Modeling of Atmospheric Chemistry
Page 13
The aerosol size distribution can be characterized by the number size distribution function nN(r) [particles cm–3 μm–1]
(3.59)
such that nN(r)dr represents the number of particles per cm3 of air in the radius size range[r, r + dr], and N is the total particle number concentration. Other related measures of the aerosol size distribution are the surface area size distribution function
ns(r) = 4πr2nN(r)
(3.60)
and the volume size distribution function
(3.61)
Plots of the aerosol size distribution generally use a log scale to account for the variation of particle sizes over typically five orders of magnitude, from 10–3 to 102 μm:
(3.62)
such that nN(logr) d (logr) represents the number concentration of particles in the size range [logr, logr + d (logr))]. Similar expressions apply for nS(logr) and nV(logr).
The integrals of these distributions yield the total aerosol number concentration N, surface concentration S, and volume concentration V:
(3.63)
(3.64)
(3.65)
Figure 3.11 shows the number, surface, and volume size distributions for a generic aerosol. There are three distinct modes, called the Aitken mode (diameter < 0.1 μm), the accumulation mode (0.1–1 μm) and the coarse mode (>1 μm). The Aitken mode is made up of freshly nucleated particles, which grow rapidly by gas condensation and coagulation to the accumulation mode. Further growth of accumulation mode particles above 1 μm is slow. The coarse mode is mostly composed of primary particles emitted mechanically from the Earth’s surface, such as soil dust, sea salt, and pollen.
Figure 3.11 Idealized size distributions for an aerosol population by number, surface, and volume.
From Seinfeld and Pandis (2006).
We see from Figure 3.11 that the number, surface, and volume size distributions for a given aerosol are very different, reflecting the r2 and r3 weighting of the surface and volume size distributions over five orders of magnitude in r. Thus the Aitken particles dominate the number size distribution but are unimportant for volume. The coarse particles contribute an insignificant number but make a major contribution to volume. Accumulation particles are important for all measures of the size distribution and especially for the surface size distribution, which is most relevant for aerosol optical properties.
3.9.2 Chemical Composition
The aerosol chemical composition is commonly classified following dominant aerosol types as (1) sulfate, (2) nitrate, (3) organic carbon (OC), (4) black carbon (BC), (5) soil dust, and (6) sea salt. There are other minor constituents, such as trace metals and pollen. Sulfate and nitrate are often associated with ammonium and one refers to sulfate–nitrate–ammonium (SNA) aerosol as the coupled thermodynamic system. Organic carbon and black carbon are sometimes grouped as carbonaceous aerosol. They are directly emitted by incomplete combustion, and OC also has an important secondary source from condensation of semivolatile organic compounds (SVOCs; Section 3.5). Figure 3.12 shows the mass concentrations of different aerosol types at surface sites in the United States. SNA and OC dominate. BC makes little contribution to mass but is of great interest because of its light-absorbing properties and effects on health. Figure 3.13 shows illustrative results from a global model simulation of different aerosol types, highlighting the dominance of different types in different regions.
Figure 3.12 Annual mean chemical composition of PM2.5 at selected sites in the USA in 2013. Values are from the Chemical Speciation Network of the US Environmental Protection Agency.
Figure produced by Eloïse Marais, Harvard University, used with permission.
Figure 3.13 Portrait of global aerosols produced by a simulation of the NASA Goddard Earth Observing System Model (GEOS) version-5 at a spatial resolution of 10 km. The figure highlights the relative abundance in different regions of the world of dust (red) lifted from arid soils, sea salt (blue) embedded in fronts and cyclones, smoke (green) from tropical wildfires, and sulfate particles (white) from fossil fuel and volcanic emissions. It also shows the influence of transport of aerosols by the atmospheric circulation.
From W. Putman, National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC): www.nasa.gov.
