Numerical Advection Schemes Research Articles

Exploring the properties of local and non-local vertical diffusion schemes in the EMEP model using <SUP align="right">222</SUP>Rn data

The simulations of hourly 222 Rn concentrations are performed with the EMEP model to validate different parameterization schemes for vertical mixing. In addition to recently evaluated non-local vertical diffusion schemes, a new scheme which is a local and based on total turbulent energy (TTE) has been implemented in the model and evaluated. Due to its properties 222 Rn is commonly used to study the model sub-grid mixing schemes, numerical advection schemes or to compare the performance of different models. The effects on simulated 222 Rn concentrations are investigated providing useful information on dynamic properties and on turbulence parameterization techniques.

Monotonic Limiters for a Second-Order Finite-Volume Advection Scheme Using Icosahedral-Hexagonal Meshes

Abstract An assessment of a recently developed (by Miura) second-order numerical advection scheme for icosahedral-hexagonal grids on the sphere is presented, and the effects of monotonic limiters that can be used with the scheme are examined. The cases address both deformational and nondeformational flow and continuous and discontinuous advected quantities; they include solid-body rotation of a cosine bell and slotted cylinder, and moving dynamic vortices. The limiters of Zalesak, Dukowicz and Kodis, and Thuburn are tested within this numerical scheme. The Zalesak and Thuburn limiters produce solutions with similar accuracy, and the Thuburn limiter, while computationally less expensive per time step, results in more stringent stability conditions for the overall scheme. The Dukowicz limiter is slightly more diffusive than the other two, but it costs the least.

On usage of CABARET scheme for tracer transport in INM ocean model

The contemporary state of ocean numerical modelling sets some requirements for the numerical advection schemes used in ocean general circulation models (OGCMs). The most important requirements are conservation, monotonicity and numerical efficiency including good parallelization properties. Investigation of some advection schemes shows that one of the best schemes satisfying the criteria is CABARET scheme. 3D-modification of the CABARET scheme was used to develop a new transport module (for temperature and salinity) for the Institute of Numerical Mathematics ocean model (INMOM). Testing of this module on some common benchmarks shows a high accuracy in comparison with the second-order advection scheme used in the INMOM. This new module was incorporated in the INMOM and experiments with the modified model showed a better simulation of oceanic circulation than its previous version.

Evaluating two numerical advection schemes in HYCOM for eddy-resolving modelling of the Agulhas Current

Abstract. A 4th order advection scheme is applied in a nested eddy-resolving Hybrid Coordinate Ocean Model (HYCOM) of the greater Agulhas Current system for the purpose of testing advanced numerics as a means for improving the model simulation for eventual operational implementation. Model validation techniques comparing sea surface height variations, sea level skewness and variogram analyses to satellite altimetry measurements quantify that generally the 4th order advection scheme improves the realism of the model simulation. The most striking improvement over the standard 2nd order momentum advection scheme, is that the southern Agulhas Current is simulated as a well-defined meandering current, rather than a train of successive eddies. A better vertical structure and stronger poleward transports in the Agulhas Current core contribute toward a better southwestward penetration of the current, and its temperature field, implying a stronger Indo-Atlantic inter-ocean exchange. It is found that the transport, and hence this exchange, is sensitive to the occurrences of mesoscale features originating upstream in the Mozambique Channel and southern East Madagascar Current, and that the improved HYCOM simulation is well suited for further studies of these inter-actions.

Numerical model to calculate microphysical influences on sound wave propagation in forests

There is a growing need for increasingly accurate and reliable numerical models to predict micrometeorological and turbulence conditions in complex sound wave propagation environments. In this paper, a prototype finite-difference computer model is developed to calculate the microphysical influences on sound speed in forested areas. Several numerical tests are conducted to assess model code capabilities using micrometeorological field data collected in June 2006. For the current analysis, the model domain and total number of grid points are greatly increased from earlier reported versions and the finite-difference numerical schemes for advection are modified to permit different inflow and outflow boundaries. Preliminary results for three cases are encouraging. Micrometeorological profiles and calculated fields of sound speed are presented. Some initial approximations of short-range acoustic transmission loss for the experimental test site are also discussed.

