Reproduced from W. Aeschbach-Hertig. Institute für Umweltphysik, University of Heidelberg, Germany.
The flux of quantity Ψ associated with the atmospheric flow of velocity v is expressed as
(8.9)
or
(8.10)
Its average, for example, over a model grid cell, is given by
(8.11)
since by definition and . The first term on the right-hand side of (8.11) is the flux associated with the resolved circulation (product of the mean values). The second term is the eddy flux arising from the covariance between the fluctuations of v and Ψ.
When the Eulerian form of the continuity equation for the density ρi of chemical species i is considered (Section 4.2),
(8.12)
the separation between spatially resolved and unresolved transport is expressed by:
(8.13)
By taking the average of each term and noting that , and , the continuity equation for the mean concentration becomes
(8.14)
The transport terms that appear in the continuity equation account for contributions by the mean (resolved) circulation and by the eddy (unresolved) motions. The eddy transport term must be parameterized in some way (Section 8.4). In the vertical direction, the eddy flux is generally much larger than the resolved flux because mechanical and buoyant turbulence dominate the motion. In the horizontal direction, the eddy flux is generally less important. In 2-D altitude–latitude models with no zonal resolution, often used for the stratosphere, the eddy terms account for the meridional transport associated with large-scale planetary wave disturbances, and the contributions of the mean and eddy transport terms tend to be equally important.
When expressed in terms of mass mixing ratio μi = ρi/ρa, for species i, the continuity equation (8.14) becomes (see (4.8)):
(8.15)
Here again ρa represents the air density [kg m–3]. If we apply the Reynolds decomposition for variables ρa, μi, v, and si, equation (8.15) becomes
(8.16)
By taking the average of each term, we obtain:
(8.17)
Ignoring the fluctuations of the air density (ρ′ = 0), (8.17) becomes:
(8.18)
Making use of the continuity equation for the mean air density, one deduces that
(8.19)
If the air density can be assumed to be constant over the entire spatial domain under consideration, as often assumed in boundary layer problems, (8.19) becomes
(8.20)
which shows that, in the absence of chemical reactions, the local change of mixing ratio is affected by the divergence of the eddy flux density . When chemical reactions are taken into account, the mean source term includes covariance terms that measure the correlation between the fluctuations in the concentrations of the different reactive species, as further discussed in Section 8.3.
The assumption of constant air density is often inadequate. To address this problem, the Reynolds decomposition procedure can be replaced by the so-called Favre decomposition (Hesselberg, 1926; Favre, 1958a, 1958b; Van Mieghem, 1973):
(8.21)
in which represents the density-weighted average over a volume V:
(8.22)
and Ψ″ is the departure from this average. Note that . Applying a Favre averaging procedure, the continuity equation takes the exact form (Kramm and Meixner, 2000)
(8.23)
This equation avoids assumptions about the fluctuation of air density. Like the averaging procedure based on the Reynolds decomposition, the Favre decomposition term representing the eddy flux must be parameterized. In what follows we will use the standard notation for the Reynolds decomposition but the equations can also be applied to the Favre averaging procedure.
8.3 Chemical Covariance
Subgrid variability affects the local chemical terms in the continuity equation through covariances between concentrations of reacting species. This can be important for modeling the evolution of chemical and aerosol plumes. It can be addressed by application of Reynolds decomposition to the chemical variables. For the simple case of a single reaction A + B → C, the mean chemical production rate for species C is
(8.24)
where ρA and ρB represent the density of chemical species A and B, respectively. The variation in the reaction rate constant kAB generally results from fluctuations in temperature and can then be expressed by
(8.25)
The term represents the chemical covariance between the concentrations of A and B, while the terms account for the covariance between species concentration and temperature. The last term is the third-order moment of the fluctuations in concentrations and temperature.
If we ignore the variability in the rate constant kAB that results from eddy variations in the temperature (k′AB = 0), the mean source term is expressed by the simpler relation
(8.26)
The mean chemical source rate is therefore the sum of a resolved source term that can be explicitly calculated from the mean concentrations and an unresolved chemical covariance term whose value must be parameterized in some way. If chemical species A and B have a common origin the covariance term is usually positive. If they have different origins it is often negative.
The segregation ratio (or intensity of segregation) provides an estimate of the degree of mixing for A and B (Brodkey, 1981):
(8.27)
A value of IAB equal to zero implies that the reactants are well-mixed so that chemical evolution can be computed from the grid mean concentrations. A value of –1 (anti-correlation) indicates that the reactants are fully segregated and the mean source term is then equal to 0. From (8.26) and (8.27) one finds
(8.28)
This expression suggests that, in theory, the effects of turbulence on chemical reactions can be accounted for by replacing the rate constants kAB by effective rate constants kAB,eff = kAB (1 + IAB) (Vinuesa and Vilà-Guerau de Arellano, 2005). However, inferring the value of IAB and its variability is not straightforward, so this approach cannot be easily implemented.