We previously saw how sulfuric and nitric acids are produced in the atmosphere by oxidation of SO2 and NOx, respectively. Ammonia has natural biogenic sources and also a large anthropogenic source from agriculture. Sulfuric acid has very low vapor pressure over H2SO4–H2O solutions and condenses immediately under all atmospheric conditions, including in the stratosphere, to produce concentrated sulfuric acid particles. Ammonia partitions into this aerosol following acid–base titration to produce ammonium bisulfate (NH4HSO4) and ammonium sulfate ((NH4)2SO4) aerosol depending on the ammonia to sulfuric acid ratio:
H2SO4(a) + NH3(g) → NH4HSO4(a)
(3.66)
NH4HSO4(a) + NH3(g) → (NH4)2SO4(a)
(3.67)
Letovicite ((NH4)3H(SO4)2) can also be produced. Here (g) denotes the gas phase and (a) the aerosol phase, which may be solid or aqueous depending on relative humidity. In aqueous aerosol the ammonium-sulfate salts dissociate to NH4+, SO42–, and H+. The thermodynamic driving force for reactions (3.66) and (3.67) is so large that complete titration of sulfuric acid is achieved if ammonia is in excess. In that case, the excess ammonia may go on to combine with nitric acid and form nitrate aerosol:
NH3(g)+HNO3(g) ⇌ NH4NO3(a)
(3.68)
with an equilibrium constant that increases with decreasing temperature and with increasing relative humidity. Again, the ammonium nitrate aerosol may be solid or aqueous (NH4+, NO3–), depending on relative humidity.
Organic carbon (OC) aerosol includes a very large number of species. Its composition is highly variable and not well understood. The traditional approach in models is to distinguish between primary organic aerosol (POA) directly emitted by combustion and secondary organic aerosol (SOA) formed in the atmosphere. SOA is formed from the atmospheric oxidation of VOCs, resulting in products with chemical functionalities (such as hydroxy, peroxy, carbonyl, carboxylic, and nitrate groups) that enable their uptake by pre-existing organic or aqueous aerosol. This uptake may be reversible or irreversible, in the latter case through subsequent oxidation or oligomerization in the aerosol.
3.9.3 Mixing State, Hygroscopicity, and Activation
Individual particles have compositions that reflect their origin and atmospheric history. Particles may originate as a specific aerosol type, such as sulfate or BC, but subsequent mixing with other aerosol types takes place as they age in the atmosphere. The degree of mixing is important for characterizing aerosol properties. It is convenient in models to consider two limiting cases: external and internal mixing. An external mixture is one in which different aerosol types do not mix, so that individual particles are of a single type; this is usually appropriate close to the source. An internal mixture is one in which all particles have the same composition; this is usually appropriate in a remote air mass. Aerosols in the atmosphere gradually evolve from an external to an internal mixture through particle coagulation, gas uptake, and cloud processing. Internal mixing usually assumes that the different chemical constituents are well-mixed within individual particles, but that does not hold if the particles are not fully liquid. For example, a common configuration for BC-sulfate internal mixing is for BC to form a solid core embedded within the aqueous sulfate solution. This core–shell model has important implications for calculating the optical properties of BC.
The hygroscopicity of an aerosol refers to its thermodynamic capacity for taking up water at a given relative humidity (RH). By Raoult’s law, an aerosol particle behaving as an ideal aqueous solution has a water molar content xH2O = RH/100, where RH is expressed as a percentage. Such dissolution of the aerosol requires from precipitation equilibrium that xH2O be sufficiently high (i.e., that RH be sufficiently high). Sulfuric acid is aqueous at all RH because it
s condensation takes place as a H2SO4–H2O binary mixture. Other aerosols are dry at thermodynamic equilibrium in a low-RH atmosphere and become aqueous when the RH exceeds the level required by precipitation equilibrium. This RH level is called the deliquescence relative humidity (DRH); it represents a sharp particle transition from non-aqueous (usually solid) aerosol to aqueous. It is 40% for NH4HSO4, 62% for NH4NO3, 75% for NaCl, and 80% for (NH4)2SO4. Additional water condenses as the RH continues to increase. Starting from high RH, a decrease in RH will similarly result in a sharp conversion of the aerosol from aqueous to non-aqueous. However, if the non-aqueous state involves crystallization (as in the case of SNA and sea salt particles) it may be retarded by the energy barrier for crystal formation. In that case, the crystallization relative humidity (CRH) is lower than the DRH, and for CRH < RH < DRH the aqueous phase is metastable.
