Parameterization by Statistical Theory on Turbulence Applied to the BAM-INPE Global Meteorological Model

Abstract

A parameterization for the planetary boundary layer (PBL) based on the statistical theory of turbulence formulated by Geoffrey Ingram Taylor is derived to be applied in the Brazilian Global Atmospheric Model (BAM). The BAM model is the operational system employed by the National Institute for Space Research (INPE), Brazil, to produce numerical weather and climate predictions. A comparison of the BAM model simulations using Taylor’s parameterization is carried out against other three turbulent representations. The forecasting from different parameterizations with BAM is evaluated with the ERA-5 reanalysis. Predictions were performed on different initial conditions, representing two types of climate seasons: dry and wet seasons, for the Southern Hemisphere. The comparison shows that Taylor’s approach is competitive with other turbulence parameterizations, especially for the dry season. It must be highlighted that the forecasting over the Amazon region—one of the regions on the planet with the most intense rainfall, where Taylor’s approach provided more effective precipitation forecasting, a particularly challenging meteorological variable to predict.

1. Introduction

Turbulence is a permanent feature in the dynamics of the atmosphere. The closest layer to the ground is the region called the planetary boundary layer (PBL), where the turbulence must be represented. The production of turbulence in the PBL is associated with wind shear or transport of thermal energy between the ground and the atmosphere. If the ground temperature is lower than the atmosphere temperature, there will be a heat flow from the atmosphere to the ground, removing energy from the PBL, a situation commonly found during the night. On the other hand, when the surface temperature is greater than the atmosphere—for instance, during the daytime—a thermal flux from the land can be established, generating a convective boundary layer [1]. Different stability conditions for the PBL can be typified by calculating the Monin–Obukhov length.

The complete understanding of turbulence is a challenge for modelers. However, a representation of the turbulent flow is essential to be included in the computer programs for numerical weather prediction (NWP). One approach is to consider the Reynolds’ assumption, where those fluid properties can be expressed as the summation of a mean value for the physical property (flowing in a similar way, such as a laminar flux) added to a stochastic forcing. The average of random forcing is assumed to be zero, but the average of the product between two of these stochastic variables is not null. Indeed, the product of two random variables is named as Reynolds tensors [2]. From this consideration, the goal becomes to express the Reynolds tensors as a function of the mean atmospheric variables.

These tensors are parameterized to represent turbulence in simulation software. Turbulence parameterization has been investigated using different closing approaches [1] and different closing orders, either first-order [3,4], or higher-orders [5].

For regional and global weather prediction systems, turbulence parameterizations are employed under permanent development. The COSMO (Consortium for Small-scale Modeling) model is a non-hydrostatic limited-area atmospheric model. The relevance and consequences of the mixing length implementation in the COSMO model were covered by [6] in a new physically based horizontal length scale for a high-resolution mesoscale model over a complex terrain.

The version of the MPAS (Global Model for Prediction Across Scales) coupled to the ocean circulation is an open-source model. The MPAS-Ocean is discretized by an unstructured Voronoi mesh, allowing a finer grid over a specified region, describing mesoscale simulation for ocean dynamics embedded in a global domain. Li and van Roekel [7] used a large eddy simulation code PALM (Parallelized Large Eddy Simulation Model) coupled with the global MPAS-Ocean model. The multiscale formulation is applied to describe features of interactions between the small-scale turbulence with the larger-scale linked to ocean dynamics.

Another representation for turbulence was introduced by Zhu et al. [8] for the FV3 (Finite-Volume Cubed-Sphere Dynamical Core) global model. A turbulence parameterization for high-resolution numerical weather prediction models is presented as a 3D scale-aware turbulence scheme based on a generalized turbulence closure applicable across scales, developed and implemented in the Hurricane Analysis and Forecast System.

The modern implementation of Taylor’s parameterizations for turbulence has relevant features, where the turbulent kinetic energy is represented by its spectrum, following Kolmogorov’s scaling law of energy distribution among the wavelengths (turbulent eddy diameters) [9] with the maximum observed energy also included. The analytical formulation for the turbulent spectra under different atmospheric stability conditions allows for the application of Taylor’s parameterization to meteorological simulators.

