The Lagrangian Approach to Dispersion Modeling: Why We Like It (and What We Did with It)

Abstract

An introduction to the Lagrangian approach for dispersion modeling and some historical notes are presented. In parallel, a brief review of our activity in Lagrangian particle models development and application, from the long range to the microscale, is proposed. Advantages and disadvantages of the Lagrangian approach are discussed, with a special highlight on the reasons why we like it and adopted it.

Questions and Answers

Questioner: Peter Builtjes

Question: Nearly all observations, meteorological (wind speed, for instance) and concentrations are Eulerian. Is that an issue, a problem, for trajectory models?

Answer: Since the Lagrangian models are grid-free, this is not an issue. Meteorological observations can be an input to Lagrangian models; when enough densely distributed in the simulation domain, they can also be treated with a diagnostic mass-consistent model to provide the Lagrangian model with gridded input data. In the Lagrangian model, the observed (or modelled) meteorological variables of interest are interpolated at the particle position to provide the due thermo-dynamical values driving its motion. Observed concentrations are compared to the output model predictions: in this case, the pollutant mass brought by the particles which arrive at the observation point in the time interval of interest are used to calculate the simulated concentration, with different approaches (3D boxes, kernels…).

Questioner: Heinke Schlünzen

Question: How do you determine how many particles have to be released to calculate a minimum concentration that is given?

Answer: The number of particles N_p that have to be emitted at each time step to estimate a given minimum concentration can be determined, for instance, by a relationship that considers the volume ΔxΔyΔz of the cell in the concentration computational grid:

$$N_{p} = \frac{{N_{c} }}{{C_{x} }}\frac{{Q\Delta t}}{{\Delta x\Delta y\Delta z}}$$

where Q is the emission rate, Δt is the time step, N_c is the minimum number of particles wanted in the 3D grid cell in order to produce a sensible concentration value, C_x (given by N_c =1) is the minimum concentration associated to a single particle found in a cell. In alternative, kernel-based methods can be used.

Question: What is the smallest distance (as time scale) a Lagrangian Model can be applied to? How close to a source are the determined concentrations reliable?

Answer: In principle, the concentrations predicted by a Lagrangian stochastic model are reliable at any distance from the source. In practice, the discrete simulation time step determines the minimum distance at which, given the driving mean wind and turbulence, a particle can be found, thus contributing to the calculation of the concentration in a volume close to the source. Clearly, the grid resolution of the meteorological input influences an effective estimation of the concentration because the particles are moved inside a single grid cell based on interpolation of the input variables.

Questioner: Andrey Vlasenko

Question: My naïve understanding of Lagrangian approach says me, that the size of a Lagrangian particle in the numerical model should be at least not larger than the Kolmogorov size. Am I right?

Answer: We need to refer to time scale. In high Reynolds-number turbulence, as in the atmospheric flows, the diffusion term of the Langevin equation is consistent with Kolmogorov theory of local isotropy in the inertial subrange, in that the correlation time scale of the particle accelerations is of the order of the Kolmogorov time scale. This last is much shorter than the Lagrangian integral time scale. In this sense, the related size of the Lagrangian particle has to be congruent with the Kolmogorov spatial scale in the inertial subrange.

Question: Why particles moving from Chernobyl towards Italy propagate in a narrow corridor? Is it because the corridor coincides with the computational domain?

Answer: It is just a graphical effect: in the figure we plotted a small percentage of the particles travelling in (and filling!) the computational domain, in order to highlight the areas that have been more affected by the plume impact during the release and simulation time.