The Atmospheric Gray-Zone (a.k.a. Terra Incognita) Problem: A Strategy Analysis from an Engineering Viewpoint

Abstract

The Terra Incognita (or gray-zone) problem seen in atmospheric flow simulations causes serious consequences: it implies, e.g., significantly incorrect flow predictions and results that often simply depend on flow simulation settings as the computational grid applied. There is definitely the need for a robust gray-zone modeling to ensure that research and technology decisions are based on reliable results. As a matter of fact, solution approaches to deal with this problem in atmospheric and engineering type simulations reveal remarkable differences. In contrast to atmospheric flow simulations, there exists a broad spectrum of solution concepts for engineering applications. Driven by these conceptual differences, the paper presents an analysis of the Terra Incognita problem and corresponding solution concepts. Specifically, the paper presents a modeling approach that overcomes the core problem of currently applied methods. A new method of providing a resolution-aware turbulence length scale (one of the major problems in atmospheric flow simulations) is presented. This approach is capable of seamlessly covering the full range of microscale to mesoscale simulations, and to appropriately deal with mesoscale to microscale couplings.

1. Introduction

Accurate microscale (a resolution of meters to a few km) and mesoscale (a resolution of a few km to hundreds of km) simulations of atmospheric flows are essential for understanding, predicting, and managing weather, climate, and environmental impacts at local and regional scales. This concerns, e.g., high-resolution weather prediction, severe weather forecasting, urban and environmental applications (air pollution dispersion at street level, wind safety for buildings and infrastructure), wildfire and smoke behavior, climate and renewable energy applications, and disaster risk reduction (chemical/biological releases).

In particular, the coupling of mesoscale and microscale simulation methods is a major challenge of wind energy research [1,2,3,4,5,6]. Microscale simulation methods are capable of providing detailed information about terrain effects and interactions with structures such as wind turbines. The atmosphere drives changes at the wind plant scale; it is therefore essential to model the variability at the mesoscale. Consequently, the appropriate coupling of mesoscale and microscale simulation methods for atmospheric boundary layer (ABL) processes is highly relevant to the evaluation of topography effects and the evaluation of the suitability of wind turbine designs and wind farm effectiveness [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. Results of an extended study of strategies to deal with this problem (performed in the frame of the U.S. Department of Energy Mesoscale-to-Microscale-Coupling project) were presented recently [6,16,17,18,19,20,21,28,29]. There is definitely progress regarding the understanding of the nature of this problem and potential ways to address this issue. But simultaneously, there is so far no indication that a convincing solution to this problem can be expected in the near future. Doubrawa and Muñoz-Esparza [22] recently concluded “the need for a robust gray-zone modeling solution is urgent to ensure that research and technology decisions are based on reliable results”.

The technical nature of these issues is the following (an overview of terminology applied in this paper can be found in

Table 1. Overview of terminology applied in this paper.

Resolving vs. nonresolving simulations: Resolving simulations performed on sufficiently fine grids enable the calculation of most of the spectrum of turbulent motions. They include turbulent fluctuations produced by the simulation. Nonresolving simulations do not include fluctuations; the turbulence is reflected by a turbulence model. Resolved vs. modeled motion (or flow): Resolved motion stands for turbulent fluctuations produced by a resolving simulation. Modeled motion stands for turbulence described by an explicit model for turbulence.

RANS vs. LES [averaged vs. filtered equations]: RANS and LES equations can be derived using, e.g., ensemble averaging or filtering in space. With respect to simulations, there is no difference if the equation structure and parametrization of turbulence length scales are the same. The typical RANS-LES difference is the inclusion of a relatively large (small) turbulence length scale or viscosity, respectively. RANS (LES) are usually performed on relatively coarse (fine) grids focusing on flow modeling (flow resolution). Typically applied RANS equations can be used for resolving LES if the turbulence length scale and grid are sufficiently small. Used on appropriate grids, typically applied RANS equations can produce fluctuations; such RANS are called unsteady RANS (URANS).

Hybrid RANS-LES vs. RANS or LES: The term hybrid RANS-LES will be used for methods aiming at the ability to be seamlessly applicable to nonresolving and resolving simulations (in contrast to RANS or LES aiming at one of these cases). This view of hybrid RANS-LES includes, e.g., DES, which switches RANS and LES length scales to transition from one regime to the other (to enable a transition between modeled and resolved flow regions). Depending on the set-up, this view of hybrid RANS-LES can include WMLES, which applies RANS elements in some flow regions. This view of hybrid RANS-LES does not include typical atmospheric LES applied in conjunction with wall-modeling. Such simulations will be considered here simply as LES because the focus is still on flow resolution, i.e., not an a transition between resolution regimes.

Mesoscale vs. microscale simulations: Atmospheric mesoscale (microscale) simulations are performed with a resolution of a few km to hundreds of km (a resolution of meters to a few km). For simplicity, mesoscale and microscale simulations will be considered here as using incompletely resolving RANS and resolving LES. Specifically, such mesoscale simulations typically represent URANS, which includes the calculation of fluctuations.

Gray-zone vs. Terra Incognita: The term gray-zone refers to simulations on intermediate grids where the basic assumptions of RANS or LES are not satisfied. This may refer to coarse LES, where the resolved fluctuations do not need to represent realistic turbulence. This may refer to URANS where the appearance of fluctuations can lead to essentially misaligned production–dissipation mechanisms of equations. The term Terra Incognita is usually applied to refer to the gray-zone problem in atmospheric simulations. Dynamic LES is no option to address gray-zone problems. Dynamic LES is still resolving LES aiming at a better performance based on dynamic coefficient adjustments. The problem of under-resolution on coarse grids cannot properly be addressed by dynamic LES.

Scale-aware vs. resolution-aware parametrizations: Scale-aware parametrizations usually include reference to the filter width ($\Delta$) in equations (like usual LES parametrizations). Under many conditions, such a reference is an insufficient reflection of the real flow resolution. Resolution-aware parametrizations (like CES) include explicit reference to the actual flow resolution in equations (by inclusion of $k_{+}=k/k_{tot}$ or $L_{+}=L/L_{tot}$; see below).

DES vs. CES simulations: Both address the gray-zone issue, but very differently. DES switches RANS and LES length scales, hoping for a seamless transition between modeled and resolved flow regions. Rather often, the latter transition happens with significant reluctance, which can lead to a significant mismatch of modeled and resolved flow. These problems are avoided in CES because of the direct interaction of modeled and resolved flow.

