Parameterizations of atmospheric turbulence in the surface layer are generally based on the similarity theory developed by Monin and Obukhov (1954), which uses scaling arguments to provide relationships between dimensionless quantities. A central parameter is the friction velocity u∗ [m s–1] that characterizes the surface momentum flux and is defined by
(8.102)
Its value is typically ~10% of the 10-m wind speed and increases with surface roughness. One also defines a potential temperature scale θ∗ and mixing ratio scale μ∗ as
(8.103)
Another key parameter in similarity theory is the Monin–Obukhov length L [m], defined as
(8.104)
where g is the gravitational acceleration, k is the von Karman constant, is the mean virtual potential temperature, and [K m s–1] the sensible heat flux at the surface. |L| represents the height above the surface at which buoyant production/suppression of turbulence by surface heating/cooling equals mechanical production of turbulence by wind shear. For a neutral boundary layer (as over the ocean), |L| → ∞ because the sensible heat flux is negligible and all turbulence is generated mechanically. In a nighttime stable atmosphere over land, L is positive (typically ~100 m) as turbulence is generated mechanically and suppressed by buoyancy. In a daytime unstable atmosphere, L is negative (typically ~ –100 m). For z < |L|, turbulence is mostly mechanical; for z > |L| it is mostly determined by buoyancy.
If we assume neutral stability of the surface layer and adopt a coordinate system in which the wind is aligned with the x-direction, one deduces from a dimensional analysis that the vertical wind shear is proportional to the friction velocity (Stull, 1988). Thus,
(8.105)
By integration over height z, one obtains a logarithmic relation for the vertical profile of the mean wind velocity:
(8.106)
Parameter z0,m [m], the aerodynamic roughness length, is the height at which the mean wind vanishes. Its value varies with the height of the physical elements that generate the surface drag (trees, ocean waves, etc.). It is typically ~3% of the height of these elements and ranges from ~10–5 m for smooth ice surfaces to ~1 m for a tall forest canopy.
Under buoyant conditions where neutral stability cannot be assumed, one replaces (8.105) by an expression written in term of a dimensionless wind shear:
(8.107)
Empirical values of Φm commonly used in models and going back to Businger et al. (1971) and Dyer (1974) are
(8.108)
with βm = 4.7 and γm = 15.0.
Expression (8.107) can be integrated to yield the vertical profile of the mean wind velocity in the surface layer:
(8.109)
where the correction term Ψm is
(8.110)
When adopting the empirical expressions (8.108), the correction term Ψm for the mean wind velocity is
(8.111)
where
(8.112)
Under this formalism, the correction factor Ψm and the wind velocity (Figure 8.9) are equal to zero at z = z0,m in all situations. In the stable case, Φm > 1 and Ψm < 0, while in the unstable case, 0 < Φm < 1 and Ψm > 0. When z ≫ z0,m one can assume x0 ≈ 1 in the above expressions.
Figure 8.9 Variation with height of the mean wind velocity in the surface layer for different static stability conditions. The wind scale is linear and the altitude scale is logarithmic.
Reproduced from Wallace and Hobbs (2006), based on Ahrens (2000) and Stull (1988).
Finally, by combining relation (8.102) written in the 1-D case (x-direction) with equation (8.107) and the eddy diffusion formulation for the momentum flux (see equation (8.90))
(8.113)
one derives an expression for the momentum eddy diffusion coefficient Km:
(8.114)
Relationships similar to (8.107) can be obtained for the virtual potential temperature and mixing ratio gradients, respectively:
(8.115)
Empirical expressions for Φh are available from Businger et al. (1971):
(8.116)
where the Prandtl number Pr = Km/Kh represents the ratio between the eddy diffusion coefficients for momentum and heat. Businger et al. (1971) estimate βh = 4.7, γh = 9.0 and Pr ≈ 0.74 for a von Karman constant k = 0.35. Hogstrom (1988) derives βh = 7.8, γh = 11.6 and Pr ≈ 0.95 for k = 0.4. It is generally assumed that the dimensionless gradients for the virtual potential temperature and chemical mixing ratios are equal, i.e., Φμ ≈ Φh. The vertical profiles of the potential virtual temperature and species mixing ratio in the surface layer are obtained by integration of (8.115) with (8.116):
(8.117)
(8.118)
where
(8.119)
Here, z0,h, and z0,μ are the roughness lengths for the virtual potential temperature and species mixing ratio, and are generally much smaller than z0,m.
The vertical eddy diffusion coefficient for heat and chemical species is given by
(8.120)
Over regions where roughness elements are packed together, (e.g., forest canopies, urban centers), the altitude above the ground at which the mean wind vanishes is shifted by a displacement height d, and expressions (8.109), (8.117), and (8.118) are replaced by:
(8.121)
(8.122)
(8.123)
Values of d are typically of the order of 65–75% of the height of the roughness elements and define the effective surface.
