8.7 Planetary Boundary Layer Processes
Section 2.10 presented an overview of the factors controlling the structure and vertical mixing of the PBL. The PBL is defined as the lower part of the atmosphere, typically 1–3 km in depth, that exchanges air with the surface on a daily basis. As discussed in Section 2.10, one can distinguish between three regimes to describe boundary layer dynamics, depending on the sensible heat flux at the surface: (1) a convectively unstable regime where heating of the surface drives strong buoyant motions, typical of land during daytime; (2) a convectively stable PBL where cooling of the surface suppresses buoyant motions, typical of land at night, and characterized by stratification of the atmosphere; (3) a neutral regime with little sensible heating of the surface, typical of marine conditions. Different formulations of turbulence must be considered for these three regimes.
Cumulus or stratus clouds are often present in the upper part of the PBL. Their formation is determined by evaporation of water from the surface (latent heat flux) and eventual condensation as air parcels rise. Cloud formation affects the dynamics of the PBL through both latent heat release and radiative effects. Clouds at night decrease nighttime stability both by suppressing radiative cooling of the surface and by radiative cooling at cloud top.
In this section we present approaches of varying complexity to describe vertical turbulence in the PBL and its implications for the concentrations and transport of chemical species. The simplest approach, often sufficient for rough estimates, is to use a box model for the convectively unstable or neutral mixed layer assuming vertically uniform concentrations through the mixing depth. Consider a horizontally uniform atmosphere with a time-varying mixing depth h(t) through which the atmosphere is assumed to be vertically well mixed. The budget of a chemical species i is defined by the mass conservation equation
(8.78)
where ρi is the mass concentration, Fi(0) is the net surface flux from emission and/or deposition, Fi(h) is the entrainment flux at the top of the mixed layer, and si is the net chemical source term. Here and elsewhere, we define vertical fluxes as positive upward. As the mixed layer grows in the morning, the entrainment flux is given by
(8.79)
where ρb is the background concentration entrained from above the mixed layer, which can represent free tropospheric air or residual boundary layer air from the previous day. When the mixing depth decreases, as in the evening, air is removed from the box and Fi(h) = 0.
Box 8.4 Vertical Flux Gradients in the Mixed Layer
Consider a chemical species i in the well-mixed convective boundary layer (mixed layer) with no in-situ production or loss, no horizontal gradient, and a constant surface flux. Assume that the mixed layer extends to the PBL top capped by a subsidence inversion, as under clear-sky daytime conditions, so that the vertical flux across the mixed layer top is zero. Further, assume that the change in air density with altitude can be neglected. The 1-D (vertical) continuity equation in Eulerian form for the chemical species is
where ρi is the density and Fz is the vertical flux. Since the surface flux is constant and the atmosphere is well-mixed, the mass mixing ratio μi must change at the same rate at all altitudes: ∂μi/∂t = α where α is a constant. If the air density ρa is fixed, then ∂ρi/∂t = α ρa is a constant too. It follows that ∂Fz/∂z is a constant and hence that the magnitude of the chemical flux varies linearly with altitude with a boundary condition Fz = 0 at the top of the mixed layer.
This result is somewhat counter-intuitive, as one might have expected the flux to be uniform with altitude in a well-mixed layer. However, that would hold only if the concentrations were constant, which cannot be the case since the flux at the top of the mixed layer is zero. This linear variation of the flux with altitude is important for interpreting vertical flux measurements from aircraft (Box 8.4 Figure 1), as these measurements will underestimate the surface flux in a predictable manner. Meteorologists refer to the surface layer as the lowest part of the atmosphere where the vertical fluxes are within 10% of their surface values. We see that this surface layer extends to 10% of the mixing depth; for a typical 1-km daytime mixing depth the surface layer is 100 m deep.
Box 8.4 Figure 1 Surface flux measurements from aircraft. The aircraft makes vertical eddy flux measurements (Section 10.2.4) on successive horizontal flight legs in the mixed layer at different altitudes, under conditions when the mixed layer extends to the PBL top (daytime, clear-sky). The line fitted to the flux measurements at different altitudes is extrapolated to the surface to derive the surface flux Fz(0).
From a more fundamental perspective, the governing equations for the mean winds, temperature, and species concentrations in the PBL can be obtained by applying the Reynolds decomposition described in Section 8.2 to the momentum, energy, and continuity equations (Stull, 1988; Garratt, 1994). The continuity equation (8.19) for the mean mixing ratio of species i in Cartesian coordinates is:
(8.80)
Again, the eddy terms must be parameterized by assuming, for example, that the local turbulent flux is proportional to the local gradient in the mean mixing ratio (see Section 8.4.1). This assumption requires that the scale of the turbulence be smaller than the characteristic spatial scale of the flow, a condition that is not met when the size of the eddies is of the same order as the vertical extent of the boundary layer. In LES models, the largest eddies that contribute most of turbulent transport are explicitly resolved, while the dissipative processes produced by the smaller eddies are parameterized. When applied to the atmospheric boundary layer, these models have a horizontal resolution of typically a few hundred meters.
