Cent. Eur. J. Geosci. • 6(3) • 2014 • 257-278 DOI: 10.2478/s13533-012-0188-6

Central European Journal of Geosciences

Dispersion modeling of air pollutants in the atmosphere: a review

Research Article

Ádám Leelossy˝ 1, Ferenc Molnár Jr.2, Ferenc Izsák3, Ágnes Havasi3, István Lagzi4, Róbert Mészáros1∗

1 Department of Meteorology, Eötvös Loránd University, Budapest, Hungary 2 Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York, USA 3 Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, Budapest, Hungary 4 Department of Physics, Budapest University of Technology and Economics, Budapest, Hungary

Received 22 October 2013; accepted 15 May 2014

Abstract: Modeling of dispersion of air pollutants in the atmosphere is one of the most important and challenging scientific problems. There are several natural and anthropogenic events where passive or chemically active compounds are emitted into the atmosphere. The effect of these chemical species can have serious impacts on our environment and human health. Modeling the dispersion of air pollutants can predict this effect. Therefore, development of various model strategies is a key element for the governmental and scientific communities. We provide here a brief review on the mathematical modeling of the dispersion of air pollutants in the atmosphere. We discuss the advantages and drawbacks of several model tools and strategies, namely Gaussian, Lagrangian, Eulerian and CFD models. We especially focus on several recent advances in this multidisciplinary research field, like parallel computing using graphical processing units, or adaptive mesh refinement. Keywords: air pollution modeling • Lagrangian model • Eulerian model • CFD; accidental release • parallel computing © Versita sp. z o.o.

1. Introduction

their release across the globe depending on their chem- ical and physical properties (e.g., chemical composition, solubility in water, or size distribution for aerosol parti- cles), and they affect the human health and result in a Chemical species including toxic materials have various long-term effect on our environment. Such incidents can emission pathways into the atmosphere. They can be have a huge economic impact. For example the eruption of emitted either from approximately point sources (e.g., acci- Eyjafjallajökull in Iceland over a period of six days in April, dental release at nuclear power plants (NPP) or volcanic 2010 caused enormous disruption to air travel in most eruptions) or from area sources (e.g., emission of photo- parts of Europe because of the closure of airspace, due to chemical smog precursors and forest fires). Moreover, time volcanic ash ejected to the atmosphere. The estimated loss interval of releases can be diversified. These air pollutants of airlines’ revenue was about US$ 1.7 billion [1]. Another can travel hundreds, even thousands of kilometres from ∗ emblematic example is the effect of the photochemical air pollution. Photochemical smog can affect human health E-mail: [email protected] and reduce crop yield due to its oxidative nature. To save 257 Dispersion modeling of air pollutants in the atmosphere: a review

2. Overview of atmospheric disper- sion modeling human lives and to reduce economic losses, computational models of air pollutant dispersion are being developed to understand and predict the outcome of these phenomena and accidents. In this section we will give a brief overview how to de- Model simulations should be fast and must have a high scribe mathematically and parameterize various important degree of accuracy to be used in real time applications (e.g., processes in the atmosphere. We will start with the at- decision support). Therefore, one of the main challenge mospheric transport equation, which describes the spatial- of atmospheric dispersion modeling is to develop models temporal change of the concentrations of air pollutants. and software that can provide numerical predictions in Next we will focus on turbulent parameterization and finally an accurate and computationally efficient way. Disaster show how to handle deposition and chemical transformation at Chernobyl NPP has stimulated the development of in the atmosphere. such accidental release and decision support software (e.g., 2.1. The transport equation RODOS). Underestimating the concentrations or doses of air pollutants (e.g., radionuclides) can have serious c health consequences. However, in case of overestimation In a fluid, the mass conservation of a component described in regions where significant dose will not be reached would ∂c by concentration −∇·canc bev~ expressedS ∇· as:D ∇c : waste valuable human and financial resources. ∂t c c The transport and transformation of air pollutants in the v~ Sc Dc atmosphere is mainly governed by advection (wind field). = ( ) + + ( ) (1) Other processes such as turbulence, chemical reactions, where is the wind vector, is the source term and radioactive decay and deposition can also play important is the diffusion coefficient. Eq. (1) represents the change roles in the dispersion of toxic substances. Therefore, of the concentration at a specified point as the sum of strategies for model development require an interaction the advective flux, the source termsSc and the diffusive flux, of researchers from different fields (e.g., meteorology, geo- respectively. Dry and wet deposition and chemical or ra- physics, chemistry, IT). dioactive decay are part of the term, while gravitational Another important issue in modeling the dispersion of air settling can be added as an extra advection component. pollutants is the analysis of the lifetime of the chemical Turbulent diffusion, however, is not represented in Eq. (1). species and the characteristic distance that species can Turbulence is usually taken into account with Reynolds’ be transported, which is correlated with the lifetime. If theory that splits the wind and concentration field into the emitted species have short lifetime (minutes-hours) in time-averaged and turbulentv~ v~ perturbationv~0 values: the atmosphere (usually reactive species or aerosols), they

cannot be transported long-range and their effects willx be = ¯ + (2) concentrated on local scale (e.g., effect of the PM – particu- 0 2 c c c late matter). However, chemical species such as NO , SO , which have longer lifetimes (hours-days) can have wider = ¯ + (3) impact zones. Therefore, in such cases regional or conti- nental modeling approaches are needed. Some pollutants For turbulent flows, the transport equation can be written 3 (e.g., CO, NH ) have long lifetimes (days-weeks) resulting in the same form as Eq. (1) with the addition of three eddy in long-range transport, so for these pollutants continental covariance terms, yielding a form that is widely used in and global transport models need to be employed. atmospheric dispersion modeling: For simulating the dispersion of air pollutants, various ∂c ∂ u0c0 − ∇· v~c Sc Dc∇ c − modeling approaches have been developed. The main aim ∂t ∂x 2 of this article is to provide a brief review of air pollution ¯ ∂ v 0c0 ∂ w0c0 ( ) = − (¯¯)− + + ¯ modeling tools and their application. The paper is struc- ∂y ∂z (4) tured as follows. Section 2 provides a general overview ( ) ( ) of the related physical problems and air pollution model- ing. The following three sections describe three families of models: Gaussian, Lagrangian and Eulerian dispersion The right hand side of Eq. (4) describes advection, source models in detail with their advantages and drawbacks. terms, molecular diffusion and horizontal and vertical turbu- Finally, Sections 6 and 7 discuss parallel computing and lent fluxes. In the atmosphere, turbulent mixing is several computational fluid dynamics models for environmental orders of magnitude more efficient than molecular diffu- modeling. sion, thus the third component of Eq. (4) can usually be neglected, except in the laminar ground layer. 258 Á Leelossy˝ et al.