). Microscale and mesoscale core techniques are well developed. Microscale simulations are usually performed by using the large eddy simulation (LES) technique in conjunction with surface parameterizations of heat and momentum fluxes on relatively fine grids that resolve energetic, turbulent eddies aside from the close proximity to the surface. Such simulations are characterized by a relatively small turbulent viscosity (a relatively small turbulence length scale). Mesoscale simulations are performed on the basis of Reynolds-averaged Navier–Stokes (RANS) equation methods in conjunction with relatively coarse grids. Turbulent fluctuations are damped out, which is managed through a relatively large turbulent viscosity (a relatively large turbulence length scale). The need to couple mesoscale and microscale simulation methods (and the use of mesoscale models at a higher resolution to improve simulation results) requires the use of LES (RANS) methods at a rather coarse (fine) resolution, which disagrees with the underlying physical principles of these techniques. For example, there is no guarantee that resolved fluctuations produced by coarse-grid LES relate to real turbulence. There is also no guarantee that mechanisms that govern RANS equations are properly functional in the presence of chaotic impacts (fluctuations). Thus, there exists a gray-zone under such conditions, a region of (at the minimum) an unclear standard of simulations. In regard to atmospheric flow simulations, the gray-zone existence was identified by Wyngaard (2004) and called the Terra Incognita problem [30].

In regard to the gray-zone problem, it is of interest that there exist significant differences between atmospheric and engineering solution approaches. In contrast to such developments in regard to atmospheric flows, a huge variety of methods have been developed in respect to engineering applications. One very popular way is wall-modeled LES (WMLES) [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46], where RANS components are involved close to solid walls. A variation in this approach is given by the development of Reynolds-stress-constrained LES (RSC-LES) [47,48,49,50,51,52,53,54,55,56,57,58]. Another very popular way is given by detached eddy simulation (DES) [59,60,61,62,63,64,65,66,67,68,69,70], where the performance of RANS models is improved by switching from the RANS turbulence length scale applied close to the wall to a much smaller LES-type length scale away from the wall. A variety of alternative methods were suggested [71,72,73,74], including unified RANS-LES (UNI-LES) [75,76,77,78,79,80,81], partially averaged Navier–Stokes (PANS) [82,83,84,85,86,87,88,89,90,91,92,93], partially integrated transport modeling (PITM) [94,95,96,97,98,99,100,101,102,103,104], and scale adaptive simulation (SAS) methods [70,105,106,107,108,109,110]. These developments were driven by needs to appropriately deal with separated turbulent flow [59,67,68] and many other flow simulations [111].

The motivation of this paper is to provide an analysis of the atmospheric gray-zone problem from an engineering viewpoint by simultaneously developing a solution concept for this problem. Section 2 deals with a closer look at the atmospheric gray-zone problem, its nature, its relevance, and currently applied solution concepts. The specific goal is twofold. First, the predominant trend of dealing with the gray-zone problem in regard to atmospheric flow simulations (to follow engineering DES concepts) is identified. Second, evidence obtained by engineering simulations is used to argue that this current trend implies a variety of issues. Section 3 presents a solution strategy to overcome the gray-zone problem in atmospheric flow simulations based on exact mathematical developments. Following CES concepts, a hybrid RANS-LES model is presented that is demonstrated to overcome the typical DES problems. This discussion includes computational evidence for conceptual CES advantages. Section 4 presents conclusions of this analysis.

2. The Atmospheric Gray-Zone Problem

Let us first have a closer look at the concrete problem considered, current solution concepts, and corresponding concepts used in engineering applications.

2.1. The Problem Considered

The atmospheric gray-zone problem is illustrated in Figure 1: Figure 1a shows a mesoscale RANS grid, Figure 1c shows an LES grid, and the grid in Figure 1b is in between the RANS and LES grids. Specifically, mesoscale RANS simulations do not attempt to resolve turbulence structures, whereas microscale LES simulations are focused on a resolution of turbulence structures. It is worth noting that mesoscale and microscale simulations are usually performed using very different equation structures (involving, e.g., one-dimensional (1D) and 3D turbulence schemes), and, first of all, by using very different turbulence length scales (a large-scale turbulence length scale L is used in mesoscale simulations, whereas an LES filter width $\Delta$ is applied in LES).

Significant questions arise in regard to simulations using intermediate grids as shown in Figure 1b. These are conditions where the grid spacings ($\Delta$) are comparable to the size of energy-containing turbulent eddies L. What can be wrong with such simulations? Such simulations are significantly affected by fluctuations (produced resolved motion), but these fluctuations cannot represent real turbulence because there is insufficient flow resolution. At the same time, there is no guarantee that mechanisms that govern RANS equations are properly functional in the presence of random disturbances (resolved fluctuations).

Explicit shortcomings of gray-zone simulations are described in Table 2. Some of the key issues are the following ones (the horizontal blocks in Table 2 are ordered according to the four following issues):

1.

Inappropriate parametrizations: (i) Turbulence schemes reveal drawbacks: the resolved part is too large (or too weak) if the turbulence scheme is used without (or with) the mass-flux scheme [112,113], (ii) schemes need to account for 3D turbulence in the gray-zone [113,114,115], (iii) there can be excessive mixing due to a large turbulence length scale at coarse resolutions, which results in the cut-off of resolved turbulence in the gray-zone [116].

2.

Spin-up need and its sensitivity: (i) Without perturbations (initiation of resolved turbulence called spin-up), it is possible that resolved turbulence does not become established at all [117]. (ii) Models require an enhancement of diffusion to give a collapse of turbulence at $\Delta x\propto h$ (h is the boundary-layer depth), whereas spin up of turbulence within the gray-zone requires a reduced diffusion [118].

3.

Grid and scheme dependence of results: When the grid spacing is similar to the dominant physical scales, then the grid influences what motions get allowed or forced, leading to artificial structures, and spurious triggering of convection or turbulence. For example, (i) mesoscale structures (rolls vs. cells, turbulence flux profiles, etc.) depend strongly on the horizontal grid resolution (and anisotropy between horizontal and vertical motions) [22,119], (ii) with respect to a tropical cyclone structure and intensity (eyewall slope, radius of maximum wind, latent heating, microphysics) simulation results vary significantly with the resolution; the results depend on convective parametrization schemes [120].

4.

