We employed direct numerical simulations to estimate the error on chemical calculation in simulations with regional chemical-transport models induced by neglecting subgrid chemical segregation due to inefficient turbulent mixing in an urban boundary layer with strong and heterogeneously distributed surface emissions. In simulations of initially segregated reactive species with an entrainment-emission configuration with an A–B–C second-order chemical scheme, urban surface emission fluxes of the homogeneously emitted tracer A result in a very large segregation between the tracers and hence a very large overestimation of the effective chemical reaction rate in a complete-mixing model. This large effect can be indicated by a large Damköhler number (

Turbulence mixes initially segregated reactive species in the boundary layer and allows chemical reactions to occur. However, for fast chemical reactions with the chemical timescale shorter than the turbulent timescale, turbulent motions mix the reactants so slowly that they remain segregated rather than reacting. This segregation can be a result of the inefficient mixing due to the state of turbulence and its driver, such as thermal stability, canopy–atmosphere interaction and cloud processes, and/or a result of the heterogeneity of surface emissions

Efforts have been made to examine and quantify this error under different turbulent and chemical regimes in a range of atmospheric environments. Earliest studies can be dated back to

Many of these LES studies focus on the convective boundary layer (CBL), in which the imbalance between updraught and downdraught transport produces a large segregation of the reactants

There are a number of factors that may increase chemical segregation. One possible factor is to increase the surface emission of the bottom-up tracer. For instance,

The aim of the present work is to investigate the effect of inefficient turbulent mixing on chemical reactions in an urban-like boundary layer with strong and heterogeneously distributed surface emissions and to account for the errors induced by neglecting the resultant subgrid chemical segregation in relatively coarse regional models. While previous studies focused on agricultural and rural conditions where the emission fluxes are relatively low (

The results from our DNS runs are then degraded to lower resolution to mimic the calculations from regional models. Previous studies often compared their LES results with a mixed-layer or complete-mixing model, which assumes the whole simulation domain to be in the same model grid. This assumption may still be reasonable for forested areas, as the global and mesoscale models in use typically have a mesh size of the order of 10

The structure of this paper is as follows. The next section introduces the DNS model adopted in this work and the settings of the simulations. Subsequently, the results of our DNS runs are first presented with cases of homogeneous emissions and then of heterogeneous emissions. This is followed by the comparison between the results from the degraded coarse-grid models and those from the DNS model to account for the errors from regional models. The implication of our results for regional models applied to urban environments is then discussed, and conclusions are provided at the end.

The relation between turbulence and chemistry in a convective boundary layer is investigated in this work by means of direct numerical simulation. There are two common approaches to numerically simulate turbulent flows, namely large-eddy simulation (LES) and direct numerical simulation (DNS). The former applies a low-pass filter and models the subgrid-scale effect on the filtered variables. The later solves the original Navier–Stokes equations but often with a molecular viscosity that is larger than the value in the real application represented

In this work, we opt for DNS because we are studying the variance and covariances of various fields in which entrainment may play an important role. It is known that the smallest resolved scales might become important for these variables in typical resolutions

We employed the direct numerical simulation tool Turbulence Laboratory (TLab) to perform our computational experiments of turbulent mixing of reactive species in the convective boundary layer. Source files with the implementation of TLab and further documentation can be found at

A no-penetration, no-slip boundary condition is imposed at the surface, and a no-penetration, free-slip boundary condition is imposed at the top boundary. Neumann boundary conditions are imposed for the buoyancy and velocity fields at both the top and the surface to maintain constant fluxes. The velocity and buoyancy fields are relaxed towards zero and

The size of the computational grid is

Schematic diagrams of the configurations of the DNS runs with homogeneous emissions

The names and simulation parameters of the DNS runs, including

We use a sixth-order compact scheme to calculate the spatial derivatives and a fourth-order Runge–Kutta scheme to advance the equations in time. The pressure Poisson equation is solved by applying a Fourier decomposition along the horizontal directions and solving the resultant set of finite-difference equations in the vertical direction to machine accuracy

The simulations are terminated after a total simulation time equivalent to 4.5

An archetypical entrainment-emission configuration, as typically used by past LES studies such as

Four emission fluxes of tracer A (

For the simulations with heterogeneous emissions, tracers A and B are emitted alternately from patches on the surface with widths of 1, 2 and 6

Note that the length of heterogeneity referred to in this study is equivalent to half of the length denoted in

Note that the adopted emission fluxes are doubled from the values in the simulations with homogeneous emissions in order to conserve the total fluxes.

