6. Atmospheric MixingPrevious chapters have dealt solely with transport in various water bodies and have presentedexamples of one-dimensional solutions to the transport equations. We now turn our attention totransport and mixing in the atmosphere, and by necessity, we will have to give more attention tothree-dimensional solutions. Because of the atmosphere’s unique composition and boundary andforcing conditions, atmospheric turbulence is more complicated than the idealized homogeneous,stationary, isotropic case. Moreover, these complications impact transport and mixing becausethey determine the values of the turbulent diffusion and dispersion coefficients. Hence, a concisediscussion of atmospheric mixing requires also studying atmospheric turbulence and the resultingmodifications in the behaviour of mixing coefficients from the idealized case.This chapter begins with an introduction to atmospheric turbulence and a review of turbu-lent boundary layer structure. The log-velocity profile for a turbulent shear flow is introduced,and the behaviour of turbulence throughout a neutrally stable atmospheric boundary layer isdescribed. Because of their importance to turbulence characteristics, the buoyancy effects ofheating and cooling within the boundary layer are discussed qualitatively. The discussion onmixing begins with a review of turbulent mixing in three-dimensional, homogeneous, stationaryturbulence. The solution for a continuous point source is derived and used to illustrate mixingin the remaining section. The chapter closes by adapting the idealized solution in homogeneous,stationary turbulence to the turbulence present in the atmosphere.Much of the material in this chapter was taken from Csanady (1973) and from Fedorovich(1999). For further reading, those two sources are highly recommended, along with the classicbooks by L umley & Panofsky (1964) and Pasquill (1962) and more recent contributions byGarratt (1992) and Kaimal & Finnigan (1994).6.1 Atmospheric turbulenceIn Environmental Fluid Mechanics, we are concerned with local mixing processes in fluids thatinteract with living organisms. For the atmosphere, this means that we are interested in mixingprocesses near the earth’s surface. Because of the no-slip boundary condition at the surface,wind in the upper atmosphere generates a near-surface boundary layer, defined by variations invelocity and often accompanied by variations in temperature (and density). Figure 6.1 showsthis situation schematically. Because of its dominant role in mixing near the earth’s surface, wepresent here a short introduction to turbulence in the atmospheric boundary layer.Copyrightc 2004 by Scott A. Socolofsky and Gerhard H. Jirka. All rights reserved.114 6. Atmospheric MixingU(z) T(z)Fig. 6.1. Schematic of the velocity and temperature variation within the atmosphere near the earth’s surface.The region of high velocity shear is called a boundary layer.APBLTroposphereTropopausez [km]Capping InversionPotential Temperature [°C]02101220 50 65StratosphereFig. 6.2. Schematic of the potential temperature profile in the earth’s troposphere and lower stratosphere showingthe atmospheric planetary boundary layer (APBL).6.1.1 Atmospheric planetary boundary layer (APBL)Fedorovich (1999) defines the atmospheric planetary boundary layer (APBL) as the subdomainof the lower portion of the earth’s planetary atmosphere (troposphere) which is in contact withthe bottom boundary (earth’s surface) and which varies in depth from several meters to a fewkilometers. Figure 6.2 provides a schematic of this definition. The figure depicts the APBL as thelower part of the trop osphere and shows that it is separated from the linearly stratified region ofthe troposphere by a strong density gradient, called the capping inversion. The capping inversionarises due to strong mixing that occurs at the ear th’s surface which results in a weaker densitygradient within the APBL than in the upper troposphere. Although the density gradient shownin the figure is for a neutral APBL (no density gradient), heating and cooling processes withinthe APBL can lead to both unstable and stable conditions, discussed below under buoyancyeffects. Above the APBL, the wind has an approximately constant velocity; hence, the APBLencompasses the full near-surface boundary layer.6.1.2 Turbulent properties of a neutral APBLFigure 6.3 shows the development of a general turbulent boundary layer over a flat surface. In theupper figure, the boundary layer is tripped at x = 0 and begins to grow in height downstream asan increasing function of x1/2. In the idealized case, the boundary layer is tripped by the edge of6.1 Atmospheric turbulence 115Outer layer: U = f(δ)- Match log-velocity profile to U0Inertial sub-layer: U = f(u*)- Use log-velocity profileUzU0VSLδ(x)δ(x) ~ x1/2x' xz(a.) Growth of a boundary layer with increasing fetch.(b.) Boundary layer structure at the section x'.Fig. 6.3. Schematic of the development of a turbulent boundary layer over a flat surface.a flat plate extending into a free turbulent flow. In nature, boundary layers start in response tochanges in friction (roughness), as when the wind blows over a long, smooth lake and suddenlyencounters a forest on the other side. The distance the wind has blown downstream of a majorchange in surface properties is called the fetch.A turbulent boundary layer at any point x contains three major zones that differ in theirturbulence characteristics (refer to Figure 6.3(b.)). The lowest layer, directly in contact withthe surface, is the viscous sub-layer (VSL). It has a depth of about 5ν/u∗(of order millimeterin the atmosphere). The VSL thickness is independent of the total boundary layer depth δ(x),and velocities in the VSL are low so that the flow is laminar. A transition to turbulence occursbetween 5ν/u∗and 50ν/u∗. Above this transition zone, and to a height of about 10-20% of thetotal boundary layer depth (of order 100 m in the atmosphere), lies the inertial sub-layer (ISL),also called the Prandtl layer in the atmosphere. The inertial sub-layer is fully turbulent, andturbulent proper ties are functions of the friction velocity only (i.e. they are independent of thetotal boundary layer depth). The mean longitudinal velocity profile in the ISL is given by thewell-known log-velocity profileU(z)u∗=1κlnu∗zν+ C (6.1)116 6. Atmospheric Mixingwhere κ ≈ 0.4 is the von Karman constant and C is an integration constant