Turbulent modeling based on Taylor’s formulation has already been applied to the atmospheric pollutant models: Lagrangian [10], Eulerian [11], and Gaussian [12], and atmospheric mesoscale models: BRAMS (Brazilian developments on the Regional Atmospheric Modeling System) [13]—see: http://brams.cptec.inpe.br/ (accessed on 6 September 2025)—and METRAS (MEsoskaliges TRAnsportund Strömungsmodell) [14]—see on the internet: https://www.mi.uni-hamburg.de/en/arbeitsgruppen/memi/modelle/dokumentation/msys-scientific-documentation.pdf (accessed on 6 September 2025).

This article describes the results with the implementation of the turbulence parameterization based on the 1922 Taylor’s theory [15] for different PBL stability conditions in the BAM model (Brazilian global Atmospheric Model)—see Figueroa et al. [16]. Here is the first time that Taylor’s theory on turbulence is applied to a global model of atmospheric dynamics. It’s particularly desired to evaluate the parameterization results for the Amazon region, given its important climatic influence in neighboring regions, especially for weather forecasts in the southeast part of Brazil [17,18]. For the Amazon region, the proposed parameterization obtained superior results for precipitation prediction among the evaluated turbulent representations.

The results with the new parameterization are compared with the schemes presented by Holtslag-Boville [4], Bretherton-Park [19], and Mellor-Yamada [5] to represent the turbulence.

2. BAM: The Brazilian Global Atmospheric Model

The Center for Weather Forecasting and Climate Studies (CPTEC) of the National Institute for Space Research (INPE) began its activities in November 1994, having implemented the global CPTEC/COLA model, originally developed from the model of the Center for Ocean-Land-Atmosphere Studies (COLA) in the USA. This model has been developed over the years and better adapted for Brazilian climate conditions for use in both diagnostics research and routine (operational) predictions [20,21], being renamed as the CPTEC Atmospheric Global Circulation Models (CPTEC-AGCM). Despite the model being able to simulate the climatological atmospheric circulation features, INPE has a permanent action to improve the weather/climate prediction system. A new model called the Brazilian Global Atmospheric Model (BAM) [16] was executed operationally, substituting the CPTEC-AGCM model [22].

The INPE is responsible for the operational execution of numerical weather forecasting, climate forecasting, and environmental forecasting models. The BAM model [16] is a three-dimensional hydrostatic code with two formulations of dynamic kernel (Eulerian and semi-Lagrangian), where the equations are spatially discretized using a spectral method, developed for numerical weather forecasting, simulations, and climate forecasts [22]. According to Figueroa et al. [16], the global model was developed entirely to be used on time scales ranging from days to seasons and with horizontal resolutions ranging from 10 km to 200 km. The model can be used for either deterministic numerical weather forecasting on the 1-to-10-day scale, probabilistic extended numerical weather forecasting on the 1-to-4-week scale (when coupled with an ocean model), or even as a comprehensive Earth system model for seasonal climate forecasting and climate change studies. During INPE’s operation, the model runs with a spatial resolution of approximately 20 km, with 64 vertical levels and a semi-Lagrangian semi-implicit scheme for time integration. Among others, the BAM model provides data to the National Electric System Operator [23] and the Cearense Foundation for Meteorology and Artificial Rainfall [24].

In the global model of atmospheric dynamics adopted by INPE for numerical weather forecasting, first- and second-order PBL parameterizations were implemented, which currently include the following three different options to be selected by the user:

1.

The Holtslag–Boville first-order representation of turbulent flows [4] takes into account a counter gradient term (computed for convective layer), modeling local and non-local schemes for vertical diffusion.

2.

The University of Washington Moist Turbulence (UWMT) developed by Christopher S. Bretherton and Bretherton–Park [19] is a first-order moist turbulence parameterization improved for numerical stability and efficiency with the long time steps used in climate models, with the original goal of providing a more physically realistic treatment of marine stratocumulus-topped boundary layers.

3.

The Mellor–Yamada second-order PBL turbulence model [5], an experience realized at Princeton University, is the result of a collection of studies carried out by the authors and others that culminated in a set of physical models based on the hypotheses of Rotta and Kolmogorov.

Three different BAM executions are carried out for each parameterization listed above for comparison with Taylor’s turbulence modeling.

