In recent years, urban canopy models (UCMs) have been used as fully coupled components of mesoscale atmospheric models as well as offline tools to estimate temperature and surface fluxes using atmospheric forcings. Examples include multi-layer urban canopy models (MLUCMs), where the vertical variability of turbulent fluxes is calculated by solving prognostic momentum and turbulent kinetic energy (TKE,

The urban canopy is a unique and complex land cover type in climate models

The development of UCMs relies on the similarity between flow through the urban canopy layer (UCL) and other more commonly investigated types of flow. For example,

The modeling of urban surfaces develops along with the capability of computational power and availability of realistic building morphology

The development of UCMs includes designing modules to predict flow properties such as wind speed

Multi-layer UCMs are fundamentally more versatile than single-layer UCMs in characterizing the urban effects

Arising from spatially averaged flow properties (that is core to MLUCM development), dispersive fluxes illustrate the transport of variables by time-mean structures smaller than the averaging grid size, constituting another unique urban phenomenon

Based on the aforementioned fundamental deficiencies embedded in the multi-layer model, a better performance of UCMs on more complex flows requires a finer and more physically based characterization of dispersive fluxes. In this research, based on a recently developed urban flow dataset

separate characterization of transport efficiency of different flow properties.

introduce a physics-based modeling of the dispersive transport of momentum.

This paper first describes the flow dataset using the Parallelized Large-Eddy Simulation Model (PALM) in Sect.

In this study, we used two methods to assess the turbulent transport behavior of the urban canopy flow, i.e., computational fluid dynamics (CFD) simulations and an offline multi-layer urban canopy model. In Sect.

Conventionally, two configurations of idealized building arrays, “aligned” and “staggered”

Simulations are conducted in the Parallelized Large-Eddy Simulation Model (PALM, version r4554)

The computational domain is discretized equidistantly using second-order central differences

The detailed design of building arrays can be found in

Dataset details for 49 urban arrays discussed in this study. Four building height arrangements (uniform, continuous, clustered, and high-rise) with two standard deviations of height configurations are considered. The maximum

In mesoscale climate models, flow through canopies is often simulated at scales much larger than the typical surface processes, such as turbulent eddies and obstacle wakes. A common approach to derive parameterizations is to apply a time averaging over an interval longer than the scale of the slowest eddies and a spatial averaging over a length larger than the typical spatial deviations in the flow

In urban canopies where the volume fraction occupied by obstacles is not negligible, the intrinsic average

The Reynolds averaging over the momentum equation yields nonlinear terms that must be parameterized to close the equation. The most common approach in parameterizing the Reynolds stress in UCMs is based on

Similar to the momentum equation (Eq. (

To date, most models assume the same value for the diffusion coefficient in the momentum, TKE, and scalar equations, possibly due to the lack of data on TKE and scalar budget terms (note that the explicit evaluation of TKE transport and diffusivity, in particular, is only possible from numerical simulations that resolve third-order moments, e.g., LES and direct numerical simulations (DNS)). However, previous LES analyses of urban flow indicated a different transport efficiency of varying flow properties

The turbulent fluxes, vertical gradient, and eddy diffusivity of momentum, TKE, and scalars are shown in Fig.

Vertical profiles of turbulent flux (first column), vertical gradient (second column), and eddy diffusivity (third column) of momentum, TKE, and scalar flux from Eq. (

Following the parameterization of length scales in

The magnitude of eddy diffusivity for TKE is

Canopy-averaged eddy diffusivity for momentum and TKE with (

The eddy diffusivity was not directly parameterized under the 1.5-order closure framework developed in

Figure

Canopy-averaged turbulent viscosity for momentum and TKE. Panel

The momentum transport in urban canopy flow is also modulated by its spatial variability (Eq.

This section explores this alternative approach to characterize the dispersive momentum flux, other than the pragmatic approach developed in

Note that, for simplicity, the entrainment process

An example of horizontal and vertical field sampled at 93.8 % of the mean building height (

Unlike the turbulent momentum flux, which shows a uniform downward transport with an indecisive correlation with the local roughness elements, a portion of the dispersive counterpart is directly connected to the local roughness, whereas the rest remains relatively homogeneous. Therefore, a proper sampling filter should be applied to enclose these regions before gathering flow properties for parameterization. Again, we adopt a common sampling approach in the development of the EDMF

The CDF can be adjusted from

The sampling filter aims to capture the building-induced component of the dispersive momentum flux that

One tail growing stronger and thinner at the third quadrant (

A relatively symmetric elongated region at the second (

Contour plot showing the two-dimensional histograms of the CDF of the dispersive velocities at 2.5, 5, 7.5, 10, 15, and 22 m. The integral of the PDF (from outside to the mean value) within the area is delineated by isolines on the logarithmic scale. The black/red squares depict the averaged dispersive flux over the sampled regions to represent the strength of the gradient and counter-gradient transport parameterized in MLUCM v3.0.

After subtracting dispersive structures from the sampled regions, the rest of the relatively homogeneous regions may safely resort back to the lumping approach and update Eq. (

We speculate the reason that dispersive flux has long been ill represented in UCMs is attributed to the inappropriate scaling with the turbulent flux (e.g.,

Vertical profiles of sampled downward and upward dispersive momentum flux (gray-scale lines) for seven building height configurations covering

To gain a direct impression of how the sampled dispersive motions scale with the pressure gradient, we present the profile of the vertical gradient of the sum of two sampled terms from Eq. (

The above parameterization retains the physical significance of upward and downward dispersive motion demonstrated in Fig.

