Numerical Advection Schemes Research Articles

A regional scale multilayer model for the calculation of long-term transport and deposition of air pollution in Europe

In this paper, we present a multi-layer Eulerian model aiming at calculating the long-term source-receptor relationships of air pollutants at the European regional scale. As a first approach, results from annual transport and deposition of sulphur dioxide and sulphate is given in this study. The model is however formulated in order to facilitate further studies of other air pollutants in the future. The model takes advantage of new detailed information on meteorology and emissions to the air in a 50 km × 50 km resolution. The model is formulated in a terrain following coordinate system, and a higher order numerical advection scheme is employed in order to reduce numerical diffusion. A simple linear parameterization scheme for sulphur dispersion and deposition is implemented. The wet scavenging is directly coupled to the release of precipitation in each model layer. The model has been employed for a full year simulations for sulphur transport and deposition in Europe. Model comparison with 60–70 measuring stations in Europe show realistic values for SO2 and sulphate in air and wet deposition of sulphate. Averaged of all stations the model deviates by only a few percent from the measured average. However, at individual stations the differences between the model and the measurements are sometimes larger, but seldom more than a factor of two difference is encountered. Further, comparison with a two-dimensional Lagrangian single layer model shows qualitatively good agreements. The most pronounced improvement encountered by use of the multi-level model are the increased wet deposition rates in remote areas, well in accordance with the measurements. This is ascribed to the improved description of the transport and wet scavenging processes by using a multi-layer rather than a single layer approach.

Numerical advection schemes, cross‐isentropic random walks, and correlations between chemical species

The advection schemes used in numerical models of chemistry and transport at fixed resolution must unavoidably cause the models to misrepresent the transport in some way. This can include failure to establish or preserve the functional relations between long‐lived chemical tracers that are often observed in the atmosphere. We show that linear functional relations will be preserved exactly by purely linear advection schemes and also, less obviously, by certain “semi‐linear” flux‐limited schemes despite the unavoidable nonlinearity introduced by the flux limiter. In practice, semi‐linear flux‐limited schemes will also preserve nonlinear functional relations better than linear centered difference or spectral schemes that suffer from dispersion errors. The reason is that the dispersion errors lead to spurious oscillations of the mixing ratio field in physical space, artificially expanding the range of mixing ratios in any neighborhood, and hence to a spurious scatter in the relation between any two mixing ratio fields that are nonlinearly related to begin with. Examples of correlations not only preserved, but established, by real and model transport are discussed in this light, including the case of stratospheric transport on timescales of years, for which we discuss and extend earlier results on the ways in which tracer functional relations can arise, for sufficiently long‐lived tracers, purely from transport. The stratospheric results are shown not to depend on the quasi‐horizontal Fickian eddy diffusivity assumption used in the earlier work. The reason is that, whenever the quasi‐horizontal (isentropic) mixing is fast enough—even if it is non‐Fickian as expected in real stratospheric surf zones—the chaotic part of the quasi‐vertical, cross‐isentropic transport has the nature of a random walk with small vertical steps.

An automatic differentiation technique for sensitivity analysis of numerical advection schemes in air quality models

Sensitivity analysis, which characterizes the change in model output due to variations in model input parameters, is of critical importance in simulation models. Sensitivity coefficients, defined as the partial derivatives of the model output with respect to the input parameters, are useful in assessing the reliability of the output from a complex model with many uncertainty parameters. Most existing sensitivity methods, however, have one or more of the following limitations: inaccuracy in the results, high cost in human effort, and difficulty in mathematical formulation and computer program implementation. To overcome these limitations, we are exploring ADIFOR, an automatic differentiation technique for systematically studying sensitivities. One can apply ADIFOR without having an intimate knowledge of the algorithms implemented in a model, so manual preparation of sensitivity code is avoided. In this paper, ADIFOR's accuracy and computational efficiency are demonstrated by calculating the sensitivity of concentration to a global perturbation of wind velocity in advection models and comparing this with results from the brute-force method of sensitivity analysis. ADIFOR-generated code can produce exact sensitivity information up to the machine epsilon, and can reduce computer CPU time requirements by up to 57% compared with the brute-force method for a single sensitivity calculation (and the savings increases with the number of parameters). Furthermore, we demonstrate the applicability of ADIFOR to models with a large number of uncertainty parameters by calculating the sensitivity of model output to initial conditions in a two-dimensional advection model.