3. Turbulence Parameterization from Taylor’s Statistical Theory

As expressed by the gas kinetic theory, the diffusion coefficient is proportional to the product between a characteristic speed ($\langle v\rangle$) and a characteristic length (ℓ) [25]. Taking this into account, the British physicist and mathematician Geoffrey Ingram Taylor published a paper entitled Diffusion by Continuous Movements ([15]), founding the basis for his statistical theory of turbulence (see the deduction in the Appendix A). In the formulation based on Taylor’s approach, the displacement x between two particles embedded in a turbulent flow obeys the following relation [26]:

$${\displaystyle \frac{d\langle x^2\left(t\right)\rangle }{dt}}=2\langle v^2\left(t\right)\rangle {\int }_0^t\rho \left(\tau \right)d\tau$$

(1)

where the self-correlation function is denoted as $\rho \left(\tau \right)\equiv \left[\langle v\left(t\right)v(t+\tau )\rangle \right]/\langle v^2\left(t\right)\rangle$, and $v\left(t\right)$ represents a random fluctuation in the velocity field. Integrating by parts Equation (1), the fundamental equation of Taylor’s formulation describing a property for turbulent flow is derived as follows:

$$\langle x^2\left(t\right)\rangle =2\langle v^2\left(t\right)\rangle {\int }_0^t(t-\tau )\rho \left(\tau \right)d\tau .$$

(2)

However, a turbulence representation is obtained by using the Fourier transform/Fourier anti-transform pair of the correlation function, seeing in Equation (2).

Therefore, an expression for the spacing of fluid parcels is obtained in the following spectral form:

$$\langle x^2\left(t\right)\rangle ={\sigma }_i^2{t}^2{\int }_0^{+\infty }{F}_{L_i}\left(n\right)\left[{\displaystyle \frac{{sin}^2\left(\pi nt\right)}{{\left(\pi nt\right)}^2}}\right]dn$$

(3)

where ${\sigma }_i^2$ is the variance of wind speed, $F_{L_i}$ is the Lagrangian non-dimensional spectrum and the frequency n is given in Hertz—instead of radians/second ($\omega =\mathrm{rad}/s$), where $n=\omega /2\pi$.

However, most observations in meteorology are collected using the Eulerian flow approach rather than the Lagrangian one. Consequently, the Gifford–Hay and Pasquill hypothesis states that ${\rho }_{L_i}\left({\beta }_i\tau \right)={\rho }_i\left(\tau \right)$, where ${\rho }_i$ is an Eulerian correlation function and ${\beta }_i$ represents the relationship between the Lagrangian and Eulerian decorrelation scales. With a change of variables and employing an expression derived by Batchelor (1949) [27] for the eddy diffusivity, written as follows:

$$K_{\alpha \alpha }={\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{1}{2}}\langle x_i^2\left(t\right)\rangle \right]$$

(4)

where $i=u,v,w$ is the wind velocity direction and $\alpha =x,y,z$ identifies the different space directions. It can be combined with Equation (3) to derive an expression for the turbulent diffusivity, written as follows [9,28]:

$$K_{\alpha \alpha }={\displaystyle \frac{{\sigma }_i^2{\beta }_i}{2\pi }}{\int }_0^{+\infty }{F}_i\left(n\right){\displaystyle \frac{sin(2\pi nt/{\beta }_i)}{n}}dn$$

(5)

where $\alpha =(x,y,z)$ is the turbulent diffusivity direction, ${\sigma }^2$ is the wind velocity variance, ${\beta }_i={\left[\left(\pi U^2\right)/\left(16{\sigma }_i^2\right)\right]}^{1/2}$ is the ratio between the scale of the Lagrangian and Eulerian spectra—being U the wind mean speed, $F_i\left(n\right)$ is the Eulerian dimensionless spectrum of kinetic energy, n is a dimensionless frequency, and t is time.

The turbulent diffusivity tensor $K_{\alpha \alpha }$ is described by the following orthotropic form:

$${\left\{K_{\alpha \alpha }\right\}}_{\alpha =x,y,z}=\left[\begin{array}{ccc}{K}_{xx}& 0& 0\\ 0& K_{yy}& 0\\ 0& 0& K_{zz}\end{array}\right].$$

(6)

The key issue is to derive an analytical formula for the normalized spectra $F_i\left(n\right)=S_i\left(n\right)/{\sigma }_i^2$, with ${\sigma }_i^2={\int }_0^{\infty }{S}_i\left(n\right)dn$. Formulas are derived from the following general expression [29,30]:

$${\displaystyle \frac{n{S}_i\left(n\right)}{v_R^2}}={\displaystyle \frac{A{f}^{m_1}}{{(1+B{f}^{m_2})}^{m_3}}}$$