The presence of roughness elements on the urban surface implicitly modifies turbulent length scales

Drag parameterization based on the equivalent drag coefficient

Accordingly, characterizing drag impacts converts to evaluating the drag coefficient

The last term parameterized is the dissipation length scale from Eq. (

The left panels (

In this section, we evaluate the predictability of two modifications to the original MLUCM developed in

Four scenarios tested in the present study with the combination of the EDMF approach (Sect.

Vertical profiles of velocity (

Figure

The prediction of streamwise velocity does not present a great variation, where 1D-

For TKE estimation, 1D-

The variation in the performance of scenarios is not pronounced across different height variability, which further consolidates the binning strategy for parameterization in the present study. To evaluate the comprehensive performance of scenarios over different height configurations, Fig.

Difference in the root mean square error (RMSE) between the original MLUCM v2.0 (

The new parameterizations with refined transport characterization represent an overall improvement over the previous multi-layer model

This study refined the characterization of the transport of flow properties for the multi-layer urban canopy model via a separate parameterization of TKE diffusion and introduced a “mass-flux” term for the dispersive transport of momentum. The updated multi-layer model demonstrates an improved performance by correcting the underestimation of turbulent exchange and velocity below the mean building height

By analyzing 49 LES simulations over staggered urban arrays of uniform and variable building height and comprehensive density coverage, we found the turbulent exchange rate of momentum is similar to that for scalars but 3.5 times lower than TKE. However, the previous model (MLUCM v2.0,

Regarding applications, the enhanced model, which addresses the underprediction of in-canopy TKE, has the potential to alleviate the overestimation of daytime air temperature and diurnal temperature range

We also revealed the distinct horizontal distribution of dispersive momentum flux, which is induced by flow heterogeneity responding to windward (quadrant 3, upward), lateral (quadrant 4, downward), and leeward (quadrant 2, downward) flow patterns. In response to this complication, we applied a sampling filter to the cumulative density function of the dispersive transport to segment the building-induced contribution (MF) that scales well with the pressure gradient and the rest of the relatively undisturbed component that can be securely lumped into the turbulent counterpart in the 1.5-order turbulent closure (ED). The EDMF framework, having successfully modeled the planetary boundary layer with flow heterogeneity induced by thermal effects, was demonstrated to be also favorable for the representation of flow heterogeneity caused by rigid building volumes. The new framework enables an explicit consideration of the impact of flow heterogeneity in MLUCM, improving our ability to calibrate the model for simulations of complex canopy flow.

With adaptations to the parameterizations of drag coefficient and dissipation length scale to account for building height variability, the model configuration that includes both the updated eddy diffusivity of TKE (

The turbulent length scales show a great variance over heights for layouts with non-uniform building height. A height-dependent parameterization of turbulent length scale and drag profile is necessary to correctly simulate features of urban flow such as the logarithmic or exponential behavior of the wind speed profile

The dispersive transport of TKE and scalars can be considered similarly following the dispersive momentum flux parameterization. In the original implementation of EDMF in PBL parameterizations

Staggered building arrays were found to be less representative of flow over realistic dense urban neighborhoods

Similar to the scarcity of evaluations of the eddy diffusivity of flow properties other than momentum, the variation of the TKE budget terms is still poorly considered in urban flow models. Accordingly, a more comprehensive analysis that addresses the relationships between the urban morphology and the TKE budget may provide further improvements.

The applicability of the 1.5-order

The sampling procedure to extract the components of building-induced dispersive fluxes was conducted as follows: (a) separating the dispersive fluxes into gradient (downward) and counter-gradient (upward) components, (b) sorting these two fluxes into a descending order in terms of the magnitude, (c) lowering the cumulative density function of the two fluxes from the largest allows a smaller subdomain to be sampled until subdomains only enclose transport events connected to buildings, and (d) spatially averaging the sampled dispersive fluxes over subdomains for both gradient and counter-gradient dispersive momentum flux. Figure

The three-dimensional distribution of the sampled dispersive structures is shown in Fig.

Sampling regions of dispersive fluxes for five (

Three-dimensional distribution of the dispersive coherent structures for cases with S-highrise-SD56 covering

The source code and the supporting data of three scenarios for the 1D Multi-layer Urban Canopy Model are publicly available at

The dataset is available in the Supplement.

The supplement related to this article is available online at:

JL, NN, MAH, ESK, and AM collectively developed and planned the study. JL ran the large-eddy simulations designed by NN and modified the model with the help of NN and AM. JL carried out the result analyses and wrote the paper with significant input and critical feedback from NN, MAH, ESK, and AM.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

Simulations were undertaken with the assistance of resources and services from the Australian Government's National Collaborative Research Infrastructure Strategy (NCRIS), with access to computational resources provided by the National Computational Infrastructure (NCI), which is supported by the Australian Government through the National Computational Merit Allocation Scheme. We also thank the anonymous reviewers for their constructive and valuable comments.

This research has been supported by the Australian Research Council Centre of Excellence for Climate Extremes (Grant CE170100023).

This paper was edited by Yongze Song and reviewed by two anonymous referees.