Equation (4.1) can be applied to air itself as a conservation equation for the air density ρa. In that case there is no local term since changes in air density are driven solely by transport. Thus
(4.2)
Replacing the mass mixing ratio μi = ρi/ρa into (4.1) and expanding the derivative of the product, we obtain
(4.3)
Replacing (4.2) into (4.3) then yields
(4.4)
This is the Eulerian advective form of the continuity equation, expressing the concentration in terms of mixing ratio rather than density. The local term si/ρa is now in mixing ratio units. The velocity vector v is outside of the gradient operator because compression of air (∇⋅v ≠ 0) does not change the mixing ratio.
Introducing now the total derivative for an air parcel moving with the flow:
(4.5)
we obtain
(4.6)
which is the Lagrangian form of the continuity equation, based on a frame of reference moving with the flow. The total derivative (4.5) is sometimes called the Lagrangian derivative. For an air parcel moving with the flow, transport does not change the mixing ratio and the only change is from the local term si/ρa. The Lagrangian form needs to be expressed as mixing ratio so that it is not affected by compression of air as the air parcel moves.
Atmospheric chemists often express the continuity equation in terms of the number density ni [molecules cm–3] and the volume (or molar) mixing ratio Ci = ni/na (where na is the air number density). These are related to the mass density by where Mi is the molecular mass of species i and is Avogadro’s number, and to the mass mixing ratio by μi = Ci(Mi/Ma) where Ma is the molecular mass of air. The Eulerian flux form of the continuity equation is then
(4.7)
where si is now in units of [molecules cm–3 s–1]. The Eulerian advective form is
(4.8)
and the Lagrangian form is
(4.9)
The transport and local terms involve a number of different processes operating in the model environment. The continuity equation is thus usefully represented for model purposes as a sum of terms describing the different processes for which the model provides independent formulations. For example, the Eulerian form may be decomposed as
(4.10)
where the terms on the right-hand side represent successively the contributions of advection, turbulent mixing, convection, wet scavenging by precipitation, chemistry, emissions, and dry deposition. We describe the formulations for each of these terms in the following subsections. The Lagrangian form using the total derivative may be similarly decomposed but without the transport terms; a separate algorithm is needed to describe the Lagrangian transport of air parcels and this is also described below.
4.2.2 Advection
Advection describes transport by the wind resolved on the model scale. The wind velocity vector v is then a spatial and temporal average over the model grid and time step. The corresponding mass flux is . Consider an elemental volume dV = dx dy dz centered at (x, y, z), and a wind velocity component u in the x-direction. The corresponding mass flux for species i is [kg m–2 s–1]. The flow rate into the volume (kg s–1) is and the flow rate out of the volume is (Figure 4.1). The change per unit time in the concentration ρi within the volume is then given by
(4.11)
Figure 4.1 Flux Fxi of species i in the x-direction through an elemental volume dV.
By adding similar contributions for the y and z directions (with wind components v and w, respectively), we obtain
(4.12)
or
(4.13)
which is the Eulerian flux form used in (4.1). It applies only to the model-resolved winds for which we have actual information on v. Smaller-scale motions are described by turbulence parameterizations presented next. Numerical methods for computing advection are presented in Chapter 7.
4.2.3 Turbulent Mixing
Fluctuating wind patterns that are not resolved on the grid scale of the model are called turbulence. By definition, the turbulent component of the wind has a mean value of zero when averaged over the model grid and time step. It can still cause significant chemical transport in the presence of a chemical gradient. Consider as analogy a commuter train operating at the morning rush hour between the city and the suburbs. The train is full as it travels from the suburbs to the city, and empty as it travels from the city back to the suburbs. The net motion of the train is zero, yet there is a net flow of commuters from the suburbs to the city. In the same way, a back-and-forth wind operating in a uniform chemical gradient will cause a net down-gradient flux from the region of high concentration to the region of low concentration. The flux is proportional to the gradient. This proportionality, called Fick’s law of diffusion, is the foundation of transport by molecular diffusion. Therefore, a simple parameterization of small-scale turbulent transport (where “small-scale” is relative to the model grid) is to treat it as a diffusive process. Diffusive transport results in mixing and we refer to the parameterization as turbulent mixing.
The corresponding equation for the change in species concentration as a result of turbulent mixing is
(4.14)
where K is an empirical tensor describing the 3-D turbulent diffusion. The elements of that tensor are the turbulent diffusion coefficients. K is generally taken to be a diagonal matrix with diagonal elements Kx, Ky, Kz describing turbulent diffusion in the horizontal (x, y) and vertical (z) directions. The term K ρa∇(ρi/ρa) is the turbulent diffusion flux by analogy with Fick’s law, and the right-hand side of (4.14) expresses the flux divergence as derived previously for the advection term. The concentration gradient ∇(ρi/ρa) must be expressed in terms of mixing ratio to account for compressibility of air; a concentration gradient driven solely by air density changes does not drive a turbulent flux.