(7)

here $f=nz/U$ is the nondimensional frequency, with z the height above the surface, and U is the main wind; $v_R^2$ is the velocity scale; A and B are constants to be determined by the following two conditions:

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}{S}_i\left(n\right)& \sim & \mathrm{Kolmogorov}’\mathrm{s}\mathrm{law}(-5/3),\hfill \\ \hfill \max S_i\left(n\right)& \sim & \mathrm{maximum} \mathrm{observed},\hfill \end{array}$$

where: $\max S_i\left(n\right)⟹d\left\{S_i\left(n\right)\right\}/dn=0$, and the velocity scale depends on atmospheric stability [9,28]:

$$v_R^2=\left\{\begin{array}{cc}{u}_{*}^2={\tau }_s/\rho \hfill & (\mathrm{stable}/\mathrm{neutral})\hfill \\ w_{*}^2=(u_{*}^2/L^{2/3}){(-\kappa /h)}^{-2/3}\hfill & \left(\mathrm{convective}\right)\hfill \end{array}\right.$$

(8)

being ${\tau }_s$ the surface shear stress, $\rho$ is the air density, $\kappa \approx 0.4$ is the von Kármán constant, h is the PBL height, and L is the Monin–Obukov length.

The Reynolds tensors are parameterized as a first-order closure by the gradient of the mean quantity, written as follows:

$$\overline{v_{\alpha }^{\prime }{\xi }_j^{\prime }}=K_{\alpha \alpha }{\displaystyle \frac{\partial \overline{{\xi }_j}}{\partial \alpha }}$$

(9)

where ${\xi }_j=(u,v,w,p,T,r,c)$, with p: pressure, T: temperature, and r: moisture. The spectral expressions are formulated by the asymptotic behavior of Equation (5) as $t\to \infty$, implying ${lim}_{n\to 0}{F}_i\left(n\right)=F_i\left(0\right)$ [9,28].

With appropriate expressions for the spectra, it is possible to describe formulations for each type of stability in the PBL [9,28]. A general expression for the dimensionless turbulent velocity spectrum in the surface boundary layer as seen in the reference [29], where the experimental constants are derived for the desired type of stratification, and also considering Kolmogorov’s law in the inertial subdomain with local similarity, the variance of the wind velocity and the dimensionless spectrum in the origin for the stable (SBL), neutral (NBL), and convective (CBL) boundary layers.

3.1. The Turbulence Parameters

For long travel times $t\to \infty$, the asymptotic behavior of Equation (5) can be derived as follows:

$$K_{\alpha \alpha }={\displaystyle \frac{{\sigma }_i^2{\beta }_i{F}_i\left(0\right)}{4}}$$

(10)

where $K_{\alpha \alpha }$ is the turbulent diffusivity in any direction, and $F_i\left(0\right)$ is the non-dimensional spectrum at the origin. Based on the relationship between the turbulence parameters $K_{\alpha \alpha }={\sigma }_i^2{T}_{L_i}={\sigma }_i{\ell }_i$ an expression for Lagrangian decorrelation scale ($T_L$) can be derived as follows:

$$T_{L_i}={\displaystyle \frac{{\beta }_i{F}_i\left(0\right)}{4}}.$$

(11)

Similarly, an expression for the mixing length is found, written as follows:

$${\ell }_i={\displaystyle \frac{{\sigma }_i{\beta }_i{F}_i\left(0\right)}{4}}.$$

(12)

The asymptotic formula for eddy diffusivity (10) associated with the respective turbulent spectra for the PBL stability condition allows for the derivation of parameterizations for different stratification in the atmospheric boundary layer. The next two sections show the steps to derive the vertical component for eddy diffusivity based on Taylor’s statistical theory. A schematic diagram to illustrate the characteristics for different PBL stability conditions is shown in Figure 1a—unstable/convective PBL (heat flux from the surface to the atmosphere), stable PBL (heat flux from the atmosphere to the surface), and neutral PBL (there is no heat flux between the surface and the atmosphere). Figure 1b,c show typical profiles for virtual temperature and vertical wind component for unstable and stable conditions, respectively.

