# Grey-Zone Turbulence in the Neutral Atmospheric Boundary Layer

## Abstract

The turbulence generated by wind shear is described at grey-zone resolutions using a theoretical neutral boundary layer based on atmospheric conditions constructed from measurements from the CASES-99 field campaign. Six-metre-resolution large-eddy simulations (LES) are performed to access the “true” resolved turbulence for two cases, corresponding to a forcing of the boundary layer by zonal geostrophic wind speeds of \(10\,\text {m}\,\text {s}^{-1}\) and \(20\,\text {m}\,\text {s}^{-1}\). The LES fields are subject to a coarse-graining procedure in order to compute turbulence diagnostics in the grey zone, with the robustness and weakness of various averaging procedures tested, for which simple top-hat averaging is found to be both suitable and accurate. In addition, the “true” resolved and subgrid-scale fluxes, variances, turbulent kinetic energy and production terms are quantified on various scales. The grey zone of turbulence is defined as the range of scales where 10–90% of turbulence is resolved, which here ranges from resolutions of 25–\(800\,\hbox {m}\) (\(0.03<\Delta x/h<1\), where \(\Delta x\) is the horizontal resolution, and *h* is the boundary-layer height). The subgrid/resolved partitioning of the variances of the velocity components depends on the geostrophic wind speed, which is not the case for the momentum-flux partitioning. Dynamic production terms show that fine-scale turbulence is isotropic (\(\Delta x/h<0.03\)) and is quasi one-directional, oriented in the direction of the geostrophic wind vector at the mesoscale (\(\Delta x/h>1\)). The turbulence parametrizations, which are tested in the Méso-NH model by running simulations at resolutions from the LES scale to the mesoscale, fail to produce the correct turbulence regardless of resolution.

## Keywords

Grey-zone turbulence Large-eddy simulation Neutral boundary layer## 1 Introduction

The grey zone of turbulence is defined by Wyngaard (2004) as the range of scales where the resolution of the model (\(\Delta x\)) is close to the size of the turbulent structures. At these resolutions, the turbulent boundary-layer structures are partly resolved and partly subgrid scale. Beare (2014) estimated the dominant length scales in the atmospheric boundary layer (ABL) from large-eddy simulations (LES), and defined the grey zone of turbulence as the scale where both the inversion depth and the dissipation length scale are of similar magnitude. Honnert et al. (2011) studied the convective boundary layer (CBL) from LES results, and estimated that the grey zone may extend from 0.2*h* to 2*h*, where *h* is the boundary-layer height plus the depth of the cloud layer above, i.e the hectometric scales.

Honnert et al. (2011) presented theoretical similarity functions describing the resolved and subgrid-scale partitioning of turbulence, comparing these functions to actual simulations with a large set of parametrizations, to demonstrate that, in the free CBL, no parametrization is able to reproduce correctly the decrease in subgrid-scale turbulence existing at finer resolutions. Therefore, parametrizations must be adapted to the grey zone of turbulence.

Since then, while modifications of parametrizations have been made to run models at resolutions within the grey zone of turbulence, most of these modifications only concern the CBL. Boutle et al. (2014) blended a tri-dimensional Smagorinsky scheme with a uni-dimensional non-local ABL scheme with the help of the similarity functions proposed by Honnert et al. (2011), which were introduced to describe the CBL. Ito et al. (2015) extended the Mellor and Yamada (1982) parametrization by modifying the length scales using statistics obtained from a coarse-graining procedure LES investigations of the dry convective mixed layer in the free convective region. Shin and Hong (2015) quantified the local and non-local turbulence in the free CBL, and investigated the CBL forced by flow at scales in the grey zone in order to adjust the vertical profiles produced from their non-local K-gradient scheme. Dorrestijn et al. (2013) proposed a stochastic scheme for the grey zone of turbulence based on data of an oceanic CBL case.

Non-local thermal turbulence structures, which are the dominant and largest structures in the CBL, are the first structures to be partly resolved when model resolution increases. Thus, the parametrization of thermal turbulence is the first parametrization modellers have to deal with at grey-zone resolutions. Honnert and Masson (2014) proved, however, that dynamic turbulence is not negligible in the free or forced CBL at grey-zone resolutions.