To calculate the eddy covariance terms in Eq. (4), the gradient transport theory was developed as an analogy of Fick’s law for molecular diffusion. Gradient transport theory is based on the assumption that the turbulent flux is proportional to the gradient of the concentration field. This approach leads to a form where turbulent fluxes are ∂c expressed as−∇· an additionalv~c S diffusionD ∇ c term:∇· K ∇c ; ∂t c c 2 ¯ K = (¯¯) + + ¯ + ( ¯) (5)

Kx Ky Kz where is a diagonal matrix of the eddy diffusivitiesK

, , . Due to the different atmospheric turbulentDc processes in horizontal and verticalK directions, cannot be assumed to be isotropic. Furthermore, while is a property of the chemical species, is a property of the flow, thus it varies in both space and time. Assuming an incompressible fluid, isotropic horizontal turbulence and neglecting the molecular diffusion, the transport equation ∂c ∂ ∂c can be written−v~ in∇c theS form: ∇ · K ∇ c K ; ∂t c h h h ∂ z ∂z ¯ ¯ ¯ ∇h= ¯ + + ( ¯) + Kh (6) Kz where is the horizontal divergence operator, is the horizontal and is the vertical eddy diffusivity. The eddy diffusivities describe the turbulence intensity in the boundary layer, considering both thermal and kinetic turbu- lence. The vertical profile of eddy diffusivities is calculated through various parameterizations based on methods briefly Figure 1. Profiles of the normalized vertical eddy diffusivity for dif- described in Chapter 2.2. ferent normalized planetary boundary layer heights under 2.2. Turbulence parameterization (a) unstable conditions and (b) stable conditions from [3], where h is the planetary boundary layer height and L is the Monin–Obukhov length, respectively (see section 2.2.3).

Near-surface atmospheric turbulence is caused by mechan- ical and thermal effects. Friction forces in a viscous flow generate mechanical turbulence, which is driven by wind shear. Therefore, three-dimensional wind field data is Turbulence intensity can be described using variousTKE mea- necessary to estimate the intensity of mechanical turbu- sures.− Using the standard Reynolds-averaged (RANS) ap- lence. However, near-surface wind shear can be estimated proach1 of turbulence, the turbulent kinetic energy ( , using the surface roughness, a parameter that represents J kg ) represents the average kinetic energy of the subgrid the strength of friction between the surface and the atmo- scale wind fluctuationsTKE [2]:u0 v 0 w0 ; sphere.  2 2 2 0 0 0 1 Thermal turbulence is driven by buoyancy and can be u v w = + + (7) characterized with atmospheric stability measures. Sur- 2 face radiation budget has a critical effect on atmospheric where , , are wind component fluctuations with stability, therefore correct estimation of sensible and la- respect to the time-averaged wind. tent heat fluxes has a key importance in the estimation of Dispersion-oriented turbulence characteristics try to es- thermal turbulence intensity. Under stable conditions, the timate the mixing efficiency instead of describing the tur- boundary layer turbulence is dominated by mechanical ef- bulence itself. Mixing efficiency is often treated as the fects, thus turbulent mixing is driven by wind shear (stable deviation of a Gaussian plume, which is widely used in boundary layer, SBL). In unstable atmosphere, turbulence Gaussian (section 3) and puff (section 4.2) models. is mostly generated by thermal convection (convective The vertical profile of vertical eddy diffusivity (Figure 1) boundary layer, CBL). shows near-zero values very close and very far from ground, 259 Dispersion modeling of air pollutants in the atmosphere: a review

Table 1. Pasquill’s stability classes: very unstable (A), unstable (B), slightly unstable (C), neutral (D), slightly stable (E) and stable (F) boundary layer. Note: neutral (D) class has to be used for overcast conditions and within one hour after sunrise / before sunset [7].

Daytime insolation Night time Surface wind speed Strong Moderate Slight Thin overcast or Min. 4 octas max. 3 octas low cloud low cloud < 1 m/s A A B E F 1 – 3 m/s A B C E F 3 – 5 m/s B C C D E 5 – 8 m/s C D D D D > 8 m/s D D D D D 2.2.1. Stability classes

Turbulent mixing in the atmosphere is driven by wind shear and buoyancy. An obvious approach to calculate turbulence parameters is to define categories regarding radiation parameters, wind speed, surface roughness and cloud cover. The widely used method of Pasquill defines six stability classes from the very unstable A to the moderately stable F based on wind speed, sun elevation and cloud

Figure 2. A typical daily cycle of the planetary boundary layer in fair cover (Table 1)[5, 6]. The advantage of Pasquill’s method is weather (after [2]). that turbulence parameters can be directly estimated from basic meteorological measurements; however, the accuracy of pre-defined values of the six discrete classes is often low2.2.2. compared Richardson to more number sophisticated parameterizations.

and a maximum in the lower part of the boundary layer [3]. In the region above the maximum diffusivity, the height The ratio of the thermal to mechanical turbulence is char- where vertical turbulent mixing becomes negligible is de- acterized by the gradient Richardson number [8]: fined as h planetary boundary layer (PBL) height. Neg- g ∂ρ − ligible vertical turbulence causes no upward transport of ρ ∂z Ri ; air pollutants, thus pollution of sources in the PBL will ∂u ∂v 0 remain in the boundary layer. Concentration values are ∂z 2 ∂z 2 =     (8) largely influenced by the volume that is available for the + dilution of released pollutants in a given period of time; this volume is defined by the PBL height and the advection ρ g velocity. 0 Intensity of boundary layer turbulence has a significant where is the reference density and is the surface diurnal and annual variability [4]: PBL height can reach acceleration of gravity. Thermal turbulence (buoyancy) 2000 m or above in convective situations, typically in is described by the numerator of Eq. (8), while a wind the summer, but can be as low as 10–50 meters on clear shear term is present in the denominator. While buoyancy nights with low wind speed. The daytime mixed layer can have both positive and negative impact on turbulence collapses shortly before sunset, when direction of surface intensity depending on the direction of the density gradient, heat fluxes is reversed and thermal turbulence is ceased the positive wind shear term is always a generator of (Figure 2). Thermal turbulence driven boundary layers turbulence. are often referred to as convective boundary layer (CBL), Negative Richardson number values indicate unstable strat- while in stable boundary layers (SBL), stable thermal ification that develops thermal turbulence, while small pos- stratification destroys mechanical turbulence. An SBL itive Richardson numbers correspond to stably stratified keeps pollutants from surface sources close to the ground atmosphere with significant wind shear and mechanical causing significantly higher concentrations than a CBL, turbulence. At large positive Richardson numbers, thermal especially during nighttime. stability destroys mechanical turbulence and results in a stable boundary layer. Typical values of Richardson number are given in Table 2. 260 Á Leelossy˝ et al.

