# A Theoretical Analysis of Mixing Length for Atmospheric Models From Micro to Large Scales

^{1}CNRM, Université de Toulouse, Météo France, CNRS, Toulouse, France^{2}IRT Saint Exupéry, Toulouse, France

A new mixing length adapted to the constraints of the hectometric-scale gray zone of turbulence for neutral and convective boundary layers is proposed. It combines a mixing length for mesoscale simulations, where the turbulence is fully subgrid and a mixing length for Large-Eddy Simulations, where the coarsest turbulent eddies are explicitly resolved. The mixing length is built for isotropic turbulence schemes, as well as schemes using the horizontal homogeneity assumption. This mixing length is tested over three boundary layer cases: a free convective case, a neutral case and a cold air outbreak case. The later combines turbulence from thermal and dynamical origins as well as presence of clouds. With this new mixing length, the turbulence scheme produces the right proportion between subgrid and resolved turbulent exchanges in Large Eddy Simulations, in the gray zone and at the mesoscale. This opens the way of using a single mixing length whatever the grid mesh of the atmospheric model, the evolution stage or the depth of the boundary layer.

## 1 Introduction

The atmospheric boundary layer (ABL) is the part of the atmosphere directly in contact with and rapidly influenced by the surface (Stull, 1988). This region is characterized by its turbulence. Wyngaard (2004) defines the gray zone of turbulence as the grid scale close to the scale of the energy containing structures. These structures characteristics depend on the ABL type, convective or neutral, for instance. Honnert et al. (2011) studied the resolved and subgrid parts of dry and cumulus-topped convective ABL (denoted hereafter CBL) simulations between 62.5 m grid spacing (i.e., large-eddy simulations or LES) and 8 km grid spacing (i.e., mesoscale regime). They found that the gray zone of turbulence ranges between 0.2 and 2 times the ABL height in convective cases (or the top of the cloud layer in cumulus-topped cases). Beare (2014) has defined an effective length scale for numerical models, *h* is the boundary-layer height). Honnert (2018), using the method of Honnert et al. (2011), showed that the gray zone ranges between 0.03*h* and *h* in neutral cases. In a nutshell, the gray zone of turbulence concerns the hectometric scales.

Numerical Weather Prediction (NWP) models are nowadays operationally used at the mesoscale (kilometric scale) and need improved turbulence schemes in the gray zone. At the mesoscale, large scale turbulence may be represented by non-local diffusion, using a non-local Eddy-diffusivity-based scheme (Lock et al., 2000) or a mass-flux scheme (Hourdin et al., 2002; Siebesma et al., 2004; Pergaud et al., 2009). However, Honnert et al. (2011) showed that these turbulence schemes may have difficulties to adapt in the gray zone of turbulence. Typical mesoscale turbulence schemes such as mass-flux schemes (e.g., PM09, Pergaud et al., 2009) produce too much subgrid mixing. Honnert et al. (2016) and Lancz et al. (2017) proposed to weaken the mass-flux scheme in order to produce resolved turbulent structures in the gray zone of turbulence. However, at the hectometric scales, non-local structures may be widely resolved whereas the rest of the turbulence, the local turbulence, cannot be considered as homogeneous and isotropic and thus cannot be directly treated by current LES schemes (Honnert and Masson, 2014). As a consequence, the local turbulence has also to be adapted to the gray zone.

The local turbulence in meteorological models is classically represented using eddy-diffusivity turbulence schemes. In a Smagorinsky scheme (Smagorinsky, 1963), for instance, the eddy-diffusivity depends on the shear and the mixing length.The latter usually represents the typical size of the energy-containing subgrid eddies.

Boutle et al. (2014) were the first to build a seamless turbulence scheme in the gray zone. They pragmatically blended a non-local scheme (Lock et al., 2000) and a 3D Smagorinsky scheme (Smagorinsky (1963), appropriate for well-resolved large eddies in LES regime) in a way that is scale-aware and related to the ratio of the grid-scale to a diagnosed length scale for the turbulence. The parameterization requires to modify the computation of the mixing length but also the subgrid flux blending the fluxes from non-local and 3D Smagorinsky schemes. This blending is effective even when it is not necessary: when the non-local turbulence is entirely resolved, fully 3D, or mainly horizontal, which happens in the gray zone at fine resolutions or over complex terrains.