8.8 Deep Convection
Deep convective motions occur in the troposphere when surface heating and latent heat release are sufficiently strong that rising air parcels can pierce through the top of the planetary boundary layer (Chapter 2). Continued latent heat release as water condenses then produces intense updrafts. Vertical winds typically reach 10 m s–1. The updrafts form large cumulonimbus clouds (Figure 8.10) with intense rainfall. Eventually they encounter a sufficiently stable layer (which could be the tropopause) to stop their ascent. They can also be weakened by entraining free tropospheric air. Outflow from convective updrafts forms an anvil where air is released to the surrounding atmosphere.
Figure 8.10 (a) Schematic representation of a deep convective system. Courtesy of Cameron Douglas Craig. (b) Photograph of a thunderstorm cell with an updraft reaching the upper troposphere.
Deep convection is particularly important as a mechanism for vertical transport in the tropics and mid-latitudes during summer, when strong surface heating occurs. Air parcels in convective updrafts are transported from the boundary layer to the upper troposphere in less than one hour. Air can be entrained into the updraft at all levels in the convective column, broadening and diluting the updraft. Detrainment (outflow) from the updraft mostly takes place in the anvil near the top of the cloud. Rapid downdrafts can take place as precipitation evaporates to cool sinking air parcels, bringing free tropospheric air down to the ground. Outside the convective cells, a slow downward flow (subsidence) compensates for the net upward flow occurring inside the cells. Water-soluble gases and aerosols are efficiently scavenged in the precipitating updrafts, suppressing their release in the outflow. Scavenging is considered in Section 8.9. Particularly strong updrafts generate lightning due to separation of electrical charge between the cloud and the ground. The resulting production of nitrogen oxides (NOx) is discussed in Section 8.10.
Convective motions can be simulated explicitly in cloud-resolving models that use a LES with horizontal resolution of less than 1 km. In coarser-resolution models they must be parameterized. The parameterization must recognize the organized nature of deep convection across levels in the model horizontal grid. An eddy diffusion parameterization would be physically incorrect because it assumes that turbulence involves scales much smaller than the model vertical grid, whereas deep convection involves rapid unidirectional upward transport across a number of vertical grid levels. Observed vertical profiles of chemical mixing ratios near convective cells often feature a “C-shape” for species originating in the boundary layer and discharged in the convective o
utflow, bypassing the intermediate levels. This cannot be reproduced by an eddy diffusion parameterization, which can only produce a monotonic down-gradient change of concentrations with altitude.
Parameterization of convection is a major area of research in atmospheric dynamics and many different schemes are used in meteorological models. Online chemical transport models apply the same convective transport equations to chemical and meteorological variables, including scavenging for water-soluble species for which analogy with scavenging of condensed water is commonly used. Offline chemical transport models must have their own convective transport module to replicate the convective motions from the parent meteorological model. This is preferably done by using archived convective mass fluxes provided by the meteorological model.
A standard assumption in convective parameterizations is that the net updraft in the subgrid convective cell is compensated by subsidence in the non-convective fraction of the grid cell, so that there is no net vertical motion of air on the grid scale. The non-convective subsiding fraction is assumed to represent the bulk of the grid cell. The change of the average mass density of species i inside a grid cell is given by:
(8.124)
where ρa(z) denotes the mean air density, which varies with altitude z, while Mu(z) and Md(z) represent the vertical fluxes [kg m–2 s–1] of air in the updrafts and downdrafts summed over the gridbox. These two fluxes are assumed to be positive when directed upwards; thus Mu is positive and Md is negative. The differences and are the excess mass mixing ratios of chemical i inside the drafts relative to the atmospheric background (grid cell mean) mixing ratio . An equivalent form of (8.124) is
(8.125)
where Me = –(Mu + Md) represents the subsidence flux on the grid scale.
If Eu(z) and Du(z) [kg m–3 s–1] represent the rates of entrainment into and detrainment from the updrafts in the convective system, the mass flux of air Mu varies with height according to
(8.126)
with Mu = Mb at the base (altitude zb) and Mu = 0 at the top of the convective cloud system. Similarly for the downdraft, we write
(8.127)
with Md = 0 at the top of the convective cloud system. Figure 8.11 shows an example of entrainment and detrainment of air derived by a simple convective parameterization (Mari et al., 2000). There is large entrainment at cloud base and large detrainment from the updraft at the top of the convective column.
Figure 8.11 (a) Schematic representation of updraft, downdraft, and compensating subsidence in a model grid column. (b) Entrainment (solid curve) and detrainment (dashed curve) fluxes derived from a mesoscale model. The arrow shows the detrainment in the anvil of the cloud.
From Mari et al. (2000).
In the absence of chemical or physical transformations occurring in the clouds, the mixing ratio of species i in the updraft (u) and in the downdraft (d) are given by
(8.128)
(8.129)
We thus see that knowledge of the vertical distribution of entrainment and detrainment rates, as well as of the mean mixing ratio (z) in the grid cells, allows us to calculate the mixing ratios μi,u(z) and μi,d(z) inside the updrafts and downdrafts, and from there to calculate the tendency in (z) associated with convection for the whole grid column.
