## High-resolution Dispersion Modelling in the Convective Boundary Layer

In this blog I will first give an overview of the representation of pollution dispersion in regional air quality models (AQMs). I will then show that when pollution dispersion simulations in the convective boundary layer (CBL) are run at $\mathcal{O}$(100 m) horizontal grid length, interesting dynamics emerge that have significant implications for urban air quality.

## Modelling Pollution Dispersion

AQMs are a critical tool in the management of urban air pollution. They can be used for short-term air quality (AQ) forecasts, and in making planning and policy decisions aimed at abating poor AQ. For accurate AQ prediction the representation of vertical dispersion in the urban boundary layer (BL) is key because it controls the transport of pollution away from the surface.

Current regional scale Eulerian AQMs are typically run at $\mathcal{O}$(10 km) horizontal grid length (Baklanov et al., 2014). The UK Met Office’s regional AQM runs at 12 km horizontal grid length (Savage et al., 2013) and its forecasts are used by the Department for Environment Food and Rural Affairs (DEFRA) to provide a daily AQ index across the UK (today’s DEFRA forecast). At such horizontal grid lengths turbulence in the BL is sub-grid.

Regional AQMs and numerical weather prediction (NWP) models typically parametrise vertical dispersion of pollution in the BL using K-theory and sometimes with an additional non-local component so that

$F=-K_z \frac{\partial{c}}{\partial{z}} +N_l$

where $F$ is the flux of pollution, $c$ is the pollution concentration, $K(z)$ is a turbulent diffusion coefficient and $z$ is the height from the ground. $N_l$ is the non-local term which represents vertical turbulent mixing under convective conditions due to buoyant thermals (Lock et al., 2000; Siebesma et al., 2007).

K-theory (i.e. $N_l=0$) parametrisation of turbulent dispersion is consistent mathematically with Fickian diffusion of particles in a fluid. If $K(z)$ is taken as constant and particles are released far from any boundaries (i.e. away from the ground and BL capping inversion), the mean square displacement of pollution particles increases proportional to the time since release. Interestingly, Albert Einstein showed that Brownian motion obeys Fickian diffusion. Therefore, pollution particles in K-theory dispersion parametrisations are analogous to memoryless particles undergoing a random walk.

It is known however that at short timescales after emission pollution particles do have memory. In the CBL, far from undergoing a random trajectory, pollution particles released in the surface layer initially tend to follow the BL scale overturning eddies. They horizontally converge before being transported to near the top of the BL in updrafts. This results in large pollution concentrations in the upper BL and low concentrations near the surface at times on the order of one CBL eddy turnover period since release (Deardorff, 1972; Willis and Deardorff, 1981). This has important implications for ground level pollution concentration predicted by AQMs (as demonstrated later).

Pollution dispersion can be thought of as having two different behaviours at short and long times after release. In the short time “ballistic” limit, particles travel at the velocity within the eddy they were released into, and the mean square displacement of pollution particles increases proportional to the time squared. At times greater than the order of one eddy turnover (i.e. the long time “diffusive” limit) dispersion is less efficient, since particles have lost memory of the initial conditions that they were released into and undergo random motion.  For further discussion of atmospheric diffusion and memory effects see this blog (link).

In regional AQMs, the non-local parametrisation component does not capture the ballistic dynamics and K-theory treats dispersion as being “diffusive”. This means that at CBL eddy turnover timescales it is possible that current AQMs have large errors in their predicted concentrations. However, with increases in computing power it is now possible to run NWP for research purposes at $\mathcal{O}$(100 m) horizontal grid length (e.g. Lean et al., 2019) and routinely at 300 m grid length (Boutle et. al., 2016). At such grid lengths the dominant CBL eddies transporting pollution (and therefore the “ballistic” diffusion) becomes resolved and does not require parametrisation.

To investigate the differences in pollution dispersion and potential benefits that can be expected when AQMs move to $\mathcal{O}$(100 m) horizontal grid length, I have run NWP at horizontal grid lengths ranging from 1.5 km (where CBL dispersion is parametrised) to 55 m (where CBL dispersion is mostly resolved). The simulations are unique in that they are the first at such grid lengths to include a passive ground source of scalar representing pollution, in a domain large enough to let dispersion develop for tens of kilometres downstream.

## High-Resolution Modelling Results

A schematic of the Met Office Unified Model nesting suite used to conduct the simulations is shown in Fig. 1. The UKV (1.5 km horizontal grid length) model was run first and used to pass boundary conditions to the 500 m model, and so on down to the 100 m and 55 m models. A puff release, homogeneous, ground source of passive scalar was included in all models and its horizontal extent covered the area of the 55 m (and 100 m) model domains. The puff releases were conducted on the hour, and at the end of each hour scalar concentration was set to zero. The case study date was 05/05/2016 with clear sky convective conditions.

