Andreas Bott

A Positive Definite Advection Scheme Obtained by Nonlinear Renormalization of the Advective Fluxes

Abstract A new method is developed to obtain a conservative and positive definite advection scheme that produces only small numerical diffusion. Advective fluxes are computed utilizing the integrated flux form of Tremback et al. These fluxes are normalized and then limited by upper and lower values. The resulting advection equation is numerically solved by means of the usual upstream procedure. The proposed treatment is not restricted to the integrated flux form but may also be applied to other known advection algorithms which are formulated in terms of advective fluxes. Different numerical tests are presented illustrating that the proposed scheme strongly reduces numerical and diffusion and simultaneously requires only small computational effort. For Corant numbers with absolute values not exceeding one, the scheme preserves numerical stability except in strong deformational flow fields where slight instabilities may occur.

A Flux Method for the Numerical Solution of the Stochastic Collection Equation

Abstract A new mass conservative flux method is presented for the numerical solution of the stochastic collection equation. The method consists of a two-step procedure. In the first step the mass distribution of drops with mass x′ that have been newly formed in a collision process is entirely added to grid box k of the numerical grid mesh with xk ⩽ x′ ⩽ xk+1. In the second step a certain fraction of the water mass in grid box k is transported to k + 1. This transport is done by means of an advection procedure. Different numerical test runs are presented in which the proposed method is compared with the Berry–Reinhardt scheme. These tests show a very good agreement between the two approaches. In various sensitivity studies it is demonstrated that the flux method remains numerically stable for different choices of the grid mesh and the integration time step. Since a time step of 10 s may be used without significant loss of accuracy, the flux method is numerically very efficient in comparison to the Berry–Re...

A Radiation Fog Model with a Detailed Treatment of the Interaction between Radiative Transfer and Fog Microphysics

Abstract A one-dimensional radiation fog model is presented which includes a detailed description of the interaction between atmospheric radiative transfer and the microphysical structure of the fog. Aerosol particles and activated cloud droplets are treated using a two-dimensional joint size distribution whereby the activation process of aerosols is explicitly modeled. For this purpose a new positive definite semi-Lagrangian advection scheme is developed that produces only small numerical diffusion and is numerically very efficient. For the radiative calculations, time dependent attenuation parameters are determined from the actual particle size distributions. The diffusional growth of the particles is calculated by considering radiative effects in the droplet growth equation. Numerical results elucidate that the strong interaction between the radiatively induced droplet growth and their gravitational settling is responsible for a distinct reduction of the liquid water content which also shows quasi-peri...

Monotone Flux Limitation in the Area-preserving Flux-form Advection Algorithm

Abstract The area-preserving flux-form advection algorithm is extended to monotonicity. For this, the nonlinear positive-definite flux limitation of the original approach is replaced by new monotone flux limiters. The monotone fluxes are derived for one-dimensional constant transport velocities. The deformation occurring in divergent flow is accounted for by adding to the monotone advection fluxes a correction term, which has been derived from the deformation of the upstream method. The final algorithm is applicable to arbitrary multidimensional transport problems. However, due to the use of the time-splitting method, it is strictly monotone only in uniform flow fields. Results of different one- and two-dimensional advection experiments are presented, demonstrating that the monotone flux limitation is an attractive alternative to the positive-definite algorithm. Amplitude and phase speed errors are somewhat larger in the monotone advection scheme. The computational effort of the new version is not much la...

PAFOG—a new efficient forecast model of radiation fog and low-level stratiform clouds

Abstract The new one-dimensional forecast model PAFOG for radiation fogs and low-level stratiform clouds will be presented. The aim of the model is to improve the local visibility forecast on airports and other traffic locations where fog and low-level stratus frequently occur. PAFOG has been developed on the basis of the microphysical fog model MIFOG of Bott et al. [J. Atmos. Sci. 47 (1990) 2153]. To obtain a numerically efficient model, the detailed spectral cloud microphysics of MIFOG has been replaced by the parameterization scheme of Chaumerliac et al. [J. Geophys. Res. 92 (1987) 3114]. Furthermore, according to Siebert et al. [Beitr. Phys. Atmos. 65 (1992a) 93], a model for low vegetation is included in PAFOG so that now fog evolution as influenced by different types of vegetation can also be accounted for. The performance of PAFOG has been tested by comparing the model results with routine observations of the German Weather Service. Nine different weather periods comprising a total of 45 days have been investigated. In 41 cases, PAFOG yields agreement with the observations in terms of occurrence or nonoccurrence of fog or stratiform clouds. During radiation fogs, the calculated and observed visibilities are quite similar. However, in the model simulations the formation of dense fogs tends to be somewhat delayed. From the case studies with stratiform clouds, it is seen that cloud evolution in time and space strongly depends on the value of the large-scale subsidence. Since this quantity is not available from measurements, it must be provided by means of a numerical weather forecast model.