Turbulent diffusion is not a mechanistic description of turbulence; it is simply a parameterization that has some physical basis and describes relatively well the effect of small-scale turbulence on a chemical concentration field. It allows for consistent treatment of turbulent transport for all chemical species, because the turbulent diffusion coefficients are generally taken to be the same for all species; this is known as the similarity assumption for turbulence. Models often apply a turbulent mixing parameterization in the planetary boundary layer below ~2 km altitude, where the turbulence tends to be small in scale. Turbulent mixing is particularly important in the vertical direction, where mean winds are weak but strong turbulent motions can be generated by surface roughness or by buoyancy.
Numerical methods for computing turbulent diffusion as formulated by (4.14) are presented in Chapter 8. The formulation introduces a second-order derivative in the continuity equations that dampens local concentration gradients. In fact, introducing a diffusion term tends to stabilize the numerical solution of the continuity equation by reducing the magnitude of local gradients. This stabilization effect is discussed in Chapter 7.
Turbulent diffusion should not be confused with molecular diffusion, which is an actual physical process describing the random motion of molecules. Vertical turbulent diffusion coefficients in the planetary boundary layer are typically in the range 104–106 cm2 s–1, whereas the molecular diffusion coefficient is of the order of 10–1 cm2 s–1. Under these conditions, molecular diffusion is completely negligible relative to turbulent mixing. The molecular diffusion coefficient varies as the inverse of atmospheric pressure (lower pressure means longer mean free paths for molecules), so that at sufficiently high altitudes molecular diffusion becomes relevant for atmospheric motions. In the thermosphere above 90 km altitude, molecular diffusion is a major process for vertical transport of chemical species and a corresponding term must be added to the continuity equation (Chapman and Cowling, 1970; Banks and Kockarts, 1973).
4.2.4 Convection
Convection refers to buoyant vertical motion of sufficiently large scale to have grid-resolved vertical structure while still remaining subgrid on the horizo
ntal scale. It is generally associated with cloud formation (wet convection), since water vapor condensation in a rising air parcel provides a local source of heat that accelerates the rise by buoyancy. Thunderstorms are a dramatic example and represent deep wet convection. Fair-weather cumuli are a more placid example and represent shallow wet convection. Dry convection refers to vertical buoyant motion not involving cloud formation, and is in general smaller in vertical extent than wet convection.
A wet convective system involves a cloud updraft, typically ~1 km in horizontal scale, that drives rapid upward transport from the base to the top of the convectively unstable column. This transport typically takes place on timescales of less than one hour, with updraft velocities of the order of 1–10 m s–1. Models with sub-kilometer horizontal resolution can describe convective updrafts as advection; these are called cloud-resolving models or large-eddy simulation (LES) models. They are limited computationally to small domains, typically of the order of 100 km. Larger-domain models cannot resolve convective updrafts and these must therefore be parameterized. As part of the parameterization, the updraft must be balanced on the horizontal grid scale by large-scale subsidence so that there is no net vertical air motion on the model grid. This subsidence is typically modeled by grid-scale sinking of the convective outflow. Additional processes in convective transport parameterizations include entrainment into the updraft, detrainment from the updraft, and subgrid-scale downdrafts. Figure 4.2 shows a general schematic of wet convective transport identifying the individual processes. Different types of convective transport parameterizations are described in Section 8.7.
Figure 4.2 Wet convective transport processes.
4.2.5 Wet Scavenging
Aerosols and water-soluble gases are efficiently scavenged by precipitation. One generally distinguishes in meteorological models between convective precipitation initiated by subgrid convection and large-scale or stratiform precipitation resolved by grid-scale motions. Scavenging takes place below the cloud through uptake by precipitation (below-cloud scavenging or washout) and within the cloud through uptake by cloud droplets and ice crystals followed by precipitation (in-cloud scavenging or rainout). These processes are represented with varying degrees of complexity in models. They are generally first-order loss processes, with rates proportional to the concentration of the species being scavenged. However, in some cases the scavenging may be complicated by nonlinear chemistry; for example, scavenging of SO2 by cloud droplets is contingent on the supply of oxidants (Section 3.8). In coupled aerosol–climate models, the aerosols may affect the precipitation resulting in nonlinear scavenging. Evaporation of precipitation below the cloud releases the water-soluble species to the atmosphere, in which case the effect of scavenging is downward transport in the atmosphere rather than deposition. Scavenging efficiencies are in general very different for liquid and solid precipitation, and the retention efficiency upon cloud freezing (riming) may be very uncertain. Figure 4.3 illustrates the physical processes involved. Details about the formulation of wet scavenging in chemical transport models are presented in Section 8.8.
Figure 4.3 Wet scavenging processes.
