Figure 9.12 Annual mean dust emission from natural and anthropogenic sources.
From Ginoux et al. (2012).
9.3 One-Way Dry Deposition
Dry deposition or surface uptake is the process by which gases and particles are transferred from the atmosphere to the Earth’s surface. It is a major sink for many atmospheric species. Except for very large particles, it does not take place by gravity, which is negligibly slow. It takes place instead by turbulent transfer to the surface followed by surface uptake. One-way deposition as described here assumes that the deposition is irreversible so that the surface is a terminal sink. Generalization to two-way exchange is presented in Section 9.4.
9.3.1 Dry Deposition Velocity
The dry deposition sink for a species i is computed as the dry deposition flux FD,i [molecules cm–2 s–1] applied to the lowest altitude z1 resolved by the model (lowest model grid point). Proper physical description requires that the dry deposition flux computed at z1 represent the flux at the actual surface. This holds if z1 is within the surface layer (Section 8.7.3), typically 50–100 m deep, where vertical fluxes can be assumed uniform. FD,i depends on the number density ni(z1) at altitude z1, the efficiency of vertical transfer from altitude z1 to the surface, and the efficiency of loss at the surface. If the loss rate at the surface has a first-order dependence on the surface number density ni(0), as is usually the case, then the deposition flux has a first-order dependence on ni(z1):
FD, i = − wD, i(z1)ni(z1)
(9.12)
Here, wD,i(z1) is the dry deposition velocity [cm s–1] of species i at altitude z1. It is called a “velocity” because of its units, but it describes in fact a turbulent process and not a simple one-way flow. One-way gravitational settling is important only for very large aerosol particles and is covered in Section 9.3.8. The flux is defined as positive when upward, thus the dry deposition flux in (9.12) is negative.
Conservation of the vertical flux in the air column below z1 is an important assumption in the computation of dry deposition using (9.12). Aside from z1 being in the surface layer, it requires that the atmospheric lifetime of the depositing species against chemical loss be long relative to the timescale for turbulent transfer from z1 to the surface. The latter timescale is of the order of minutes for z1 in the range 10–100 m. Shorter-lived species require finer vertical resolution near the surface to compute dry deposition, although one might be able to assume in those cases that dry deposition is negligible relative to chemical loss.
9.3.2 Momentum Deposition to a Flat Rough Surface
Insight into the deposition of chemical species can be gained from similarity to deposition of momentum, Consider the simple case of momentum deposition to a flat rough surface (Figure 9.13). Momentum is transported to the surface by turbulence. Turbulent eddies in the surface layer are sufficiently small that an eddy diffusion parameterization is adequate (Section 8.7.3). Let ρau be the mean scalar horizontal momentum where ρa is the air density and u is the mean horizontal wind speed. The momentum deposition flux Fm is related to the vertical gradient of the horizontal momentum by:
(9.13)
where we neglect the small variation of ρa with altitude. The eddy diffusion coefficient Kz has units [cm2 s–1] and needs to be empirically specified. Dimensionality considerations are helpful here. Kz can be viewed as the product of a length scale [cm] and a velocity scale [cm s–1]. We expect Kz to increase with distance from the surface as eddies become less restricted by the surface boundary. Thus z is an appropriate length scale. We also expect Kz to increase as the momentum deposition flux increases, and this can be expressed in terms of the friction velocity u* = (|Fm|/ρa)1/2 introduced in Section 8.7.3. Therefore:
Kz = ku*z
(9.14)
where k = 0.35 is the von Karman constant. Replacing (9.14) and the definition of the friction velocity into (9.13), we get:
(9.15)
and by integration,
(9.16)
where c is an integration constant. We see from the form of (9.16) that the mean wind speed must die out (u = 0) at some distance above the surface called the roughness height for momentum z0,m. Applying this boundary condition to (9.16) we obtain the log law for the wind (equation (8.106)):
(9.17)
Field observations show that this relationship is generally well obeyed. Plots of ln z vs. u from experimental data can be fitted to a straight line, and the values of u* and z0,m can be derived from the slope and intercept. The thin layer [0, z0,m] close to the surface is viewed as a quasi-laminar boundary layer in which molecular diffusion plays an important role.
