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ATMOSPHERIC ELECTRICITY IN THE PLANETARY BOUNDARY LAYER 160 are vertically directed and that their sum, the total current density, is height independent, leading to great simplifications from a modeling point of view. About this mean state, of course, are fluctuations caused by turbulent eddies. (A parallel may be drawn here to the prevalence of one-dimensional models of the mean meteorological structure of the PBL, in which small-scale phenomena such as turbulence enter only in terms of their horizontally averaged effects.) The primary goal of electrical modeling of the boundary layer to date has been to understand the mean profiles of the electrical variables resulting from currents driven by the global circuit. Thus, theoretical developments have focused on the sources of charge within the PBL and the phenomena produced by the vertical turbulent transport of that charge. After an introduction to the modeling of turbulent mixing, we shall examine the electrode layer and the convection currents resulting from electrode-effect space charge. Turbulence Modeling The simplest and most pervasive mathematical description of turbulent mixing is the gradient-diffusion model. Applied to the vertical transport of space-charge density by vertical velocity fluctuations Ï', it states that the mean turbulent flux (or convection current) is proportional to the local mean gradient: Here K(z) is the eddy-diffusion coefficient, usually taken to be proportional to height above the surface. Values at 1 m generally lie between 0.01 and 1.0 m2/sec. This model is a form of first-order closure of the conservation equation for space charge. It is generally believed to provide an acceptable description of mixing near the surface, where shear production predominates over buoyant production of turbulent kinetic energy. Gradient diffusion is often applied throughout the PBL, sometimes with a stability-dependent coefficient and a decrease at the top to represent an inversion, although it is known to provide poor results in the interior of an unstable mixed layer. In atmospheric electricity its use is best restricted to modeling of the electrode effect. There are many more sophisticated (and more complex) approaches to turbulent transport modeling. A popular one is second-order closure, in which conservation equations are derived for evaluating the second moments, such as (Î¸u is the virtual potential temperature), in terms of each other and of mean variables like . Such models allow the mean charge flux at a given height to depend on the dynamics of the entire PBL, not just on local properties. This capability is essential for the correct modeling of unstable mixed layers, where local flux- gradient relationships are known to break down. Before leaving this discussion of turbulent transport modeling, some cautionary mention should be made of secondary flows. There are many circumstances when the largest scale of motion in the boundary layer is organized into a nonrandom structure. The most familiar example is the stationary roll vortices, which sometimes form over the tropical ocean. Such organized motions can carry a large fraction of the vertical transports; and since they cannot be adequately represented by turbulence models of the types discussed above, these transports must be described explicitly with a dynamical model of two or more dimensions. The Electrode Effect The electrode effect has already been defined, and its simplest manifestation has been described for laminar flow and uniform ionization in aerosol-free air (Figure 11.3). In this case the electrode layer has a thickness of only a few meters, over which the electric-field magnitude decreases by about a factor of 2. The importance of the phenomenon lies in its ability to separate substantial amounts of charge near the Earth's surface. In conditions of low turbulence this leads to high space-charge densities in shallow layers, which can produce high electrical noise levels as intermittent eddies move this charge around. In strongly unstable conditions, on the other hand, it provides a source of charge to be carried deep into the interior of the PBL by convection currents. The theory of the nonturbulent electrode effect is fully developed and has been verified over water and, at least qualitatively, over land. The simplest case, the so-called ''classical" electrode effect, is illustrated schematically in the first frame of Figure 11.14. The thickness of the layer is determined by the lifetime of the small ions and their drift velocity in the ambient electric field. Since aerosol particles act as recombination sites for small ions, they reduce the ion lifetimes and, hence, the thickness of the nonturbulent electrode layer, as illustrated in the second frame of Figure 11.14. A shallow layer of enhanced ionization, which can arise from surface radioactivity or trapping of radioactive emanations from the soil, can cause a reversed electrode effect, as illustrated in the third frame of the figure. Here a stratum of negative space charge is developed owing to the sweeping of negative ions upward out of the highly ionized layer, causing the electric-field magnitude to in