Similarly, Ito et al. (2015) proposed an extension of Mellor-Yamada-Nakanishi-Niino (MYNN) model where the length scale is modified in order to maintain the turbulence kinetic energy (TKE) dissipation invariant to the grid spacing. The modification uses the partition function proposed by Honnert et al. (2011) in order to provide an appropriate subgrid TKE contribution. By using a coarse-graining approach on LES data, Kitamura (2015) estimated the length scale dependency on the grid spacing at the hectometric scale. He assumed that the turbulent fluxes have the form of a TKE-based model (Deardorff, 1980) and proposed two scale-aware mixing lengths, one vertical and the other horizontal. Recently, Kurowski and Teixeira (2018) defined a new mixing length by merging the existing mixing lengths formulations from LES and NWP model.

This mixing length may ‘overshoot’: when the large scale and the LES mixing lengths are close one to the other (which may happen in the gray zone), the resulting mixing length may be larger than each of the two mixing lengths.

At the same time, Efstathiou et al. (2018) worked on the dissipation of the Smagorinsky scheme at fine resolutions by adapting the mixing length to the growth of a CBL. The turbulence spin-up was improved in simulations where the resolution was coarser than in LES. However, the technique seems too expensive for operational applications at the hectometric scales. For the coarser resolutions of the gray zone of turbulence, Ďurán et al. (2020) modified the Bougeault and Lacarrère (1989) mixing length based on budget of TKE for coarse-grained LES. However, they did not intend to reach the LES scales.

NWP models cover a range of scale from 300 m to the mesoscale (2 km and above). They could even reach 100 m resolution in the foreseeable future. For this range of scale, turbulent eddies in the stable boundary layers are still entirely subgrid and thus will not be treated in this article. Indeed, even for a weakly stable boundary layer, a 3vm grid spacing (or even less) is necessary to be considered resolved (Beare et al., 2006). Furthermore, Beare et al. (2006) have shown the sensitivity to Smagorinsky mixing length, backscatter or dynamic model at a grid length of 6.25 m. The sensitivity to subgrid model can be considered as a hallmark of the gray zone. Although, authors dealing with the stable boundary layer gray zone do not call it the gray zone, just an unresolved LES, the gray zone of turbulence in stable boundary layers can be handled by stochastic parameterizations (Brown et al., 1994). or dynamic (Esau, 2004; Khani and Waite, 2014; Khani, 2018). However, depending on the grid-spacing and the boundary layer depth, models do represent eddies in neutral and convective boundary layers either as LES, gray zone or uniquely vertical subgrid motions. Most of the previous-mentioned attempts to modify the turbulence schemes in the gray zone of turbulence are restricted to a part of the scale range from LES to mesoscale.

The objective of this paper is then to propose a new unique mixing length adapted to the gray zone of turbulence for both convective and neutral ABL, pragmatic but based on theoretical foundations and allowing a robust use from LES (typically 50–100 m) to NWP models (at any scale). The paper is organized as follows. A new adaptive formulation for *l* in the gray zone of turbulence is presented in Section 2. This new mixing length is tested over three boundary layer cases. Their descriptions and the evaluation method are detailed in Section 3. Results are presented in Section 4. A discussion and some future directions are discussed in Section 5. At the end, a summary of findings is given in Section 6.

## 2 Mixing Length for all Scales

### 2.1 Mesoscale Mixing Lengths

In climate, NWP or mesoscale research models, non-local mixing lengths are commonly used. Figure 1 represents a schematic eddy created by a non-local mixing length in a large scale model. At large scales, the horizontal resolution (larger than 1 km) is coarser than the vertical resolution closed to the surface (usually finer than a few tens of meters). A non-local mixing length represents large ABL eddies included in the horizontal grid but larger than the vertical one. Non-local mixing lengths have been developed to represent coherent structures, particularly for convective cases (Bougeault and Lacarrere, 1989; Holtslag and Boville, 1993; Lenderink and Holtslag, 2004). For instance, Bougeault and Lacarrère (1989) (hereafter named as BL89) is used in operational NWP models such as AROME (Seity et al., 2011) or WRF^{1} (Shin and Hong, 2016). The size of the largest ABL eddies in BL89 is given by:

