Advection scheme

The transport of heavy metals in the atmosphere is described by means of a monotone version of Bott s advection scheme. Pressure-based s-coordinate in the vertical makes possible to take into account an effect of the underlying surface elevation. Vertical eddy... 

The advection scheme of the regional model is improved to take into account the surface orography. Terrain following vertical structure of the model domain with higher resolution was incorporated. Wet removal of heavy metals from the atmosphere was enhanced by developing newparameterizations of precipitation scavenging. Both in-cloud and sub-cloud wet removal were modified on the basis of the up-to-date scientific literature data. 

Dabdub, D., and J. H. Seinfeld, Numerical Advective Schemes Used in Air Quality Models—Sequential and Parallel Implementation, Atmos. Environ., 28, 3369-3385 (1994b). 

Hwang D, Byun DW, Odman MT (1997) An automatic differentiation technique for sensitivity analysis of numerical advection schemes in air quality models. Atmospheric Environment, 31(6) 879-888. 

In these equations fi is the coluirm mass of dry air, V is the velocity (u, v, w), and (jf) is a scalar mixing ratio. These equations are discretized in a finite volume formulation, and as a result the model exactly (to machine roundoff) conserves mass and scalar mass. The discrete model transport is also consistent (the discrete scalar conservation equation collapses to the mass conservation equation when = 1) and preserves tracer correlations (c.f. Lin and Rood (1996)). The ARW model uses a spatially 5th order evaluation of the horizontal flux divergence (advection) in the scalar conservation equation and a 3rd order evaluation of the vertical flux divergence coupled with the 3rd order Runge-Kutta time integration scheme. The time integration scheme and the advection scheme is described in Wicker and Skamarock (2002). Skamarock et al. (2005) also modified the advection to allow for positive definite transport. 

Baklanov A, Sprensen JH (2001) Parameterisation of radionuclide deposition in atmospheric long-range transport modelling. Phys Chem Earth (B) 26(10) 787-799 Bott A (1989a) A positive definite advection scheme obtained by non-linear renormalization of the advective fluxes. Mon Weather Rev 117 1006—1015 Bott A (1989b) Reply. Mon Weather Rev 117 2633-2636... 

Nair R, Machenhauer B (2002) The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere. Mon Weather Rev 130(3) 649-667... 

Further tests and previous experiences in other geographic locations confirmed that minimum Kz is a relevant (and often neglected) parameter to model properly the dispersion during weak wind and very stable conditions. Unfortunately no general value for minimum Kz can be defined, while proper values depend on season and local climatology, as well as on numerical diffusion in the advection scheme. 

In the case of the GRG coupled system, both the IFS and the CTM simulate atmospheric transport processes. Different advection schemes or spatial and temporal resolutions may lead to different concentration fields in the IFS and the CTM. Thus, the applied CTM tendencies can be inconsistent with the concentration fields in the IFS. The most annoying consequence would be negative concentration values in the IFS, due to un-balanced loss processes. 

Calculate tendencies due to semi-Lagrangian advection scheme... 

Bott A (1989) A Positive Definite Advection Scheme Obtained by Nonlinear Renormalization of the Advective Eluxes. AMS 117 1006-1015... 

The Enviro-HIRLAM (see its description and validation in a separate paper by Korsholm et al. in this volume) is a fully online NWP-ACT integrated system. The following steps towards Enviro-HIRLAM are being incorporated or have already been achieved (1) nesting of models for high resolutions, (2) improved resolution of boundary and surface layer characteristics and structures, (3) urbanisation of the model, (4) improvement of advection schemes, (5) implementation of chemical mechanisms, (6) implementation of aerosol dynamics, (7) realisation of feedback mechanisms, (8) assimilation of monitoring data (ongoing). 

Test of CISL advection scheme (Lauritzen 2005 Kaas 2008)... 

Tests of new advection schemes in Enviro-HIRLAM (Univ. Copenhagen, PhD and MSc studies)... 

Numerical advection schemes in the meteorological part are not meeting all the requirements for ACTMs, so they should be improved and harmonized in NWP and ACT models. 