Until now, the grey zone of turbulence of dynamic origin has not been investigated. The dynamic production of turbulence is of a different nature from the thermal production: turbulent coherent structures produced by free convection are large and vertical (CBL thermals), whilst the structures produced by wind shear are tilted and three-dimensional. These differences may affect both the range of scales concerning the grey zone and the parametrizations of the turbulence at these particular scales. Therefore, as shear-driven turbulence has to be studied at sub-kilometre scales, it is difficult, if not impossible, to separate dynamic and thermal turbulence in LES investigations based on real cases. Observations of purely dynamic turbulence are rare and perhaps unrealistic, which explains the necessity of using an idealized LES approach of neutral cases for the investigation of dynamic turbulence in the grey zone.

Investigated here is the grey zone in a neutral atmospheric boundary layer by means of a LES approach. In Sect. 2, the benchmark cases from the CASES-99 field campaign are presented. In Sect. 3, the subgrid-scale velocity variances, turbulent kinetic energy (TKE), and momentum fluxes are diagnosed in the grey zone of turbulence of a neutral case, as well as the dynamic turbulence-production terms. Sub-kilometre simulations are performed and compared with the LES reference case in order to test parametrizations in Sect. 4. The last section is dedicated to discussions and conclusions.

## 2 Benchmark Simulations

### 2.1 Reference Experimental Data

The simulations are based on the CASES-99 experiment, which took place from 1 to 31 October 1999 in Kansas, USA, on a flat and homogeneous site. While the experiment was originally designed to investigate the stable ABL, and the morning and evening transition periods (Poulos et al. 2002), measurements were also made in near-neutral conditions, which can be used to simulate purely dynamic turbulence. The experimental data chosen here are described in Drobinski et al. (2007).

The prescribed roughness length is \(0.1\,\hbox {m}\), which generates a friction velocity of about \(0.42\,\text {m}\,\text {s}^{-1}\) at the prescribed wind speed. A constant (\(293.15\,\hbox {K}\)) potential temperature is imposed up to an altitude of \(750\,\hbox {m}\) (Drobinski et al. 2007), and then a constant adiabatic gradient is applied up to \(1500\,\hbox {m}\) in height. A stable layer suppresses turbulence above an altitude of \(750\,\hbox {m}\) (initially) and limits the size of the eddies. The heat and humidity fluxes at the surface are zero during the simulations, and there is no water vapour in the simulation. The buoyancy flux is, therefore, zero inside the ABL during the simulations, and the production of turbulence is purely dynamic. The zonal geostrophic wind speed is prescribed for two sets of simulations of geostrophic wind speeds of \(10\,\text {m}\,\text {s}^{-1}\) and \(20\,\text {m}\,\text {s}^{-1}\), which are denoted as slow and fast simulations, respectively.

### 2.2 Large-Eddy Simulations

Figure 1a, b shows, respectively, the vertical profiles of the potential temperature, and the hodographs of the velocity vector of the two sets of simulations. Even if the temperature of the fast simulations slightly increases during the simulation, the profile remains neutral and quasi-constant in the ABL over the 5-h simulation period. However, the top of the ABL, however, varies and depends on the geostrophic wind speed. Figure 1b shows the hodographs of the fast and slow simulations after 5 h of simulation time, presenting a regular rotation of the wind vector with height and an increase in wind speed typical of an Ekman layer. If the ABL height is defined as the lowest altitude where the hodograph arrows are parallel to the geostrophic wind direction, the ABL heights of the fast and slow simulations are \(h \approx 1200\,\hbox {m}\) and \(h \approx 985\,\hbox {m}\), respectively.