The nirvana scheme applied to one‐dimensional advection

The NIRVANA project is concerned with the development of a nonoscillatory, integrally reconstructed, volume‐averaged numerical advection scheme. The conservative, flux‐based finite‐volume algorithm is built on an explicit, single‐step, forward‐in‐time update of the cell‐average variable, without restrictions on the size of the time‐step. There are similarities with semi‐Lagrangian schemes; a major difference is the introduction of a discrete integral variable, guaranteeing conservation. The crucial step is the interpolation of this variable, which is used in the calculation of the fluxes; the (analytic) derivative of the interpolant then gives sub‐cell behaviour of the advected variable. In this paper, basic principles are described, using the simplest possible conditions: pure one‐dimensional advection at constant velocity on a uniform grid. Piecewise Nth‐degree polynomial interpolation of the discrete integral variable leads to an Nth‐order advection scheme, in both space and time. Nonoscillatory results correspond to convexity preservation in the integrated variable, leading naturally to a large‐Δt generalisation of the universal limited. More restrictive TVD constraints are also extended to large Δt. Automatic compressive enhancement of step‐like profiles can be achieved without exciting “stair‐casing”. One‐dimensional simulations are shown for a number of different interpolations. In particular, convexity‐limited cubic‐spline and higher‐order polynomial schemes give very sharp, nonoscillatory results at any Courant number, without clipping of extrema. Some practical generalisations are briefly discussed.

Two Modified Versions of Bott's Positive-Definite Numerical Advection Scheme

Abstract Two modified versions of Bott's forward-in-time, positive-definite numerical advection scheme are described. One version is a conservative flux form; the other is a nonconservative advective form. Both versions demonstrate improved stability in tests involving a strongly deformational flow field. The modified versions are identical to Bott's original scheme for cases of uniform fluid density and purely rotational flow. Additional forms of the odd-order versions of the scheme are presented that exhibit lower phase-speed errors than the original odd-order versions.

Comments on “A Comparative Study of Numerical Advection Schemes Featuring a One-Step Modified WKL Algorithm”

Comments on “A Comparative Study of Numerical Advection Schemes Featuring a One-Step Modified WKL Algorithm”

Numerical Weather Simulations with Different Formulations for the Advection of Humidity and Cloud Water

This study investigates the effect on short-range weather prediction of using different numerical advection schemes for humidity and cloud water. Comparisons are made between predictions using the basic centered and upstream schemes and the more sophisticated Smolarkiewicz and Bott schemes. The main purpose of these alternative schemes is to reduce numerical diffusion and dispersion errors encountered in the basic schemes and thereby preserve the shape of features such as fronts. At the same time those schemes are efficient enough to be a realistic choice for operational models. The simulations are made with a numerical weather prediction model with 150-km horizontal resolution and ten levels. Frontal movements over large distances have been investigated in a domain that extends from eastern North America through Europe. The simulations yield little sensitivity to the choice of advection scheme for cloud water. On the other hand, a large sensitivity to the treatment of the humidity advection is found. This is connected to a substantial feedback with the model dynamics through release of latent heat. Larger numerical errors are identified with the basic schemes than with the alternative schemes, which yield a better shape preservation of the frontal zones. The centered scheme tends to give the heaviest precipitation and the deepest cyclones, while the upstream scheme gives least precipitation. Exaggerated low-level cloudiness is found with the Bott scheme and, to some extent, the Smolarkiewicz scheme. This appears to be caused by insufficient adjustment of other parts of the model to the new, more accurate, transport formulation. The Bott scheme has less numerical diffusion than the Smolarkiewicz scheme, but is somewhat more expensive computationally.

A Comparative Study of Numerical Advection Schemes Featuring a One-Step Modified WKL Algorithm

Abstract A fourth-order Crowley-type advection scheme based on the multistep Warming-Kutler-Lomax (WKL) scheme is proposed in this study. This scheme utilizes a free parameter to minimize dispersion and dissipation and can be used to represent the advection of positive-definite scalars (such as moisture). Linear Fourier component analyses indicate that the fourth-order Crowley-type scheme can reproduce the features of other modified Crowley-type schemes of third order, such as the scheme of Schlesinger and the quadratic upstream interpolation. Using the free parameter, the scheme may illustrate the limitation of the Crowley-type schemes for which diffusion is required for numerical stability of advective quantity. For these schemes, formulations that preserve amplitude are inevitably associated with smaller time steps. Adding the first cross-space term into these schemes could eliminate marginal instability or overshooting in linear advection. Linear and nonlinear advection tests show that the performance...

The influence of numerical advection schemes on the results of ocean general circulation models

The dependence of results from coarse-resolution models of the North Atlantic circulation on the numerical advection algorithm is studied. In particular, the sensitivity of parameters relevant for climate simulations as e.g., meridional transport of mass and heat and main thermocline thickness is investigated. Three algorithms were considered: (a) a central difference scheme with constant values for horizontal and vertical diffusion, (b) an upstream scheme with no explicit diffusion, and (c) a flux-corrected transport (FCT) scheme with constant and strictly isopycnal diffusion. The temporal evolution of the three models on time scales of centuries is markedly different, the upstream scheme resulting in much shorter adjustment time whereas the central difference scheme is slower and controlled by vertical diffusion rather than advection. In the steady state, the main thermocline structure is much less diffusive in the FCT calculation which also has much lower heat transport. Both horizontal circulation and overturning in the meridional-vertical plane are strongest in the upstream-model. The results are discussed in terms of the effective vertical (diapycnal) mixing in the different models. A significant increase in vertical resolution would be required to eliminate the high sensitivity due to the numerical algorithms, and allow physically motivated mixing formulations to become effective.