Incorrect simulation predictions: Errors in regard to precipitation extremes, boundary layer depth, diurnal heating, and cloud fraction often increase in gray-zones. For example, (i) we see overestimations and erroneous precipitation locations or an erroneous simulation of heavy precipitation at high resolution [120,121,122], or (ii) the neutral ABL can be ill-represented (erroneous ABL height and wind-direction rotation, strong effect of resolution) [123].

Table 2. Failures in gray-zone atmospheric simulations. GZC refers to gray-zone coverage (usually through varying horizontal resolution). The horizontal blocks follow the summary paragraph in Section 2.1.

Cases and Resolution	Gray-Zone Performance and Requirements
Honnert et al. (2011) [112]; Honnert (2016) [113]: CBLs (five dry and cloudy cases); GZC; several mesoscale parametrizations	Turbulence schemes reveal drawbacks: resolved part is too large (or too weak) if the turbulence scheme is used without (or with) the mass-flux scheme; the turbulence scheme does not mix the boundary layer well enough, schemes need to account for 3D turbulence in gray-zone
Honnert (2018) [123]: neutral ABL; $3.2\times 3.2\times 1.5$ km3; GZC; different turbulence and length scale models	Neutral ABL is ill-represented (erroneous ABL height and wind-direction rotation, strong effect of resolution); most of the turbulence schemes are incorrect
Honnert et al. (2021) [124]: three boundary layer cases (a free convective case, a neutral case and a cold air outbreak case); GZC; mesoscale parametrizations; use of DES-type turbulence length scale	With new DES-type mixing length, the turbulence scheme produces improved proportions between subgrid and resolved turbulent exchanges in LES, in the gray zone and at the mesoscale
Juliano et al. (2022) [114]: idealized cases (growing CBL over homogeneous land surface, sea-breeze front, mountain–valley circulation); GZC; 3D ABL scheme	New 3D ABL scheme is very promising; turbulent length scale model (and master length scale concept) need improvements, in particular for convective and stable cases
Efstathiou and Beare (2015) [116]: 3D dry CBL simulations; domain of at least $9.6\times 9.6$ km2; ranging from the LES ($\Delta x=25$ m) to the mesoscale limit ($\Delta x=3200$ m)	Excessive mixing due to the large values of mixing length at coarse resolutions results in the cut-off of resolved turbulence in the gray zone; the damping of resolved motions comes earlier for wind shear runs
Kealy et al. (2019) [117]: 3D LES of day 33 of the Wangara experiment (a widely studied CBL case); domain of $9.6\times 9.6\times 2.5$ km3; ranging from the LES ($\Delta x=20$ m) to gray-zone runs ($\Delta x=200-800$ m)	Without perturbations (initiation of resolved turbulence called spin-up), resolved turbulence does not become established at all; (1) applying structured perturbations, and (2) using a dynamically evolving Smagorinsky coefficient are shown to encourage faster spin-up
Beare and Efstathiou (2025) [118]: 3D CBL LES; case of Sullivan and Patton [125]; domain of $5120\times 5120\times 2048$ m3, ranging from the LES ($\Delta x=50$ m) to gray-zone runs ($\Delta x=500-1000$ m)	Models require an enhancement of diffusion to give a collapse of turbulence at $\Delta x\propto h$ (h is the boundary-layer depth), whereas spin up of turbulence within the gray zone requires a reduced diffusion
De Roode et al. (2022) [119]: clear CBL, $12.8\times 12.8\times 1.594$ km3; GZC; LES	Mesoscale structures (rolls vs. cells, turbulence flux profiles, etc.) depend strongly on horizontal grid resolution (and anisotropy between horizontal and vertical motions)
Sun et al. (2013) [120]: sensitivity of Typhoon Shanshan (2006) simulations to changes in horizontal grid spacing and to choices of convective parametrization schemes at gray-zone resolutions (1–7.5 km); WRF model is applied	Tropical cyclone structure and intensity (eyewall slope, radius of maximum wind, latent heating, microphysics) vary significantly with resolution; results depend on convective parametrization schemes
Park et al. (2022) [121]: convective cell-related heavy rainfall in South Korea; effect of a scale-aware convective parametrization scheme (timescale and entrainment rates are adjusted based on the horizontal grid spacing) is compared with simpler schemes; gray-zone simulations	Simpler schemes imply overestimations and erroneous precipitation locations or an erroneous simulation of heavy precipitation at high resolution; scale-aware schemes can improve simulations although model parameter settings play a crucial role
Tomassini et al. (2023) [122]: several global cases (e.g., convection–circulation over Africa, Hurricane Dorian and Typhoon Goni, Darwin mesoscale convective systems case, Atlantic Extratropics) using 5 km resolution	Very similar observations as reported by Park et al. (2022) [121]: a scale-aware turbulence scheme and a carefully reduced and simplified mass-flux convection scheme outperform simpler schemes

Table 2. Failures in gray-zone atmospheric simulations. GZC refers to gray-zone coverage (usually through varying horizontal resolution). The horizontal blocks follow the summary paragraph in Section 2.1.