The effect of turbulent fluctuations on a chemical reaction rate can be described by the Damköhler number (Damköhler, 1940, 1947):
(8.29)
which represents the ratio between the timescale τturb associated with turbulence in the flow and the timescale τchem associated with chemical evolution. Different formulations are available to estimate the turbulence timescale. For example, in the convective atmospheric boundary layer, the time constant for transport in buoyant eddies can be expressed as the ratio between the mixed layer depth h and the convective velocity scale w* defined as
(8.30)
For typical daytime values h = 1000 m and w* = 1–2 m s–1 (Stull, 1988), the turbulent timescale is of the order of 10–15 minutes. It is considerably longer in a stable atmosphere such as at night. In the slow chemistry limit, in which the chemical timescale is long in comparison with the turbulent timescale (Da ≪ 1), the chemical species are well mixed and the concentrations can be approximated by their mean values. The chemical covariance terms can be ignored. In the opposite situation, referred to as fast chemistry limit (Da ≫ 1), chemical transformation is limited by the rate at which turbulence brings reactants together. In this case, the covariance between the fluctuating components of the concentrations is large in comparison to the product of the mean concentrations and must be estimated from closure relations.
8.4 Closure Relations
In order to solve the continuity equations including eddy contributions that arise from Reynolds decomposition, closure relations that relate the eddy flux and covariance terms to mean quantities must be formulated. These closure relations are effectively parameterizations. Different formulations are possible. Local closure schemes express unknown eddy quantities at a given model grid point as a function of known mean quantities or their derivatives at that grid point. Non-local closure schemes relate the unknown eddy quantities at a grid point to known mean quantities at other grid points. In fir
st-order closure, the mean quantities are the only dependent variables solved by the continuity equations. In higher-order closure, additional equations for the higher moments are solved together with the equations for the mean quantities. For example, in second-order closure schemes, prognostic equations are expressed for covariance terms such as and closure formulations must then be adopted for the third moments .
8.4.1 First-Order Closure
A simple first-order closure relation assumes that the eddy flux of species i is proportional to the gradient of the mean mixing ratio . This amounts to assuming analogy of turbulent mixing and molecular diffusion (Fick’s law):
(8.31)
Such parameterization of turbulent mixing as molecular diffusion is grounded in the observed near-Gaussian dilution of plumes emanating from point sources. It is a good assumption when eddy scales are small relative to the model grid scale. The turbulent (or eddy) diffusion matrix K has elements Kij that describe turbulent diffusion in the three spatial directions (x, y, z). These elements Kij are called turbulent diffusion coefficients and are derived from empirical relations. One generally ignores the off-diagonal terms Kxy, Kyz, etc. that allow for the existence of counter-gradient fluxes, and retain only the diagonal terms Kxx, Kyy, and Kzz.
We pointed out above that turbulent mixing is generally most important in the vertical direction where mean winds are slow. If only the vertical direction is considered, the eddy flux of species i is written
(8.32)
If we neglect inhomogeneities in air density ( = 0), we have
(8.33)
Here Kz is the vertical turbulent (or eddy) diffusion coefficient. The same value of Kz is assumed to apply to all chemical species, and is often derived from turbulent diffusion of momentum or specific heat: This is the so-called similarity assumption. Kz depends on the intensity of turbulence. Standard semi-empirical formulations of Kz for the PBL are presented in Section 8.7. The Kz formalism is also used in conceptual 1-D (vertical) models of the global atmosphere, in which case Kz values are chosen to fit observed gradients of atmospheric tracers (Liu et al., 1984). Values of Kz in the boundary layer are of the order of 100 m2 s–1 in the daytime (unstable atmosphere) and 0.1 m2 s–1 at night (stable atmosphere). Values in the free troposphere are of the order of 10 m2 s–1 and values in the stratosphere are of the order of 0.1–1 m2 s–1. The time constant that describes diffusive transport over a length scale L is L2/2Kz, by analogy with Einstein’s equation for molecular diffusion. For example, a 1-km thick daytime boundary layer with Kz = 100 m2 s–1 mixes vertically on a timescale of 5 × 103 s or 1.5 hours. This timescale is much longer than the transport time h/w* in buoyant updrafts introduced in Section 8.3. Thus species emitted at the surface can be injected rapidly to the top of the boundary layer in buoyant updrafts, but thorough vertical mixing of the boundary layer takes a longer time.
8.4.2 Higher-Order Closures
The first-order closure formalism has the advantage of being computationally expedient. It is not suited to strongly convective environments, where the transport is mostly accomplished by the largest eddies (Wyngaard, 1982; Vilà-Guerau de Arellano, 1992) instead of the small eddies assumed in the turbulent diffusion closure. One can address this problem by using higher-order closure approaches. This adds other equations (Stull, 1988; Garratt, 1994; Stensrud, 2007) that describe the evolution of higher order moments (e.g., eddy fluxes, covariances, turbulent kinetic energy). The equations for the mean and turbulent components are established by applying Reynolds decomposition to the dependent variables in the different prognostic equations (continuity, momentum, energy). Equations for the mean quantities are obtained by averaging all terms in the equations. Equations for the turbulent components are obtained by subtracting the equations for the mean quantities from the governing equations. From there, predictive equations can be established for the different eddy fluxes and covariances. Box 8.1 gives an example.