3.1.1. Stable Boundary Layer (SBL) and Neutral Boundary Layer (NBL)

Using the so-called Monin–Obukhov local similarity theory adjusted for SBL and NBL, NIEUWSTADT [31] adapted the similarity theory for the stable boundary layer, where the wind shear ($\tau$), heat flux ($\overline{w^{\prime }{\theta }^{\prime }}$), and local Monin–Obukov length ($\Lambda$) are expressed by the following:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\tau }{{\tau }_0}}={\displaystyle \frac{U_{*}^2}{u_{*}^2}}& =& {\left(1-z/h\right)}^{{\alpha }_1};\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\overline{w^{\prime }{\theta }^{\prime }}\left(z\right)}{{\left(\overline{w^{\prime }{\theta }^{\prime }}\right)}_0}}& =& {\left(1-z/h\right)}^{{\alpha }_2};\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\Lambda }{L}}& =& {\left(1-z/h\right)}^{3{\alpha }_1/2-{\alpha }_2}.\hfill \end{array}$$

(15)

Here, the potential temperature is given by $\theta \left(z\right)=T\left(z\right){\left(p_0/p\left(z\right)\right)}^{R/c_p}$, where the temperature T is in Kelvin, R is the gas constant for air, $c_p$ is the specific heat capacity at constant pressure, and $\overline{w^{\prime }{\theta }^{\prime }}\left(z\right)$ is the local heat flux. Using the formulation presented in Equation (10), the turbulent diffusivity for the SBL and NBL is given by:

$${\displaystyle \frac{K_{\alpha \alpha }}{u_{*}h}}={\displaystyle \frac{\sqrt{\pi }}{16}}{\displaystyle \frac{0.64{\left(2.33c_i\right)}^{1/2}}{{\left(f_m\right)}_{n,i}^{4/3}{c}_{\epsilon }^{-1/3}}}{\displaystyle \frac{[1-{(z/h)}^{{\alpha }_1/2}(z/h)]}{1+3.7(z/h)(h/\Lambda )}}$$

(16)

where h is the PBL height, $u=\sqrt{{\tau }_s/{\rho }_{air}}$ is a velocity scale (friction velocity), $\Lambda$ is the local Monin–Obukhov length, ${\left(f_m\right)}_{n,i}$ is the spectral maximum under the neutral condition, $c_{\epsilon }=1.25$ [9], $c_i=a_1{b}_1{\left(2\pi \kappa \right)}^{-2/3}$—with $a_1=3/2$, and $b_1=0.5\pm 0.05$. For the neutral boundary layer, the exponents ${\alpha }_1=2$, and ${\alpha }_2=3$ were estimated from the Minnesota experiment [32], values ${\alpha }_1=3/2$ and ${\alpha }_2=1$ are applied to the stable boundary layer, results from the Cabauw experiment [33]. The PBL height h depends on the stability conditions, and schemes used to calculate it are shown in Section 3.2.

By combining the appropriate turbulence parameters with the experimental data, the vertical diffusion coefficient for turbulence in the SBL and NBL can be obtained as follows [9,34]:

$${\displaystyle \frac{K_{zz}}{u_{*}h}}={\displaystyle \frac{3\sqrt{\pi }}{16}}{\displaystyle \frac{{[1-(z/h)]}^{{\alpha }_1/2}(z/h)]}{1+3.7(z/h)(h/\Lambda )}}.$$

(17)

3.1.2. Convective Boundary Layer (CBL)

Using a convective velocity length based on the heat flux ${\left(\overline{w^{\prime }{\theta }^{\prime }}\right)}_0$ between the surface and the atmosphere, the Monin–Obukhov length is calculated for the CBL. Also, a spectrum model with the convective velocity scale calculated from the general expression for the dimensionless turbulent velocity spectrum alongside Kolmogorov’s law for the inertial subdomain formulated in terms of convective similarity parameters is used to derive the spectrum for the CBL. From the definition given in Equation (10), using the variance of the wind velocity and the dimensionless spectrum at the origin of the CBL, an expression for turbulent diffusivity $K_{\alpha \alpha }$ for the CBL is seen in the equation below [28]:

$${\displaystyle \frac{K_{\alpha \alpha }}{w_{*}h}}={\displaystyle \frac{1.5^{5/6}{\kappa }^{1/3}{c}_i^{1/2}\sqrt{\pi }}{16}}{\displaystyle \frac{{\psi }^{1/3}{\left({\lambda }_m\right)}_i^{4/3}}{h^{1/3}}}$$

(18)

where $w_{*}=\left[(g/{\theta }_0){\left(\overline{w^{\prime }{\theta }^{\prime }}\right)}_{0}h\right]$ is a convective velocity scale, $\kappa =0.4$ is the von Kármán constant, $c_i=a_1{b}_1{\left(2\pi \kappa \right)}^{-2/3}$ as before in Equation (16), $\psi =[\left(\epsilon h\right)/\left(w_{*}^3\right)]$ is a dimensionless dissipation function, ${\lambda }_m$ the wavelength value for the spectral peak, and h is the PBL height.

