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8
Parameterization of Subgrid-Scale Processes
8.1 Introduction
Meteorological variables affecting chemical concentrations vary on all scales down to the millimeter Kolmogorov scale, below which turbulence dissipates by molecular diffusion (Kolmogorov, 1941a, 1941b). The smallest scales cannot be represented deterministically in atmospheric models and must be parameterized in some way.
Processes that usually require parameterization include near-surface turbulence driving surface fluxes, turbulent eddies in the planetary boundary layer (PBL), and deep convective transport in updrafts and downdrafts (Figure 8.1). Clouds offer a vivid illustration of the variability of atmospheric motions on small scales and their link to large-scale effects (Figure 8.2). Parameterization of cloud processes is of importance in chemical transport models not only because of their dynamical role, but also because of their effect on radiation, precipitation scavenging, aqueous-phase chemistry, and aerosol modification.
Figure 8.1 Schematic representation of subgrid physical processes affecting atmospheric transport.
Courtesy of S. Freitas, INPE, CPTEC, Brazil.
Figure 8.2 Observation of clouds from aircraft provides a vivid illustration of the wide range of scales of atmospheric variability. This photograph shows cirrus clouds in the upper troposphere, the anvil of a cumulonimbus associated with deep convection, and shallow cumulus clouds at the top of the boundary layer.
Direct numerical simulations (DNS) provide a deterministic representation of chemical behavior in turbulent flows, with no need for parameterization, by solving the Navier–Stokes (momentum) and continuity equations on an extremely fine grid down to the dissipative Kolmogorov scales. This allows explicit simulation of small eddies but the simulation domains are very limited. The computational burden can be reduced by applying a low-pass filter to the Navier–Stokes and continuity equations, retaining only the larger eddies. In the large-eddy simulation (LES) method, introduced in 1963 by Joseph Smagorinsky for atmospheric flows, the effects of the unresolved smaller scales on large eddies are parameterized by a subgrid model. This provides a practical approach for studying processes on the scale of the PBL with sub-kilometer horizontal resolution. Another method, called Reynolds decomposition, expresses model variables as the sums of time-averaged and fluctuating values, and solves the corresponding Navier–Stokes and continuity equations for the time-averaged values resolved by the model. Covariance between the fluctuating values appears in the resulting equations and must be parameterized to provide “closure.” In statistical methods, momentum and chemical concentrations are treated as random variables defined by coupled probability density functions (PDFs). The momentum PDFs may be sampled stochastically at individual time steps, and the resulting concentration PDFs are constructed from a large ensemble of random realizations of the flow.
Here we refer to subgrid-scale processes (or simply subgrid processes) as the ensemble of processes driven by transport on scales smaller than the spatial/temporal resolution of the model. Although “grid” has an Eulerian connotation, consideration of subgrid processes equally applies in Lagrangian models for scales that are not explicitly resolved. These subgrid processes affect the larger-scale composition of the atmosphere and must therefore be parameterized. The parameterizations may be based on empirical information from atmospheric observations or on the analysis of results from finer-scale models. The type of parameterization adopted depends on the resolution of the model, as this defines the scales that are explicitly resolved. As model resolution increases, the subgrid parameterizations must usually be revised.
This chapter presents general approaches to parameterization of subgrid processes in chemical transport models. Section 8.2 presents the Reynolds decomposition and averaging procedure. We then discuss methods to describe chemical covariances (Section 8.3), and present closure relations that relate eddy terms to mean quantities, including turbulent diffusion formulations (Section 8.4). Stochastic statistical approaches are discussed in Section 8.5. Numerical algorithms to solve the diffusion equation are presented in Section 8.6. We apply these concepts to the PBL (Section 8.7) and further discuss parameterizations of deep convection (Section 8.8), wet scavenging (Section 8.9), lightning (Section 8.10), gravity wave breaking (Section 8.11), mass transfer through dynamical barriers (Section 8.12), and long-lived free tropospheric plumes (Section 8.13). Gaussian plume models for boundary layer point sources were presented in Section 4.12. Near-surface turbulence driving mass transfer between the atmosphere and the surface (dry deposition, two-way exchange) is covered in Chapter 9.