On Near-Diffusion-Free Advection over Spherical Geodesic Grids

Abstract A forward trajectory advection scheme has been designed for its use in an icosahedral–hexagonal grid model. The scheme has been evaluated with two-dimensional test cases: solid-body rotation and deformational flow; both depict important characteristics of atmospheric flows. The main motivation of this study is to achieve good accuracy without using higher-order interpolations in a numerical advection scheme, so that it may become viable in fine-resolution GCMs. The computation of the error norm shows its gradient as constant and the scheme is approximately first-order accurate. The other interesting feature of this study is that its downstream search algorithm reduces the complexity from O(n2) to O(n).

Investigation of some numerical issues in a chemistry‐transport model: Gas‐phase simulations

Many numerical strategies have been specifically developed for chemistry‐transport models. Since no exact solutions are available for 3‐D real problems, there are only few insights to choose between alternative numerical schemes and approximations, or to estimate the performance discrepancy between two approaches. However it is possible to assess the importance of numerical approximations through the comparison of different strategies. We estimated the impact of several numerical schemes for advection, diffusion and stiff chemistry. We also addressed operator splitting with different methods and operator orders. The study is performed with a gas‐phase Eulerian model from the modeling platform Polyphemus. It is applied to ozone forecasts mainly over Europe, with focus on a few key species: ozone, nitric oxide, nitrogen dioxide, sulfur dioxide and hydroxy radical. The outcome is a ranking of the most sensitive numerical choices. It stresses the prominent impact of the advection scheme and of the splitting time step.

ŒDIPUS: a new tool to study the dynamics of planetary interiors

We present a new numerical method to describe the internal dynamics of planetary mantles through the coupling of a dynamic model with the prediction of geoid and surface topography. Our tool is based on the simulation of thermal convection with variable viscosity in a spherical shell with a finite-volume formulation. The grid mesh is based on the ‘cubed sphere’ technique that divides the shell into six identical blocks. An investigation of various numerical advection schemes is proposed: we opted for a high-resolution, flux-limiter method. Benchmarks of thermal convection are then presented on steady-state tetrahedral and cubic solutions and time-dependent cases with a good agreement with the few recent programs developed to solve this problem. A dimensionless framework is proposed for the calculation of geoid and topography introducing two dimensionless numbers: such a formulation provides a good basis for the systematic study of the geoid and surface dynamic topography associated to the convection calculations. The evaluation of geoid and surface dynamic topography from the gridded data is performed in the spectral domain. The flow solver is then tested extensively against a precise spectral program, producing response functions for geoid as well as bottom and surface topographies. For a grid mesh of a reasonable size (6 × 64 × 64 × 64) a very good agreement (to within ∼1 per cent) is found up to spherical harmonic degree 15.

Coupling stabilized finite element methods with finite difference time integration for advection–diffusion–reaction problems

We present some numerical schemes for the unsteady advection–diffusion–reaction linear problem, in one space dimension. We investigate two possible different ways of combining the discretization in time and in space (where the sequence of the discretizations is interchanged). Discretization in time is performed by using the Crank–Nicolson finite difference scheme, while for the space discretization we consider three classical stabilized finite element schemes and the more recent Link-Cutting Bubble strategy proposed in [F. Brezzi, G. Hauke, L.D. Marini, G. Sangalli, Link-cutting bubbles for the stabilization of convection–diffusion–reaction problems, Math. Models Methods Appl. Sci. 13 (2003) 445–461]. Numerical experiments are presented to assess and valuate the capabilities of the proposed methods. An L 1-error analysis of the Link-Cutting Bubble strategy for solving the steady problem is included.

Efficiency of high order numerical schemes for momentum advection

The usefulness of high order numerical schemes for treating the advection of momentum in the primitive equations is examined. It is shown that the use of high order advection schemes for the momentum advection can improve the dynamics of potential vorticity for length scales similar to the deformation radius. This is illustrated using numerical experiments in academic and realistic configurations. In particular, a model of the North Sea is used to evaluate the effect of a high order momentum advection scheme. It is found that a fourth order scheme at 4 km resolution gives results close to a second order scheme at 2 km resolution, as far as mean and eddy kinetic energy, position of currents, size eddies are concerned. It is concluded that computing the advection of momentum with a fourth order scheme is more cost effective than doubling their resolution.

