SMC grid has been implemented in the WAVEWATCH III model and validated with altimeter and buoy observations, except for the Arctic part, which could not be fully tested due to a lack of observations as the polar region is still covered by sea ice. Here, an idealised ice-free Arctic case is used to test the Arctic part and it is compared with a reference case with real ice coverage. The comparison indicates that swell wave energy will increase near the ice-free Arctic coastlines due to increased fetch. An expanded Arctic part is used for comparisons of the Arctic part with available satellite measurements. It also provides a direct model comparison between the two reference systems in their overlapping zone.

A Lagrangian-Eulerian advection scheme with moment-of-fluid interface reconstruction

ABSTRACTA Lagrangian–Eulerian advection scheme (LEAS) with Moment-of-Fluid (MoF) interface reconstruction is presented for interfacial flows. In MoF method, in addition to the volume fraction field, material centroid is used for interface reconstruction. Physically material centroid indicates the material location inside a mixed cell. Therefore, MoF method reconstructs linear interfaces exactly and is second order accurate for curved interfaces. Despite being second order accurate for static interface reconstruction, MoF reconstruction is inaccurate when centroid is not properly advected. An accurate centroid advection method based on Barycenter of centroid is discussed here. A comparison of this centroid advection with the pure Lagrangian advection based on Runge–Kutta integrator is demonstrated for two different advection tests. The superiority of MoF is established by comparing the volume error norm with other interface reconstructions. Significant improvement in accuracy is observed when LEAS material advection is used in conjuction with accurate centroid advection for MoF reconstruction.

An accurate momentum advection scheme for a z-level coordinate models

In this paper, we focus on a conservative momentum advection discretisation in the presence of z-layers. While in the 2D case conservation of momentum is achieved automatically for an Eulerian advection scheme, special attention is required in the multi-layer case. We show here that an artificial vertical structure of the flow can be introduced solely by the presence of the z-layers, which we refer to as the staircase problem. To avoid this staircase problem, the z-layers have to be remapped in a specific way. The remapping procedure also deals with the case of an uneven number of layers adjacent to a column side, thus allowing one to simulate flooding and drying phenomena in a 3D model.

Forward Semi-Lagrangian Advection with Mass Conservation and Positive Definiteness for Falling Hydrometeors

Abstract A semi-Lagrangian advection scheme is developed for falling hydrometeors in hopes of replacing the conventional Eulerian scheme that has been widely used in the cloud microphysics scheme of numerical atmospheric models. This semi-Lagrangian scheme uses a forward advection method to determine the advection path with or without iteration, and advected mass in a two-time-level algorithm with mass conservation. Monotonicity is considered in mass-conserving interpolation between Lagrangian grids and model Eulerian grids, thus making it a positive definite advection scheme. For mass-conserving interpolation between the two grid systems, the piecewise constant method (PCM), piecewise linear method (PLM), and piecewise parabolic method (PPM) are proposed. The falling velocity at the bottom cell edge is modified to avoid unphysical deformation by scanning from the top layer to the bottom of the model, which enables the use of a large time step with reasonable accuracy. The scheme is implemented and tested in the Weather Research and Forecasting (WRF) Single-Moment 3-Class Microphysics Scheme (WSM3). In a theoretical test bed with constant terminal velocity, the proposed semi-Lagrangian algorithm shows that the higher-order interpolation scheme produces less diffusive features at maximal precipitation. Results from another idealized test bed with mass-weighted terminal velocity demonstrate that the accuracy of the proposed scheme is still satisfactory even with a time step of 120 s when the mean terminal velocity averaged at the departure and arrival points is employed. A two-dimensional (2D) squall-line test using the WSM3 scheme shows that the control run with the Eulerian advection scheme and the semi-Lagrangian run with the PCM method reveal similar results, whereas behaviors using the PLM and PPM are similar with higher-resolution features, such as mammatus-like clouds.