Cases and Resolution	Gray-Zone Performance and Requirements
Honnert et al. (2011) [112]; Honnert (2016) [113]: CBLs (five dry and cloudy cases); GZC; several mesoscale parametrizations	Turbulence schemes reveal drawbacks: resolved part is too large (or too weak) if the turbulence scheme is used without (or with) the mass-flux scheme; the turbulence scheme does not mix the boundary layer well enough, schemes need to account for 3D turbulence in gray-zone
Honnert (2018) [123]: neutral ABL; $3.2\times 3.2\times 1.5$ km3; GZC; different turbulence and length scale models	Neutral ABL is ill-represented (erroneous ABL height and wind-direction rotation, strong effect of resolution); most of the turbulence schemes are incorrect
Honnert et al. (2021) [124]: three boundary layer cases (a free convective case, a neutral case and a cold air outbreak case); GZC; mesoscale parametrizations; use of DES-type turbulence length scale	With new DES-type mixing length, the turbulence scheme produces improved proportions between subgrid and resolved turbulent exchanges in LES, in the gray zone and at the mesoscale
Juliano et al. (2022) [114]: idealized cases (growing CBL over homogeneous land surface, sea-breeze front, mountain–valley circulation); GZC; 3D ABL scheme	New 3D ABL scheme is very promising; turbulent length scale model (and master length scale concept) need improvements, in particular for convective and stable cases
Efstathiou and Beare (2015) [116]: 3D dry CBL simulations; domain of at least $9.6\times 9.6$ km2; ranging from the LES ($\Delta x=25$ m) to the mesoscale limit ($\Delta x=3200$ m)	Excessive mixing due to the large values of mixing length at coarse resolutions results in the cut-off of resolved turbulence in the gray zone; the damping of resolved motions comes earlier for wind shear runs
Kealy et al. (2019) [117]: 3D LES of day 33 of the Wangara experiment (a widely studied CBL case); domain of $9.6\times 9.6\times 2.5$ km3; ranging from the LES ($\Delta x=20$ m) to gray-zone runs ($\Delta x=200-800$ m)	Without perturbations (initiation of resolved turbulence called spin-up), resolved turbulence does not become established at all; (1) applying structured perturbations, and (2) using a dynamically evolving Smagorinsky coefficient are shown to encourage faster spin-up
Beare and Efstathiou (2025) [118]: 3D CBL LES; case of Sullivan and Patton [125]; domain of $5120\times 5120\times 2048$ m3, ranging from the LES ($\Delta x=50$ m) to gray-zone runs ($\Delta x=500-1000$ m)	Models require an enhancement of diffusion to give a collapse of turbulence at $\Delta x\propto h$ (h is the boundary-layer depth), whereas spin up of turbulence within the gray zone requires a reduced diffusion
De Roode et al. (2022) [119]: clear CBL, $12.8\times 12.8\times 1.594$ km3; GZC; LES	Mesoscale structures (rolls vs. cells, turbulence flux profiles, etc.) depend strongly on horizontal grid resolution (and anisotropy between horizontal and vertical motions)
Sun et al. (2013) [120]: sensitivity of Typhoon Shanshan (2006) simulations to changes in horizontal grid spacing and to choices of convective parametrization schemes at gray-zone resolutions (1–7.5 km); WRF model is applied	Tropical cyclone structure and intensity (eyewall slope, radius of maximum wind, latent heating, microphysics) vary significantly with resolution; results depend on convective parametrization schemes
Park et al. (2022) [121]: convective cell-related heavy rainfall in South Korea; effect of a scale-aware convective parametrization scheme (timescale and entrainment rates are adjusted based on the horizontal grid spacing) is compared with simpler schemes; gray-zone simulations	Simpler schemes imply overestimations and erroneous precipitation locations or an erroneous simulation of heavy precipitation at high resolution; scale-aware schemes can improve simulations although model parameter settings play a crucial role
Tomassini et al. (2023) [122]: several global cases (e.g., convection–circulation over Africa, Hurricane Dorian and Typhoon Goni, Darwin mesoscale convective systems case, Atlantic Extratropics) using 5 km resolution	Very similar observations as reported by Park et al. (2022) [121]: a scale-aware turbulence scheme and a carefully reduced and simplified mass-flux convection scheme outperform simpler schemes

2.2. Solution Concepts

Thorough reviews of ongoing activities to overcome the obvious shortcomings of dealing with gray-zone treatments in atmospheric simulations can be found elsewhere [22,115]; see also the references reported in Table 2. Specifically, there is work along the following lines:

1.

Incorporation of turbulence anisotropy: When the horizontal grid spacing is relatively coarse, horizontal (momentum, heat, and scalar) gradients are relatively small compared to vertical gradients, such that 1D (vertical) turbulence models are appropriate. But within the gray-zone (using finer grids), such horizontal gradients become nonnegligible and 3D effects of turbulence should be considered [114,123]. Thus, the inclusion of anisotropy is clearly an opportunity to substantially improve simulation results. A valuable extension of previously applied simulation methods has been presented recently by Juliano et al. [114]. Specifically, the authors presented a 3D ABL parametrization based on the level 2.5 model of Mellor and Yamada [126] and demonstrated the advantages in comparison with simpler methods. Additional attempts are described elsewhere [115].

2.

Turbulence length scale transport models: More specifically in regard to the inclusion of an appropriate turbulence length scale, it is worth noting the following. The predominant present approach to address this problem is the definition of appropriate algebraic length scales. An approach that can provide more generally applicable turbulence length scales is the use of two-equation turbulence models which enable (in one or the other way) the inclusion of length scales subject to local turbulence production and dissipation processes. Only a few attempts exist to address the problem in this way, see, e.g., Refs. [127,128,129,130,131,132,133,134].

3.

Turbulence length scale merging: Mesoscale and microscale regimes differ essentially through the inclusion of relatively large and small length scales, respectively. As is well-known, the inclusion of a length scale that appropriately describes the structure of the turbulence in the gray-zone represents a highly relevant challenge: the simple use of usually applied RANS or LES length scales is proven to be inappropriate. Based on existing concepts [135], a valuable extension of existing methods has been presented recently by Honnert et al. [124]. The basic approach is an appropriate merging of RANS and LES length scales, which enables significant simulation performance advantages. A discussion of a promising alternative way to address this question, which is based on a continuous length scale blending suggested by Senocak et al. [136], can be found elsewhere [27].

4.

Scale-aware parametrizations: A substantial disadvantage of using usually applied mesoscale parametrizations in the gray-zone is their scale independence. The latter assumes turbulent kinetic energies and length scales that are often way too large to properly represent gray-zone turbulence variables. It is now commonly accepted that scale-aware turbulence schemes (which include reference to the grid scale $\Delta$) are very efficient to overcome this problem. Several works in this regard are cited in Table 2, see Refs. [120,121,122]. We note that varying scale-awareness concepts are currently in use, e.g., McWilliams et al. require the following: “a scale-aware parameterization needs to set the proper energy or scalar transfer between the resolved and subgrid fields” [137].

5.

Initiation of resolved turbulence: In gray-zone simulations, the development of resolved flow structures is certainly not trivial. As is well known, the presence of excessive modeled motion can block (suppress) the development of resolved flow, which can cause significant shortcomings in applications [73]. Valuable attempts to overcome this issue have been reported, for example, in Refs. [117,118], where spin-up techniques have been presented to support the initiation of resolved turbulence. It has to be noted, however, that the success of these techniques turned out to be sensitive to the treatment of diffusion [118].