Another important property of aerosol particles is their ability to serve as cloud condensation nuclei (CCN) for activation of cloud droplets under supersaturated conditions (RH > 100%). This involves overcoming the surface tension for growth of the gas–droplet interface. Particles larger than 0.1 μm and at least partly wettable are effective CCN at typical atmospheric supersaturations (100% < RH < 101%). Models often distinguish between hydrophobic particles as non-wettable and hydrophilic particles as wettable. For example, freshly emitted BC is typically hydrophobic and thus an ineffective CCN, but becomes hydrophilic in the atmosphere on a timescale of a day as it ages and mixes with other aerosol types.
3.9.4 Optical Properties
Aerosol particles interfere with the propagation of radiation by scattering and absorption. The resulting attenuation of radiation along a path through the atmosphere is described by Beer’s law:
I = Io exp [−τs]
(3.69)
where Io is the incident radiation, I is the transmitted radiation through the path, and τ is the optical path. One can express this optical path as τ = βext L where L [m] is the physical path length and βext [m–1] is an extinction coefficient characteristic of the aerosol. The extinction coefficient is the sum of a scattering coefficient βscat [m–1] and an absorption coefficient βabs [m–1]. For a monodisperse population of spherical particles of radius r with number density N [m–3], the extinction coefficient is a function of the dimensionless extinction efficiency Qext (defined as the probability that a photon incident on the particle will be absorbed or scattered), as given by
βext = QextNπr2
(3.70)
The extinction efficiency of a particle can be viewed as the sum of a scattering efficiency Qscat and an absorption efficiency Qabs. The single-scattering albedo ω is defined as the ratio
(3.71)
and measures the relative contributions of scattering and absorption to extinction. Absorption depends on the chemical properties of the molecule, while scattering depends on the particle size parameter
(3.72)
Scattering is most efficient for particles of radius equal to the radiation wavelength (α = 2π). Since most of the aerosol particle area is typically in the 0.1–1 μm size range, we see that aerosols are efficient scatterers of solar radiation. See Figure 5.8 in Section 5.2.4 for dependences of Qscat and Qabs on particle size and refractive index.
The aerosol optical depth in the atmospheric column is defined as the optical path for radiation propagating vertically from the top of the atmosphere to the surface. It is given by
(3.73)
Here again, the total optical depth can be expressed as the sum of two terms that account for absorption and scattering: τ = τscat + τabs. Figure 3.14 shows the global distribution of the total aerosol optical depth in the visible light as measured from space in different seasons. Elevated values are due to desert dust, biomass burning, anthropogenic pollution, and sea salt.
Figure 3.14 Average column-integrated aerosol optical depth at λ = 558 nm measured by the Multi-angle Imaging Spectro-Radiometer (MISR) satellite instrument from December 2001 to November 2002.
Source: National Aeronautics and Space Administration/GSFC/LaRC/JPL, MISR Team.
The detailed calculation of the radiative effects of aerosols is complicated. One distinguishes between three regimes: (1) the Rayleigh scattering regime in which the particles are much smaller than the wavelength of the incident radiation (α ≪ 1), (2) the Mie scattering regime in which the size of the particles is of the same order of magnitude as the wavelength (α ≈ 1), and (3) the geometric scattering regime, in which the particles are much larger than the wavelength (α ≫ 1). For small particles (Rayleigh regime), one can show that the scattering efficiency varies as λ–4, while the absorption efficiency varies as λ–1. Thus, light at shorter (bluer) wavelengths is scattered more effectively than light at longer (redder) wavelengths. White light passing through a layer of small aerosol particles becomes redder as it propagates toward an observer.
References
Brasseur G. P., Orlando J. J., and Tyndall G. S. (eds.) (1999) Atmospheric Chemistry and Global Change, Oxford University Press, Oxford.
Brasseur G. P., Prinn R. G. and Pszenny, A. A. P. (eds.) (2003) Atmospheric Chemistry in a Changing World, Springer, New York.