Viscous liquid sloshing damping in cylindrical container using a volume of fluid method

Liquid sloshing is a kind of very complicated free surface flow and exists widely in many fields. In order to calculate liquid sloshing damping precisely a volume of fluid method based on finite volume scheme is used to simulate free surface flows in partly filled cylindrical containers. A numerical method is presented to simulate the movement of the free surface flow, in which a piecewise linear interface construction scheme and an unsplit Lagrangian advection scheme instead of Eulerian advection scheme are used. The damping performance of liquid sloshing in cylindrical containers under fundamental sloshing mode is investigated. There are four factors determining the surface-wave damping: free surface, boundary-layer, interior fluid and contact line. In order to study different contributions from these four factors to whole damping, several examples are simulated. No-slip and slip wall boundary conditions on both side wall and bottom wall of the cylindrical containers are studied to compare with the published results obtained by solving Stokes equations. In the present method the first three main factors can be considered. The simulation results show that the boundary-layer damping contribution increases while the interior fluid damping contribution decreases with increase of Reynolds number.

Using the material‐point method to model sea ice dynamics

The material‐point method (MPM) is a numerical method for continuum mechanics that combines the best aspects of Lagrangian and Eulerian discretizations. The material points provide a Lagrangian description of the ice that models convection naturally. Thus properties such as ice thickness and compactness are computed in a Lagrangian frame and do not suffer from errors associated with Eulerian advection schemes, such as artificial diffusion, dispersion, or oscillations near discontinuities. This desirable property is illustrated by solving transport of ice in uniform, rotational and convergent velocity fields. Moreover, the ice geometry is represented by unconnected material points rather than a grid. This representation facilitates modeling the large deformations observed in the Arctic, as well as localized deformation along leads, and admits a sharp representation of the ice edge. MPM also easily allows the use of any ice constitutive model. The versatility of MPM is demonstrated by using two constitutive models for simulations of wind‐driven ice. The first model is a standard viscous‐plastic model with two thickness categories. The MPM solution to the viscous‐plastic model agrees with previously published results using finite elements. The second model is a new elastic‐decohesive model that explicitly represents leads. The model includes a mechanism to initiate leads, and to predict their orientation and width. The elastic‐decohesion model can provide similar overall deformation as the viscous‐plastic model; however, explicit regions of opening and shear are predicted. Furthermore, the efficiency of MPM with the elastic‐decohesive model is competitive with the current best methods for sea ice dynamics.

A numerical method for solving a scalar advection-dominated transport equation with concentration-dependent sources

A numerical scheme, upstream biased Eulerian algorithm for transport equations with sources (UpBEATES), is developed for solving a scalar advective-dominated transport equation with concentration-independent and -dependent source terms. A control-volume method is used for spatial discretization. Time integration is invoked to yield a discrete system of integrated-flux-integrated-source form equations. The Bott's upstream-biased Eulerian advection scheme [Moneatry and Weakly Review 117 (1989a) 1006; Moneatry and Weakly Review 117 (1989b) 2633] is employed for approximating advective fluxes. A two-level time weighting scheme is employed for the dispersive fluxes. An upstream-biased Eulerian algorithm is proposed for the concentration-dependent source term. Flux and source limiters are developed to ensure non-negative evolution of the scalar concentration field. Numerical experiments were presented to illustrate its performance in comparison with theoretical solutions and those of conventional methods. The proposed scheme is mass-conservative, produces non-negative concentration values, exhibits low numerical dispersion, and is efficient for advection-dominated problems with concentration-dependent source terms. Like other Eulerian schemes, the Courant–Friedrich–Levy (CFL) stability criterion has to be met.

Validation of the Eulerian pollution transport model PolTran on the Kincaid data set

The study represents the first performance of the model PolTran on the Kincaid data set. PolTran is a typical representative of the Eulerian models: two dimensional Eulerian advection scheme and numerical calculation of the vertical diffusion. The meteorological pre-processor uses the Kincaid measurements of wind and temperature at 10, 50 and 100 m and provides the dispersion model with wind and turbulence exchange profiles. The model performance is comparable to other models applied to the Model Validation Kit, that could be considered as a good result for a local scale application of a model of Eulerian type.