where *z* is the altitude,

**FIGURE 1**. Schematic turbulent eddy simulated by a non-local mixing length over the grid of a large scale model.

Local formulations of the mixing lengths can be based on the local vertical wind shear (Hunt, 1988; Tjernström, 1993; Schumann and Gerz, 1995; Grisogono and Enger, 2004; Grisogono and Belušić, 2008; Grisogono, 2010; Venayagamoorthy and Stretch, 2010). Recently, Rodier et al. (2017) (hereafter RM17) have improved BL89 to make it physically more accurate for all stratification conditions. Indeed, BL89 is infinite for neutral cases. However, in case of strong wind shear, the ABL turbulence is disorganized and the size of the turbulent eddies is reduced compared with a free-convective case. In order to reproduce the shear-induced vertical disorganization of the free-buoyancy eddies, RM17 integrates the local vertical wind shear within the original BL89 formulation. The size of the largest ABL eddies in RM17 becomes:

where

The mixing length is then:

Redelsperger et al. (2001)correction is applied in the first 25 m of the model to improve the representation of the wind profile in the surface boundary layer. Moreover, Rodier et al. (2017) promoted the use of a dissipation constant (

### 2.2 Large-Eddy Simulation Mixing Lengths

Large-Eddy simulations (Lilly, 1967; Deardorff, 1980; Pope, 2000) resolve most of the turbulence scales. LES are commonly used as a state-of-the-art numerical tool in order to study turbulence or to develop and evaluate parameterizations (Couvreux et al., 2010; Rio et al., 2010).

In LES, the grid is much smaller than the largest turbulent eddies and the nearly cubic grid fits the residual isotropic subgrid turbulence (see Figure 2). In this case, the most energy-containing subgrid eddies have the size of the grid box:

where

This mixing length (hereafter

where *N* is the Brunt-Väïsälä frequency.

### 2.3 A New Mixing Length From Micro to Large Scales

None of the above-described mixing lengths are adapted to the gray zone of turbulence. Indeed,

**FIGURE 3**. Schematic turbulent eddy in the gray zone of turbulence when the mixing length depends on **(left)** or not **(right)**.

With

where *α* is a tuning parameter set to 0.5, according to the numerical experiments detailed in Section 4.3. This formulation is continuous and does not overshoot. Usually,

The

## 3 Model, Case Studies and Evaluation Method Description

### 3.1 Model

The model used in the present work is the atmospheric non hydrostatic research model Meso-NH (Lac et al., 2018). It is used here from LES to hectometric scales in idealized frameworks. The spatial discretization is based on the C grid of Arakawa with Cartesian coordinates. The advection scheme for momentum variables is a centered scheme of 4th order in space, and a Runge–Kutta time-splitting of 4th order in time, well adapted to LES as the effective resolution is *l*:

where *ϕ* and *e* is the TKE.

At mesoscale resolutions (horizontal mesh larger than 1 km), it can be assumed that the horizontal gradients and the horizontal turbulent fluxes are much smaller than their vertical counterparts: they are therefore neglected (except for the advection of TKE) and the turbulence scheme is used in its 1D version, as in AROME (Seity et al., 2011). At finer resolution, the whole subgrid equation system is considered in its 3D version, as it is the case in the present work. The microphysical scheme is a mixed-phase one moment scheme (Pinty and Jabouille, 1998) which considers three solid (ice crystal, snow, and graupel) and two liquid (cloud droplets and raindrops) hydrometeors. The radiation parameterizations come from ECMWF, and include the Rapid Radiation Transfer Model (Mlawer et al., 1997) for long-wave and Fouquart and Bonnel (1980) for shortwave radiations.