Figure 9.13 Log law for the horizontal wind speed over a flat rough surface.
9.3.3 Big-Leaf Model for Dry Deposition
The formulation of momentum deposition to a flat rough surface (Section 9.3.2) provides a simple basis for parameterizing deposition of chemical species to a complex canopy. This parameterization is called the big-leaf model or resistance-in-series model (Hicks et al., 1987). Figure 9.14 gives a schematic. From the atmospheric perspective, the canopy is modeled as a flat rough surface based at a displacement height d above the ground (Section 8.7.3). Depositing species are delivered to the surface by turbulent and quasi-laminar transfer (Section 9.3.2) and penetrate into the surface medium, where they are eventually removed. Think of the “big leaf” as a porous medium above which the airflow follows atmospheric dynamics for deposition to a flat rough surface (Section 9.3.2), and below which some combination of surface processes leads to the actual uptake. Vertical transport to the big-leaf surface takes place by atmospheric turbulence down to altitude d + z0,m, and final transport to the surface through the quasi-laminar boundary layer is facilitated by molecular diffusion. The log law for the wind (9.17) needs to be adjusted for the displacement height (Section 8.7.3):
(9.18)
where the altitude z is relative to the actual Earth surface below the canopy. Typically d is about 2/3 of the canopy height, z0,m is about 1/30 of the canopy height, and u* is about 1/10 of the wind speed. Assuming similarity between turbulent transport of chemicals and momentum, a similar log law applies to the vertical concentration profile of species i, but with a non-zero concentration as boundary condition at altitude d + z0,c:
(9.19)
Figure 9.14 Schematic of the big-leaf model for one-way dry deposition. Vertical axis is not to scale.
Here, z0,c is the roughness height for depositing species (assumed to be the same for all species) and d + z0,c is the effective height of the big-leaf surface. The quasi-laminar boundary layer is thus defined as the layer [d + z0,m, d + z0,c], from the point where the wind dies out down to the effective surface. In one-way deposition the surface is a terminal sink for the depositing species, and this is enforced by a boundary condition ni,o = 0 within the big-leaf medium (Figure 9.14).
Downward vertical transfer in the [z1, d + z0,m] column takes place by turbulence; thus we write for that column:
(9.20)
where Ci is the mixing ratio of species i. The turbulent flux is proportional to the mixing ratio gradient in the eddy diffusion formulation, but we can neglect the vertical dependence of the air density na within the surface layer and write the flux as proportional to the number density gradient. Integration of equation (9.20) yields
(9.21)
where RA [s cm–1] is the aerodynamic resistance to deposition:
(9.22)
The term “resistance” reflects the analogy with electrical circuits, taking ni(z1) – ni(d + z0,m) as the analog of a difference in potential and FD,i as the analog of a current intensity.
Following on the analogy with electrical circuits, we can define a quasi-laminar boundary layer resistance RB,i [s cm–1] (commonly called boundary resistance) to describe vertical transport through the quasi-laminar boundary layer:
(9.23)
and a surface resistance RC,i [s cm–1] to describe the uptake at the surface:
(9.24)
where n
i(d + z0,c) is the concentration in contact with the surface. We combine (9.21), (9.23), and (9.24) to eliminate ni(d + z0,m) and ni(d + z0,c), and obtain:
(9.25)
where Ri = RA + RB,i + RC,i [s cm–1] is the total resistance to dry deposition and is the inverse of the dry deposition velocity (9.12). Thus:
(9.26)
We see by analogy to Ohm’s law that Ri is the sum of three resistances in series describing resistance to turbulent transport through the surface layer (RA), resistance to diffusion through the quasi-laminar boundary layer (RB,i), and resistance to surface uptake (RC,i), as illustrated in Figure 9.14. By calculating the individual resistances we can derive the dry deposition velocity, and by comparing the magnitudes of the individual resistances we can determine the process limiting dry deposition. In the following subsections we describe the calculation of the individual resistances.