Table 2. Typical values of the Richardson number and the Monin–Obukhov-length at each stability class [15]. locations. Dry deposition is a continuous process, while wet removal can be realized only in the presence of pre- cipitation. Both dry and wet depositions depend on the Stability RichardsonRi < − : Monin–Obukhov-length class number− : ≤ Ri < − : (m) properties of the gases or particles, and removal processes A − : ≤0 86Ri < − : –2 to –3 are governed by several environmental factors. The effect − : ≤ Ri < : B 0 86 0 37 –4 to –5 of both dryk andd wet depositions cankw be incorporated in C :0 37≤ Ri < : 0 10 –12 to –15 dci models asdt a first−k orderw/dci reactionci taking into account dry D :0 10≤ Ri 0 053 Infinite i E 0 053 0 134 35 to 75 deposition ( ) and wet deposition ( ) coefficients through reaction = , where is the concentration of F 0 134 8 to 35 2.3.1. Dry deposition 2.2.3. Monin–Obukhov theory the component of the system.

The exchange of trace gases between the surface and The most widely used approach to describe vertical pro- the atmosphere is mainly governed by turbulent diffusion. file of turbulence intensity was developed by Monin and The deposition can be estimated by the widely used „big- Obukhov in 1954. They combined the turbulent vertical leaf” model, in which the deposition velocity is defined fluxes of heat and momentum into a parameter with dimen- as the inverse of the sum of the atmospheric and surface sion of length, the L Monin–Obukhov-length:0 0 −u w − L − 3 ; resistances [16–18v ]: R R R ; g 2 d a b c κ T 0w
0 T 1 = (9) Ra Rb R=c ( + + ) (10) T 0 u0 w0 T κ where , , and are the aerodynamic resistance, the where , , are the temperature, horizontal and vertical quasi-laminar boundary layer resistance, and the canopy wind turbulent fluctuations, is the temperature and is resistance, respectively. Each term is given by more or less ∗ the von Kármán constant, usually taken equal to 0.4 [u2]. detailed parameterization in different models. The deposi- Hardly measurable turbulentq fluxes can be estimated with tion of numerous trace gases (e.g., ozone, nitrogen oxides, ∗ the definition of two parameters:u theq friction velocity ammonia) can be estimated using this simple model. In con- and the turbulent heat flux . There are different methods trast to the previous case, the dry deposition of particles is for the determination of and using net radiation highly dependent on their characteristic size. Smaller par- components and (potential) temperature gradient, which ticles are deposited similarly as gases, while the removal largely differ in the convective and in the stable boundary of larger ones is mainly governed by gravitational settling layer [9, 10]. z (sedimentation). Dry deposition processes for particles The Monin–Obukhov length in Eq. (9) is a reference scalez/L with a size between 0.1 and 1 µm are less efficient. The for boundary layer turbulence, thus any height can dry deposition velocity can be written then as [19]: vd vg ; be given with the dimensionless stability parameter . Ra Rb vgRaRb Monin and Obukhov defined universal functions for neu-z/L 1 = + (11) tral, stable and unstable cases that describe the vertical vg + + profiles of wind and temperature as a function of the Ra Rb height [11]. The eddy diffusivities in Eq. (6) are param- where is the gravitational settling velocity (or terminal eterized based on the vertical profiles obtained from the settling velocity), and the resistance terms , and are Monin–Obukhov-theory (Figure 1)[3]. the same asRc those for gases (see above). As we assume The Monin–Obukhov theory was validated against nu- that all deposited particles stick tokd the surface, the surface merous measurement and advanced computational fluid resistance ( ) is zero for particles. dynamics datasets, and although some weaknesses were The dry deposition coefficient, , can be derived from shown, it still provides the most reliable basis for all fields the deposition or settling velocity simply dividing it by a 2.3.2. Wet deposition of atmospheric modeling [3, 9, 10, 12–14]. Typical values characteristic height, where the deposition occurs. of2.3. the Monin–Obukhov-length Deposition are given in Table 2.

Precipitation cleanses the air by capturing pollutants and depositing them onto the surface. The efficiency of this The removal of gases and particulates from the atmosphere process can be expressed by the fractional depletion rate can occur by dry or wet deposition (Figure 3). The im- of pollutant concentrations in the air. This rate is gener- portance of each deposition form varies with species and ally characterized by the so-called scavenging coefficient. 261 Dispersion modeling of air pollutants in the atmosphere: a review

Figure 3. Dry and wet deposition processes in the atmosphere.

During the parameterization of wet deposition (or wet scav- network. This reaction network can be interpreted as a set enging), in-cloud scavenging (rainout) and below-cloud of reaction mechanisms. A reaction mechanism consists of scavenging (washout) are usually distinguished. However, elementary reactions by which overall chemical change oc- for a more complete estimation of wet deposition, explicit curs. Chemical reactions can be described and calculated dc i information about clouds and precipitation rate at all model by a set of ordinary differentiali k equationscαn ; (ODEs) dt n i levels would be required. Instead of these complex pa- n i rameterization schemes [19], the wet deposition is often− X Y = th (13) estimated by a simple method in chemical transport models.1 kn n i The species dependent scavenging coefficient (kw in s ) αn can take the simple form: B where i is the chemical rate coefficient for the reaction kw AP ; and is the order of reaction with respect to chemical P = − A (12)B species . 1 Radioactive decay differs from chemical reactions due to where A is theB rate of rainfall (in mm h ) and and its nature, because during the radioactive decay the atomic are constants specific for gas or aerosol particle. In some properties are changed. However, from the mathematical models and vary also as a function of scavenging type point of view this process can be interpreted as a first dc (rainout, washout), or in case of aerosol particles as a order chemical reaction: i −kc ; dt i 2.4.function Chemical of particle radius reactions [20]. and radioactive de- cay = (14)

The solution of Eq. (14) is a monotonically decreasing exponential function with time. The radioactive decay Chemical reactions and transformations (e.g., radioactive constant k is an inherent property of the radionuclides. decay) of air pollutants can be very complex in the atmo- In models, consecutive decays can be easily taken into

sphere, where thermal and photochemical reactions can account and theS solutionc can be expressed analytically. simultaneously occur. Photochemical reactions usually In environmental models Eqs. (13)–(14) are included in produce radicals that have high affinity to react with each the source term ( ) of the atmospheric transport equation other and with other chemical species in the atmosphere. [Eq. (10)]. Eulerian models can easily handle this mean These radicals can initiate various reactions, and reactions field description by using (quasi-continuous) concentration in the atmosphere can be described as a complex reaction field. 262 Á Leelossy˝ et al.