### 3.2 Case Studies and LES Specifications

Boundary-layer turbulence comes from boundary-layer convection (thermal-induced turbulence) and wind shear (dynamical-induced turbulence). Three cases combining differently one or the other source of turbulence are evaluated here, in dry or cloudy conditions. For all cases, the reference LES is run with

##### 3.2.1 Free Convective Boundary Layer

The first case is a free convective boundary layer (CBL). It is based on radio-soundings collected during the International ^{−1}). The weak vertical shear guarantees a free convective case. The well-mixed boundary layer reaches 1.2 km in the beginning of the afternoon. This case is chosen as it presents a relatively uniform site topography and a development of a typical continental CBL. The numerical domain is a 16 × 16 km^{2} area. Lateral boundary conditions are periodic. Seven hours of dynamics are reproduced. The horizontal grid mesh for the LES is 50 m. The vertical grid is stretched, but the spacing is always ＜100 m in the ABL, so that the grid boxes are close to cubes. The numerical diffusion is small, the time at which the

##### 3.2.2 Neutral Boundary Layer

The second case is a neutral boundary layer corresponding to the incoming flow conditions of case 2681829 of the Mock Urbain Setting Test experiment (hereafter must, Yee and Biltoft, 2004). This near full-scale experiment was conducted during September 2001 in Utah’s West desert, at US Army Dugway Proving Ground. The site is located 1310 m above the mean sea level and can be considered as flat and homogeneous. MUST, starting the September 25, 2001 at 1830 LT (local time = UTC^{−1} at *z* = 4 m height) and a shear-induced turbulence. This case is chosen as it represents a typical continental neutral boundary layer. Moreover, this case is of particular interest for further developments of the Immersed Boundary Method (IBM) version of the Meso-NH (MNH-IBM, Auguste et al., 2019). The numerical domain is a 40 km side square. The vertical elevation of the domain goes up to 1500 m high. The first vertical layer measures 10 m high. The vertical grid size increases linearly with a geometric ratio of 1.09 and reaches it maximum value (^{−1} at *z* = 10 m and the wind direction has an angle of −41° with respect to the *x*-axis. The case is purely neutral. The

##### 3.2.3 Cold Air Outbreak

The third case is a cold air outbreak. Cold-air outbreaks are large-scale cold air masses over relatively warm waters. This phenomenon is common in the North Atlantic during winter (Field et al., 2014). The CONSTRAIN campaign (Field et al., 2014) studied cold air outbreaks in the North Sea involving a complex microphysics with mixed phases of ice and liquid water and mesoscale 2D (cloud streets) or 3D (open cells) structures. Several models inter-comparisons have been conducted on this study-case: an inter-comparison of global (Tomassini et al., 2016), limited area models (Field et al., 2017) and LES simulations (de Roode et al., 2019). The present case is described in de Roode et al. (2019). It corresponds to a transition from stratocumulus to cumulus as the cloud deck is advected over an increasingly warmer sea surface temperature. This simulation combines turbulence from thermal and dynamical origins, as well as presence of clouds. Similarly to de Roode et al. (2019), the top of the LES domain is at 5 km, with a vertical grid spacing of 25 m between the surface and 3 km and stretched above. The simulations covers an area of 96 × 96 km^{2} and last 14 h (from 00 to 14UTC). In the present paper, the LES is run at 62.5 m (instead of 250 m in de Roode et al. (2019)) to insure LES conditions. In de Roode et al. (2019), due to the Louis (1979) parameterization, Meso-NH produced too strong heat surface fluxes compared with other simulations of the inter-comparison. Here, this point is improved by using the iterative method proposed by Fairall et al. (2003) to produce sea surface fluxes. Moreover, the present microphysics considers the mixed phase while most of de Roode et al. (2019) simulations were based on the warm case. In this paper, several mixing lengths (

### 3.3 Simulations at Gray Zone Resolutions and Their Evaluation

According to Honnert et al. (2011), the gray zone of turbulence ranges between 0.2 and 2 times the boundary-layer height for CBL. The boundary-layer height being about 1.5 km for both IHOP and CONSTRAIN experiments, horizontal resolutions between 100 and 500 m are therefore considered as in the gray zone of turbulence. For neutral cases, Honnert (2018) has shown that the gray zone of turbulence ranges between 0.03 and 1 times the boundary-layer height. For MUST, the boundary layer is of 1500 m in height and horizontal resolutions between 50 and 400 m are considered as in the gray zone of turbulence.