Box 8.1 Second-Order Closure Equations for the Turbulent Flow, Eddy Fluxes, and Covariances of Chemical Species
The governing equations that describe the interactions between chemistry and turbulent mixing of species include second moments such as eddy fluxes or chemical covariance . The derivation of the equations for such quantities requires long algebraic manipulations. We consider here a simplified case assuming horizontal homogeneity (no derivative along horizontal directions) and no air subsidence (mean vertical wind component equals zero). We start by writing the vertical projection of the momentum equation in which we assume that the friction term is proportional to the Laplacian of the velocity
Here ν stands for the kinematic viscosity coefficient. We now apply the Reynolds decomposition with
If we assume that the atmospheric mean state follows hydrostatic equilibrium conditions and if we further neglect density variations in the inertia term ∂w/∂t but retain it in the gravity term (Boussinesq approximation), we find the turbulent momentum equation
In this equation, we have replaced the density variations by virtual potential temperature variations as deduced from the equation of state. The virtual potential temperature (potential temperature that accounts for the buoyancy effects related to humidity – see Section 2.4) is related to the value of the potential temperature θ by θv = θ (1 + 0.61 rw) if rw represents the water vapor mixing ratio by mass.
We now consider the simplified continuity equation for the mixing ratio μ that includes a molecular diffusion term
where D represents the diffusion coefficient. If we apply again a Reynolds decomposition with
we obtain the following equation
in which subsidence has been ignored and the mean vertical velocity
Applying the averaging operator to each term of the equation, noting that, by continuity, ∂w′/∂z = 0, we find the equation for the mean mixing ratio
where the second term reflects the effect of turbulence on the vertical distribution of the mean mixing ratio. In this expression, molecular diffusion is often neglected because it is much smaller than the eddy flux term. By subtraction, we obtain the equation for the eddy component of the mixing ratio
We then derive the equation for the mean vertical eddy flux of the tracer mixing ratio by multiplying the momentum perturbation equation by μ′ and the tracer perturbation equation by w′. We then take the Reynolds average of both equations and add them together. After some manipulations that include the transformation of the turbulent flux divergence term into its flux form, we derive the equation for vertical turbulent tracer flux:
The terms on the right side of the equation represent the flux source/sink terms associated with the vertical gradient in the mean mixing ratio, the vertical turbulent transport of the flux, the buoyant production, the pressure covariance, tracer flux dissipation, and chemical transformations. The dissipation term F must be parameterized. For a chemical scheme that includes N photolysis reactions and M second-order reactions, the chemical term can be expressed as
with m < n in the second summation. Factor sign is equal to +1 if the reaction constitutes a production and to –1 if it is a loss. J and k represent photolysis coefficients and reaction rate constants, respectively.
The chemical covariance that appears in the source term of the continuity equation for reactive species can be derived from a covariance budget equation (Garratt, 1994; Verver et al., 1997)
Here, the first two terms on the right-hand side of the equation represent the production of chemical covariance by the concentration gradients, the third term accounts for the vertical turbulent transport of the second moment, and the fourth term represents the dissipation by molecular diffusion. The last term accounts for the chemical influence on the covariances. For example, if k denotes the rate constant of a reaction between two species m and n, the corresponding loss term is
Additional moments including the wind variance , the concentration variance , and the covariance between species concentrations and the virtual temperature are provided by Stull (1988) an
d Garratt (1994). The triple correlation terms that appear in the second-order equations must be determined by higher-order closure equations or empirical closure expressions (Verver et al., 1997).
Second-order closure formulations include predictive equations for covariances between wind components, wind and temperature, wind and humidity, and wind and chemical concentrations. Triple correlation terms appear in the equations and lead to a new closure problem. In principle, additional differential equations can be written to describe the evolution of third moments, but in this case fourth-order moments will appear. In most practical applications, the system is closed either by neglecting these high-order moments or by introducing diagnostic expressions that include adjustable empirical parameters.
8.5 Stochastic Representation of Turbulent Reacting Flows
An alternative approach to treat turbulence is through stochastic methods (Pope, 2000). In this case, the flow velocity v(r, t) = (v1, v2, v3)T and the vector of concentrations for N interacting chemical species Ψ(r, t) = (Ψ1, …, ΨN)T at location r are viewed as random variables with respect to time t. Their dynamical behavior is fully described by a joint velocity-composition PDF, denoted here pv,Ψ, that describes the likelihood for the continuous random variables v and Ψ to take given values. Specifically, the probability that the random velocity v and the random chemical concentration Ψ fall into the infinitesimal intervals [u, u + du] and [ϕ, ϕ + dϕ], respectively is given by
Modeling of Atmospheric Chemistry Page 43