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Lagrangian models (LMs) track the movement of fluid parcels in their moving frame of reference. As such, scientists using LMs are forced, in a way, to imagine themselves moving with the parcel and experiencing the effects of advection, turbulence, and changes in the parcel’s environment.
LMs have advanced in sophistication over recent decades, allowing them to be used increasingly for both scientific and societal purposes. For example, it is common practice now for researchers around the world to apply LMs to examine a wide spectrum of geophysical phenomena. Atmospheric chemists can track intercontinental transport of pollution plumes [Stohl et al., 2002] or airborne radioactivity [Wotawa et al.,2006]. By running LMs backward in time [Flesch et al., 1995; Lin et al., 2003], instrumentalists can establish the source regions of observed atmospheric species with high computational efficiency [Ryall et al., 2001]. Therefore, LMs are being used increasingly to quantify sources and sinks of greenhouse gases by combining simulations with observations in an inverse modeling framework [Trusilova et al., 2010]. Such “top-down”emissions estimation is receiving growing acceptance as an independent tool to test the veracity of emissions inventories and to verify adherence to treaties.
A recent indication of the tremendous societal importance of LMs was their role in predicting the spread of volcanic ash from the eruption of Eyjafjallajökull volcano inIceland. Figure 1 demonstrates the power ofLMs to accurately track the multiday dispersion of a plume as it eventually transforms into a complicated filamentary structure. The example further demonstrates the great potential of applying LMs in combination with data assimilation and inverse modeling to improve source estimates and the simulation of hazardous plumes.
As Lagrangian modeling increases in complexity and popularity, it is imperative to reexamine the physical foundations and implementation aspects of LMs used today.From this, scientists can build a road map of further steps needed to move Lagrangian modeling forward and to ensure its successful application in the future.
As opposed to Eulerian models (which use grid cells that are fixed in place), LMs are known to create minimal numerical diffusion and thus are capable of preserving gradients in tracer concentration. Additionally,Lagrangian integration is numerically stable, meaning that models can take bigger time steps. Furthermore, the Lagrangian framework is a natural way to model turbulence,as it is a closer physical analog to the pathways traced by eddies.
These advantages served as the inspiration from which Lagrangian particle dispersion models (LPDMs) have evolved, in which air parcels are modeled as infinitesimally small particles that are transported with random velocities representing turbulence. LPDMs often track many thousands to millions of particles in three dimensions and are more sophisticated than simple trajectory or puff models. With the availability of computational resources, full three-dimensional LPDM simulations that were expensive to run just a decade ago are now routinely carried out.