However, if the dispersion of chemical species are consid- ered as a transport of small particles (e.g., in Lagrangian models), application of the chemical and decay rate con- stants are meaningless, because these quantities represent “bulk” properties of the system, and they are not related to property of individual particles. This problem can be overcome using a probabilistic description of the processes (radioactive decay, first order chemical reaction and depo- sition). The probability that individual particles during a time step will transform, decay or deposit can be calculated by the following expressionp − e−k t ; ∆ Figure 4. Schematic figure of a Gaussian plume. The effective stack k = 1 (15) height He and the crosswind and vertical deviation of the profile are the key parameters of the model. where is either the “macroscopic” first order rate con- stant (reaction rate constant, radioactive decay constant) or wet/dry deposition constant. In this way microscopic quantities can be coupled to macroscopic properties. 3. Gaussian dispersion models similar) for each receptor point instead of solving differen- 3.1. Theory and limitations of Gaussian mod- tial equations. This calculation is almost immediate even on common computers, however, meteorological data pre- els processing and sophisticated turbulence parameterizations can increase the computational cost. Gaussian models are widely applied in decision support software where robust model setup and fast response time is a key priority [22]. The transport equation [Eq. (6)] is a partial differential equation that can be solved with various numerical methods; ; h Gaussian models provide poor results in situations with described in section 5. Assuming a homogeneous, steady- low wind speeds, where the three-dimensional diffusion state flow and a steady-state point source at (0 0 ), is significant. Unfortunately, these situations have proven Eq. (6) can also be analytically integrated and results in to be the most dangerous ones in real-life atmospheric the well-known Gaussian plume distribution (see derivation dispersion problems as they are often connected to a in [21]): stably stratified atmosphere or low-level inversions [23]. Inspired by the fast increasing computational capacity and Q −y − z − h c x; y; z serious environmental incidents, Gaussian models have πσ σ u σ σ y z y2 !   z 2  been developed to increase their accuracy and take into ( ) ¯( ) = −expz h 2 exp 2 account some of the unrepresented physical processes. In 2 ¯ 2 ; 2 σ 2 state-of-the-art Gaussian models like AERMOD, CTDM  z  ( + ) or ADMS, the impact of complex terrain [10] or convective + exp 2 (16) c 2 boundary layer turbulence [24] is parameterized to provide Q x y more accurate prediction in environmental applications. where ¯ is the time-averagedz concentration atu a given One of the most widely used atmospheric dispersion model position, is the source term, is the downwind, ish is the open-source AERMOD, developed by the US Envi- the crosswind and is theσy verticalσz direction and ¯ is the ronmental Protection Agency (EPA). AERMOD includes time-averaged wind speed at the height of the release . modules to handle complex terrain and urban boundary The standard deviations and describe the crosswind layer [9], as well as the Plume Rise Model Enhancements and vertical mixing of the pollutant. Eq. (16) describes a (PRIME) algorithm to estimate downwash effects near mixing process that results in a Gaussian concentration buildings. AERMOD calculates eddy diffusivities based distribution both in crosswind and in vertical directions, on the Monin–Obukhov-theory (see section 2.2.3). centered at the line downwind from the source (Figure 4). Despite the advanced features of AERMOD, it still uses The last term of Eq. (16) expresses a total reflection from a steady-state approximation for the flow and the source, the ground. thus it can be used only within a distance of 10–100 km Gaussian models have an extremely fast response time, from the source. It is mostly used for long-term statistical because they only calculate a single formula (Eq. (16) or impact studies to estimate air pollution around industrial 263 Dispersion modeling of air pollutants in the atmosphere: a review

sites [26, 27], environmental protection or agricultural areas equations (PDEs) in the original dispersion problem, which [28, 29]. For mountain regions, EPA has also developed the is computationally an easy task, and avoids spatial trunca- Complex Terrain Dispersion Model (CTDM) that provides tion errors and numerical diffusion. The final distribution tools to estimate concentrations above complex terrain of a large number of particles gives an estimation of the without the costly simulation of the mesoscale wind field concentration field. [10]. Computational cost of Lagrangian models is independent Besides AERMOD, the British ADMS model is also widely of the output grid resolution, therefore this approach is ex- used for air quality simulations. It provides a range of ceptionally efficient for short-range simulations compared modules specified for different locations like urban, coastal to gridded computations with very fine resolution. How- or mountain areas, and is able to calculate deposition and ever, long-range simulations require the calculation of a radioactive doses [30]. ADMS has become very popular large number of single trajectories that rapidly increases in Europe for environmental impact studies and urban air the computational cost. Nested models use Lagrangian quality prediction [31]. For the latter purpose, ADMS- approach near the source and interpolate the concentration Urban module includes pre-defined emission scenarios and field to an Eulerian grid at a given distance to perform a chemistry model to calculate interactions between plumes large-scale Eulerian simulation [45, 46]. of several point and line sources in an urban area [32]. The trajectory equation for a single particle is an ordinary d~r There are several other Gaussian models available like differential equation v~ v~ ; CALINE3 for highway air pollution, OCD for coastal areas, dt 1 BLP and ISC for industrial sites or ALOHA for accidental ~r = + v~ (17) and heavy gas releases (see [33] for a detailed list). They are widely used by authorities, environmental protection where is the positionv~ of the particle, is the determin- organizations and industry for impact studies and health istic particle speed (consisting of advection, settling and 1 risk investigations [34, 35]. Their short runtime allows the buoyancy), while is the turbulent wind fluctuation vector. users to make long-term statistical simulations [36, 37] or The turbulent fluctuation is often estimated as a random detailed sensitivity studies [5, 38], and provide immediate walk, described by the Langevin equation [47]: Wt σw first-guess information in case of an accidental release. dwt − dt dW : TL s TL2 They are often coupled with GIS software to create an 2 efficient decision support tool for risk management [22, 39]. wt = + TL (18) While the earlier industrial incidents in Seveso (Italy) and σw Bhopal (India) and other air pollution episodes had been Where is the verticaldW turbulent fluctuation, is a La-

concentrated on a local scale, the Chernobyl accident in grangian integration time step, isd thet vertical turbulent 1986 had serious consequences in several countries and velocity fluctuationTL and represents a white noise pro- the radioactive 131I gas was measured globally [40]. It was cess with mean zero and variance . The Lagrangian clear that the steady-state assumption of Gaussian models timescale is a key parameter and is often given ex- could not handle continental scale dispersion processes, plicitly [48], or computed from velocity fluctuations [14]. however, existing Eulerian and Lagrangian models provided Numerical representation of the white noise process in precious information in the estimation of the impact of the Eq. (18) requires an efficient random number generator accident [40, 41]. The fast development of computers and that can significantly increase the computational cost of NWP (Numerical Weather Prediction) models allowed 4.1.the model. Puff models researchers to create more and more efficient dispersion simulations using gridded meteorological data. Eulerian and Lagrangian models, presented in sections 4 and 5, are state-of-the-art tools of recent atmospheric dispersion Puff models show similarities with both Gaussian and simulations [42–44]. Lagrangian dispersion models. They treat the pollution 4. Lagrangian models as a superposition of several clouds, “puffs” with a given volume, and calculate the trajectories of these puffs. Puff models separate the model physics by scale: the sub- puff scale processes are treated with Gaussian approach Lagrangian models calculate trajectories of air pollutants (Eq. (16) or similar), while above the puff size, Lagrangian driven by deterministic (wind field and buoyancy) and trajectories are calculated [48]. The result is a Gaussian stochastic (turbulence) effects. plume transported along a wind-driven trajectory instead of These trajectories are calculated based on ordinary differ- a straight centerline as in the standard Gaussian approach ential equations (ODEs) instead of using partial differential (Figure 5). The final concentration field is given as a 264 Á Leelossy˝ et al.