In the context of an urban environment, the second-order chemical reaction imposed in our simulations can be considered analogous to the reaction between NO and peroxyl radical derivatives (

The initial Damköhler numbers

Two numbers are mainly employed to quantify the effect of turbulent mixing on chemical reaction, namely the Damköhler number

To quantify the chemistry–turbulence interaction at the molecular diffusion spatio-temporal scale, the Kolmogorov Damköhler numbers are also calculated in some of simulations. The definition of Kolmogorov Damköhler number is adopted from

The second number, the effective chemical reaction rate (

The error induced by neglecting such chemical segregation in a complete-mixing model, in which both tracers A and B are assumed to be evenly distributed throughout the boundary layer, can be written as

When one considers a confined volume within a horizontal layer at a specific height

For both VV05 runs, all the tracers are relatively well mixed in the mixed layer. Tracer A is largely consumed in the mixed layer, while tracer B is in excess so that its concentration remains essentially constant. The reaction between tracers A and B can then be considered a pseudo first-order reaction, and the corresponding production term

Colour maps of the distribution of the production term (

Vertical profiles of the horizontally averaged vertical flux of tracer C

We use the results of the two VV05 runs to compare with those presented in the LES study of

Unfortunately we cannot compare the vertical profiles of our horizontally averaged effective chemical reaction rate with the vertical profiles of the segregation coefficient presented in

Vertical profiles of the horizontally averaged normalised effective chemical reaction rate

When the emission fluxes of tracer A increase to urban values beyond 0.25

The shift of the reaction from tracer-A limiting to tracer-B limiting can also be seen from the profiles of the vertical fluxes of tracer C in Fig.

For the runs with urban emission fluxes, the boundary-layer-averaged effective chemical reaction rate

With the emissions of tracers A and B heterogeneously distributed on the surface, the segregation is maximum at the surface with its magnitude decreasing with increasing altitude, as seen from the vertical profiles of the horizontally averaged segregation coefficient (

Vertical profiles of the horizontally averaged normalised effective chemical reaction rate

When shifting from slow to fast chemistry (dotted red line), the magnitude of the segregation further increases, resulting in a boundary-layer-averaged value of

To discuss the effect of the length of heterogeneity (

Converting the conclusions of

Similar to the cases with homogeneous emissions, the increased segregation with increasing surface emission flux and imposed chemical reaction rate can be indicated by the relatively large values of the final Damköhler numbers. Since the boundary-layer-averaged Damköhler numbers of tracers A and B are statistically the same in the simulations with heterogeneous emissions, here we take the mean of the two numbers to get the averaged final Damköhler number

Fitted curves of the normalised boundary-layer-averaged effective chemical reaction rate

Plot of the vertical grid spacing (

In the previous sections, the boundary-layer-averaged effective chemical reaction rate is compared with the imposed rate in a complete-mixing model that assumes the tracers to be completely well mixed in the whole boundary layer. However, the horizontal resolutions in regional chemical-transport models are comparatively high (of the order of a few kilometres) with multiple vertical levels within the boundary layer. Such models are often employed when modelling urban areas. To evaluate the importance of the subgrid chemistry–turbulence interaction in these regional chemical-transport models, we degrade our DNS model to coarse-grid models with two horizontal resolutions (1 and 3

Plot of the model errors

The tracer concentration fields obtained from the DNS runs are interpolated from the high-resolution DNS model grids to the lower-resolution coarse-grid model grids. The volumetric averages of tracer concentrations in each model grid are then calculated. The statistics of these resolution-degraded concentration fields are then calculated as in Sect.

Vertical profiles of the horizontally averaged normalised effective chemical reaction rate

Plot of model errors in percentage, categorised with different lengths of heterogeneity (

We take a closer look at the results of the simulations with homogeneous emissions as described in Sect.

Figure

The errors from the four coarse-grid models (

For the simulations with

Resultant model errors in percentage from the complete-mixing model (

Although the DNS runs in this work are only conducted in idealised conditions, they can provide insights into and estimations of the errors induced by neglecting the subgrid chemical segregation due to inefficient turbulent mixing in larger-scale models. The simulations with homogeneous emissions (Sect.

There are additional points to note in our estimations. When arriving at our conclusion of the dependency of

By increasing the emission fluxes to urban values in Sect.

Unlike other studies (e.g.

The work in Sect.