Tracking accuracy of a semi‐Lagrangian method for advection–dispersion modelling in rivers

AbstractThere is an increasing need to improve the computational efficiency of river water quality models because: (1) Monte‐Carlo‐type multi‐simulation methods, that return solutions with statistical distributions or confidence intervals, are becoming the norm, and (2) the systems modelled are increasingly large and complex. So far, most models are based on Eulerian numerical schemes for advection, but these do not meet the requirement of efficiency, being restricted to Courant numbers below unity. The alternative of using semi‐Lagrangian methods, consisting of modelling advection by the method of characteristics, is free from any inherent Courant number restriction. However, it is subject to errors of tracking that result in potential phase errors in the solutions. The aim of this article is primarily to understand and estimate these tracking errors, assuming the use of a cell‐based backward method of characteristics, and considering conditions that would prevail in practical applications in rivers. This is achieved separately for non‐uniform flows and unsteady flows, either via theoretical considerations or using numerical experiments. The main conclusion is that, tracking errors are expected to be negligible in practical applications in both unsteady flows and non‐uniform flows. Also, a very significant computational time saving compared to Eulerian schemes is achievable. Copyright © 2006 John Wiley & Sons, Ltd.

Effective diffusivity in the middle atmosphere based on general circulation model winds

The mixing of a passive tracer in the stratosphere and lower mesosphere is studied on the basis of the effective diffusivity, which is obtained in the framework of the tracer‐based coordinate system. This characteristic is proportional to the average diffusion flux over Lagrangian contours and inversely proportional to the mean tracer gradient. The tracer distribution used in the calculation of the effective diffusivity is obtained after integration of the advection‐diffusion equation using general circulation model winds and a new numerical advection scheme with small numerical diffusivity. Using some theoretical and experimental arguments, it is shown that the interpretation of the seasonal variability of the effective diffusivity field cannot be done on the basis of the momentary wind field alone, but some flow history should be taken into account. The climatology of the effective diffusivity for different months is presented up to the lower mesosphere and compared with previous studies. In the stratosphere some new features of the effective diffusivity distribution are obtained. For example, there is a local maximum of the effective diffusivity at midlatitudes of the Northern Hemisphere of the summer middle stratosphere. The effective diffusivity fields in the lower mesosphere show a strong increase of the mean effective diffusivity from the upper stratosphere to the lower mesosphere and the existence of a complex latitudinal structure of the effective diffusivity at mesospheric heights. In the lower mesosphere there is a marked interannual variability during the Southern Hemisphere easterly wind development. A possible explanation for the obtained structure is discussed on the basis of in situ Rossby wave generation and Rossby‐wave‐breaking effects.

Secondary flows induced by wind forcing in the Rh�ne region of freshwater influence

Secondary flows induced by the blocking effect of a river plume on a transverse upwelling are investigated in a microtidal region of freshwater influence (ROFI). A nested version of the SYMPHONIE primitive-equation free-surface model for 3-D baroclinic coastal flows has been developed for the Rhône ROFI. The main characteristics of the model are a generalized sigma coordinate system in finite differences, using a time splitting for external and internal modes and high-order numerical advection schemes for density fields in combination with an modified turbulence closure scheme. The nesting system consists of two grids forced by the high-resolution ALADIN model atmospheric data. The coarse grid of 3 km resolution for the whole Gulf of Lions allows the forcing of the Liguro-Provençal large-scale current when the fine mesh of 1-km resolution is centred on the river mouth of the Grand Rhône. Documented field experiments from the Biodypar 3 field campaign performed during March 1999 are used for validation. Numerical results, CTD profiles and a SPOT TSM visible image are in good agreement concerning the shape and structure of the river plume. Other coastal flow features can be observed from satellite imagery. Computations of realistic situations recover these main secondary structures. Complementary process-oriented runs give an explanation of how the coastal upwelling induced by an inhomogeneous offshore wind is destabilized by the combination of the river plume and along-shelf current-blocking effects. In the end, a factor-separation analysis provides evidence that the locally non-linear effects in momentum contribute to the occurrence of secondary vortices.