Using the convective diffusivity from Equation (18) and the appropriate convective experimental data, the vertical diffusion seen in Equation (19) models the turbulence for the CBL [28]:

$${\displaystyle \frac{K_{zz}}{w_{*}h}}=1.16{\psi }^{1/3}{\left[1-exp\left(-4{\displaystyle \frac{z}{h}}\right)-0.0003exp\left(8{\displaystyle \frac{z}{h}}\right)\right]}^{4/3}.$$

(19)

3.2. Modeling Different PBL Stability Conditions

To calculate the three different PBL stability conditions supported by the Taylor parameterization, an update to the model was made by implementing a decision based on the stability parameter, consisting of the height z divided by the Obukhov length L. Given the definition:

$$L=-{\displaystyle \frac{u_{*}^3}{\kappa {\beta }_c{q}_{*}}}$$

(20)

where $q_{*}$ is the vertical heat flux, an unstable PBL can be defined when the Obukhov length is negative, and a stable PBL when the Obukhov length is positive. Theoretically, neutral conditions occur when there is no vertical heat flux, i.e., when $q_{*}$ approaches zero, causing the Obukhov length L diverge to infinity. To avoid issues with infinite values and division by zero, a threshold is adopted, defining neutral conditions as occurring when $\{z/L\}$ is within the range −0.001 and $0.001$. The parameter $\kappa =0.4$ is the von Kármán constant, and ${\beta }_c=g/{\overline{\theta }}_v$, with $g\approx 0.98$ being the Earth’s gravity acceleration and ${\overline{\theta }}_v$ being the mean virtual potential temperature (potential temperature where the moisture mixing ratio is taken into account).

The PBL height is calculated according to the stability condition. For stable and neutral conditions, the following expression is employed to compute the h [35]:

$$h=B_v{u}_{*}^{3/2}$$

(21)

being $B_v=2.4\times {10}^3$ m^−1/2 s^3/2.

For convective conditions, the PBL height h is obtained by using the Richardson number ($R{i}_g$)—see [36], calculated as follows:

$$R{i}_g={\displaystyle \frac{(g/{\theta }_{v_s})({\theta }_{v_h}-{\theta }_{v_s})(h-z_s)}{{(u_h-u_s)}^2+{(v_h-v_s)}^2}}$$

(22)

with $z_s=0.1,h$, and ${\theta }_v$ denoting the virtual potential temperature, the subscripts h and s indicate the height levels at which the quantities are calculated. The value $R{i}_g$ represents the Richardson number between the atmospheric levels $z_s$ and h.

4. Experiment and Model Configuration

To evaluate the Taylor-based PBL parameterization, a set of simulations was conducted using the BAM model, and the results were compared with the reanalysis data from ERA5 hourly data on single levels from 1959 to the present [37]. The ERA5 data uses a 31 km horizontal resolution and 137 vertical levels. The simulations contemplated two different initial conditions, one in January (the wet season in the Amazon region), and the second in September (the dry season in the Amazon region). For each initial condition, two different simulations were conducted: one 7-day (168 h) simulation using the TQ62 triangular truncation horizontal resolution (approximately 215 km) and a second 15-day (360 h) simulation in the TQ126 triangular truncation horizontal resolution (approximately 106 km), both using 28 vertical sigma-layers. All atmospheric simulations were executed using the BAM model in Eulerian mode. Four different PBL parameterizations were used in the simulations: Holtslag–Boville, Bretherton–Park, Mellor–Yamada, and the Taylor parameterization, presented in this paper. The summary of the experiment elements (variables used for comparison) is shown in

Table 1. Experiment introduction with initial condition data, parameterization methods used, and simulations range, and resolution.