One important aim of the study of chemistry–turbulence interaction is to provide a correction to the error in large-scale models induced by neglecting subgrid chemical segregation. While our work suggests a correction of

Due to computational limitations of the DNS runs, some other conditions that may be important for turbulent mixing in the urban boundary layer were neglected in our work. First of all, the growth of the boundary layer is driven by a constant buoyancy flux, so the boundary layer height gradually increases. Hence, the simulation can only approximate the time when the boundary layer grows from sunrise to mid-afternoon. This also raises the issue of the time required for statistical equilibrium to be attained. In some cases, this time may be longer than the duration of daylight, after which the atmosphere is no longer convective. This poses a greater problem to cases with strong emission fluxes, as the time required to attain statistical equilibrium is even longer. It may not be practical to run longer than the duration of daylight even though statistical equilibrium is not reached. Second, our simulations only address a convective boundary layer under clear-sky conditions. We do not take other scenarios with different weather conditions (such as cloud-top boundary layer, e.g.

An important source of surface forcings in an urban boundary layer is undoubtedly from the urban structures (buildings and streets in the urban canopy). The structure of turbulent flow can be significantly altered in the street canyons due to the perturbation in radiation and the exchange of heat and momentum with the urban structures (e.g.

We explore in this work the effect of chemical segregation due to inefficient turbulent mixing on chemical reactions in an urban boundary layer, and we estimate the resultant error in the regional chemical-transport models by conducting direct numerical simulation (DNS). As past studies mainly examined scenarios in forestal areas, we focus on urban conditions, specifically with strong emission fluxes and heterogeneous emissions, as both factors can potentially increase chemical segregation.

With homogeneous emissions, our simulations give similar results as past studies using large-eddy simulations when the emission flux of the surface-emitted tracer A is of rural value, in spite of the increase in resolution of our DNS model. On the other hand, increasing the emission flux of tracer A to urban values depletes the entrained tracer B, so its availability limits the reaction. In this situation, the segregation between tracers A and B becomes very large, which results in significant overestimation of the effective chemical reaction rate

To evaluate the errors induced by neglecting the chemical segregation in regional chemical-transport models with higher resolution than a complete-mixing model, we degraded our DNS model to coarse-grid models with two horizontal and vertical resolutions commensurable to regional models. With homogeneous emissions, all the coarse-grid models give smaller errors than the complete-mixing model. Yet the errors from the coarse-grid models remain high for simulations with urban emission fluxes. The improvement is more significant for the increased vertical resolution instead of horizontal resolution, as the initial segregation between tracers A and B is in the vertical direction. All coarse-grid models give the largest overestimations of the height-dependent effective chemical reaction rate near the top of the surface layer and the entrainment zone, indicating that high resolution is most important in these areas. With heterogeneous emissions, the coarse-grid models perform worse than the complete-mixing model when the coarse horizontal model grid cannot resolve the emission heterogeneity. With higher vertical resolution, the respective coarse-grid model gives an even larger error. This illustrates that increasing the model resolution may not improve the model performance when the enhanced model resolution still fails to resolve the emission heterogeneity. For the coarse-grid models which can resolve the emission heterogeneity, the model improvement of the coarse-grid model is more significant for increased horizontal than vertical resolution, as in these cases the initial segregation is in the horizontal direction. This suggests that whether the model improvement is more sensitive to the increase in the horizontal or vertical resolution depends on the direction of initial segregation between the reactants.

Our results from the DNS runs are based on the data in the fully developed turbulent regime in the simulation that is established
after the initial transient phase. In that regime, the initial conditions have been sufficiently forgotten, and the parameters

The rate equations of the species are non-dimensionalised by introducing the characteristic scales for the mixing ratios.
The characteristic scale for tracer A is controlled by its emission flux

Source files of the DNS model TLab and further
documentation can be found at

CWYL wrote this article, designed the research, conducted the simulations and performed the result analysis of this work. GPB and HS provided guidance and supervision to CWYL and contributed ideas to the work. JPM provided the DNS model TLab, added the chemical tracers into the original DNS model, and provided technical support on modelling issues and technical information of the DNS model in this article. In addition, GPB, HS and JPM contributed to the editing of this article.

The authors declare that they have no conflict of interest.

This work has been financially supported by the Max Planck Institute for Meteorology as part of the doctoral thesis of CWYL

The article processing charges for this open-access publication were covered by the Max Planck Society.

This paper was edited by Stefano Galmarini and reviewed by Jordi Vila-Guerau de Arellano and two anonymous referees.