2.3. Solution Concepts: An Engineering Perspective

The orientation of current attempts to improve atmospheric gray-zone modeling by accounting for anisotropy of turbulence can be seen as being specific for the ABL. Such anisotropy modeling can be certainly involved in engineering simulations, but the usual focus is on the generation of meaningful resolved motion in separated turbulent flow simulations, having in mind that the use of simple models can be compensated by the inclusion of a sufficient amount of resolved motion. Apart from the inclusion of anisotropy, the current attempts to deal with the atmospheric gray-zone problem, basically, follow the engineering development of DES methods, usually without explicitly mentioning this fact. More specifically, by considering points 2–5 presented in Section 2.2, it is worth noting the following. The calculation of a turbulence length via a scale transport equation (via $k-ϵ$ or $k-\omega$ models) is standard in engineering simulations, and the merging of RANS and LES length scales is the core concept of DES methods. Scale-aware parametrizations are applied depending on the concrete problem considered. The problem of dealing with blocked resolved motion or inappropriate contamination of RANS regions via resolved motion is a standard issue of DES methods.

What is the problem with such following of engineering DES concepts? The advantage of DES is its conceptual simplicity. However, the DES concept faces significant issues, as explained in the following:

P1.

Core problem: The core idea of DES methods is to enable a seamless transition between modeled (RANS) and resolved (LES) regimes based on a switch of RANS and LES length scales. This idea sounds perfectly fine as long as such a seamless transition is actually realized in simulations. Unfortunately, this is not always the case: the switch of length scales does often not imply a corresponding switch of simulation regimes. Specifically, a desired flow resolution imposed by the LES length scale does often not translate into a corresponding real flow resolution [73]. A usual practical problem is the blocking of resolved motion through modeled motion or an inappropriate contamination of RANS regions via resolved motion.

P2.

Uncertainty of proper approach and predictions: The DES concept can be applied with respect to most regularly implied turbulence models. It can be applied in a variety of versions including delayed DES (DDES), improved delayed DES (IDDES), in zonal and in non-zonal set-ups. Usually, the results significantly depend on the choice of these modeling options. DES is known to sometimes require schemes for the initialization of resolved motions at case-dependent locations. DES is known to be rather sensitive to different mesh distributions (using the same number of grid points) and model parameter settings. This modeling uncertainty often translates into a relatively large uncertainty range of simulation results [73,138].

P3.

Balanced simulation performance and computational cost: Usually, the uncertainty range of DES is used to set up simulations such that specific flow characteristics (like pressure or skin-friction distributions) are well characterized at the cost of the simulation performance in regard to other flow characteristics (like velocity and stress profiles). The fact that the real flow resolution accomplished by DES methods can be significantly below expectations [139,140] also has implications for the computational cost of DES methods. It is common practice to compensate simulation shortcomings by an increase in computational cost (finer grids), such that DES simulations can be very expensive [139,140,141].

3. Toward a Gray-Zone Problem Solution

Perspectives to overcome the gray-zone problem, which were developed in regard to engineering applications, will be presented next. First, we will address turbulence and turbulence length scale modeling. Then, an explicit solution concept for the gray-zone problem will be presented in conjunction with related computational evidence.

3.1. Turbulence Model

For clarity purposes, the following development will focus on two-equation, eddy viscosity-based turbulence models. However, this is not at all a required limitation. A model that includes the anisotropy effect (which generalizes the well-known Mellor–Yamada hierarchy [126]) is presented in Appendix A. Relevant facts to note in this regard are the following ones. First, the latter model is based on a realizable stochastic turbulence model. Second, the model implies significant implications for the master length scale concept suggested by Mellor and Yamada. Third, this model is fully compatible with length scale modeling as presented in Section 3.2 and Section 3.3.

The typical structure of a microscale model for the stratified ABL [13,142,143,144,145] is the following one. The i-th component of the spatially filtered velocity is denoted by ${\tilde{U}}_i$, and $\tilde{\mathrm{\Theta }}$ refers to a corresponding scalar (the potential temperature). We have the incompressible continuity equation $\partial {\tilde{U}}_i/\partial x_i=0$, and the momentum and potential temperature equations read

$${\displaystyle \frac{D{\tilde{U}}_i}{Dt}}=-{\displaystyle \frac{\partial (\tilde{p}/\rho +2k/3)}{\partial x_i}}+2{\displaystyle \frac{\partial (\nu +{\nu }_t){\tilde{S}}_{ik}}{\partial x_k}}-{ϵ}_{ikn}{f}_k{\tilde{U}}_n-{\beta }_{\theta }\Delta \theta g_i,$$

(1)

$${\displaystyle \frac{D\tilde{\mathrm{\Theta }}}{Dt}}={\displaystyle \frac{\partial }{\partial x_k}}\left[\left({\displaystyle \frac{\nu }{Pr}}+{\displaystyle \frac{{\nu }_t}{P{r}_t}}\right){\displaystyle \frac{\partial \tilde{\mathrm{\Theta }}}{\partial x_k}}\right]+S_{\theta }.$$

(2)

Here, $D/Dt=\partial /\partial t+{\tilde{U}}_k\partial /\partial x_k$ denotes the filtered Lagrangian time derivative and the sum convention is used throughout this paper. In addition, we have here the filtered pressure $\tilde{p}$, $\rho$ is the constant mass density, k is the modeled energy, $\nu$ is the constant kinematic viscosity, and ${\tilde{S}}_{ij}=(\partial {\tilde{U}}_i/\partial x_j+\partial {\tilde{U}}_j/\partial x_i)/2$ is the rate-of-strain tensor. The modeled viscosity is given by ${\nu }_t=C_{\mu }{k}^{1/2}L$. Here, $C_{\mu }$ is a model parameter with standard value $C_{\mu }=0.09$, and L is a characteristic turbulence length scale. The latter can be related to the dissipation rate $ϵ=k/\tau$ of modeled kinetic energy, or the dissipation time scale $\tau$, or turbulence frequency $\omega =1/\tau$ by $L=k^{3/2}/ϵ=k^{1/2}\tau =k^{1/2}/\omega$, respectively. On top of that we involve the gravity vector $g_i$, ${\beta }_{\theta }$ is the thermal expansion coefficient (the inverse reference temperature), $\Delta \theta$ is the difference between a reference temperature and $\tilde{\mathrm{\Theta }}$, $f_k$ is the Coriolis vector, and ${ϵ}_{ijk}$ is the Levi-Civita symbol. In the $\tilde{\mathrm{\Theta }}$ equation we have the molecular and turbulent heat diffusivities $\nu /Pr$ and ${\nu }_t/P{r}_t$, where $Pr$ and $P{r}_t$, are the Prandtl number and turbulent Prandtl number, respectively. The effect of radiation can be taken into account via the source term $S_{\theta }$.

