Evaluation of footprints in homogeneous and inhomogeneous terrain with a Lagrangian stochastic particle model embedded into a large eddy simulation model

Recently, several review papers, that deal with the current state of the art in the field of footprint modelling, have reported on severe deficiencies of currently used footprint models such as a lack of validation and a lack of general applicability. Those papers have pointed out that the technique of large eddy simulation (LES) is a potentially powerful tool for the validation and advancement of currently used footprint evaluation methods, such as analytical or Lagrangian footprint models. For instance, turbulence statistics derived from LES runs might be invaluable for footprint evaluations especially under complex, but realistic conditions, such as heterogeneously heated boundary layers, at forest edges or in urban canopies. Measurements in areas as such are getting more and more common and reliable footprints would be helpful for the interpretation and quality control of the recorded data. While standard footprint models are always completely based on externally derived parameterisations of the turbulent flow, that are often valid only under horizontally homogeneous conditions, the LES technique requires only the parameterisation of statistically similar small scale turbulence elements and allows the explicit resolution of the dominant large scale turbulence elements even under heterogeneous conditions. In spite of the great potential of the LES method for footprint evaluations only a few studies on LES derived footprints have been published so far. While the earlier studies made use of an Eulerian approach for the calculation of the dispersion of a scalar constituent with subsequent footprint evaluation, recent papers presented suggestions how to couple a Lagrangian stochastic (LS) particle model offline with an LES model for the purpose of dispersion and footprint modelling, respectively.

Here, an LS particle model embedded into the parallelised LES model PALM is used for dispersion and footprint evaluations. Both, footprints of flux and concentration measurements of a passive scalar are determined. Contrary to previous LES-LS approaches the new scheme applies an online coupling between the LES and the LS model, while the physics of the LS model are strictly based on already published suggestions. The online coupling allows to drive the LS model with spatially and temporally higher resolved LES data. The application of a sorting algorithm, that rearranges particles concerning their position in the LES grid, accelerates the execution of the particle code, so that a considerable reduction in the consumption of CPU time can be achieved.

Comparisons with prior numerical simulations, tank experiments and additional simulations with PALM using a Eulerian approach reveal that the new LES-LS model is capable to reproduce concentration distribution patterns, that are typical of a near-surface source in a convective boundary layer, provided that subgrid scale turbulence is parameterised even in the LS model. Neglecting the contribution of the subgrid scale turbulent kinetic energy (TKE) to the total TKE, i.e. switching off the stochastic part of the Lagrangian model, leads to a considerable overestimation of the near surface concentration of the scalar.

Moreover, comparisons with the results of a study that presented an intercomparison of several footprint models show that our model successfully evaluates footprints. The fact that our footprint approach results in locations of the footprint maximum that are situated less far upstream than those reported in the intercomparison study, is attributed to the fact that our model takes into account streamwise dispersion, while those in the intercomparison study do not. Our assumption is supported by the good agreement of our results with those of a standard Lagrangian footprint model that considers streamwise dispersion. Moreover, our results show that in a convective boundary layer even areas with negative values of the footprint function exist for flux measurements in some distance from the sensor position. These negative values are closely related to the concentration pattern that is typically observed in the convective boundary layer.

The wide range of applicability of our LES-LS approach is shown by its application to footprint evaluations in a neutrally and in a stably stratified boundary layer. As in the convective case a strong decoupling between the source areas for different measurement heights is also observed for these thermal stratifications. Moreover, the footprints are evidently affected by the fact that the wind direction turns considerably with height.

Finally, we show that our LES-LS model with its LS model in forward mode, can also be used for the evaluation of footprints of measurements in complex terrain, i.e. in terrain with horizontally heterogeneous surface conditions. We show results from footprint evaluations for measurements in a thermally heterogeneous domain as well as results from footprint evaluations for measurements in an aerodynamically heterogeneous domain. The thermal inhomogeneity is realised by a one-dimensional, sinusoidal variation of the near-surface heat flux, while the aerodynamically inhomogeneous terrain is realised as an array of cubic buildings. Evaluations of footprints under conditions as such require the tracking of hundreds of millions of particles in order to get statistically firm results. Our results emphasize that the footprint function is considerably impacted by the prescribed inhomogeneity, which let the application of standard footprint models under inhomogeneous conditions appear questionable. E.g. the location of the sensor with respect to the heat flux maximum is essential for the extension of the footprint area, as the mean values as well as the variances of the horizontal velocity components change with distance from the heat flux maximum. On the basis of our results it becomes evident that there is a need to advance footprint models towards an applicability under inhomogeneous conditions. Due to the vast amounts of CPU time that have been needed for the simulations shown here, it is obvious that our LES-LS approach cannot be regarded as a standard tool for the evaluation of footprints. However, it has indicated its value for the validation of standard footprint models.