Box 8.5 illustrates LES modeling with an application to marine boundary layer chemistry. These models can provide information on the chemical segregation of species within the mixed layer and the implications for chemical reaction rates. As discussed in Section 8.3, the fate of an atmospheric species emitted at the surface and reacting in the mixed layer is characterized by its Damköhler number, which is the ratio between the integral timescale of turbulence (mixing) and the timescale for chemical loss. Under convectively unstable conditions, a chemical species released at the surface may have much higher concentrations in buoyant rising plumes (updrafts) than outside (see Box 8.5). While in an updraft, the species does not mix with other species present outside the updraft. As a result of this segregation, reaction rates may be very different than in a well-mixed atmosphere (Section 8.3). A classic example for the continental mixed layer is that of isoprene, a major biogenic hydrocarbon emitted by vegetation with a mean lifetime of less than one hour against oxidation by the OH radical. The mean lifetime of isoprene is shorter than the typical timescale for overturning of the mixed layer, so that one would expect a strong vertical gradient of concentrations. In fact, isoprene in updrafts may be sufficiently concentrated to deplete OH and thus reach the top of the mixed layer with minimal chemical loss.
Box 8.5 LES Simulation of the Marine Boundary Layer
An LES performed by Kazil et al. (2011) illustrates the complexity of PBL processes with the example of sulfate particle formation from oceanic dimethylsulfide (DMS). The model accounts for the coupling between dynamical, chemical, aerosol, and cloud processes (Grell et al., 2005). Its resolution is 300 m in the horizontal and 30 m in the vertical. The figure shows an instantaneous cross-section of a 60-km South Pacific domain with three cloudy regions: a decaying convective zone in the west, a broad active convective cell in the center, and a localized convective updraft in the east. DMS is uniformly emitted from the surface and has a lifetime of hours against oxidation by OH to produce SO2. It is rapidly transported in the localized updrafts of the convective cells. OH concentrations are particularly high at cloud tops because of scattered radiation, resulting in fast DMS oxidation and SO2 production. SO2 produced near cloud top is oxidized by OH in the gas phase to produce gaseous sulfuric acid [H2SO4(g)] and in clouds to produce aqueous sulfate. H2SO4(g) initiates nucleation of new particles as shown in the up
per part of the PBL around cloud tops.
Box 8.5 Figure 1 Large-scale eddy simulation in the ocean boundary layer.
Lagrangian stochastic models are often used to simulate the transport and dispersion of trace species in the PBL (Thomson and Wilson, 2012). In the simplest of these models, the path of air particles is calculated by a sequence of random increments in position (random walk). In more sophisticated formulations that draw on the idea of Brownian motions and represent turbulent motions by a Markov chain process, the trajectory of air particles is obtained by integrating a sequence of random increments in velocity (see Section 4.11.2 for more details). Stochastic time-inverted Lagrangian transport (STILT) models are used to derive surface sources and sinks of trace species from atmospheric concentration data at a receptor location (Lin et al., 2003). Lagrangian approaches are particularly effective at dealing with such receptor-oriented problems.
8.7.1 Mean Atmospheric Wind Velocity and Temperature
The solution to the continuity equation (8.80) requires that the advection terms be either specified from a meteorological analysis or calculated from the Navier–Stokes equations. The mean wind velocity is derived from the momentum equation to which the Reynolds decomposition and averaging is applied:
(8.81)
(8.82)
Here Fx and Fy represent the influence of viscous stress and can generally be neglected in comparison to other terms in the equations. The geostrophic components of the winds and are related to the pressure gradients by
(8.83)
(8.84)
Here, f denotes the Coriolis parameter [s–1]. Assuming steady-state and horizontal homogeneity with no significant subsidence in (8.81) and (8.82), the deviation of the wind from its geostrophic value in the extra-tropical boundary layer is proportional to the turbulent momentum flux divergence (Holton, 1992):
(8.85)
(8.86)
The Reynolds decomposition can also be applied to the energy conservation equation to derive the mean potential temperature :
(8.87)
where Q represents the net radiative heating rate, Lv (2.5 × 106 J kg–1 at 273 K) the latent heat associated with gas–liquid water phase change, E the mass of water vapor produced by evaporation by unit volume and unit time, Cp = 1004.67 J kg–1 K–1 the specific heat for moist air at constant pressure, and Fθ the effect of thermal diffusivity (often neglected).
8.7.2 Boundary Layer Turbulence Closure
The closure relation required to solve the continuity equation for chemical species i in the boundary layer is often represented by a first-order local diffusion formulation:
(8.88)
where Kz is an eddy diffusion coefficient. The resulting equation becomes
(8.89)
Similar local closure relations for the turbulent momentum and heat fluxes are expressed by
(8.90)
where Km is the eddy viscosity coefficient and Kθ is the eddy diffusivity of heat. Simple formulations for Km and Kθ have been developed as a function of the wind shear, atmospheric stability, and PBL height (see e.g., Holtslag and Boville, 1993).