Finlayson-Pitts B. and Pitts J. N. (2000) Chemistry of the Upper and Lower Atmosphere: Theory, Experiments and Applications, Academic Press, New York.
Lee C., Martin R. V., van Donkelaar A., et al. (2009) Retrieval of vertical columns of sulphur dioxide from SCIAMACHY and OMI: Air mass factor algorithm development, validation, and error analysis, J. Geophys. Res. 114, D22303, doi: 10.1029/2009JD012123
National Research Council (1991) Rethinking the Ozone Problem in Urban and Regional Air Pollution, National Academy Press, Washington, DC.
Seinfeld J. H. and Pandis S. N. (2006) Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Wiley, New York.
Warneck P. (1999) Chemistry of the Natural Atmosphere, Academic Press, New York.
Waters J. W., Froidevaux L., Read W. G., et al. (1993) Stratospheric ClO and ozone from the Microwave Limb Sounder on the Upper Atmosphere Research Satellite. Nature, 362, 597–602.
Wu S., Mickley, L. J., Jacob D. J., et al. (2007) Why are there large differences between models in global budgets of tropospheric ozone?, J. Geophys. Res., 112, D05302, doi:10.1029/2006JD007801
4
Model Equations and Numerical Approaches
4.1 Introduction
Atmospheric chemistry focuses on understanding the factors that control the concentrations of chemical species in the atmosphere. These factors include processes of emissions, transport, chemical production and loss, and deposition. Here we present the general mathematical foundations for atmospheric chemistry and the corresponding model frameworks.
We begin in Section 4.2 by introducing the continuity equation, which is the fundamental mass conservation equation for atmospheric chemistry. The continuity equation expresses how the concentration of a chemical species changes with time in response to a sum of individual forcing terms describing emissions, transport, chemistry, and deposition. The continuity equation for aerosols also includes terms to describe microphysical growth of particles; this is presented in Section 4.3. The continuity equation is a differential equation in space and time, and its integration solves for the evolution of chemical concentrations as controlled by the ensemble of driving processes. Analysis of timescales over which the individual processes operate can be very useful to identify dominant terms; this is presented in Section 4.4. Computing transport terms requires solving the conservation equations for atmospheric dynamics that serve as foundations for meteorological models. These equations are presented in Section 4.5.
The continuity equations for atmospheric chemistry cannot be solved exactly (except for highly idealized problems) because of the complexity of the flow and because of chemical coupling between species. Numerical methods are needed that provide the foundations for atmospheric chemistry models. We present in this chapter the general frameworks for these metho
ds as implemented in models including different coordinate systems, dimensionality, and grid geometries (Sections 4.6–4.7), as well as different approaches including Eulerian, Lagrangian, and statistical (Sections 4.8–4.13). Standard modeling strategies of operator splitting, numerical filtering, and remapping are presented in Sections 4.14–4.16. This chapter sets the stage for the following chapters where specific numerical algorithms will be presented to compute chemistry (Chapters 5 and 6), transport (Chapters 7 and 8), and emission and deposition (Chapter 9).
4.2 Continuity Equation for Chemical Species
4.2.1 Eulerian and Lagrangian Formulations
The continuity equation expresses mass conservation within an elemental volume of fluid. For a chemical species i with concentration measured by its mass density ρi [kg of species i per m3 of air], the continuity equation is expressed in an Eulerian framework (Chapter 1) as
(4.1)
where v = (u, v, w)T is the wind velocity vector,∇ ⋅ (ρiv) is the flux divergence (flux out of the volume minus flux in), which represents the transport term, and si is the net local source of the species, which represents the local term. Equation (4.1) is the Eulerian flux form of the continuity equation. The transport term includes contributions from advection, which describes the flow by large-scale winds resolved on the scale of the model, and turbulence, which is not resolved on the model scale and must be represented stochastically. The turbulent component of the transport term is separated further into turbulent mixing, which is effectively random on the model scale, and convection, which has organized vertical structure on the model scale. The local term si includes contributions from chemistry, emissions, and wet and dry deposition. Surface exchange by emissions and dry deposition represents a flux boundary condition for the continuity equation but in a gridded model environment it is treated as a local term for the lowest model level.