4.2. Trajectory models

Trajectory models perform stochastic simulation of numer- ous single point particles, each of them corresponding to a fraction of the released mass of pollution. The trajectory of a single particle is driven by advection, buoyancy and tur- bulence (random walk), calculated through Eqs. (17)–(18). Trajectory models provide a fine estimation of turbulent mixing without solving partial differential equations, on the other hand, a large number of single trajectories have to be calculated that largely increases the computational cost. With the fast increase of computational capacity, trajec- tory models have become state-of-the-art tools of regional Figure 5. Schematic representation of Gaussian plume and puff to global scale atmospheric dispersion simulations with models. Puff models still estimate a Gaussian dispersion, but are able to take into account temporal and spatial wind calculation of millions of long-range trajectories [42, 66]. changes. Two of the most popular models are the NAME, developed by the UK Met Office, and the open-source FLEXPART. NAME and FLEXPART have an impressive background of validation studies and environmental applications [66–69] and provided valuable information for authorities in critical superposition of the concentration field of each puff. With situations like the 2001 foot-and-mouth disease epidemic this two-way approach, puff models can handle spatial [49], the Eyjafjallajökull eruption [42] or the Fukushima nu- and temporal changes of wind direction with keeping the clear accident [44, 70]. We note that the HYSPLIT model, computational cost reasonably low. presented in section 4.1 as a hybrid puff-trajectory model, Puff models include the effect of turbulence in two different can also be used in trajectory mode. ways: with a stochastic random-walk approach in the Besides the prediction of air quality and dispersion path- trajectories of the puffs (Eqs. (17)-(18)); and through the ways, trajectory models are often used in backward mode deviation of a normal distribution inside each puff. It can to identify source regions of air pollution [63, 71, 72]. be interpreted as a separated treatment of large-scale and Backward simulations provide a source-receptor sensitivity sub-puff-scale turbulence. field that corresponds to probabilities of possible source Several puff models are available with a wide range of locations [71]. applications in risk management and environmental pro- 5. Eulerian models for solving atmo- tection [45, 49, 50]. The Danish RIMPUFF model is part of the RODOS decision support system, the European spheric transport equation framework for nuclear security [51]. The US EPA offers the CALPUFF system as a preferred model for environmental 5.1. General principles protection and public health studies. CALPUFF has been used worldwide for scientific and regulatory purposes [52– 55]. The RAPTAD puff model was specifically designed for simulations around complex terrain [56, 57] with both The main idea in the Eulerian models is to solve numer- industrial and urban applications [58, 59]. ically the atmospheric transport equation [Eq. (1)] in a One of the most popular dispersion models is HYSPLIT, fixed coordinate frame. Mathematically, it is a second a hybrid puff-particle model developed by the NOAA Air order partial differential equation, and its solution with Resources Laboratory. HYSPLIT has been used in nu- the appropriate initial and boundary conditions provides merous studies to estimate consequences of an accidental the spatiotemporal evolution of the concentration . There release or identify source regions of pollution [60–63]. It are several numerical methods to solve PDEs. In general, can be run in both single trajectory and puff mode, or these methods consist of two steps: (i) spatial discretiza- with a mixture of a vertical trajectory and a horizontal puff tion, (ii) the temporal integration of the derived ordinary simulation [64, 65]. differential equations. The spatial discretization, which is mostly performed on a grid, reduces the PDE to a system of ODEs depending on a single (time) variable. The system of ODEs then can be solved as an initial value problem, 265 Dispersion modeling of air pollutants in the atmosphere: a review

Table 3. Principles in the time integration.

n n n n n {ci;j } → {ci;j } {ci;j ; ci;j } → {ci;j } explicit implicit n−k n+1 n n−k +1 n +1 n {ci:j ; : : : ; ci;j } → {ci;j } {ci:j ; : : : ; ci;j } → {ci;j } One-step +1 +1 +1 Multistep

and a variety of powerful methods and software tools exist boundary conditions, namely, we assume that the concen- for this purpose. tration is zero at the boundary of the physical domain, There are several Eulerian models developed and used by which is a good approximation if the boundaries are far a large number of scientific communities. The GEOS–Chem 5.2.away from A simple the source. approach: the finite difference model is a global three-dimensional dispersion model using method assimilated meteorological observations from the Goddard Earth Observing System of the NASA Global Modeling and Assimilation Office. This model can be used to solve various atmospheric composition problems on global scale In this simple approach, all the derivatives are approxi-x × y [73]. WRF-Chem is the Weather Research and Forecast- mated by finite differences.c The spatial derivatives are dis- ing (WRF) model coupled with chemical transformations cretized on a grid with uniform rectangles of size ∆ ∆ . occurring in the atmosphere. The model takes into ac- n We assume that is sufficientlyci;j smooth (differentiable). count the emission, transport, turbulence and reactions of th Thisc n t; is xi justified; yj byn the presence of the diffusionxi term.; yj chemical compounds in gas phase and aerosols coupled We use the notation to approximate the values of to the meteorology. The model can be used for investiga- ( ∆ ) at the time level and at grid points ( ). tion of regional-scale air quality problems and cloud-scale As a feasiblecn choice− cn we can use centralcn − finitecn differences i ;j i− ;j K i ;j i− ;j interactions between clouds and chemistry [74]. The Com- x x x munity Multi-scale Air Quality (CMAQ) modeling system +1 1 +1 1 and 2 (20) has been developed to solve multiple air quality issues, 2∆ ∂ (∆∂ ) ∂x Kx ∂x such as tropospheric ozone, aerosols and acid deposition. 2 xi; yj CMAQ is a multi-platform model that can be used for ei- 2 to approximate the derivatives andO x , respectively in ther urban or regional scale air quality modeling. For this the grid point ( ) on the right hand2 side of Eq. (20). In purpose governing equations in the model are expressed in n both cases, the truncation errorci;j is (∆ n) . Additionally, the a generalized coordinate system. CMAQ simulates various approximationsi; of j the derivatives should be modified such chemical and physical processes that are important in tro- that the boundary condition for all is incorporated for pospheric trace gas transformations. The model contains those indices ( ) which lie on the boundary. To simulate three types of modeling components: a meteorological n T the time evolution in Eq. (20),ci;j we firstn recognize∈ ; that modeling system for the description of atmospheric states t the above proceduren results; ;::: in an ODE system, where and motions, emission models for man-made and natural ∆
the unknowns are the values with 0 . To emissions and a chemistry-transport modeling system for approximate it for = 1 2 we should apply time simulation of the chemical reactions [75]. integration, or in other words, we have to use an ODE To illustrate the concept of Eulerian models, we will present solver. This can be either explicit or implicit and can some techniques for a two-dimensional atmospheric trans- n contain one or more time steps. {ci;j } port equation∂c ∂c ∂ c ∂ c n −u Kx Ky ; u > Expliciti methodsj are easy to implement and can{ci;j be} calcu- ∂t ∂x ∂x2 ∂y2 n lated quickly: here using the values for all+1 possible{ci;j } 2 2 = + + 0 (19) indices and one can calculate the values . It+1 is also possible that for the calculation of the values on the physical domain Ω. This equation describes a sim- we already use them, which results in a system of equa- plified situation, where the spread of a pollutant occurs by tions to be solved for these unknowns. These possibilities advection (wind) and turbulent diffusion. Here we adjust are depicted in Table 3. the coordinate system to the direction of the wind, which is As an example, we give a finite difference scheme for assumed to be steady. To have a well-posed problem, we Eq. (20), where second order differences are used for the need to complete Eq. (19) with initial and boundary condi- spatial approximation and the time integration is executed tions. In the derivations, we use homogeneous Dirichlet using the so-called Crank–Nicolson method: 266 Á Leelossy˝ et al.

