(8.148)
This implies the following physical picture for the decay of an initially thick plume in stretched flow. The thickness of the plume decreases in time due to the stretching of the flow until W = rb, at which point stretching and diffusion precisely match so that the plume thickness does not decrease further but the mixing ratio continues to decay. Replacing W = rb in (8.148) implies α = λ, so that the decay rate is equal to the Lyapunov exponent of the flow (Chella and Ottino, 1984; Balkovsky and Fouxon, 1999). The decay rate of a stretched plume thus approaches a limit that is independent of the numerical diffusion.
We can now understand the numerical decay of the plume shown in Figure 8.15. Under a uniform flow the characteristic timescale for plume decay is
(8.149)
where W is the width of the plume. On the other hand, in a divergent flow this decay timescale is
(8.150)
where λ is the Lyapunov exponent of the flow. We see that τu > τd when W > (D/λ)1/2 = rb. A plume thicker than rb decays faster than a simple estimate from numerical diffusion would suggest. Ultimately the plume decays at a rate that is determined by the Lyapunov exponent.
Let us consider the consequences for the sensitivity of plume decay to grid resolution. A straightforward analysis demonstrates that the numerical diffusivity near sharp boundaries is D ~ uΔx where Δx is the grid spacing (Rastigejev et al., 2010). This is the case even with a higher-order advection algorithm, as the higher order of accuracy is contingent on adequate resolution of gradients on the grid scale, which fails when the boundaries are sharp (i.e., when the plume is resolved by only a few grid cells).
With D ~ uΔx we find
(8.151)
Thus the length scale rb is roughly the geometric mean of the grid spacing Δx and the length scale u/∇u over which the velocity field varies. The length scale below which numerical diffusion is important is not the grid resolution but a much larger (flow-dependent) length scale. For the flow field in Figure 8.15, which varies over ~104 km and with horizontal grid resolution ~100 km, this crossover scale is ~1000 km.
These arguments imply that the decay rate of a plume with initial width W > rb is initially set by numerical diffusion. Ultimately, the plume will be stretched so that W = rb, at which point the decay rate approaches the Lyapunov exponent of the flow. Increasing the grid resolution of the model delays the attainment of this regime, but only moderately so as rb ~ Δx1/2. Improving resolution of plumes by a factor of 2 would require a factor of 4 increase in grid resolution. The situation is in fact worse because stretching of the flow increases as the grid resolution increases and smaller eddies are resolved (Wild and Prather, 2006). Numerical tests by Rastigejev et al. (2010) indicate that rb ~ Δx1/4 because of this effect. Increasing the resolution of plumes by a factor of 2 would thus require a factor of 8 increase in grid resolution.
Lagrangian models perform much better than Eulerian models in preserving plumes during long-range transport. As discussed in Section 8.12, Lagrangian models have been used in the stratosphere to improve the simulation of transport across dynamical barriers. However, Eulerian models are generally preferred in global applications for several reasons, including better representation of area sources, ability to describe nonlinear chemistry and aerosol evolution, and completeness and smoothness of the solution. One possible approach is to use embedded Lagrangian plumes within the Eulerian framework, as is sometimes done in regional air quality models to describe Gaussian plumes originating from point sources (Section 4.12). Another approach is to use an adaptive grid model where localized increases in grid resolution are triggered by strong concentration gradients (Box 4.6). These approaches have yet to be implemented in global models.
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9
Surface Fluxes
9.1 Introduction
Solving the continuity equation for the concentrations of atmospheric species requires boundary conditions at the Earth’s surface. The surface can act either as a source or a sink. The boundary condition can be expressed as a vertical flux or as a concentration. The surface flux is called an emission when upward and a deposition when downward. Direct deposition of gas molecules and aerosol particles to the surface is called dry deposition, to distinguish it from wet deposition driven by precipitation scavenging (Chapter 8). Many species can be both emitted and dry deposited, and the difference between the two represents the net surface flux.
Emission processes include volatilization of gases from the surface, mechanical lifting of particles by wind action, and forced injection of volatile and particulate material from combustion and volcanoes. Injection may take place at significant altitudes above the surface (smokestacks, volcanoes, large fires, aircraft) and this is implemented in atmospheric models as sources at the corresponding vertical model levels.
Modeling of Atmospheric Chemistry Page 48