In kilometric-scale simulations, non-local schemes such as mass-flux schemes (Hourdin et al., 2002; Siebesma et al., 2004; Pergaud et al., 2009) are used to represent the non-local turbulence of CBLs. However, Honnert et al. (2011) proved that mesoscale mass-flux schemes are too diffusive in the gray zone of turbulence and cannot be used at hectometric scales. In addition, the presence of a non-local turbulence scheme would distort the new mixing length impact analysis on the turbulence scheme. This is the case even if the influence of the non-local scheme would be very reduced as in non-local schemes dedicated to the gray zone (Boutle et al., 2014; Honnert, 2016). In the present paper, all simulations are therefore undertaken without any additional non-local boundary-layer scheme. We will focus on resolutions between LES and 500 m (upper gray-zone). The

This study does not use direct observations. However, as seen in the previous section, the test cases come from intercomparison studies, meaning that they are close but simplified compared with the reality. We have respected the intercomparison setup where the LES simulations have been validated compared with the observations, allowing the use of those LES as a reference for the evaluation of the new mixing length simulations.

Quantification of the resolved and subgrid fluxes are key points for the parameterization evaluation in the gray zone. Honnert et al. (2011) use the coarse-graining of LES fields in order to quantify the ‘true’ resolved and subgrid part of the turbulence in the gray zone. It appears that the partitioning between subgrid and resolved fluxes in the mixed layer of CBLs follows one unique function depending only on

## 4 Results

### 4.1 Free Convective Boundary Layer

##### 4.1.1 Boundary-Layer Height

Figure 4 shows the boundary-layer height computed from the IHOP simulations at resolutions ranging from 50 to 500 m. Three different mixing lengths are used:

**FIGURE 4**. Time evolution of the boundary-layer height for the IHOP case run with RM17, DEAR and

During the first 5 h of dynamics and for all the configurations, the boundary layer height growth rate is similar to the reference LES (dark triangles in Figure 4) but with slightly lower values. Then, the growth of the CBL slows in the LES, which is not the case in high resolution simulations using

**TABLE 1**. Bias and standard deviation (separated by a

##### 4.1.2 Partial Similarity Functions

At the mesoscale, the turbulence is entirely subgrid and the boundary-layer eddies are smaller than the resolution of the model. In LES, the turbulence is mainly resolved, as it explicitly represents the largest and most energetic turbulent eddies. In-between, in the gray-zone of turbulence, the subgrid/resolved partitioning of the TKE follows the partial similarity function from Honnert et al. (2011). Figure 5 presents this subgrid/resolved TKE partitioning for the different IHOP simulations undertaken as a function of

**FIGURE 5**. Subgrid TKE as a function of the resolution normalized by the ABL height for IHOP simulations undertaken with RM17 (in blue), DEAR (in green) or

The black line in Fig. 5 is the partial similarity function of the TKE. The dashed lines are the first and the last vigintiles of the coarse-grained LES data. They represent the allowed margin of error. The data obtained with simulations using

It has been found (not shown in Figure 5) that the subgrid TKE evolution as a function of

##### 4.1.3 Spectral Analysis

In order to complete this analysis, Figure 6 presents the energy spectra after 6 h of dynamics, using

**FIGURE 6**. IHOP horizontal TKE spectral analysis, inside the boundary layer and at 13 local hour of dynamics, for simulations undertaken with

In Fig. 6, the gray dashed line is the Kolmogorov

### 4.2 Neutral Boundary Layer: Resolved TKE Vs. Coarse-Grained LES

The neutral MUST case allows to examine the behavior of the mixing lengths in the fine resolution part of the gray zone of turbulence. Figure 7 shows the vertical profiles of total (left column) and resolved (right column) TKE obtained for the MUST case. The reference LES (50 m grid spacing with DEAR) is represented by the solid black lines. Simulations are undertaken at four horizontal grid spacing (50, 100, 200, and 400 m) with three mixing lengths,