A turbulence model requires two main ingredients: a model for the kinetic energy of turbulence, and a model for the scale of turbulence. The first ingredient is usually provided by a transport equation for the modeled energy k. The second ingredient can be provided, e.g., on the basis of a transport equation for the length scale L (the dissipation rate $ϵ$, or the turbulence frequency $\omega$). Compared to corresponding algebraic models, the use of a transport equation for turbulence scale variables offers an essential advantage: L can be a realistically calculated subject to a potentially significant nonequilibrium between production and dissipation of turbulence, i.e., the calculation of L is not constrained by idealistic equilibrium ideas. Compared to $k-ϵ$ or $k-\omega$ model developments, however, the development of $k-L$ models [107,108,126,146,147,148,149,150] is far behind; generally accepted suggestions do not exist so far. One reason for that is the large variety of suggested model formulations, which sometimes neglect the influence of production on L [149], require empirical extra terms to be consistent with the log-law [108,151], and often involve rather different diffusion contributions.

3.2. Hybrid Length Scale Model

According to the author’s knowledge, the implementation of any transport equation for a scale variable in atmospheric flow codes represents a major exception, and only $k-ϵ$ models were considered so far [133]. In line with that, personal communication indicates that the implementation of a $k-ϵ$ model in regularly applied atmospheric flow codes is far from trivial. At least in regard to atmospheric flow applications, the consideration of a transport equation for L offers essential advantages: (i) there are homogeneous wall boundary conditions since both k and L vanish at solid surfaces, and (ii) there is direct access to the turbulence length-scale, which is revealing with respect to turbulent flow analysis. Given the potential benefits, $k-L$ models will be considered in the following with the understanding that the related analysis will enable it to overcome the problems reported in the last paragraph of the preceding subsection.

For reasons to be discussed in the following paragraph, we consider here the following $k-k^{1/2}L$ model as basis for the further development,

$${\displaystyle \frac{Dk}{Dt}}=P+P_b-ϵ+D_k,{\displaystyle \frac{D\sqrt{k}L}{Dt}}=C_1{\displaystyle \frac{L}{\sqrt{k}}}\left(P+P_b-\alpha ϵ\right)+D_{\sqrt{k}L}.$$

(3)

We note that $k^{1/2}L$ is proportional to the turbulent viscosity, i.e., the boundary conditions for this model are as simple as for the $k-L$ model. The diffusion terms read $D_k=\partial [{\nu }_t\partial k/\partial x_j]/\partial x_j$, and $D_{\sqrt{k}L}=\partial [({\nu }_t/\sigma )\partial \left(\sqrt{k}L\right)/\partial x_j]/\partial x_j$, where $\sigma$ is a model parameter. The shear production of kinetic energy is given by $P={\nu }_t{S}^2$, where $S={\left(2{\tilde{S}}_{mn}{\tilde{S}}_{nm}\right)}^{1/2}$ is the characteristic shear rate, and the buoyancy production is given by $P_b=-RiP/P{r}_t$ where $Ri={\beta }_{\theta }g\partial \tilde{\mathrm{\Theta }}/\partial x_3{S}^{-2}$ is the gradient Richardson number [152,153,154,155]. The buoyancy term in the $\sqrt{k}L$ equation is sometimes multiplied with a coefficient $C_b$. This buoyancy coefficient $C_b$ is characterized by a large uncertainty: its values range from $-1.4$ to $+1.45$ [156,157] including $C_b=1$ [126]. For simplicity, we assume the buoyancy coefficient $C_b$ is equal to unity in the following in line with the recommendation of Mellor and Yamada [126]. It is worth noting that the turbulence model considered is consistent with the $\widetilde{u_i{u}_j}$ Equation (A3) presented in Appendix A based on $k=\widetilde{u_i{u}_i}/2$. In regard to deriving the k equation from Equation (A3), the effect of the Coriolis term is neglected; the Coriolis effect is known to possibly affect the anisotropy of turbulence but not the kinetic energy [126,158]. The derivation of the $\sqrt{k}L$ equation from a $k-ϵ$ model reveals the relationships [159]

$$C_1=2-C_{{ϵ}_1}=0.56,C_1\alpha =2-C_{{ϵ}_2}=0.08,$$

(4)

which follows from using the $k-ϵ$ model standard values $C_{{ϵ}_1}=1.44$ and $C_{{ϵ}_2}=1.92$.

Let us come back to the question of why the consideration of an $k^{1/2}L$ equation is beneficial compared to the direct consideration of a L equation. The exact $k-L$ model implied by Equation (3) is presented in Appendix B. It is shown there that this approach faces technical questions given by the switch of dissipation terms into production terms, and the need to provide dissipation through quadratic energy gradient terms. These problems are avoided by defining the transport of L via an equation for $k^{1/2}L$.

Equation (3) is in line with Rotta’s derivation of a turbulence length scale equation [146,150], see Refs. [107,108]. These equations are also in line with the consideration of a generic turbulence model [159]. But there are two major problems related to Equation (3).

• The first problem is that the equation structure is not completely specified as long as the concrete production–dissipation structure is unknown. The latter requires the specification of model parameters like $\alpha$ as functions of L or other variables. There are certainly questions about this. Mellor and Yamada [108,126,151] applied a version of this model which includes an empirical quadratic length scale ratio in $\alpha$ to ensure consistency with the log-law. A discussion of possibilities to actually specify this quadratic length scale ratio can be found elsewhere [160,161]. Rotta suggested a modification of the production term by involving the third-order velocity derivative [146,150]. Menter and Egorov [108] criticized Rotta’s approach. They suggested the implementation of another empirical quadratic length scale ratio (involving the von Kármán length scale) in the production coefficient $C_1$. But there are different lines to argue regarding the concrete model formulation: a linear dependence on the corresponding second velocity derivative was proposed first [107], whereas a quadratic formulation has been used later [108].
• The second significant problem related to Equation (3) is the model’s scale and resolution independence. The model as is is incapable of seamlessly covering LES and RANS regimes, i.e., the model cannot properly cover the gray-zone regime.