Figure 6. Uniform grid (left) and examples of h-refinement (center) and r-refinement (right). Reprinted with permission from [76].

n n n / ci;j ci;j ci;j +1 n +1n2 n / n / n / ci;j − ci;j c+i ;j − ci;j ci− ;j = Kx +1 t 2+1 2 x+1 2 +1 2 +1 1 n / 2 n +/ n / = ci;j − c2i;j ci;j− Ky ∆ +1 2 (∆ ) y+1 2 +1 2 +1 1 n / n / c c 2 2 + (21) +− u i ;j i− ;j +1 2 x +1 2(∆ ) +1 1

2∆

n This method is implicit, since we have used values at the n time{ci;j level} + 1 for the spatial discretization. In this way, Figure 7. H-refinement applied to simulating photochemical air pollu- we+1 obtain a system{ ofci;j equations} for the set of unknowns tion in Central Europe. The time evolution of the adaptive . It turns out that we arrive at the same scheme grid: (a) to, (b) to + 12 h, (c) to + 24 h, (d) to + 36 h, (e) to + 48 h, (f) t + 60 h, (g) t + 72 h, (h) t + 84 h. if the cell averages ¯ are the unknowns (this is a so- o o o called finite volume discretization) and a trapezoidal rule 5.3.is utilized Further for the numerical time integration. details

warranting increased resolution, and subsequently increase All of the methods above can be further developed by ap- grid element concentration in areas where the greatest plying a time-dependent discretization [76]. In practice, inaccuracies occur [81, 82].h In thisr technique, the total one uses a variable grid with finer resolution at loca- number of grid points remains constant. Figure 6 illustrates tions where the numerical error is high (usually where the the conceptual differences of - and -refinement. Figure 7 concentrationh gradient is large); this is called adaptive and 8 show these two types of adaptive gridding strategies gridding.h There are two groups of adaptiver gridding tech- for simulating the dispersion of air pollutants from a single niques called -refinement (“refine grid locally” and where point source. H stands for the element size) and -refinement (“relocate Another generalization is provided by the discontinuous a grid”). Galerkin method, which differs from the finite element -refinement dynamically changes the total number of approach in the sense that the basis functions are not grid elements within a computational domain for which necessarily continuous. the original structure remains fixed. This technique is In each case,ci;j we arrive at a large linear system of sparse also known as mesh enrichment or local refinement, andh matrices to solve, the size of which is the number of un- it is carried out by subdividing grid elements into smaller knowns . Typical shape of such matrices is depicted in self-similar components [16, 77–80]. An example of - Figure 9. One has to apply such a solver that can exploit refinementR is shown in Figure 6. Higher refinement levels this property. Also, it is especially useful to apply parallel can be achieved by further subdivision of cells. solvers. Several software packages include feasible solvers. -refinement techniques, commonly referred to as mesh The main message of the mathematical analysis is that the moving or global refinement, relocate mesh nodes to regions discretization parameters cannot be chosen independently. 267 Dispersion modeling of air pollutants in the atmosphere: a review

is discussed in detail in [85] and a good source on finite volume methods is [86]. A practical exposition of finite element methods can be found in [87], while the application of the discontinuous Galerkin methods for atmospheric 5.4.problems Operator is discussed splitting in [88].

From the mathematical point of view, the basis of air pollution models is a set of partial differential equations that are coupled to each other. The right-hand side of Figure 8. R-refinement applied to simulating air pollution dispersion from a point source. Reprinted with permission from [76]. these equations contains several terms describing different physical and chemical sub-processes such as advection, turbulent diffusion, reactions, deposition and emission. The definition of the sub-process is quite flexible, for instance, the advection can be treated as one sub-process, and the horizontal and vertical advection can be interpreted as two separated sub-processes. The numerical solution of these transport equations is a difficult and challenging computational task, especially in large-scale global models. In these models the num- ber of gridpoints can reach a few hundred thousand, and the number of chemical species can be up to several hun- dred. Usually these partial differential equations cannot be efficiently solved numerically in one integration step, Figure 9. The location of the non-zero elements (thick dots) in the matrix that arises in the time-integration corresponding to especially in case of complex reaction mechanisms that the scheme Eq. (21). The number of unknowns is × , the take into account the chemical transformation of species at size of the matrix is × , which contains 82 non-zero different time scales. This time scale separation of chemical elements. On the horizontal and the vertical axis show4 5 the number of the rows20 and the20 columns, respectively. processes causes the stiffness of the system, which means that a stable and convergent numerical solution can be only obtained using a very small time step (i.e., the numer- ical integration scheme cannot provide solution within a reasonable time). Solving a set of equations that describes In practice, for an up-to-date forecast it is favorable to all sub-processes at the same time is challenging and, in choose a large time step. At the same time, for any explicit most cases, impossible. However, the physical background method there is a strict constraint for the time step to of the sub-processes has been thoroughly investigated, and ensure the convergencet O ofx the numerical solution to the efficient numerical methods exist for solving the models real one. In general, if diffusion2 terms are present then of these sub-processes (e.g., diffusion and advection equa-
the condition ∆ = (∆ ) should be satisfied, which tions) as well as to solve the system of stiff equations for – at a precise spatial grid – allows only very tiny time chemical reactions. This provides the idea of operator split- steps. Many of the implicit methods do not require such ting: division of the right-hand side of the original system constraints, but in each time step a large linear system into several simpler terms and solution of the correspond- has to be solved. Therefore, one has to evaluate which ing systems – which are coupled to each other through approach needs less computational resources in every con- the initial conditions – successively in each time step of crete situation. the numerical solution [89–93]. In this way, the original In practice, we mostly face advection dominated problems, model is replaced with a model in which the sub-processes

where the average displacementu  Kx of thev  pollutantKy by ad- take place successively in time. This de-coupling method vection is much more than the displacement caused by allows the solution of a few simpler systems instead of the diffusion, meaning that and . The finite original and more complicated system. element solution of these problems needs special care. To illustrate the concept of the operator splitting, we Numerical methods for PDEs have a vast literature. Many present a splitting method used in the Danish Eulerian of the important principles and methods can be found in Model [93–95]. This is a long-range air quality model that general textbooks [83, 84]. The finite difference method has several subsystems describing the horizontal advection 268 Á Leelossy˝ et al.