## 3 Results

### 3.1 Partial Similarity Functions

*h*defined by

The value of \(P_{{ SGS}_e}\) is small at fine resolutions, and close to one at the mesoscale. The grey zone of turbulence is defined here as that covering the range of scales for which the resolved turbulence is 10–\(90\%\) of the total TKE. For the large scale (\(\Delta x>800\,\hbox {m}\), or \({\Delta x}/{h}>1\)), Fig. 2 shows that the total TKE is mainly composed of subgrid-scale TKE (resolved TKE less than \(90\%\)). In this case, the TKE is mainly resolved (more than \(90\%\)) for resolutions finer than \(25\,\hbox {m}\) (or \({\Delta x}/{h}<0.03\)) in the slow and fast simulations. For the intermediate resolutions, the subgrid-scale turbulence increases with the grid resolution. The grey zone of turbulence of the neutral ABL can then be defined as the range of scales between \(0.03<{\Delta x}/{h}<1\) (approximately 25–\(800\,\hbox {m}\)) in the slow and fast simulations. The resolved turbulence (not shown) equals the subgrid-scale turbulence at \({\Delta x}/{h}\approx 0.27\) (about \(\Delta x\approx 250\,\hbox {m}\)). In the grey zone, the datasets present a large variability because the resolved TKE depends on the altitude, the size of the eddies, and the anisotropy, among other factors. Outside the grey zone, the variability is smaller.

To study the anisotropy of the turbulent motions in the ABL, the partial similarity functions of the variance of the velocity components are analyzed to give information on the size of the structures: at a given resolution, the larger the field structures, the more they are resolved. When the subgrid-scale turbulence is large, the structures are smaller than when the resolved turbulence is large. Figure 2 shows that the zonal velocity component (the free atmosphere) has larger structures than the meridional component, which has in turn larger structures than the vertical velocity component. The partitioning of the variances depends on the imposed geostrophic wind speed in the slow and fast simulations, with the latter structures rapidly resolved, while the subgrid-scale turbulence of the slow simulation is significant, even at high resolutions, which shows that the eddies are larger in the fast than in the slow simulation.

In contrast, the partitioning of the momentum fluxes (cf. Fig. 2) is the same for both the slow and fast simulations. At a given resolution, the intensity of the subgrid-scale exchange of momentum inside the ABL does not depend on the geostropic wind speed. The partitioning of the vertical fluxes (\(\overline{u'w'}^{\Delta x}\) and \(\overline{v'w'}^{\Delta x}\)) is similar, while the subgrid-scale flux \(\overline{u'v'}^{\Delta x}\) is smaller. Thus, the fields of \(\overline{u'w'}\) and \(\overline{v'w'}\) show structures of the same size; the \(\overline{u'v'}\) field presents larger structures, as this flux is more resolved at a given resolution.

### 3.2 Dynamic Production

*x*, the sum of the terms containing a gradient along

*y*forms the meridional production, whereas the vertical production is the sum of terms containing a vertical gradient.

Figure 3 shows the “true” percentages of the zonal, meridional and vertical dynamic production of turbulence as a function of the scaled resolution \(\left( {\Delta x}/{h}\right) \). Boxplots present the dispersion of the production terms in each class of \({\Delta x}/{h}\). Although the boxplots summarize the production by the two LES in one unique group, the production terms are also shown by lighter colours in Fig. 3 (slow: triangles; fast: plus signs).

For fine resolutions (\(\Delta x < 25\ \hbox {m}\), \({\Delta x}/{h}<0.03\)), the zonal, meridional and vertical production terms of turbulence are of the order of \(33\%\), implying that the turbulence is isotropic regardless of the geostrophic wind speed. For resolutions between 25 and \(100\,\hbox {m}\) (\(0.03<{\Delta x}/{h}<0.1\)), the proportion of the vertical and meridional production of turbulence decreases, whilst the zonal production increases. For coarser resolutions (\(\Delta x > 100\,\hbox {m}\)), the zonal production is predominant, because it lies along the geostrophic wind direction. At a resolution of \(3.2\,\hbox {km}\) in the slow simulation, zonal production is responsible for about \(70\%\) of the total production. The turbulence is, therefore, isotropic for fine resolutions, while it is anisotropic for scaled resolutions coarser than 0.03, with stronger dynamic production in the wind direction.