For offline ACTMs the choice of advection schemes is an independent and critically important problem. Additionally using NWP input data with nonconservative and not harmonized schemes (due to different schemes, grids, time steps in NWP and ACT models) offline ACTMs can get a dramatic problem with an explosive solution for the chemical part. For onhne coupling it is not a problem if one uses the same mass-conservative scheme for chemicals, cloud water and humidity. 

It would be good to find an optimal advection scheme for integrated ACT-HIRLAM, however, estimations of ACT and NWP modellers can be different, because they can use different requirements/criteria for the best scheme for NWP and ACTM. At least it is reasonable to analyze and compare different schemes used in the HIRLAM community (for NWP and ACT including the semi-Lagrangian, CISL, Bott, Easter, Chlond, Walcek, Galperin and Kaas). 

Even the continuity equation (advection) scheme improvement is a more general problem in HIRLAM-A, it would be beneficial to coordinate this work of the HIRLAM Advection team with Hirlam-countries ACTM developers (e.g., the CISL and other new schemes tests, inter-comparisons and implementation). 

There are many choices for numerical schemes for the advection of tracers. Generally, a compromise between numerical accuracy and computational effort is chosen. Especially, for ecosystem models it is important, that the advection scheme is positive definite. Positive... 

A third variant of the VOF method calculates the interface tension force by the CSS method and perform an independent FLIC reconstruction of the interface to improve the design of the advection schemes. In this way the tailored advection discretization schemes prevent numerical smoothing of the interface [149]. 

The truncation error associated with convection/advection schemes can be analyzed by using the modified equation method [205]. By use of Taylor series all the time derivatives except the 1. order one are replaced by space derivatives. When the modified equation is compared with the basic advection equation, the right-hand side can be recognized as the error. The presence of Ax in the leading error term indicate the order of accuracy of the scheme. The even-ordered derivatives in the error represent the diffusion error, while the odd-ordered derivatives represent the dispersion (or phase speed) error. 

Another method for analyzing the truncation error of advection schemes is the Fourier (or von Neumann method) [135, 174, 136]. This method is used to study the effects of numerical diffusion on the solution. 

Let us temporary assume that the velocity and density fields are known, hence F and Fg can be determined. By that means (12.88) reduces to a transport equation for the property with only one unknown variable. However, in order to solve the convection-diffusion equation we need to approximate the transport property V at the e and w faces. A few classical convec-tion/advection schemes are outlined in the subsequent sections. 

High-order convection/advection schemes are widely used in meteorological applications solving h3rperbolic equations. For example, in European weather forecast models the explicit non-flux-based modified methods of characteristics have been very popular as they are very fast. Typical examples of this type of schemes are the semi-Lagrangian advection schemes of Bates and McDonald [9], McDonald [129] and McDonald [130]. These methods have an unrestricted time step advantage, but also an important disadvantage that they are not strictly conservative due to their non-flux-based formulation. 

Bates JR, McDonald A (1982) Multiply-Upstream, Semi-Lagrangian Advective Schemes Analysis and Application to a Multi-Level Primitive Equation Model. Mon Wea Rev 110 1831-1842... 

Dabdub D, Seinfeld JH (1994) Numerical Advective Schemes Used in Air Quality Models - Sequential and Parallell Implementation. Atm Env 28(20) 3369-3385... 

Leonard BP, Lock AP, MacVean MK (1996) Conservative Explicit Unrestricted-Time-Step Multidimensional Constancy-Preserving Advection Schemes. Mon Wea Rev 124 2588-2606... 

McDonald A (1984) Accuracy of Multiply-Stream, Semi-Lagrangian Advective Schemes. Mon Wea Rev 112 1267-1275... 

Smith GD (1985) Numerical Solution of Partial Differential Equations Finite Difference Methods. Third edition. Clarendon Press, Oxford Smolarkiewicz PK (1983) A simple positive definite advection scheme with small implicit diffusion. Mon Wea Rev 11 479-486... 

Thuburn J (1995) Dissipation and Cascades to Small Scales in Numerical Models Using a Shape-Preserving Advection Scheme. Mon Wea Rev 123(6) 1888-1903... 

Thuburn J (1996) Multidimensional Flux-Limited Advection Schemes. Journal of Computational Physics 123 74-83... 

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