(22), horizontal turbulent diffusion (23), chemical reactions For numerical applications, such as the dispersion simula- with emissions of air pollutants (24), deposition (25) and tion of air pollutants, parallelization is usually considered vertical turbulent diffusion (26) by the following set of on the level of processes and threads that run on dif- partial differential equations:∂ uci ∂ vci ferent processing units (processor cores, machines) and ∂ci − (1) − (1) ; ∂t(1) ∂x  ∂x  perform different parts of the computation at the same time. Depending on the memory structure, parallel computing = (22) systems can be classified as distributed or shared memory ∂ci ∂ ci ∂ ci Kx Ky ; systems. The distinction is fundamental in determining ∂t(2) ∂x2 (2) ∂y2 (2) the type of computation that best fits these systems, i.e., 2 2 = + (23) which atmospheric dispersion model fits which system. ∂ci Ei x~ Ri x;~ c ; c ; : : : ; cm ; The most distributed and heterogeneous form of parallel ∂t(3)  (3) (3) (3) computing is called grid computing [99, 100], in which 1 2 = ( ) + (24) regular standalone computers in numbers ranging from a ∂ci −k ci − k ci ; few dozen up to several thousand are connected by con- ∂t(4) (4) (4) dry wet ventional networks or the Internet. There is high commu- = (25) nication latency between computers, therefore in general, ∂ wci ∂ci ∂ ci these systems best fit for computations that do not require − (5) Kz ; ∂t(5)  ∂z  ∂y2 (5) any cooperation (sharing data) between different parts. i ; : : : ; m = m + 2 (26) Clusters are another, large family of parallel computing Ei Ri systems. Clusters are “homogeneous”, consisting of identi- for = 1 , where is the number of chemical species, cal computing units, called nodes. A node can be either a and are emission and reaction terms, respectively. regular desktop computer [101], or a mere computing board The original system of PDEs is split into five simpler with basic hardware (CPU, RAM, and network interface) sub-systems, which canc ; be : : : solved ; cm one after the other in [102]. Nodes are connected by a dedicated high-speed each time step in the following way. We assume that the network, and data transfer is managed by specialized soft- 1 concentration vector ( ) at the beginning of the ware frameworks, e.g. MPI [103] or PVM [104]. Although time step is known. System (22) is solved by using this clusters have distributed memory, these frameworks enable vector as an initial condition (starting vector). The obtained applications to see the entire cluster as a shared memory solution is an initial vector for the next system (23), and system. so on. The solution of the fifth system is considered as Multicore workstations [105] and the recently developed an approximation to the concentration vector at the end of technology of using graphics processors (GPUs) for gen- the time step, i.e, the numerical solution of the full system. eral purpose computation (GPGPU, [106]) are examples This procedure has a competitive advantage, because the of shared memory systems. GPU computing has been sub-systems are easier to solve numerically than the orig- successfully applied in atmospheric dispersion modeling inal system. In special cases one of the sub-problems can [107, 111] as well as in closely related fields [112–116]. be even solved analytically. However, it should be noted Depending on the type of computation, the GPU’s speedup that the operator splitting may cause inaccuracy in the over a single CPU can be one or two orders of magnitude solution, since the original model is replaced by a model [117, 118], and accordingly, a single high-end video card in which the sub-processes take place one after the other, can compete with a smaller cluster of computers. and this error is called splitting error [96–98]. It is important to note, that theoretically the achievable 6. Parallel computing speedup using parallel computation, regardless of the num- ber of parallel processors used, is limited by the ratio of 6.1. Introduction to parallel computation the non-parallel parts of the computation to the entire run time (Amdahl’s law [119]), whenever the size of the computation is fixed. However, the size of the computa- tional task is usually scaled up as the parallel performance The main limitation of all atmospheric dispersion models increases, therefore the speedup tends to increase with beyond their theoretical capabilities is the computational increasing number of processes (Gustafson’s law [120]). In- effort needed to calculate predictions with them. There is deed, in practice, we tend to increase system size or model always a trade-off between speed and accuracy, however, complexity if increased parallel performance is available, both are requirements for reliable decision support. Natu- because the time allowed for an atmospheric dispersion rally, the solution to increase computational performance prediction is usually fixed, while the demand for accuracy is parallelization. keeps increasing. 269 Dispersion modeling of air pollutants in the atmosphere: a review

6.2. Parallelization of atmospheric dispersion models fore, grids are not suited well for Eulerian models; low la- tency clusters [126–130] and shared memory systems such as multi-core workstations and GPUs [110–112, 114, 116] The basic question in parallelization of an atmospheric are the best options. For clusters, communication is coor- dispersion calculation is how to split the task to smaller dinated by message passing between neighbour nodes; for parts that fits well to a certain parallel computing system. shared memory systems each thread waits for neighbours In certain cases, parallelization is not necessary at all to make a local copy of the boundary of the spatial domain [121]. The simplicity (hence, shortcomings) of Gaussian before updating it in the next iteration. The procedure is models are reflected in their small computational needs: illustrated in Figure 11. Note, that some clusters (e.g., Gaussian models only require simple equation evaluations [102]) feature multidimensional connection topologies be- for a prediction, since these models employ analytical ap- tween neighbour nodes. Following the same topology for proximation formulas. Generally, they do not need parallel splitting the spatial domain of computation greatly en- computation. hances communication efficiency and increases the overall In Lagrangian models, the computational task is the time- simulation performance. integration of SDEs or ODEs describing the motion of 7. Computational Fluid Dynamics particles. Parallelization is based on exploiting the in- models dependence of particle trajectories. Any number of tra- jectories can be computed independently in parallel, and communication is required only to distribute tasks and gather results. Such a computation is called an embar- In urban air pollution problems, source and receptor points rassingly parallel problem, and fits well to any parallel are often located within a few hundred meters from each computing architecture, including grids [122, 123], clusters other, surrounded by a very complex geometry. NWP of any size [124, 125], and GPUs [107–109]. models have far too coarse resolution to represent the Most commonly, implementations follow the master-slave wind field within an urban area, thus existing Eulerian and paradigm. The parallel computation consists of three steps: Lagrangian dispersion models cannot be used. First, a master thread assigns particles to available com- As urban air quality has become more and more important puting units (threads), then each slave thread calculates part of environmental and health protection, computational trajectories in parallel, and finally the master collects par- fluid dynamics (CFD) models have been introduced in tial results from each slave and calculates a statistical environmental modeling. CFD tools are flexible, efficient average to provide the air pollution prediction (see Fig- ∂v~ general purpose solversv~∇ v~ of− the∇ Navier–Stokesp − g~ ν ∇ v; equation~ [2]: ure 10). By repeating these steps multiple times, one can ∂t ρ T 2 reduce statistical errors and increase the precision of the 1 prediction. v~ + ( ) = ρ + p (27)