Stability of explicit advection schemes. The balance point location rule

AbstractThis paper introduces the balance point location rule, providing specific necessary and sufficient conditions for constructing unconditionally stable explicit advection schemes, in both semi‐Lagrangian and flux‐form Eulerian formulations. The rule determines how the spatial stencil is placed on the computational grid. It requires the balance point (the center of the stencil in index space) to be located in the same patch as the departure point for semi‐Lagrangian schemes or the same cell as the sweep point for Eulerian schemes. Centering the stencil in this way guarantees stability, regardless of the size of the time step. In contrast, the original Courant–Friedrichs–Lewy (CFL) condition requiring the stencil merely to include the departure (sweep) point, although necessary, is not sufficient for guaranteeing stability. The CFL condition is of limited practical value, whereas the balance point location rule always gives precise and easily implemented prescriptions for constructing stable algorithms. The rule is also helpful in correcting a number of misconceptions that have arisen concerning explicit advection schemes. In particular, explicit Eulerian schemes are widely believed to be inefficient because of stability constraints on the time step, dictated by a narrow interpretation of the CFL condition requiring the Courant number to be less than or equal to one. However, such constraints apply only to a particular class of advection schemes resulting for centering the stencil on the arrival point, when in fact the sole function of the stencil is to estimate the departure (sweep) point value—the arrival point has no relevance in determining the placement of the stencil. Unconditionally stable explicit Eulerian advection schemes are efficient and accurate, comparable in operation count to semi‐Lagrangian schemes of the same order, but because of their flux‐based formulation, they have the added advantage of being inherently conservative. Copyright © 2002 John Wiley & Sons, Ltd.

Test of the Volume-of-Fluid Method on Simulations of Marine Boundary Layer Clouds

Abstract The impact of using grid-averaged thermodynamic properties (i.e., neglecting their subgrid variability due to partial cloudiness) to represent forcings for condensation or evaporation has long been recognized. In particular, numerical difficulties in terms of spurious oscillations and/or diffusion in vicinity of a cloud–environment interface have been encountered in most of the conventional finite-difference Eulerian advection schemes. This problem is equivalent to the inability of models to accurately track the cloud boundary within a grid cell, which eventually leads to spurious production or destruction of cloud water at leading or trailing edges of clouds. This paper employs a specialized technique called the “volume-of-fluid” (VOF) method to better parameterize the subgrid-scale advection process that accounts for the transport of material interfaces. VOF also determines the actual location of the partial cloudiness within a grid box. Consequently, relevant microphysical parameterizations in...

A model for the evaluation of long-term airborne pollution transport at regional and continental scales

The paper contains the description of numerical model of the airborne pollution transport developed by a scientific team of M. Galperin. During several years the model was used for the calculations in Meteorological Synthesizing Centre – East, within the scope of Co-operational Programme for Monitoring and Evaluation of the Long-Range Transmission of air Pollutants in Europe (EMEP). The paper presents the extensive description of the current version of the model and outlines the results of its quality assurance. The model is based on an original Eulerian advection scheme with variable time step, which can be classified as a “pseudo-Lagrangian” scheme. Vertical concentration profiles are parameterized by truncated Gaussian function with dynamic parameters calculated at every model time step. The model includes two types of chemistry and deposition units: acid compounds and ammonia, and heavy metals carried with aerosol particles. One of the model advantages is the ability to make the evaluation of source–receptor pollution exchange for substances with non-linear atmospheric chemistry and deposition. The model evaluation was carried out by the comparison of multi-annual calculated concentrations with measured values for several acidifying substances.

The Use of TVD Limiters for Forward-in-Time Upstream-Biased Advection Schemes in Ocean Modeling

Abstract This paper explores the use of the constant grid flux form forward-in-time upstream-biased advection schemes for the advection of temperature and salinity in ocean modeling. The constant grid flux form schemes are shown to be an improvement over the traditional central differencing commonly used in ocean models. In addition, nonoscillatory versions of the scheme, which employ flux limiters, are explored. The limiters are based on total variation diminishing concepts and are applied to higher-order (in space) versions of the constant grid flux form scheme. The constant grid flux form schemes are Crowley-type upstream-biased Eulerian advection schemes. They are mass conserving and possess small amplitude and phase errors. The flux limiters prevent the under- and overshooting associated with the numerical dispersion of the unlimited schemes. The limited schemes are easy to implement, efficient, and nonoscillatory. Of these schemes the third-order and fifth-order versions employing the PDM limiter ar...