4.2.6 Chemistry
Concentrations of chemical species change as a result of chemical production or loss in a manner determined by the rate laws for the elementary reactions. For example, if a molecule XY is photolyzed by solar radiation into fragments X and Y, the loss rate for XY and the corresponding production rate for X and Y is proportional to the concentration of XY, and the proportionality factor is the photolysis frequency JXY [expressed in s–1]. JXY depends on the actinic photon flux (flux integrated over 4π solid angle, i.e., photons coming from all directions), on the absorption cross-section measuring the probability that a photon intercepted by molecule XY will be absorbed, and on the quantum yield measuring the probability that photon absorption will result in photolysis. As another example, if two chemical species A and B react to form product species C and D, the rate of destruction of A and B (equal to the rate of formation of C and D) is proportional to the collision frequency and to the probability that collision will result in reaction. The rate of reaction is kA+B[A][B] where kA+B is the reaction rate constant (also called rate coefficient) and [A] and [B] are in units of density so that their product is proportional to the collision frequency.
In the general case, the chemical tendency equation for species i takes the form
(4.15)
where the indices j and k refer to other species that produce or react with i. Equation (4.15) can be written more concisely as
(4.16)
where pi [kg m–3 s–1] and ℓi [s–1] represent the overall production rate and loss rate constant of species i, summing over all individual processes. If pi and ℓi are independent of the density ρi, equation (4.16) is linear and has a simple exponential solution. However, pi and ℓi may depend on ρi due to coupling with other species in the chemical mechanism. One then needs to solve equation (4.16) as part of a system of coupled equations, one for each species in the mechanism. Numerical methods for this purpose are presented in Chapter 6.
4.2.7 Surface Exchange
Exchange with the surface by emission and deposition represents a vertical flux boundary condition to the continuity equation, with flux F given by
Fi = Ei − Di
(4.17)
where Ei is the emission flux of species i generally provided as external input to the model, and Di is the deposition flux which is generally first-order dependent on the atmospheric concentration:
Di = wD , i(z) ρi(z)
(4.18)
Here, wD,i [m s–1] is the deposition velocity of species i determining the deposition flux, Di is often called the dry deposition flux, and wD,i the dry deposition velocity, to distinguish them from wet deposition which operates by precipitation through the atmospheric column (Section 4.2.5), wD,i is called a “velocity” because of its dimensions. If calculated at the actual surface (z = 0), wD,i depends solely on the chemical properties of i for uptake or reaction at the surface. In practice, however, the model does not resolve the concentration at the actual surface. The deposition flux must be calculated from knowledge of the concentration ρi(z1) at the lowest model level z1, and in that case the calculation of wD,i(z1) must account for turbulent transfer between z1 and the surface. Model representation of emission and deposition processes is described in Chapter 9.
The surface flux boundary condition to the continuity equation is practically implemented in chemical transport models as a tendency term in the lowest model level. In an Eulerian model with a lowest model layer of thickness Δz, this is expressed as
(4.19)
and
(4.20)
In a Lagrangian model this is expressed as source and sink terms for particles brought in sufficiently close contact with the surface. Emissions injected aloft such as from smokestacks, buoyant fires, aircraft, lightning, or volcanoes are handled by applying the tendency terms to the corresponding model levels. In the case of gravitational settling of very large particles, the deposition velocity represents an actual downward settling velocity that needs to be applied to all model levels. Again, this is readily done as a tendency term following the above formulation.
4.2.8 Green Function for Lagrangian Transport
Lagrangian models track the transport of individual air parcels within which local source and sink terms operate to describe chemistry, emissions, and deposition. Wind information to describe the transport of the individual parcels is provided as input, typically from a gridded meteorological data set. Subgrid turbulence must be described by an additional stochastic (probabilistic) motion applied to the air parcels.
Consider a Lagrangian model from which we wish to obtain a 3-D field of mixing ratios μi(r, t) for a specified set of points r = (x, y, z)T at time t. This can be derived as the superimposition of mixing ratios produced by an ensemble of pulses applied at points r′and times t′ < t, with local source/sink terms applied over the [t�
��, t] trajectory. We define the Green function G(r, t; r′, t′)as the normalized time-evolving spatial distribution of the species mixing ratio at points r and time t resulting from the injection of a pulse at location r′ and time t′. From an Eulerian perspective, the Green function is the solution of the continuity equation (4.4) in which the source rate si/ρa is replaced by a Dirac function in space and time:
(4.21)
and v includes turbulent components that can be described using the turbulent diffusion and convective parameterizations of Sections 4.2.3 and 4.2.4. The Green function as defined here represents the transition probability density that a parcel initially located at point r′ at time t′will move to point r at time t. It is also sometimes called an influence function. It has the unit of inverse volume.
For an inert chemical (no local sources or sinks), the Green function provides a solution to the evolution of the mixing ratio over time [0, t], starting from initial conditions defined over the model domain V:
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