where g is gravity, T is temperature, and Tv is virtual temperature, which includes the density effects of water vapor on the temperature. The parameter g/T is the buoyancy parameter, which is the thermal expansion coefficient of air times the acceleration of gravity. Since the buoyancy flux can also be negative, this term may also act to dissipate turbulence. The total turbulence energy production is the sum of (11) and (9). Averaged over the entire PBL, the energy production must be equal to the turbulence dissipation, which is the loss of turbulence energy due to the viscous forces that occur predominantly at very small scales (i.e., less than a few centimeters). In effect, the viscous forces convert the kinetic energy of turbulence into thermal energy, and thus heat the air (although the temperature increase is insignificant).
Near the surface, the negative ratio of energy production by buoyancy to production by shear is given by u\/kz
The length at which this ratio is unity, called the Obukhov length, is given by
This is a measure of the stability of the surface layer and is used as a scaling height to normalize the observation height. Similarly, velocity, temperature, and scalar variables in the surface layer can be normalized by uv T^ = —(wT)i)/ulf, and S*. In this way, normalized surface layer variables as functions of height can be expressed as universal functions in both the unstably and stably stratified surface layer. This is a powerful technique for relating surface layer measurements to a universal surface layer structure in diabatic (nonzero surface buoyancy flux) PBLs with enough mean wind to generate a well-defined w*. For example, (4) and (5) can be extended to the diabatic surface layer by including stability functions in the formulations, and dS s (Z\
where the stability functions <f>m and 4>h have been obtained empirically from carefully designed field studies. For a neutral PBL, (f> = 1; for an unstable PBL, 0 < 1; and for a stable PBL, (f> > 1. These expressions can be integrated as in (6), (7), and
(8) to relate the fluxes to measurements of velocity and scalar differences at two heights in the surface layer.
A similar procedure is used in the mixed layer, with the scaling height being the depth of the CBL, zi and the velocity scale being the Deardorff velocity,
However, in the mixed layer a further complication is that the behavior of mixed-layer variables depends not only on surface fluxes but also on fluxes through the top of the CBL, the entrainment fluxes. Therefore, for scalar fluxes in the mixed layer both the surface flux Fs0 and the entrainment flux Fszi need to be incorporated in generalized formulations. For the scalar flux-gradient relationship, this can be expressed as where goiz/zj) and gzi(z/zj) are the normalized mixed-layer gradient functions. Thus far, these gradient functions have not been measured in the atmosphere; however, they have been estimated from detailed numerical simulations of the CBL.
Usually it is the density of the trace constituent that is measured since most sensors respond to the number of molecules in a particular volume of air. In estimating the flux of a species, we normally calculate the quantity ws with the assumption that W = 0 at the surface. This is not strictly true even over a horizontally homogeneous surface if the water vapor and temperature fluxes are not zero. This arises from the constraint that the flux that is most realistically zero at the surface is the mass flux of dry air, ~pw = 0. Intuitively, we can see that in the case of a heated surface, rising parcels of air will be on average warmer and lighter, and consequently contain fewer molecules per unit volume than their surroundings, while descending parcels will be colder and denser, and contain more molecules than their surroundings so that for zero species flux at the surface, ws < 0. This is known as the Webb effect. To obtain the correct flux, it is necessary to correct for W ^ 0 by incorporating terms proportional to the fluxes of humidity and temperature. This correction becomes significant if W5/5" is less than about 0.01 m/s. Alternatively, if instead of measuring the constituent density, we measure its mixing ratio with respect to dry air, there is no Webb correction. In subsequent discussion we disregard this correction, but note that it can be important for surface fluxes of relatively long-lived atmospheric species such as CO2, CH4, or N20.
Since the boundary layer is a conduit for transport of trace species between the surface and the overlying free troposphere, measuring species fluxes within the PBL is a standard approach for estimating their sources or sinks at the surface, as well as w* = (F60zf)
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