9.3.4 Aerodynamic Resistance
Equation (9.22) expresses RA as a function of the eddy diffusion coefficient Kz. For flow over a flat rough surface, we have Kz = ku*z (Section 9.3.2), and correcting for the displacement height yields Kz = ku*(z – d). Replacing into (9.22):
(9.27)
This expression applies for neutral buoyancy conditions when the log law for the wind holds. The atmosphere can be assumed neutral when mechanical turbulence dominates over buoyant turbulence, that is when z1 ≪ |L| where L is the Monin–Obukhov length (Section 8.7.3). When this condition is not satisfied, a stability correction factor Ψm must be introduced in the formulation of the vertical wind profile as given in Section 8.7.3 (see also expression (9.18)):
(9.28)
and the expression for the aerodynamic resistance becomes:
(9.29)
Correction formulas are generally applicable up to z ≈ |L|. At higher altitudes, the parameterization of turbulence becomes more complicated as buoyant plumes dominate and the surface layer assumption of uniformity of vertical fluxes may not be valid. Values of |L| generally exceed 100 m so that a lowest model level z1 < 100 m is adequate. Very unstable conditions can have smaller values of |L|, but the aerodynamic resistance in the [|L|, z1] column is then negligibly small and RA in (9.29) can be calculated by replacing z1 with |L|. Very stable conditions at night can lead to ground-based inversions and very small positive values of L. In that case the aerodynamic resistance computed at z1 > L is very large, deposition is restricted to the shallow layer [0, L], and the concentration at z1 is decoupled from that in surface air. It may be best from the model perspective to ignore deposition under such conditions as it operates only on a small atmospheric mass. One should not expect then for the model to be able to reproduce surface observations.
9.3.5 Quasi-Laminar Boundary Layer Resistance
The quasi-laminar boundary layer resistance (boundary resistance) RB,i in the big-leaf model measures the resistance to transfer from the zero-momentum point at altitude d + z0,m to the big-leaf surface at altitude d + z0,c. Even though turbulence technically dies out at d + z0,m in the eddy diffusion parameterization for momentum, there is in reality still some turbulence to carry species down to the surface. A first estimate of RB,i can thus be made from (9.27):
(9.30)
Molecular diffusion also plays a significant role in the thin quasi-laminar boundary layer, and the corresponding rate depends on the molecular diffusion coefficient Di. This can be accounted for by the semi-empirical correction of Hicks et al. (1987):
(9.31)
where the Schmidt number Sci = ν/Di is the ratio between the kinematic viscosity of air (ν = 0.15 cm2 s–1 at standard temperature and pressure) and the molecular diffusion coefficient Di. The Prandtl number Pr is the ratio of the kinematic viscosity to the thermal diffusivity of air (Pr = 0.72 at standard temperature and pressure). The term ln(z0,m/z0,c) is roughly 2 for vegetated canopies and 1 for bare surfaces and water. The boundary resistance computed in this manner is a very crude approximation, but is of little importance for computing the deposition velocity for gases since comparison of (9.31) to (9.27) indicates that RA ≫ RB,i.
In the case of aerosol particles, the molecular diffusion coefficient must be replaced by the Brownian diffusion coefficient describing the random motion of particles. The Brownian diffusion coefficient is inversely dependent on particle size. For particles larger than ~0.1 μm, Brownian diffusion is very small, but transfer to the surface is then facilitated by interception (when particles carried by the airflow hit the surface) and inertial impaction (when particles deviate from the airflow as it curves around surface elements). There is detailed theory for these aerosol processes (Slinn, 1982; Seinfeld and Pandis, 2006) though practical application is limited by complexity of the canopy. Interception and impaction are generally the limiting factors for dry deposition of >0.1 μm aerosol particles, but the corresponding resistance is generally referred to as surface resistance in the literature. Thus for aerosol particles of diameter Dp the deposition velocity is typically computed as wD(Dp) = 1/(RA + RC(Dp)) where the surface resistance RC(Dp) accounts for Brownian diffusion, interception, and impaction. Seinfeld and Pandis (2006) give a detailed discussion of these processes. The parameterization for RC(Dp) by Zhang et al. (2001) is frequently used in models.