8.2 Reynolds Decomposition: Mean and Eddy Components
Fluid motions are categorized as either smooth, steady laminar flows or irregular, fluctuating turbulent flows (Figure 8.3). The regime associated with fluid motions is characterized by the dimensionless Reynolds number Re, defined as the ratio between the nonlinear field acceleration (v · ∇)v that generates turbulence in the Navier–Stokes equation (see Section 4.5.2 and Box 4.2) and the viscosity term v∇2v that tends to suppress it. Here ν denotes the kinematic viscosity of the fluid (1.3 × 10–5 m2 s–1 for dry air at 1 atm and 273 K) and v the velocity of the flow. Thus, we write from dimensional considerations
(8.1)
where U and L represent characteristic velocity and length scales of the flow, respectively. The characteristic velocity scale can be taken as a mean or typical wind speed, and the characteristic length scale can be taken as the distance of interest over which that wind speed varies. The transition between laminar and turbulent flow gradually takes place for Reynolds numbers of the order of 103 to 104. For a wind velocity of 10 m s–1 and a length scale of 1000 m the Reynolds number is of the order of 109, well into the turbulent regime. This can be generalized to any atmospherically relevant values of U and L: atmospheric flow is turbulent under all conditions, even when it is dynamically stable.
Figure 8.3 Schematic representation of laminar (a) and turbulent (b) flows between two fixed boundaries.
Reproduced from W. Aeschbach-Hertig, Inst. Fuer Umweltphysik, University of Heidelberg.
Turbulence can be generated mechanically by wind shear, and amplified or suppressed by buoyancy. The dimensionless gradient Richardson number, which expresses the ratio between the buoyant suppression and mechanical generation of turbulence, provides an indicator of the dynamical stability of the flow:
(8.2)
Here is the mean wind speed, the mean potential temperature, g the acceleration of gravity, and z the height. The sign of the gradient Richardson number is determined by the atmospheric lapse rate. A positive value of Ri, associated with a positive vertical gradient in the mean potential temperature, corresponds to a stably stratified atmosphere. A negative value of Ri characterizes an unstable layer with the presence of convective motions. The case for which Ri = 0 corresponds to neutral stratification. Theory shows that, even in the presence of a mean wind shear, turbulence is suppressed if Ri exceeds a critical value Rc of about 0.25.
In Reynolds decomposition, resolved (or mean) and unresolved (or turbulent) processes are separated by expressing the model variables (such as temperature, wind velocity, humidity, chemical concentrations) as the sum of a slowly varying mean value and a rapid fluctuation Ψ′ around this mean. Thus,
(8.3)
Component Ψ′, called the eddy term, is associated with the irregular and stochastic nature of the motions around a mean state (Figure 8.4). By definition, its mean value is equal to zero (Ψ ′ = 0).
Figure 8.4 Time series of wind velocity measured at a fixed location in a turbulent boundary layer for 90 seconds. The Reynolds number was 2 × 107. Fluctuations occur over a range of timescales.
Data recorded by B. Dhruva of Yale University and reproduced from Ecke (2005).
The mean term represents for example an average at a given point of the atmosphere over a time period T:
(8.4)
The deviation Ψ ′ (t) captures the rapid fluctuations around the mean value . These fluctuations are characterized by various timescales or frequencies ν. The corresponding spectru
m can be derived by a Fourier transform
(8.5)
Similarly, we can define the mean value of function Ψ(x) over a given spatial length L and derive a spectrum of the turbulent component Ψ ′ (x) as a function of wavenumber k = 2π/λ, where λ is the wavelength in radians on the spherical Earth:
(8.6)
Turbulent motions are often described by the mean turbulent kinetic energy of the flow per unit mass [TKE in m2 s–2]:
(8.7)
where u′, v′, and w′ represent the fluctuations of the three wind components (u, v, w).
The spectrum of atmospheric variability to be described by eddy terms may range from mesoscale weather patterns not resolved by global models down to the millimeter scale resolved only by the DNS approach. The largest turbulent elements receive their energy from the mean flow. Through a cascade process (Richardson, 1922), this energy propagates to smaller scales and is eventually dissipated by viscosity as heat. An example of the spectral distribution E(k) of the turbulent energy covering the scales from 10 m to 0.1 mm is shown in Figure 8.5. In the so-called inertial subrange, characterized by isotropic turbulence (no dominant direction), the energy decreases with increasing wavenumber according to Kolmogorov–Obukhov’s –5/3 law (Kolmogorov, 1941a, 1941b; Obukhov, 1941). The total specific turbulent kinetic energy is derived by integrating E(k) over all wavenumbers k:
(8.8)
Figure 8.5 Turbulence energy spectrum E(k) showing the cascade of energy from larger to smaller scales (smaller to larger wavenumbers k) with different spectral regions characterizing energy input, energy cascade, and dissipation. In a large portion of the spectrum, the turbulent kinetic energy spectrum varies as k–5/3 (Kolmogorov, 1941a,1941b).
Modeling of Atmospheric Chemistry Page 42