Temporal Accumulation of First-Order Linearization Error for Semi-Lagrangian Passive Advection

Abstract The tangent linear model (TLM) is obtained by linearizing the governing equations around a space- and time-dependent basic state referred to as the trajectory. The TLM describes to first-order the evolution of perturbations in a nonlinear model and it is now used widely in many applications including four-dimensional data assimilation. This paper is concerned with the difficulties that arise when developing the tangent linear model for a semi-Lagrangian integration scheme. By permitting larger time steps than those of Eulerian advection schemes, the semi-Lagrangian treatment of advection improves the model efficiency. However, a potential difficulty in linearizing the interpolation algorithms commonly used in semi-Lagrangian advection schemes has been described by Polavarapu et al, who showed that for infinitesimal perturbations, the tangent linear approximation of an interpolation scheme is correct if and only if the first derivative of the interpolator is continuous at every grid point. Here, t...

A Comparative Study of High-Order, Quasi-Conservative, Semi-Lagrangian, and Eulerian Advection Schemes

Abstract Quasi-conservative high-order semi-Lagrangian advection schemes are compared with several positive-definite Eulerian schemes in flux form, including Bott’s scheme. In this study, the conventional equipartition method is modified as a posterior iterative mass correction algorithm to restore conservation for semi-Lagrangian transport. The performance comparisons between semi-Lagrangian and Eulerian schemes are evidence that the fifth-order (seventh-order) Bott’s area-preserving algorithm without flux limitation is practically equivalent to the quintic (seventh-order) semi-Lagrangian scheme in the rotational flow, with the maximum directional Courant number smaller than 0.5. For positive-definite advection, Bott’s algorithm with the flux limitation obtains slightly better (worse) amplitude preservation in the rotational flow tests compared to semi-Lagrangian schemes of same order with (without) the mass correction algorithm. In the nonlinear deformational flow where the maximum directional Courant n...

A Mass-Conservative, Positive-Definite, and Efficient Eulerian Advection Scheme in Spherical Geometry and on a Nonuniform Grid System

Abstract The flux-form advection scheme of Bott is modified for the spherical coordinates, combined with the expanded-polar-zone (EPZ) technique to improve the overall performance of the advection calculations. With the EPZ technique, this Eulerian scheme has comparable efficiency as semi-Lagrangian methods for advection of nonreactive tracers on a sphere but with somewhat better overall numerical accuracy. The conservation of global tracer mass and the, positive definiteness of the algorithm are achieved to machine precision. For the test problem of solid body rotations on a sphere, this scheme shows small numerical diffusion, almost undetectable phase errors, and very little artificial deformation of the test shape even for cross-polar transport. In comparison with some semi-Lagrangian schemes and other high-order Eulerian methods, it shows very competitive performance. Numerical tests also indicate that, without any modifications, it performs just as well on slightly nonuniform Gaussian grid as on unif...

Semi-Lagrangian Advection on a Gaussian Grid

The treatment of advection is related to the stability, accuracy and efficiency of models used in numerical weather prediction. In order to remain stable, conventional Eulerian advection schemes must respect a Courant-Friedrichs-Lewy (CFL) criterion, which limits the size of the time step that can be used in conjunction with a given spatial resolution. In recent years, tests with gridpoint models have shown that semi-Lagrangian schemes permit the use of large time steps (roughly three to six times those permitted by the CFL criterion for the corresponding Eulerian models), without reducing the accuracy of the forecasts. This leads to improved model efficiency, since fewer steps are needed to complete the forecast. Can similar results be achieved in spectral models? This paper examines the semi-Lagrangian treatment of advection on the Gaussian grid used in spectral models. Interpolating and noninterpolating versions of the semi-Lagrangian scheme are applied to the problem of solid body rotation on the globe, and their performance is compared with that of an Eulerian spectral treatment. It is shown that the semi-Lagrangian models produce stable, accurate integrations using time steps that far exceed the CFL limit for the Eulerian spectral model.