Atmospheric Dispersion with a Large-Eddy Simulation
265
Richardson’s law, which governs relative dispersion theory. When dealing with this
problem, the usual diffi culty is to generate a PBL with a suffi cient extended inertial
range of scales; the use of the LES technique allows us to overcome this problem.
Furthermore, we applied a nonstandard technique (FSLE) coming from the study of
dynamical systems to isolate Richardson’s law. Even though this was the fi rst time
in which the FSLE has been applied in the context of boundary layer physics, we
have obtained good results. In fact, with this new tool of analysis, we have observed
a clean region of scaling that shows the occurrence of Richardson’s law fi rst in a
neutral boundary layer where we did not use any subgrid model. In this preliminary
case, it has therefore been possible to estimate the Richardson constant. We have
found that for our simulation, its value is C 2 0.5. This estimate is compatible with
recent results that fi x its value within the [0.1 − 1] range. In particular, especially the
most recent experiments and 3-D DNS (Ott and Mann, 2000; Boffetta and Sokolov,
2002) have found similar values.
9.5 CONCLUSIONS
The dispersion of contaminants in the PBL is commonly investigated in both Eulerian
and Lagrangian frameworks. The Lagrangian approach considers the trajectory of
marked fl uid particles in the fl ow. Lagrangian particle models are very useful for
describing the turbulent dispersion of passive contaminants because they can take
into account essential aspects of turbulence, although they are limited to a simplifi ed
set of reacting species. The Eulerian approach, on the other hand, is based on the
mass conservation equation and can incorporate the various second and high-order
chemical kinetic equations necessary to describe photochemical smog generation,
which is a challenging open problem. In both approaches, the understanding of the
turbulent structure of the PBL is crucial for constructing realistic models. In this
context, LES represent a very powerful method for calculating 3-D turbulent structures and are fundamental for describing any dispersion phenomena in the PBL. In
this chapter, we used an LES to study dispersion properties of the PBL, considering
both classical approaches mentioned above.
From a Eulerian point of view, in order to numerically solve the conservation
equation, we used a splitting technique developed in the 1970s by Soviet mathematicians. The advective terms, which present many complications in their numerical
discretization, are solved with a cubic-spline technique. Such a scheme can be easily
adapted to a domain with an irregular grid-spacing. To test the method, we simulated
dispersion from an elevated and low continuous point source in a CBL, and from an
elevated continuous source in a neutral boundary layer. In all cases, the nontrivial
characteristics of dispersion are adequately captured, and the crosswind integrated
concentration distribution closely resembles the well-established numerical and laboratory experiments found in the literature. This means that coupling the LES with
a dispersion equation can provide a realistic description of the dispersion processes.
This combined approach is very promising and actually represents the state of the art
in numerical investigations of PBL dispersion, although it is still affected by the lack
of high-performance computations. Our results confi rm that the LES constitutes an
alternative for fi eld experiments: it can provide databases of dispersion data on which
© 2010 by Taylor and Francis Group, LLC
265
Richardson’s law, which governs relative dispersion theory. When dealing with this
problem, the usual diffi culty is to generate a PBL with a suffi cient extended inertial
range of scales; the use of the LES technique allows us to overcome this problem.
Furthermore, we applied a nonstandard technique (FSLE) coming from the study of
dynamical systems to isolate Richardson’s law. Even though this was the fi rst time
in which the FSLE has been applied in the context of boundary layer physics, we
have obtained good results. In fact, with this new tool of analysis, we have observed
a clean region of scaling that shows the occurrence of Richardson’s law fi rst in a
neutral boundary layer where we did not use any subgrid model. In this preliminary
case, it has therefore been possible to estimate the Richardson constant. We have
found that for our simulation, its value is C 2 0.5. This estimate is compatible with
recent results that fi x its value within the [0.1 − 1] range. In particular, especially the
most recent experiments and 3-D DNS (Ott and Mann, 2000; Boffetta and Sokolov,
2002) have found similar values.
9.5 CONCLUSIONS
The dispersion of contaminants in the PBL is commonly investigated in both Eulerian
and Lagrangian frameworks. The Lagrangian approach considers the trajectory of
marked fl uid particles in the fl ow. Lagrangian particle models are very useful for
describing the turbulent dispersion of passive contaminants because they can take
into account essential aspects of turbulence, although they are limited to a simplifi ed
set of reacting species. The Eulerian approach, on the other hand, is based on the
mass conservation equation and can incorporate the various second and high-order
chemical kinetic equations necessary to describe photochemical smog generation,
which is a challenging open problem. In both approaches, the understanding of the
turbulent structure of the PBL is crucial for constructing realistic models. In this
context, LES represent a very powerful method for calculating 3-D turbulent structures and are fundamental for describing any dispersion phenomena in the PBL. In
this chapter, we used an LES to study dispersion properties of the PBL, considering
both classical approaches mentioned above.
From a Eulerian point of view, in order to numerically solve the conservation
equation, we used a splitting technique developed in the 1970s by Soviet mathematicians. The advective terms, which present many complications in their numerical
discretization, are solved with a cubic-spline technique. Such a scheme can be easily
adapted to a domain with an irregular grid-spacing. To test the method, we simulated
dispersion from an elevated and low continuous point source in a CBL, and from an
elevated continuous source in a neutral boundary layer. In all cases, the nontrivial
characteristics of dispersion are adequately captured, and the crosswind integrated
concentration distribution closely resembles the well-established numerical and laboratory experiments found in the literature. This means that coupling the LES with
a dispersion equation can provide a realistic description of the dispersion processes.
This combined approach is very promising and actually represents the state of the art
in numerical investigations of PBL dispersion, although it is still affected by the lack
of high-performance computations. Our results confi rm that the LES constitutes an
alternative for fi eld experiments: it can provide databases of dispersion data on which
© 2010 by Taylor and Francis Group, LLC