**FIGURE 7**. Vertical profiles of total **(left column)** and resolved **(right column)** TKE obtained in the MUST neutral configuration from LES (black), RM17 (yellow),

Focusing on the coarse-grained LES results, the TKE is more resolved at the finer resolutions. For instance, TKE is twice more resolved at 100 than at 400 m grid spacing. For all resolutions, no resolved TKE is obtained with RM17. In this neutral case, where sources of instabilities are smaller than in convective cases, the subgrid mixing of RM17 inhibits the resolved movements, even at finest resolutions. This highlights the need of using an appropriate mixing length in the gray zone.

At 50 m grid spacing, the good agreement between LES and

The results are similar for the total TKE (Figs 7c, 7e, 7g).

### 4.3 Sensitivity Tests to *α*

The new mixing length

The effective resolution of a model (Skamarock, 2003; Ricard et al., 2013) is the minimum wavelength correctly seen by the model. Due to the physical and numerical diffusion, it is always larger than the grid size (

In IHOP and MUST, the value of α was set to 0.5. The sensitivity of these cases results to α is here investigated. Figure 8 shows the subgrid TKE as a function of

**FIGURE 8**. Sensitivity test to the α from 0.33 to 1 in IHOP **(left)** and MUST **(right)**. The subgrid TKE is represented as a function of the resolution normalized by the ABL height. The reference partial similarity function for the TKE (Honnert et al., 2011) and the uncertainty margins are represented by the full and dashed black lines, respectively.

Concerning IHOP, Figure 8 left shows that, for the small values of *α* result in diffusive simulations. The subgrid turbulence is too large and the simulation never enters the LES regime because the turbulence is not resolved enough. At the meso-scale (i.e., for large values of

For the MUST case (see Figure 8 right), similarly to IHOP in LES regimes and in the gray-zone, large values of α (

Globally, extreme values of α should be avoided. For MUST, the best agreement with the reference partial similarity function is obtained with

### 4.4 Cloudy Evolving Boundary Layer

In the previous section, α has been tuned on idealized cases of convective (IHOP, in Section 4.1) and neutral (MUST, in Section 4.2) boundary layers. In this section,

##### 4.4.1 Ability to Reproduce the Cloud Types and Large Scales

The CONSTRAIN case is more critical as it combines turbulence from thermal and dynamical origins, as well as presence of mixed clouds. CONSTRAIN was held in the North of Scotland. The simulation begins at 00 UTC - 1 local hour.

Figure 9 shows the cloud top and base temporal evolution in simulations undertaken with DEAR, RM17 and

**FIGURE 9**. Temporal evolution of the cloud base and cloud top in CONSTRAIN with DEAR, RM17 and

**FIGURE 10**. Vertical profiles of mean cloud fraction for CONSTRAIN simulations at 2 h (left column) and 12 h (right column) at the grid spacing of 125 m (top row), 250 m (middle), 500 m (bottom row) with DEAR (in red), BL89 (in orange), RM17 (in green),

Figure 10 shows the vertical profiles of mean cloud fraction produced by the different mixing lengths in the stratocumulus stage (2 h local hour) and the cumulus stage (12 h local hour). One can see a good transition from stratocumulus to cumulus at 12 h for all simulations, with a decreasing cloud fraction and an increasing cloud depth (see Figure 9). Moreover, the different simulations produce a correct maximal cloud fraction in the stratocumulus stage despite a too low maximal cloud fraction at large scales (cf. Figure 11a). The maximal cloud fraction is also well-reproduced by

**FIGURE 11**. Vertical profiles of total **(A)** and 12 h (**B**).The gray area represents the cloud layer in the LES.

The main problem is that the coarser the resolution, the lower the cloud top (Figure 9). This can be mainly explained by the fact that the simulations do not have any non-local turbulence scheme such as a mass-flux scheme, limiting the cloud development.