A Flux Method for the Numerical Solution of the Stochastic Collection Equation: Extension to Two-Dimensional Particle Distributions

Abstract In the present paper a new method is introduced for the numerical solution of the stochastic collection equation in cloud models dealing with two-dimensional cloud microphysics. The method is based on the assumption that the probability for the collision of two cloud drops only depends on the water mass of each and not on the mass of the aerosol nuclei. With this assumption it is possible to reduce the two-dimensional solution of the stochastic collection equation to a one-dimensional approach. First, the two-dimensional particle spectrum is integrated over the aerosol mass yielding a one-dimensional drop spectrum in the water mass grid. For this intermediate drop distribution the stochastic collection equation is solved. The resulting new drop spectrum is redistributed into the two-dimensional aerosol–water grid. Numerical sensitivity studies are presented demonstrating that the flux method yields very good results. In the two-dimensional aerosol–water grid the drop distributions move from initi...

Electromagnetic energy within dielectric spheres

We present exact and approximate analytic expressions for the time-averaged electromagnetic energy within dielectric spheres on the basis of rigorous Mie theory. Such information is of importance for the study of photochemical reactions within atmospheric water spheres. Numerical results show that on the average the energy inside a cloud droplet is enlarged by a factor exceeding 2 compared with that of a sphere of the same radius of the surrounding medium. In regions of resonance peaks the electromagnetic energy may be increased by more than 2 orders of magnitude.

Interaction of radiation fog with tall vegetation

Abstract A one-dimensional radiation fog model is presented. It is coupled with a second model to include the effects of tall vegetation. The fog model describes in detail the dynamics, thermodynamics, and microphysical structure of a fog, as well as the interactions with the atmospheric radiative transfer. A two-dimensional joint size distribution for the aerosol particles and activated fog droplets is used, the activation of aerosol particles is explicitly modeled. The implications of the presence of tall vegetation on the state of the atmosphere and on the evolution of radiation fog are stated. It is shown that the existence of tall vegetation impedes the evolution of radiation fog. The life cycle of radiation fog is discussed. The input of fog water and associated aerosol particles onto the vegetation surfaces via fog water interception processes is assessed.

Multiphase chemistry in a microphysical radiation fog model—A numerical study

A microphysical radiation fog model is coupled with a detailed chemistry module to simulate chemical reactions in the gas phase and in fog water during a radiation fog event. In the chemical part of the model the microphysical particle spectrum is subdivided into three size classes corresponding to non-activated aerosol particles, small and large fog droplets. Chemical reactions in the liquid phase are separately calculated in the small and in the large droplet size class. The impact of the chemical constitution of activated aerosols on fogwater chemistry is considered in the model simulations. The mass transfer of chemical species between the gas phase and the two liquid phases is treated in detail by solving the corresponding coupled differential equation system. The model also accounts for concentration changes of gas-phase and aqueous-phase chemical species which are induced by turbulence, gravitational settling and by evaporation/condensation processes. Numerical results demonstrate that fogwater chemistry is strongly controlled by dynamic processes, i.e. the vertical growth of the fog, turbulent mixing processes and the gravitational settling of the particles. The concentrations of aqueous-phase chemical species are different in the two droplet size classes. Reactands with lower water solubility are mainly found in the large droplet size class because the characteristic time for their mass transfer from the gas phase into the liquid phase is essentially longer than the characteristic time for the formation of large fog droplets. Species with high water solubility are rapidly transferred into the small fog droplets and are then washed out by wet deposition before these particles grow further to form large droplets. Thus, the concentrations of the major ions (NO3−, NH4+) are much higher in small than in large droplets, yielding distinctly lower pH values of the small particles. In the present study the reaction of sulfur with H2O2 and the Fe(III)-catalysed autoxidation of S(IV) are the major S(VI) producing mechanisms in fog water. Most of the time the sulfur oxidation rates are higher in the large than in the small droplets. Fogwater deposition by gravitational settling occurs mainly in the large droplet size class. However, since in the small droplets the concentrations of chemical species with very good water solubility are relatively high, in both droplet size classes the total wet deposition of these reactands is of the same order of magnitude.

Modelling of radiation quantities and photolysis frequencies in the aqueous phase in the troposphere

Abstract In order to provide information about photolysis frequencies in the aqueous phase for chemical transport models including wet chemistry a parameterization which can be added to gaseous-phase photolysis models was developed. The actinic fluxes inside cloud droplets are calculated on the basis of rigorous Mie theory taking into account the effect of dissolved particulate aerosol material and 10 representative: cloud droplet size distributions. The results show that the actinic flux inside cloud droplets are on the average more than twice as large as compared to the interstitial air. The newly developed parameterization has been applied together with the model STAR (System for Transfer of Atmospheric Radiation). A;part from the parameters influencing gas-phase photolysis frequencies the radiation quantities inside the cloud droplets and therefore the photolysis frequencies in the aqueous phase depend on the droplet size distribution, the mixing ratio of dry aerosol particulate material to cloud droplet water, and the amount of light absorbing material in the droplets. In-droplet radical source strengths have been calculated for the most important photolysic sources of OH and S04.