Adaptive biorthogonal spline schemes for advection–reaction equations

In this paper, based on Eulerian–Lagrangian localized adjoint method (ELLAM), we use biorthogonal spline wavelets to develop numerical schemes for multidimensional advection–reaction equations. The derived schemes produce accurate numerical solutions even if large time steps are used. These schemes are explicit but unconditionally stable. They also have the ability to carry out adaptive compression while preserving the total mass. Numerical experiments including observations on their convergence rates are presented to show the strong potential of these methods.

Numerical simulation of unsteady multidimensional free surface motions by level set method

AbstractThis paper presents a numerical method that couples the incompressible Navier–Stokes equations with the level set method in a curvilinear co‐ordinate system for study of free surface flows. The finite volume method is used to discretize the governing equations on a non‐staggered grid with a four‐step fractional step method. The free surface flow problem is converted into a two‐phase flow system on a fixed grid in which the free surface is implicitly captured by the zero level set. We compare different numerical schemes for advection of the level set function in a generalized curvilinear format, including the third order quadratic upwind interpolation for convective kinematics (QUICK) scheme, and the second and third order essentially non‐oscillatory (ENO) schemes. The level set equations of evolution and reinitialization are validated with benchmark cases, e.g. a stationary circle, a rotating slotted disk and stretching of a circular fluid element. The coupled system is then applied to a travelling solitary wave, and two‐ and three‐dimensional dam breaking problems. Some interesting free surface phenomena are revealed by the computational results, such as, the large free surface vortices, air entrapment and splashing of the water surge front. The computational results are in excellent agreement with theoretical predictions and experimental data, where they are available. Copyright © 2003 John Wiley & Sons, Ltd.

Testing of numerical advection schemes and splitting techniques used in pollution dispersion modelling on an analytic solution

In the real nature the processes of pollutants advection, their release in and removal from the atmosphere take place simultaneously. In dispersion modelling, these processes are considered separately, one after another within a time step, which causes the generation of some specific errors. Two types of numerical advection schemes, typical Eulerian and semi-Lagrangian one, are considered. The numerical results are compared to the stationary analytic solution of the 1D advection equation with incorporation of emission and deposition. Several approaches to reduce the errors of the numerical solutions are discussed. Analytic solutions for the air concentration and deposition for some characteristic cases (different emission and deposition space distributions) are prepared in a shape, convenient to be used as a kit for testing the advection schemes and splitting techniques.

Evaluation of stratosphere–troposphere exchange and the hydroxyl radical distribution in three‐dimensional global atmospheric models using observations of cosmogenic 14CO

A new method for the quantitative evaluation of global atmospheric transport and the hydroxyl radical (OH)‐based oxidation in three‐dimensional (3‐D) atmospheric chemistry transport models (CTMs) and general circulation models (GCMs) is developed. The method is based on a cosmogenic 14CO climatology that has been previously derived from a large number of 14CO observations. Using 14CO measurements to constrain model OH distributions and the simulated stratosphere–troposphere exchange (STE) provides a challenging test for 3‐D atmospheric models. Here, the evaluation method is applied to the CTMs MATCH and TM3. Whereas MATCH overestimates the STE in both hemispheres, TM3 does reproduce the STE in the Southern Hemisphere (SH) but underestimates it in the Northern Hemisphere (NH). The STE phase in MATCH is 1 month too early, whereas no significant phase shift for TM3 is revealed. These characteristic deficiencies in both models were consistently determined, i.e., with the same boundary conditions (OH distribution and 14CO source distribution). The robustness of the results is tested by various sensitivity studies, involving the 14CO source distribution and strength, the tropospheric OH distribution, the stratospheric OH abundance, and the applied numerical advection scheme. Consistency is further checked by comparison of the model simulated vertical 14CO profiles to 14CO observations from a number of aircraft campaigns. The 14CO simulations do not support an interhemispheric asymmetry in the OH abundance with an on average higher concentration in the SH.