### Puff Releases

Figure 2 shows vertical cross-sections of puff released tracer in the UKV and 55 m models at 13-05, 13-20 and 13-55 UTC. At 13-05 UTC the UKV model scalar concentration is very large near the surface and approximately horizontally homogeneous. The 55 m model concentrations however are either much closer to the surface or elevated to great heights within the BL in narrow vertical regions. The heterogeneity in the 55 m model field is due to CBL turbulence being largely resolved in the 55 m model. Shortly after release, most scalar is transported predominantly horizontally rather than vertically, but at localised updrafts scalar is transported rapidly upwards.

By 13-20 UTC it can be seen that the 55 m model has more scalar in the upper BL than lower BL and lowest concentrations within the BL are near the surface. However, the scalar in the UKV model disperses more slowly from the surface. Concentrations remain unrealistically larger in the lower BL than upper BL and are very horizontally homogeneous, since the “ballistic” type dispersion is not represented. By 13-55 UTC the concentration is approximately uniform (or “well mixed”) within the BL in both models and dispersion is tending to the “diffusive” limit.

It has thus been demonstrated that unless “ballistic” type dispersion is represented in AQMs the evolution of the scalar concentration field will exhibit unphysical behaviour. In reality, pollution emissions are usually continuously released rather than puff released. One could therefore ask the question – when pollution is emitted continuously are the detailed dispersion dynamics important for urban air quality or does the dynamics of particles released at different times cancel out on average?

### Continuous Releases

To address this question, I included a continuous release, homogeneous, ground source of passive scalar. It was centred on London and had dimensions 50 km by 50 km which is approximately the size of Greater London. Figure 3a shows a schematic of the source.

The ratio of the 55 m model and UKV model zonally averaged surface concentration with downstream distance from the southern edge of the source is plotted in Fig. 3b. The largest difference in surface concentration between the UKV and 55m model occurs 9 km downstream, with a ratio of 0.61. This is consistent with the distance calculated from the average horizontal velocity in the BL ($\approx$7 ms-1) and the time at which there was most scalar in the upper BL compared to the lower BL in the puff release simulations ($\approx$ 20 min). The scalar is lofted high into the BL soon after emission, causing reductions in surface concentrations downstream. Beyond 9 km downstream distance a larger proportion of the scalar in the BL has had time to become well-mixed and the ratio increases.

## Summary

By comparing the UKV and 55 m model surface concentrations, it has been demonstrated that “ballistic” type dispersion can influence city scale surface concentrations by up to approximately 40%. It is likely that by either moving to $\mathcal{O}$(100 m) horizontal grid length or developing turbulence parametrisations that represent “ballistic” type dispersion, that substantial improvements in the predictive capability of AQMs can be made.

References

1. Baklanov, A. et al. (2014) Online coupled regional meteorology chemistry models in Europe: Current status and prospects https://doi.org/10.5194/acp-14-317-2014
1. Boutle, I. A. et al. (2016) The London Model: Forecasting fog at 333 m resolution https://doi.org/10.1002/qj.2656
1. Deardorff, J. (1972) Numerical Investigation of Neutral and Unstable Planetary Boundary Layers https://doi.org/10.1175/1520-0469(1972)029<0091:NIONAU>2.0.CO;2
1. DEFRA – air quality forecast https://uk-air.defra.gov.uk/index.php/air-pollution/research/latest/air-pollution/daqi
1. Lean, H. W. et al. (2019) The impact of spin-up and resolution on the representation of a clear convective boundary layer over London in order 100 m grid-length versions of the Met Office Unified Model https://doi.org/10.1002/qj.3519
1. Lock, A. P. et al. A New Boundary Layer Mixing Scheme. Part I: Scheme Description and Single-Column Model Tests https://doi.org/10.1175/1520-0493(2000)128<3187:ANBLMS>2.0.CO;2
1. Savage, N. H. et al. (2013) Air quality modelling using the Met Office Unified Model (AQUM OS24-26): model description and initial evaluation https://doi.org/10.5194/gmd-6-353-2013
1. Siebesma, A. P. et al. (2007) A Combined Eddy-Diffusivity Mass-Flux Approach for the Convective Boundary Layer https://doi.org/10.1175/JAS3888.1
1. Willis. G and J. Deardorff (1981) A laboratory study of dispersion from a source in the middle of the convectively mixed layer https://doi.org/10.1016/0004-6981(81)90001-9