Prandtl (1925) introduced the concept of mixing length l in a neutral buoyancy environment by considering an air parcel that moves upwards by a distance z′ from a reference level z in a field where the mean mixing ratio and wind velocity increase linearly with height. If no mixing occurs, the mixing ratio in the air parcel is conserved during its motion and differs from its value in the surrounding environment by a value
(8.91)
Similarly, we write
(8.92)
For the parcel to move upwards, it must have a turbulent velocity w′. We assume that w′ is proportional to the horizontal fluctuation u′ so that
(8.93)
where c is a proportionality constant. The resulting mean eddy flux is
(8.94)
Let us define the mixing length l [m] as
(8.95)
where is the variance of the displacement distance. The mixing length measures the ability of the turbulence to mix air parcels. Its value varies with the size of the eddies, so that the overall (mean) mixing length should be provided by an integration over the spectrum of all eddy sizes. Typical values of the mean mixing length in the boundary layer range between 500 m and 1 km (Stull, 1988). It is often assumed that the mixing length varies with altitude as l = k z, where k is the von Karman constant taken to be 0.35. From (8.94) and (8.95) together with (8.88), one defines an eddy diffusion coefficient [m2 s–1]:
(8.96)
In this formulation, the value of Kz increases with the vertical wind shear (a measure of the intensity of the turbulence) and the mixing length (a measure of the mixing produced by turbulence), but is not a function of the static stability of the layer. Other formulations include a dependence of the eddy diffusion coefficient on the gradient Richardson number Ri as given by (8.2). For example (Blackadar, 1979, Stull, 1988):
(8.97)
with
l = k z for z < 200 m and l = 70m for z > 200m
It is assumed that, in the convectively stable situation in which the vertical gradient of is positive, turbulence is generated only if Ri becomes smaller than a critical value Rc equal to about 0.25. Idealized vertical profiles of the mean wind components in the boundary layer can be derived by introducing the empirical closure relations in the simplified momentum equations (8.85) and (8.86), as shown in Box 8.6. The local approach is best suited when eddies are smaller than the length scale for turbulence, as can be usually assumed for neutral or stable conditions. It underestimates vertical transport under convectively unstable conditions when large eddies of the dimension of the mixed layer become important. In this case, vertical transport has a non-local character and the eddy diffusion parameterization is inadequate. This can be corrected with a non-local term added to the eddy diffusion parameterization (Deardorff, 1966, 1972; Troen and Mahrt, 1986; Holtslag and Boville, 1993):
(8.98)
where γc reflects the contribution of non-local turbulent transport. To better account for the entrainment of air from the free troposphere into the mixed layer, Hong et al. (2006) include an additional flux component as:
(8.99)
where h is the depth of the mixed layer (mixing depth) and is the entrainment flux at level z = h. The entrainment flux is commonly computed as , where is the mixing ratio difference across the top of the mixed layer (between the mixed layer and the free troposphere above), and we is an entrainment velocity derived from observations for a species or other variable such as heat for which the flux is known. A value of 0.5 cm s–1 is typical for we.
Different formulations have been introduced to express parameters Kz and γc as a function of other atmospheric quantities. Holtslag and Boville (1993) and Hong et al. (2006) express the eddy diffusion in the mixed layer as
(8.100)
where k is the von Karman constant, z the geometric height above the surface, h the mixing depth, and ws the mixed layer velocity scale dependent on stability (Stull, 1988). Kz is of the order of 100 m2 s–1 in the daytime convective mixed layer, 0.1–1 m2 s–1 under stable nighttime conditions, and 10 m s–2 under neutral conditions such as over the oceans for both day and night. The correction γc to the local gradient is given by
(8.101)
where F(0) is the surface flux. The value of the dimensionless proportionality factor b is about 6.5 (Troen and Mahrt, 1986).
Box 8.6 Mean Horizontal Wind in the Boundary Layer: The Ekman Spiral
We consider the simplified momentum equations (8.85) and (8.86), and assume that the geostrophic wind is directed along the x-axis (vg = 0). We express the eddy fluxes of momentum by the first-order closure relations (8.90) in which Km is assumed to be constant
with the adopted boundary conditions
The solution is given by (Ekman, 1905):
where γ = [f/(2Km)]1/2. The wind vector turns with height as a spiral, diminishing in amplitude toward the surface where it is directed 45 degre
es to the left of the geostrophic wind vector aloft. Whereas the geostrophic wind follows isobars, the wind in the boundary layer tilts toward the direction of low pressure. It becomes parallel to the geostrophic wind at the altitude h = π/γ. This height is often used as an estimate of the depth of the boundary layer.
Box 8.6 Figure 1 Idealized vertical structure of the horizontal wind velocity (Ekman spiral) in the atmospheric boundary layer.
8.7.3 Surface Layer
The lowest part of the mixed layer is called the surface layer. It is commonly defined as the vertical extent of the atmosphere for which vertical fluxes of conserved quantities are within 10% of their surface values (Box 8.4). It is typically 10–100 m in depth. Eddy sizes in the surface layer are constrained by the proximity to the surface. Mechanical eddies driven by surface roughness are typically more important than buoyant eddies driven by surface heating. Understanding the dynamics of the surface layer is of critical importance as it determines the rate at which chemicals are removed by dry deposition (see Chapter 9).
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