The sensitivity of a model's stratospheric tape recorder to the choice of advection scheme

AbstractThe sensitivity of a climate model's transport to the choice of numerical advection scheme is investigated using simulations of the stratospheric tape recorder. Significant differences are found between tracers advected with the Heun scheme (with centred spatial differencing) and tracers advected with a flux‐limited total variation diminishing (TVD) scheme. The tape recorder simulated by the TVD scheme propagates upwards unrealistically fast: about three times faster than the tape recorder of the Heun scheme, even though the winds are identical. In contrast, the amplitude of the TVD scheme's tape recorder is more realistic than that of the Heun scheme, which is too strong in the middle stratosphere.Further off‐line tracer experiments, using a family of conservative, upwind advection schemes, demonstrate how the tape recorder's phase speed decreases as the order of the polynomial representing the tracer's subgrid variation is increased. It is found that schemes with more implicit vertical diffusion than a quasi third‐order scheme produce a tape recorder with unacceptably fast upwards propagation. The sensitivity can only be reproduced in a one‐dimensional model when winds with strong variability are used, highlighting a limitation of the traditional advection tests that use constant winds. The experiments also show that the numerical oscillations produced by non‐monotonic methods, including the Heun scheme, can artificially strengthen the tape‐recorder signal resulting in a unrealistic lack of attenuation with height. Finally, a full climate simulation using a higher‐order, monotonic advection scheme is found to produce a tape recorder that is reasonably realistic, both in speed and amplitude. Copyright © 2002 Royal Meteorological Society

Effects of numerical advection schemes and eddy parameterizations on ocean ventilation and oceanic anthropogenic CO 2 uptake

Several different numerical advection schemes and eddy parameterizations were implemented in an ocean general circulation model to investigate the effect on ocean ventilation and the oceanic uptake of anthropogenic CO 2. To assess the models, simulations of natural Δ 14 C and excess CO 2 (defined as the increase in carbon in the ocean caused by the uptake of anthropogenic CO 2 from the atmosphere) were compared to observations. For both natural Δ 14 C and excess CO 2 the Southern Ocean (SO) was the region most sensitive to model numerics and eddy parameterizations. To produce realistic distributions of Δ 14 C and excess CO 2 the simulated oceanic anthropogenic CO 2 uptake tracked the minimum range of the modeled anthropogenic CO 2 uptakes. Models with higher uptake estimates produced too much excess CO 2 in the ocean and over-ventilated the deepwater of the SO. For the year 1990, our best estimate of the oceanic uptake of anthropogenic CO 2 was 1.8±0.1 Gt C/yr.

An Evaluation of Eulerian and Semi-Lagrangian Advection Schemes in Simulations of Rotating, Stratified Flows in the Laboratory. Part I: Axisymmetric Flow

A series of numerical simulations of steady, thermally stratified flow of a Boussinesq, incompressible fluid in a rotating, cylindrical fluid annulus were carried out over ranges of spatial resolution, grid stretch, and rotation rate. A range of different numerical advection schemes were used for the representation of heat transport, including a conventional conservative second-order Eulerian scheme and three different variants of a semi-Lagrangian scheme used either for temperature advection alone, or for both thermal and momentum advection. The resulting simulations were compared both with each other, and with high precision measurements of velocity, temperature, and total heat transport in the laboratory. The performance of the semi-Lagrangian scheme was found to be quite strongly sensitive to the spatial interpolation algorithm. A basic tensor cubic scheme generally produced good simulations of steady 2D and 3D flows, although the somewhat more accurate tensor quintic scheme (which is, however, also significantly more expensive) appeared to offer some detectable improvements in accuracy and performance in some cases. A split cubic scheme (which is computationally cheaper but formally less accurate) gave generally poor results in practice and is not recommended. In all cases considered, both the fully Eulerian and most forms of the semi-Lagrangian schemes gave good quantitative agreement with the laboratory measurements when extrapolated to very high resolution. Some significant systematic errors in the simulated heat transport and zonal momentum were found with all schemes, however, when run at moderate (though by no means very low) resolution. The semi-Lagrangian schemes had a tendency to overestimate heat transport relative to the laboratory measurements compared with the Eulerian schemes, but the latter tended to overestimate zonal momentum relative to the laboratory flows compared with the fully semi-Lagrangian simulations.