9.3.6 Surface Resistance
The surface resistance RC,i in the big-leaf model describes the physical and chemical uptake taking place on the ensemble of canopy surfaces. For aerosol particles, collision with surfaces takes place by Brownian diffusion, interception, and impaction (Section 9.3.5). For gases, the uptake involves surface adsorption or absorption followed by chemical reaction. Deposition of gases can take place to the stomata (open pores) of leaves, within which gases diffuse to eventually react in the leaf mesophyll. It can also take place to the waxy surfaces of leaves, called cuticles, and to the ground and other surfaces.
The overall surface resistance is commonly decomposed into processes representing uptake by different canopy elements and parameterized as an ensemble of resistances in parallel and in series. Figure 9.15 from Wesely and Hicks (2000) shows a standard scheme. In that scheme, uptake by the canopy takes place in parallel to the canopy leaves, the lower canopy, and the ground. Uptake by canopy leaves takes place in parallel to the stomata and to the cuticles, and uptake by the stomata is described by two resistances in series representing diffusion through the stomata and reaction at the mesophyll. Uptake to the lower canopy and to the ground involves aerodynamic resistance to transfer through the canopy. The overall surface resistance RC,i is computed from this network of resistances by adding resistances in series, and adding conductances (inverses of resistances) in parallel, as one would do for electrical resistances. Wesely and Hicks (2000) and other literature provide estimates of each resistance in Figure 9.15 for different canopy types, gas chemical properties (e.g., effective Henry’s law constant, oxidant potential), and meteorological variables.
Figure 9.15 Surface resistance model from Wesely and Hicks (2000) separating contributions from vegetation, lower canopy, and the ground. The leaf mesophyll, lower canopy vegetation, and ground all have their own internal concentrations (ni,M, ni,LC, ni,GR) as boundary conditions instead of the single concentration ni,o given in Figure 9.14. For one-way deposition these boundary conditions are all set to zero.
Adapted from Wesely and Hicks (2000).
Table 9.3 gives surface resistances for SO2 and ozone computed with the Wesely and Hicks (2000) model for a deciduous forest in summer (full canopy) and winter (no leaf canopy), during day and night. SO2 and ozone are commonly used as reference species for dry deposition because relatively large observational databases are available for both. Uptake of SO2 is driven by its effective water solubility while uptake of ozone is driven by its reactivity as an oxidant. Uptake is particularly efficient at the leaf stomata, where water-soluble gases dissolve in the leaf water and oxidants react with unsaturated organic compounds. The overall surface resistances for SO2 and ozone are much smaller in summer than in winter and much smaller in daytim
e than at night, reflecting the importance of the stomata, which are open only during daytime.
Table 9.3 Surface resistances RC,i (s cm–1) for a deciduous forest canopy
Species Day Night
SO2 summer 1.3 10
winter 9.8 10
O3 summer 1.1 9.5
winter 6.1 30
An illustrative back-of-the-envelope estimate of the surface resistance can be made for deposition of a highly water-soluble or reactive species to a leafy canopy in the daytime. In that case, leaves account for most of the total depositing surface in the canopy and reaction at the mesophyll is fast (RM,i ≈ 0 in Figure 9.15). Thus RC,i ≈ RS,i, where RS,i is the stomatal resistance for the whole leaf canopy. Measurements of the stomatal resistance Rs,w for water vapor exchange per unit area of leaf indicate a typical value of 2 s cm–1. The corresponding stomatal resistance Rs,i for species i per unit area of leaf scales to that of water by the inverse ratio of molecular diffusion coefficients, and the molecular diffusion coefficients are in turn inversely proportional to the square root of the molecular weights. Thus we have:
(9.32)
where Λ is the LAI introduced in Section 9.2.1, and Mi and Mw are the molecular weights of species i and water vapor, respectively. Taking ozone as an example and a typical mid-latitudes forest LAI of 3, we obtain RC,O3 = 1.1 s cm–1, which is the value in Table 9.3.
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