During the first hours of the LES simulation *t* = 2 h, see Figure 10a) and cells of cumulus (at *t* = 12 h, see Figure 10b). As explained in de Roode et al. (2019), the two minima of

##### 4.4.2 Resolved TKE Vs. Coarse‐Grained LES

Figure 12 shows the resolved TKE produced by the different mixing lengths in the stratocumulus stage and the cumulus stage of CONSTRAIN. Focusing on the results of the coarse-graining LES (in black), it is observed that, similarly to the neutral case, the finer the resolution, the larger the resolved TKE. Contrary to the buoyancy where only a small sensitivity to the resolution has been observed, TKE is half less resolved at 500 m than at 125 m grid spacing.

**FIGURE 12**. Vertical profiles of TKE for CONSTRAIN simulations at 2 h (left column) and 12 h (right column) at the resolutions of 125 m (top row), 250 m (middle), 500 m (bottom row) with DEAR (in red), BL89 (in orange), RM17 (in green),

By construction, the **f**).

In the stratocumulus stage (see Figures 12a,**c**,**e**), the

In the cumulus part (see Figures 12b,d,f),

## 5 Discussion

As seen in this paper,

Over complex terrain, Cuxart (2015) underlines that hectometric scales using a 3D turbulence scheme are not in the LES regime, since the resolved motions fall outside the inertial sub-range and follow the definition of Smolarkiewicz and Prusa (2002) of the “Very-Large Eddy simulation” (VLES). In the same vein, Efstathiou et al. (2018) identified a regime called *near gray zone*, where most of the TKE is resolved but for which these coarse LES simulations should not be considered as LES converging because their grid length is not fine enough to present a clear inertial sub-range (see Sullivan and Patton, 2011). In a way, this paper deals with these VLES or near-gray zone regimes and

The turbulence is not isotropic (as in LES) in the gray zone of turbulence, due to the residual traces of non-local turbulence. This is why Kitamura (2015) proposed two mixing lengths in the gray zone, one horizontal and one vertical. In the present paper, as in LES, the turbulence scheme is 3D isotropic at all resolutions by construction.

At kilometric-scales,

The value of the α tuning parameter has to be valid whatever the boundary layer stability as RM17 is supposed to adapt to the BL stability. However, when *α* around 0.5, the determination of the α value should be undertaken as any other tuning parameter of the given model.

## 6 Conclusion

This work seeks to improve the turbulence scheme at the gray zone of turbulence scales, particularly when the resolution is too coarse and the meshes too anisotropic for the LES configuration to give a good subgrid/resolved partitioning of the TKE. This article describes a new mixing length,

where α is a tuning parameter that may depend on the boundary layer characteristics (not on the resolution). The sensitivity test to α performed herein establishes

## Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

## Author Contributions

RH and VM had the idea of this article. RH run IHOP experiment, plot the IHOP and CONSTRAIN images, write the article, TN run the MUST experiment, plot the associated images and write the part on MUST and CL run the CONSTRAIN experiment and write the part on the Méso-NH model. All authors have had a substantial contributions to the conception or design of the work and the acquisition and analysis of data for the work; They all revised the draft critically for important intellectual content. They all agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

## Funding

Tim Nagel postdoctoral position was funded by the FCS-STAE foundation and the IRT Saint-Exupéry, Toulouse, under the PPM project.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Acknowledgments

The authors would like to thank Robert Schoetter for his critical review of the performances of this new mixing length and Amélie Sterle for her technical support.

## Footnotes

^{1}The BL89 mixing length is activated with the BOULAC option in WRF.

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Keywords: mixing length, gray zone, turbulence, boundary layer, parameterization, large-eddy simulations

Citation: Honnert R, Masson V, Lac C and Nagel T (2021) A Theoretical Analysis of Mixing Length for Atmospheric Models From Micro to Large Scales. *Front. Earth Sci.* 8:582056. doi: 10.3389/feart.2020.582056

Received: 10 July 2020; Accepted: 20 October 2020;

Published: 11 January 2021.

Edited by:

Ismail Gultepe, Environment and Climate Change, CanadaReviewed by:

Sina Khani, University of Washington, United StatesEric Hendricks, National Center for Atmospheric Research (UCAR), United States

Copyright © 2021 Honnert, Masson, Lac and Nagel. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Rachel Honnert, rachel.honnert@meteo.fr