Improvements of Upstream, Semi-Lagrangian Numerical Advection Schemes

Abstract Semi-Lagrangian methods are common numerical techniques for the computation of the advection term in a nondivergent flow. They combine high accuracy and moderate computational requirements. Three aspects of these methods will be discussed. First, the condition of nondivergence must be carefully obeyed; examples of possible error sources are given. Second, the determination of the starting point of the backward trajectory can be improved considering a linear variation of the wind field around a grid point instead of the assumption of homogeneous velocity. It is shown both theoretically and by a suitable test that this improvement is necessary in a deformational flow in order to obtain satisfactory results. Examples are presented using cubic splines and a Fourier representation for interpolation of the advected quantity. Third, an improved method of making the scheme positive definite is introduced, which also removes the positive parts of 2Δx oscillations and avoids the potential instability of a ...

Sulfate Production and Deposition in Midlatitude Continental Cumulus Clouds. Part I: Cloud Model Formulation and Base Run Analysis

Abstract We have developed a 1.5 dimensional Eulerian cumulus cloud model which incorporates: (i) two in-cloud regions, (ii) a bulk water parameterization including the ice phase, (iii) an entrainment formulation based on the turbulent kinetic energy, (iv) a sustained subcloud forcing, and (v) a flux-corrected transport algorithm in the numerical advection scheme. This model formulation was chosen in order to provide a simple dynamic framework which permits simultaneous up- and downdrafts in-cloud. The model is compared against aircraft and radar data for clouds observed during the Cooperative Convective Precipitation Experiment (CCOPE) held in south-central Montana during the summer of 1981. The overall dynamic, thermodynamic and microphysical agreement is good. Sensitivity analysis of adjustable parameters indicates that model response is not overly sensitive to the characteristic entrainment mixing length used, but fairly sensitive to parameters used in the ice phase microphysics.

The numerical simulation of atmospheric gravity currents. Part I: Neutrally-stable environments

Abstract A two-dimensional numerical model for a gravity current of cold air moving into a neutrally-stable layer of warm air at rest is described. The ability of the model to capture the main observed features of gravity currents in the atmosphere and in the laboratory is demonstrated. The model has some novel features compared with earlier ones: it includes a stretched vertical grid to give high resolution near the ground, uses a sophisticated monotonic numerical advection scheme, and in one form has a turbulent closure scheme with variable eddy diffusivity related to the local deformation of the flow and to the local Richardson number. The sensitivity of model simulations to the surface boundary condition, the turbulent closure, the numerical integration scheme and the grid resolution is explored also. The model provides a basis for the study in Part II of the effect of ambient low-level stable layers on the evolution of gravity currents, a common situation in the atmosphere.

An evaluation of several numerical advection schemes

Three categories of numerical advection techniques are tested: SHASTA, a flux-corrected onedimensional algorithm; FCT, a multi-dimensional flux-corrected technique; and the BIQUINTIC method, a polynomial approximation algorithm. The tests include the transport of a finite amount of material through a 25 × 25 cell grid driven by three different flow fields: one-dimensional linear flow, two-dimensional linear flow, and rotational flow. For each of these tests, two initial distributions of material are used: a rectangular blockshape and an ellipse-shape. In addition, a test with a non-divergent flow field with a homogeneous material field was performed to check for inherent divergence in a particular method. Results show that the FCT and BIQUINTIC methods maintain the integrity of an initial distribution of material better than SHASTA through a simulation. While SHASTA is the fastest of the methods tested computationally it also produces the greatest amount of spurious numerical diffusion. The FCT method has a tendency to flatten the top of a peaked distribution, and the BIQUINTIC method tends to produce a peak in a top-hat distribution during numerical transport. The BIQUINTIC scheme requires more computer time to execute than the other methods tested.

A numerical advection scheme with small phase speed errors

AbstractIt is demonstrated that, for linear advection in one space dimension, a simple modification to the well‐known Lax‐Wendroff integration scheme leads to a substantial reduction in phase speed errors. The modification is designed in such a way that no additional restriction is placed on the timestep used in an integration. The improved performance is comparable or superior to that of previously proposed advection schemes, yet it is achieved without recourse to expensive or cumbersome computations.The new scheme is readily extended to two space dimensions and has been used in numerical weather prediction and sea wave modelling in the Meteorological Office. An example is given of an improved numerical forecast of the 500 mb height field which was obtained by applying the new scheme in the operational 10‐level numerical weather prediction model.

A numerical advection scheme with small phase speed errors

A numerical advection scheme with small phase speed errors