## 4 Model Faults

*h*is computed using Eq. 4 as the height at which the total TKE is 5% of the average total TKE below

*h*. This diagnostic approach assumes that the total TKE is large in the ABL and approximately null in the free atmosphere; however, the TKE is not zero in the free atmosphere (see Fig. 5). In the CBL, the lack of mixing is compensated by the resolved turbulence (see Honnert et al. 2011), but the total TKE in the neutral ABL is too large and too close to the total TKE in the free atmosphere at large scales. The limit between the turbulence in the ABL or in the free atmosphere is, therefore, unclear, as this diagnostic does not detect the value of

*h*, but the top of the model, or overestimates the value of

*h*. At a resolution finer than \(400\,\hbox {m}\), on the contrary, the Deardorff mixing length produces excessive turbulence, especially close to the surface, and as a consequence, the ABL height is too small compared with the LES results, and is not defined for resolutions smaller than \(200\,\hbox {m}\).

The conclusions are similar with reference to the rotation of the wind vector in the ABL (not shown). A rotation of the wind vector is shown in Fig. 1b, and is well-reproduced by the simulations with the BL89 mixing length regardless of the resolution. With the Deardorff mixing length, however, the coarser the resolution, the smaller the magnitude of the rotation, and, when the resolution is finer than \(200\,\hbox {m}\), the wind direction becomes chaotic.

Furthermore, the simulations at several resolutions have been compared with the averaged fields of the LES results. Despite the good representation of the turbulence in the spectral space, and since the ABL height and the wind rotation are ill-represented at all resolutions with the Deardorff mixing length, this mixing length is considered as inapplicable in the grey zone. However, results are similar with the BL89 mixing length regardless of the averaging procedure (see Appendix 1), geostrophic wind speed, or resolution: the BL89 mixing length never produces resolved turbulence.

## 5 Conclusion

Grey-zone turbulence of dynamic origin is studied for two LES cases corresponding to a neutral ABL forced at the domain top by two different geostrophic wind speeds. The simulations are subject to a coarse-graining procedure (see Appendix 1). The “true” subgrid-scale/resolved partitioning of the TKE, variances of the velocity components, momentum fluxes, as well as the dynamic production terms, are computed, with results showing that, below a 25-m resolution (about \({\Delta x}/{h}=0.03\)), the turbulence is mainly resolved and the subgrid-scale turbulence is isotropic. In contrast, the turbulence is entirely found at the subgrid scale at resolutions coarser than 800-m resolution (about \({\Delta x}/{h}=1\)). There is a grey zone of turbulence in a neutral ABL, which extends from a 25-m to a 800-m resolution where the turbulence is partly resolved but non-isotropic. The size of the structures, and thus the definition of the grey zone, depends on the geostrophic wind speed: the greater the wind speed, the larger the structures. When the resolution increases, the larger structures are the first to be resolved and they enter the grey zone. At a given resolution, the magnitude of the subgrid-scale momentum fluxes inside the ABL does not depend on the geostrophic wind speed. The horizontal flux is resolved at coarser scales than the vertical momentum fluxes.

In order to test current parametrizations in the grey zone for horizontal dynamic turbulence, simulations are systematically made for nine horizontal resolutions from \(12.5\,\hbox {m}\) to \(3.2\,\hbox {km}\) with the BL89 and Deardorff mixing lengths and with one- and three-dimensional schemes, with results showing that the neutral ABL is ill-represented, especially at large scales. While this may not currently affect many numerical weather prediction models, as their resolutions are coarser than the grey zone of the neutral ABL, the neutral ABL is an interesting theoretical state. In the grey zone and at a resolution of a few hundred metres, the subgrid-scale turbulence is (partly) dynamic even in the free CBL since ABL thermals are (partly) resolved. It has been proven, however, that, at those resolutions, the turbulence is not isotropic, and most of the turbulence schemes are incorrect. An adaptation of three-dimensional isotropic-turbulence schemes to the grey zone of turbulence is, therefore, necessary at hectometric scales.

## Notes

### Acknowledgements

I would like to acknowledge Guillem Coquelet, Julien Leger and Clément Blot for their contributions, and Pascal Masquet for his careful comments.

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