A way to overcome these issues related to existing gray-zone solution approaches in regard to atmospheric flow simulations will be presented next on the basis of continuous eddy simulation (CES) methods developed recently [73,138,139,140,144,145,159,162,163,164,165,166,167,168,169,170]. A rudimentary expectation for a model capable of providing a transition between resolved LES and modeled RANS regimes is its applicability to a wide ranges of scales. The latter can be accomplished by a modification of Equation (1), where the constant $\alpha$ is replaced by a variable model parameter ${\alpha }^{\ast }$,

$${\displaystyle \frac{Dk}{Dt}}=P+P_b-ϵ+D_k,{\displaystyle \frac{D\sqrt{k}L}{Dt}}=C_1{\displaystyle \frac{L}{\sqrt{k}}}\left(P+P_b-{\alpha }^{\ast }ϵ\right)+D_{\sqrt{k}L}.$$

(5)

Using variational analysis, it is possible to determine an exact expression for ${\alpha }^{\ast }$ given by

$${\alpha }^{\ast }=1+L_{\ast }^2({\alpha }_0-1),$$

(6)

where $L_{\ast }=L/L_0$ refers to the ratio of a model length scale (L) to a reference length scale $L_0$. The details of this derivation can be found elsewhere [159]. No assumption regarding the reference length scale $L_0$ has to be made at this point, which can be chosen, e.g., according to DES concepts or as von Kármán length scale $L_{vK}=\kappa |S/S^{\prime }|$ (where $\kappa$ is the von Kármán constant, $S={\left(2{\tilde{S}}_{mn}{\tilde{S}}_{nm}\right)}^{1/2}$ is the characteristic shear rate and $S^{\prime }={[{\partial }^2{\tilde{U}}_i/\partial x_k^2\times {\partial }^2{\tilde{U}}_i/\partial x_j^2]}^{1/2}$). Equation (6) is of a purely technical nature, and a corresponding expression can be obtained by hybridizing the k-equation by introducing a variable model parameter in front of $ϵ$ [159].

In regard to the two major problems described above, we observe the following:

• As discussed above, current turbulence modeling focusing on L-type equations faces significant uncertainty because of the unknown required structure of model parameters involved. These problems are solved by the approach presented which mathematically determines the appearance of a quadratic length scale ratio in the turbulence model considered. The model obtained is consistent with log-law requirements as long as the model parameter relationship ${\kappa }^2=C_{\mu }^{1/2}\sigma (C_2-C_1)$ is satisfied, with $\kappa$ being the von Kármán constant and $C_2=C_1{\alpha }_0$.
• The expression ${\alpha }^{\ast }=1+L_{\ast }^2({\alpha }_0-1)$ is simply a consequence of ensuring the validity of the turbulence model considered in between a reference state (characterized by $L_0$ and the constant ${\alpha }_0$) and a partially or fully resolving state characterized by L and ${\alpha }^{\ast }$. The model functioning depends essentially on the setting of the reference length scale $L_0$, which spans the hybridization range in between $L_0$ and L. Variants of performing this are discussed in detail elsewhere [159] by including DES, SAS, Mellor–Yamada-type, and LES scaling variants. However, the most natural, most effective setting of $L_0$ is $L_0=L_{tot}$, where $L_{tot}$ refers to the total length scale that accounts for resolved and modeled contributions (the calculation of $L_{tot}$ and $L_{+}=L/L_{tot}$ is addressed in the Appendix C). This setting enables a seamless transition between LES and RANS regimes.

3.3. Advantages Compared to Existing Approaches

Given $L_0=L_{tot}$, what are the essential advantages compared to DES type concepts used currently to address the atmospheric gray-zone problem? By taking reference to the DES problems P1–P3 reported in Section 2.3, we observe the following (see also the summary in

Table 3. Overview of significant DES and CES differences.

	Detached Eddy Simulation (DES)	Continuous Eddy Simulation (CES)
Core idea	switch of RANS, LES length scales	resolution-aware modeling (model knows and appropriately responds to actual flow resolution)
Advantage	simple idea; simple to implement (simpler than CES implementation)	exact math-based; direct actual resolution measurement; no grid influence; no DES problems
Disadvantage	Always: no implied resolution switch, resolved and modeled motion mismatch, many different versions of fixes, significant grid dependence, imbalanced predictions, significant cost to fix problems	Only once: modeling approach requires calculation of actual flow resolution indicators on the fly, implementation of scale transport equation in general required
Functionality	spatial switch between RANS-type and LES-type flow regions (no CES transitionality)	full hybrid model transitionality from modeled to resolved simulations

).

P1.

Core problem: The DES core problem does not exist because there is no switch of RANS and LES length scales. More specifically, there is no switch at all, and there is no discrepancy between resolution driving parameters (LES and RANS length scales) and the actual flow resolution. Instead, the actual flow resolution is measured (in analogy to the corresponding energy ratio $k_{+}=k/k_{tot}$) via $L_{+}=L/L_{tot}$, and the model is directly informed about the actual flow resolution via ${\alpha }^{\ast }$. The latter enables the model to properly respond to variations in the actual resolved motion via increasing or decreasing its contribution to the simulation. Thus, there is no mismatch between resolved and modeled motion.

P2.

Uncertainty of proper approach and predictions: The variety of different DES versions that exist is implied by the shortcomings of DES: these DES modifications were presented to overcome these problems. This relates, e.g., to the main DES shortcoming to block resolved motion or to contaminate modeled flow regions with fluctuations, and the DES problem to sensitively depend on the mesh organization. These problems simply do not appear in the approach described here. First, there is no discrepancy between resolved and modeled motion because of the model adjustment mechanism. Second, there is no explicit grid dependence of the model presented; the grid-related filter width $\Delta$ does not enter the model.

P3.

Balanced simulation performance and computational cost: All previous applications of CES methods presented [138,139,140] reveal a very essential feature: the simulation predictions are well balanced. This means that predictions of a variety of relevant flow characteristics like separation zone characteristics, pressure and skin-friction distributions, profiles of mean velocities and stresses agree very well (and better than the results of other methods) with experimental data: there is simply no wiggle room for flow-dependent model adjustments as given in DES methods. This fact has remarkable implications for the computational cost of CES methods, which are found significantly below the cost of other methods [139,140].