In Eulerian models, the task is computing numerical so- νT g~ lutions of PDEs. Here the common technique used is where is the wind field, is the density, is the pres- domain decomposition, where the discretized spatial do- sure, is the eddy viscosity and is the gravitational main is divided to subdomains and assigned to different acceleration vector. Eq. (27) is very similar to the one computing units, and time evolution in these subdomains solved in NWP models, however, the coordinate system, are calculated in parallel. In principle, a functional de- grid, boundary conditions and physical parameterizations composition is also possible for highly complex models are completely different [59]. exploiting operator-splitting: each processor can calculate Main parts of a CFD model are the mesh generator, the different operators at the same time. However, in this case solver and the turbulence model. As CFD models are often the entire domain must be shared between processing units used around complex geometry, a fine grid resolution is after each iteration, leading to prohibitive communication required to explicitly calculate turbulence up to a very overhead for distributed memory systems. small scale, which results in extremely high computational The main issue in domain decomposition is calculation of costs. However, subgrid-scale turbulence still hask toε be discretized spatial derivatives on the boundaries of the estimated, which in most cases is assumed to be isotropic. domains, because it requires information from neighbour- Among the various turbulence modelsk forε CFD, the - ap- ing domains, assigned to different threads. Therefore, proach has become the most popular for both engineering data must be shared regularly with neighbours in a well- and atmospheric applications [131]. - turbulence model coordinated fashion, and the time spent with transferring uses the Reynolds-averaged (RANS) approach, solving information must be significantly less than the time spent time-averaged equations of motion and considering turbu- with computation, for high efficiency and scalability. There- lent fluctuations as eddy viscosity and diffusivity terms. In 270 Á Leelossy˝ et al.

Figure 10. Schematic figure of parallelization in Lagrangian models. Each thread (processors or nodes on clusters and grids, multiprocessors on GPUs) computes trajectories for a batch of particles, and results are then combined to give a complete dispersion prediction.

Figure 11. Schematic figure of parallelization in Eulerian models. The simulated space is split between processors (nodes on clusters, multi- processors on GPUs). In each tile, processors alternate between computing time evolution and transferring updated information to neighbours.

k ε order to estimate these, the - closure introduces two new and its results are often used as a verification dataset for transport equations for the turbulent kinetick ε energy and its other models [13, 135]. dissipation rate. However, in the atmosphere, anisotropy CFD-based street canyon models are widely used to pre- of turbulence can cause large errors in - results [132], dict air quality in urban and industrial areas [59, 134–139]. thus modified atmosphere-oriented turbulence closures are However, several weaknesses were pointed out that made applied [59, 133]. A state-of-the-art solution for turbulence CFD results less reliable as the scale increased [132, 133]. modeling is Large Eddy Simulation (LES), which filters Although nesting CFD into NWP models is difficult due to the large scale (anisotropic) and small scale (isotropic) their different coordinates, governing equations and vari- eddies, and performs direct simulation on the former [134]. ables, [140] presented promising results using WRF and a Despite its huge computational cost, LES has become a three-dimensional CFD code. popular tool for planetary boundary layer case studies, 271 Dispersion modeling of air pollutants in the atmosphere: a review

Table 4. Recommended approaches for different scales and applications of atmospheric dispersion modeling.

Application < 1 km 1–10 km 10–100 km 100–1000 km Online risk management (short runtime is important) – Gaussian Puff Eulerian Complex terrain CFD Lagrangian Lagrangian Eulerian Reactive materials CFD Eulerian Eulerian Eulerian Source-receptor sensitivity CFD Lagrangian Lagrangian Lagrangian Long-term average loads – Gaussian Gaussian Eulerian Free atmosphere dispersion (volcanoes) – Lagrangian Lagrangian Lagrangian Convective boundary layer (CFD) Lagrangian Eulerian Eulerian Stable boundary layer CFD Lagrangian Eulerian Eulerian Urban areas, street canyon CFD CFD Eulerian Eulerian 8. Summary

[59], and the extension of a CFD software for atmospheric studies hold promise, and these are current topics of re- We provided here a brief overview of the problems of air pol- search in both the meteorological and environmental engi- lution modeling in the atmosphere. We discussed several neering fields [132, 134, 140]. possible modeling strategies and tools that can be used to Another promising research avenue is in coupling disper- support decision makers. The models can be successfully sion models to meteorological (or climatological) models used for different purposes, however, choosing an appro- [142] and to ensemble forecasting. Ensemble forecasting priate model in a particular case is key to handling the has been operationally used since 1990’s and can incorpo- problem and successfully estimating the dispersion of air rate the stochastic nature of weather processes by using pollutants. Atmospheric dispersion modeling approaches either different physical parametrizations or varying initial and their applicable scales are summarized in Table 4. conditions [143]. Such amodeling approach can reduce the Validation is a necessary step in development in order uncertainty in the dispersion pattern of air pollutants. to test capabilities of models and use models in decision Finally, we should emphasize that the further develop- making. There are several opportunities to validate models. ment of dispersion models will shift towards multi-scale For instance, the model performance can be compared to modeling approaches that can contain sophisticated param- the European Tracer Experiment (ETEX) inventory [141]. eterization of cloud physical processes and biogeochemical ETEX consisted of two releases to atmosphere of tracers interactions. (perfluorocarbons) sampled for three days after the be- Acknowledgments ginning of the emission using a sampling network (more than 100 measuring sites) spread over Europe. During these campaigns about 30 modeling research groups could test and analyze the performance and capability their Authors acknowledge the financial support of the Hun- emergency response models. This also contributed to the garian Research Found (OTKA K77908, K81933, K81975, developments of model parameterizations and showed the K104666, K109109, K109361, NK100296 and PD104441), importance of the application of high resolution meteoro- the Zoltán Magyary Postdoctoral Fellowship and the Eu- logical data. Moreover, validation of the model results can ropean Union and the European Social Fund (TÁMOP be done against remote sensing data (e.g., sulfur dioxide) 4.2.4.A-1). F. M. acknowledges partial support by the NSF or imission data obtained from measuring sites (e.g., ozone, through Grant No. DEB-0918413. nitrogen oxides). Appendix: list of acronyms There are several challenges and possible future develop- ments for air quality modeling. As we discussed earlier NWP and CFD models provide reliable wind data for all scales of atmospheric dispersion simulations, but their ADMS Atmospheric Dispersion Modelling System connection proved to be difficult. Recent developments of AERMOD American Meteorological Society / Environmental Protection Agency Regulatory Model NWPs achieve more and more detailed resolution, CFD ALOHA Areal Locations of Hazardous Atmospheres models with enhanced computational capacity, parallel BLP Buoyant Line and Point Source Dispersion Model computing and LES simulation for anisotropic turbulence CALINE California Line Source Dispersion Model are becoming even better for planetary boundary layer CALPUFF California Puff Model simulations. Both the modification of an NWP model to CBL convective boundary layer CFD computational fluid dynamics perform microscale simulations (the domain size is ~100 m) 272 Á Leelossy˝ et al.

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