From a numerical perspective, the clouds inside a model grid column are treated as a 1-D (vertical) system and the above equations are discretized as a function of altitude (Figure 8.12). Updraft and downdraft pipes extend from the bottom to the top of the convective system. They are isolated from the gridboxes through which they extend vertically, and exchange mass with those gridboxes only through entrainment and detrainment. There can be several updraft and downdraft pipes within a grid column, representing convective systems of different vertical extent. A downdraft pipe may be associated with each updraft pipe and represents in general a small fraction of the corresponding updraft flux. Chemical mass fluxes through each pipe are calculated as a balance between updraft (or downdraft) and entrainment/detrainment terms. The fluxes at the base of each updraft pipe, and at the top of each downdraft pipe, are initialized by entraining air from the grid cell. Mean concentrations in individual grid cells are modified by convection through entrainment, detrainment, and large-scale subsidence as follows:
(8.130)
where the fluxes (E, D, and M) are defined at the edges of the grid cell (k – ½, k + ½, etc.), while the mean mixing ratio is calculated at its center (k – 1, k, k + 1). Using an upstream scheme,
(8.131)
the flux divergence associated with the subsidence flow is:
(8.132)
Scavenging of water-soluble species is done by adding a loss term to the updraft mass balance equation (8.128):
(8.133)
where w is the updraft velocity and ki is a first-order loss constant [s–1] describing the scavenging. Further discussion of scavenging is presented in the next section.
Figure 8.12 Discretization of the tendency equation that accounts for the effect of convection on the budget of species i and on the mean mixing ratio noted here at level k of a model. Processes include the updraft Mu, the downdraft Md, the compensating subsidence flux Me, and the entrainment E and detrainment D associated with the updrafts (u) and downdrafts (d), respectively.
8.9 Wet Deposition
Wet deposition is a general term to describe the removal of gases and particles by precipitation. Scavenging refers to the process by which wet deposition takes place. One refers to large-scale scavenging or convective scavenging depending on whether the precipitation in the meteorological model results from grid-scale motion of water vapor or from a subgrid convective parameterization. The distinction is important because convective scavenging must be applied to the subgrid convective updrafts, thus requiring coupling of convective transport and scavenging (8.133). By contrast, large-scale scavenging can be decoupled from grid-scale transport through operator splitting. Note that large-scale precipitation is still often subgrid (even though it is diagnosed from grid-scale motion), in which case its scavenging affects only the precipitating fraction of the grid cell.
It is also important to distinguish between in-cloud scavenging followed by precipitation (a process called rainout), and below-cloud scavenging by precipitation (called washout). Rainout and washout involve different physical processes. In-cloud scavenging followed by evaporation of precipitation below the cloud can release species at lower altitudes, resulting in downward motion rather than actual deposition. Cirrus precipitation is an important example of this effective downward motion (Lawrence and Crutzen, 1998).
The reader is referred to Seinfeld and Pandis (2006) for a detailed description of scavenging mechanisms for aerosols and gases. Here we limit our attention to the practical implementation of scavenging in chemical transport models. This typically involves consideration of three processes: (1) scavenging in convective updrafts associated with convective precipitation, (2) rainout and washout applied to large-scale precipitation and to convective anvils (convective precipitation outside the updraft), and (3) partial or total release below cloud as precipitation evaporates. Items (2) and (3) can be treated within the same algorithm, but (1) requires a separate algorithm.
8.9.1 Scavenging in Convective Updrafts
Scavenging in convective updrafts must be computed as part of the convective transport algorithm to prevent soluble species from being detrained at the top of the cloud without having experienced scavenging. It must be applied as air rises in the updraft from one model vertical level to the next in order to properly account for entrainment and detrainment fluxes at different levels. It must allow for different phases of cloud condensate, often ranging from 100% liquid at the bottom (warm cloud) to 100% solid at the top (cold cloud). There is typically an intermediate stage, with temperatures ranging from about –10 °C to –40 °C, at which the cloud contains both liquid and solid condensate (mixed cloud). In the mixed cloud, liquid cloud droplets will freeze upon contact with ice crystals (riming), a
nd the resulting production of large ice crystals drives precipitation formation.
Let us consider the most general case of a mixed cloud, where both liquid and ice condensate are present. Warm and cold clouds can be treated as limiting cases. The scavenging rate constant ki [s–1] for species i in the updraft (8.133) is expressed as
ki = (εifi, L + fi, I)k
(8.134)
where k [s–1] is the rate constant for conversion of cloudwater to precipitation and is typically 10–3 – 10–2 s–1 (Kain and Fritsch, 1990); fi,L and fi,I are the fractions of the species in the air parcel present in the liquid and ice water respectively; and εi ≤ 1 is the retention efficiency of the species as liquid water is converted to precipitation. For a warm cloud, εi = 1 because freezing does not take place. For a mixed cloud, εi accounts for exclusion from the ice matrix as droplets freeze, and is highly dependent on species type as well as on the freezing mechanism and rate (Stuart and Jacobson, 2006). As air is lifted in the updraft from one model level to the next, the fraction Fi of species i scavenged from the updraft is computed as:
(8.135)
where Δz is the distance between level midpoints and w is the updraft velocity, which may be provided by the meteorological model or need to be assumed (typically 5–10 m s–1).
Modeling of Atmospheric Chemistry Page 46