The statements made in the preceding paragraph will be illustrated by applications of CES and DES methods to periodic hill flow simulations at a high $Re=37$ K (the highest $Re$ case for which experimental data are available for model validation). An illustration of this flow (a channel flow involving periodic restrictions) is provided by Figure 2 [138]. This flow considered involves features such as separation, recirculation, and natural reattachment [171,172]. It is used a lot for the evaluation of turbulence models [73] because of its reasonable balance of relative simplicity and complexity. DES applications were presented by Saini et al. [173]. Two variants of IDDES methods were considered: $k-\omega$-SST IDDES (an implementation of IDDES in Menter’s shear-stress transport (SST) $k-\omega$ model [174]), and Spalart–Allmaras (S-A) IDDES (an implementation of IDDES in the Spalart–Allmaras turbulence model [175]). Two grids were involved in these simulations: a coarse 0.3 M grid, and a fine 0.9 M grid. CES applications were presented by Heinz et al. [138]. Three hybrid variants of a $k-\omega$ model were considered: CES-KOS (hybridization in the $\omega$ scale equation), CES-KOK (hybridization in the k equation, and CES-KOKU (a hybridization according to unified RANS-LES). At the bottom and top, the channel is constrained by solid walls: no-slip and impermeability boundary conditions were used at these walls, and wall functions were not employed. Periodic boundary conditions are employed in the streamwise and spanwise directions. A significant range of $Re$ ($Re=[37,250,500]$ K) and a variety of coarse and finer grids ($G_{500}$, $G_{250}$, $G_{120}$ involving 500 K, 250 K, and 120 K grid points, respectively) were considered. It is worth noting that these grids cover the full range of almost modeled to almost completely resolving simulations.

Table 4. Errors of reattachment point calculations by CES [138] and IDDES [173] methods. The experimentally determined reattachment point is $x/h=3.76$ [171].

Method	Reattachment $(\mathit{x}/\mathit{h})$	Error (%)
CES ($G_{500}$)	3.78	0.5
CES ($G_{120}$)	3.7	$-1.6$
S-A IDDES (coarse 0.3M grid)	3.9578	5.3
S-A IDDES (fine 0.9M grid)	4.2922	14.2
$k-\omega$-SST IDDES (coarse 0.3M grid)	3.9578	5.3
$k-\omega$-SST (fine 0.9M grid)	4.7078	25.2

shows that DES is not as accurate as CES in regard to the calculation of separation zone characteristics (in particular the calculation of the reattachment point). We note that the CES grids are coarser than the DES grids, respectively. A disturbing fact is that the accuracy of DES significantly reduces if the grid is refined. What are the reasons for that? Figure 3 shows the ${\tilde{f}}_d$ distribution, which controls the transition from URANS to the LES regime, for the coarse grid for both models considered at $x/h=4$. The URANS region is defined by ${\tilde{f}}_d=1$, the scale-resolving region (SRR) is defined by ${\tilde{f}}_d\approx 0$, and the shaded intermediate region represents the gray-zone. It may be seen that the extend of the gray-zone is strongly influenced by the model applied. The expectation is that the corresponding distribution of $k_{+}=k/k_{tot}$ reflects this transition of ${\tilde{f}}_d$ from unity to zero. But Figure 4 (which shows coarse grid results) shows that this is not the case. The $x/h=4$ curves show little variations, there is basically no difference between gray-zone and SRR behavior, this means the $k_{+}=$ distribution is not controlled by the ${\tilde{f}}_d$ distribution (it is of interest that the same can be seen in other approaches like PITM and PANS [138] where the resolution control is decoupled from the actual flow resolution). The fact that the $k_{+}$ values are rather small may lead to the expectation that flow simulation results are excellent because the flow is apparently very well resolved. Figure 5 shows that this is not the case. The velocity profiles appear to be reasonable, but stress profiles show significant shortcomings: there is a significant model and grid dependence, and the profiles can be qualitatively incorrect.

The CES features are very different from DES features. By design, there is no mismatch between the resolution control and the actual resolution. Figure 6 shows that the $k_{+}$ distributions are almost constant over the core of the flow. There are minor variations with the model considered and grid applied. As expected, a grid refinement implies a systematic reduction of $k_{+}$. It is of interest to see the coincidence between $k_{+}$ and $L_{+}^{2/3}$. The latter can be explained by the exact relationship $k_{+}=L_{+}^{2/3}{ϵ}_{+}^{2/3}$ in conjunction with the fact that ${ϵ}_{+}$ is equal to unity except very close to the wall. Figure 7 shows implications for flow simulations. The streamwise velocity and stresses are very well predicted (the experimental stresses are known to be slightly underpredicted). It is remarkable that a grid coarsening has very little effect.

It is of interest that the conclusions of the CES-DES comparison reported for the periodic hill flow case considered here are fully in line with corresponding conclusions obtained for other flow simulations, e.g., the NASA wall-mounted hump flow and the axisymmetric transonic bump flow [139,140,162]. Flow-adjusted specific DES variants can provide good predictions of some flow characteristics but fail in regard to other flow characteristics (e.g., the separation zone characteristics or friction coefficient distributions). On the other hand, CES methods perform very well in regard to all relevant flow characteristics.

4. Conclusions

This paper presented an analysis of ongoing efforts to deal with the gray-zone problem in atmospheric simulations. The main conclusions can be summarized as follows.

1.

Significance and trend: There is currently no indication that simple fixes can overcome the gray-zone problem. The existing problems are serious. Without substantial changes in solution approaches, this problem has to be expected to significantly hamper future atmospheric simulations (possibly for decades to come). Specifically, current (DES-type) concepts to address this problem in regard to atmospheric simulations follow, basically, developments made by the engineering community years ago. However, there are clear indications that these developments significantly fall behind expectations: see the problems P1–P3 reported in Section 2.3.

2.

New modeling approach: There exist new (CES-type) simulation concepts developed in the context of engineering applications. Both theoretical features and applications provide at least strong indications that such CES-type simulations have the potential to overcome problems as seen in the context of DES applications. Specific technical features of the new modeling approach are (i) independence of the main LES restriction (i.e., the inclusion of the filter width $\Delta$), (ii) inclusion of an inherent actual resolution indicator ($L_{+}$), and (iii) identification of conceptual issues of alternative methods (where different $L_0$ settings are used).

3.

Resolution-aware and anisotropy-aware modeling: One of the most essential problems of atmospheric simulations is to provide appropriate (gray-zone applicable) turbulence length scales. The analysis presented identifies a resolution-aware length scale equation where significant uncertainty of the structure of previous $k-L$ type modeling efforts (e.g., regarding the model parameter structure) is excluded. In addition, as described in the Appendix A, a significant model extension covering the structure of the Mellor–Yamada hierarchy is well possible. An attractive feature of the model presented is the identification of length-scale relationships, which generalize the Mellor–Yamada master length scale concept.