Structure Of The Boundary Layer

Boundary layer processes are dominated by the diffusive character of turbulence. This diffusion can be described as a flux of a constituent, which is the rate of transport of the constituent across a surface per unit time and per unit area. Alternatively, it can be expressed as a constituent density times a velocity. Both velocity components and scalars are defined as sums of a mean and a fluctuation,

= C/; + Uj and £f = S + s. The horizontal wind components are commonly defined as ^ = ^ and aU1 = 'V, while the vertical component is = $). The average of the fluctuations, u,■ = s = 0, where the overbar is the common convention to denote an average of a turbulence variable over a time period or a length long enough to give a stable estimate of its mean.

In a turbulent fluid, the scalar flux is defined as the sum or integral of fluctuations in the velocity normal to a surface times the concurrent fluctuations in constituent density divided by the time period or length over which the sum or integral is calculated. Normally in the PBL, only the vertical component of the flux is of interest. Thus, ws dt = vra

where w and s are fluctuations in the vertical wind component and a scalar density, respectively, and T is the averaging time period. Equation (1) defines the eddy correlation technique for measuring flux.

The PBL can be further broken down into sublayers: The lowest few tens of meters is called the surface layer. In this region, which is less than 10% of the depth of the PBL, the fluxes can be considered constant with height, but for variables with large surface fluxes, the mean vertical gradients of these variables are large relative to the rest of the PBL. It is common to relate the flux in the surface layer to a gradient by a diffusivity,

Since transport by turbulent eddies in the surface layer is roughly about 105 times more efficient than transport by molecular diffusion, for scalars we call Ks the eddy diffusivity and for momentum the eddy viscosity. Near the surface, the typical maximum horizontal dimension of the eddies making important contributions to the flux is roughly about 200 times the height above the surface. Since the efficiency of turbulent transport scales with the size of the eddies, the eddy diffusivity increases approximately linearly with height. Equivalently, since the flux is approximately constant in the surface layer, the gradient decreases approximately inversely with height near the surface.

In the CBL, the layer above the surface layer and below the region near the PBL top is called the mixed layer. Since this encompasses the bulk of the PBL, the fluxes show considerable variability. Here the gradients are small because the mixing process is efficient. The individual turbulent eddies extend throughout the mixed layer and are called thermals (or plumes). The typical maximum size of eddies making important contributions to the flux changes more slowly with height than in the surface layer, and is roughly about 40 times the depth of the CBL. The turbulence energy (the sum of the three component velocity variances) reaches a maximum at about one third the height of the mixed layer.

Above the mixed layer is the entrainment layer. In this region, a sharp interface typically occurs between the CBL and the overlying nonturbulent free atmosphere. This interface is generated by turbulent eddies that protrude into the nonturbulent layer, engulf or capture volumes of nonturbulent air, and fall back into the mixed layer. Smaller-scale turbulence within these larger eddies then commingles this entrained air with CBL air so that it becomes part of the mixed layer. Details of the clear (cloud-free) CBL structure are shown in Figure 1.

If clouds form at the top of the CBL, one of two scenarios occurs: If the density decrease across the top of the CBL is small enough that condensation and mixing processes decrease the density to a value below the overlying air, the cloud can penetrate through the top of the CBL and form a cumulus. In that case, the CBL top is approximately at cloud base, and the clouds penetrate into the overlying air until mixing with their environment limits their growth and they lose their buoyancy. This venting process injects CBL air into the overlying atmosphere, which is a way to increase humidity above the CBL and introduce trace constituents originating at the surface or within the CBL into the free atmosphere. Compensating downward motion can dilute the concentration of pollutants in the CBL. If the decrease in density across the top of the CBL is large enough that phase changes and mixing of overlying air with CBL air do not decrease the density sufficiently to lower the density below the overlying air, the cloud layer is contained within the CBL and a stratus or stratocumulus cloud layer may exist. In this case, the CBL maintains a sharp interface. Radiative cooling at cloud top can also generate CBL turbulence and contribute to the efficiency of mixing in the cloud-capped CBL.

In the SBL, the layer above the surface layer is a region of decreasing intensity and increasing intermittency of turbulence. Here the individual turbulent eddies may not extend throughout the SBL, but occur in sublayers that develop locally enhanced shear that may intermittently break down into turbulence. The turbulence is then dissipated with the net result that the gradients in the sublayer are reduced by the transient turbulence event. The process may very well be repeated over time. Since the turbulence is more local in nature, the scales and structure are less well defined than in the CBL, and velocity and scalar variances decrease with height throughout the SBL. Relatively large gradients of both wind and scalars, and multiple sublayers that are only intermittently coupled may exist. The top of the SBL generally does not have a well-defined lid. Because of the intermittency and smaller length scales of the mixing process, the shallower and less well defined structure of the SBL, and the presence of gravity waves that produce velocity fluctuations but no flux, trace constituent fluxes are much more problematic and much less frequently measured in the SBL than the CBL. For these reasons, most of the subsequent discussion deals solely with the CBL.

4 SCALES AND PROCESSES

Wind shear is the rate of change of wind with height, dU dV 9z 3z

where U and V are the averaged horizontal wind components. Because of drag induced by Earth's surface, the horizontal wind approaches zero at the surface. Since the eddy viscosity increases approximately linearly with height and the kinematic momentum flux (Ttw) is approximately constant [and equal to (wvP)0] in the surface layer, the wind shear decreases roughly inversely with height very near the surface. Further above the surface, the wind shear, as well as scalar gradients in the surface layer depend also on the stability—that is, if the surface buoyancy flux is positive, the magnitudes of the wind shear and scalar gradients decrease with height less rapidly than the inverse of height, and if the buoyancy flux is negative, they decrease more rapidly than the inverse of height. Furthermore, the direction of the mean wind is assumed to be constant with height in the surface layer, so the coordinate system can be defined such that V = 0. Thus near the surface, or in a surface layer with zero surface buoyancy flux (a neutrally stratified PBL), dU

where ul = —(uw)0 is the friction velocity and k is the von Karman constant 0.4). A typical range of values for itt over a treeless vegetated surface in moderate winds would be 0.2 to 0.8 m/s. A similar relation holds for scalar quantities in the surface layer, dS

Yz kz

where St = ~Fs{i/ut and Fs0 is the surface-layer flux of ■'/'. These equations can be integrated to obtain vertical profiles,

K Zj

Since U 0 at the surface, we define the roughness length z0 as the height at which the extrapolated wind profile goes to zero so that tij , z £/(z) = f ln-k z0

The roughness length is approximately ^ the height of individual surface roughness elements. It ranges from about 10~4 m over calm water to about 0.5 m over a forest.

The production of turbulence energy by wind shear is given by the product of mean wind shear and (kinematic) momentum flux,

_fdU dV

Near the surface, or in a neutral PBL, inserting (4) into (9) reduces (9) to

Production (dissipation) of turbulence by convection can be expressed in terms of a buoyancy flux,

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