Initial Conditions	Parameterizations Methods	Simulations
Wet Season: 15 January 2012 at 12:00 PM	Holtslag–Boville (First-Order) Bretherton–Park (First-Order)	7 days (168 h) using TQ62 (∼215 km) resolution
Dry Season: 15 September 2012 as 12:00 PM	Mellor–Yamada (Second-Order) Taylor (First-Order)	15 days (360 h) using TQ126 (∼106 km) resolution
Data: ERA5 hourly data on single levels from 1959 to the present

. The output expressions for BAM and ERA5 are listed in Table 1. In addition, we applied an interpolation for matching the horizontal and vertical resolutions of the BAM model and ERA5 data.

Taylor-based formulation uses Equation (17) for stable or neutral conditions (the difference is on the exponent values in the expression), or Equation (19) for unstable conditions. The atmospheric stability condition is identified by the numerical value for the Monin–Obukov length—see Section 3.2, written as follows:

$$L=\left\{\begin{array}{cc}\left|L\right|\ge 500& \mathrm{neutral},\hfill \\ L>0& \mathrm{stable},\hfill \\ L<0& \mathrm{convective}.\hfill \end{array}\right.$$

(23)

Beyond the PBL, other parameterizations are present in the model, including:

• Gaseous absorption parameterization for shortwave (SW) radiation and long-wave (LW) radiation: CLIRAD [38];
• Cumulus Parameterization: Arakawa-Schubert [39];
• Ocean model: slab (forced fixed condition in which all quantities are assumed to be completely and instantaneously homogenized [40]);
• Snow model: Simplified Simple Biosphere SSiB [41];
• Soil–vegetation–atmosphere scheme: Integrated Biosphere Simulator Model IBIS [42].

The goal is to evaluate the performance of the new parameterization based on Taylor’s theory. One particular interest is focused on the weather in the Brazilian rainforest, with the relevance of this region to the Brazilian climate. The initial conditions were generated from ERA5 hourly data on two different dates, typified as the rainy (summer) and dry (winter) seasons in the Amazon region:

• dry season: 15 September 2012;
• wet season: 15 January 2012.

Following the dry and wet initial conditions, 16 simulations were executed, with two different lengths/resolutions for two different seasons and four different PBL parameterizations. Five simulation variables were chosen to compare the results from the BAM model with the ERA5 reanalysis; the name and its corresponding physical unit are shown in

Table 2. Simulation variable names and their units for the BAM model and their equivalent ERA5 reanalysis counterparts.

BAM		ERA5
Variable Name	Unit	Variable Name	Unit
Time mean temperature at 2 m	K	Temperature at 2 m	K
Planetary boundary layer height	m	Boundary layer height	m
Outgoing Longwave at the Top	W ${\mathrm{m}}^{-2}$	Top net thermal radiation	J ${\mathrm{m}}^{-2}$
Cloud cover	ND	Total cloud cover	[0, 1]
Total precipitation	kg (${\mathrm{m}}^2\times$ day)−1	Mean total precipitation rate	kg (${\mathrm{m}}^2\times$ s)−1

. All the variables that present different units from the ERA5 reanalysis to the BAM model were converted before any comparison to ensure unit consistency.

All simulations were evaluated over three different geographic areas: the Global (GBL), encompassing the entire world; the Full South America (FSA), covering latitudes from −60° to 15° and longitudes from −85° to 30°; and the Amazon (AMZ), covering latitudes from −12.5° to 0° and longitudes from −70° to −50°.

Each of the five BAM variables was compared with its respective ERA5 variable using three different error calculations, where a represents the ERA5 variable and b represents the BAM variable, written as follows:

• Mean Relative Error (MRE): percentage difference in relation to ERA5 data:

  $$MRE(a,b)={\displaystyle \frac{1}{N}}\sum _{i=0}^N{\displaystyle \frac{|a_i-b_i|}{|a_i|}}.$$

  (24)
• Mean Difference (MD): sub-estimation or the super-estimation BAM output in relation to ERA5 data using the same physical units:

  $$MD(a,b)={\displaystyle \frac{1}{N}}\sum _{i=0}^N(a_i-b_i).$$

  (25)
• Root Mean Square Error (RMSE): real (absolute) error in the same physical units of the original data. This is the chosen error calculation to evaluate most of the experiment:

  $$RMSE(a,b)={\left[{\displaystyle \frac{1}{N}}\sum _{i=0}^N{(a_i-b_i)}^2\right]}^{{\displaystyle \frac{1}{2}}}.$$

  (26)

5. Results and Discussion

For the dry and wet seasons, the results obtained with the simulations configured with different horizontal/temporal resolutions and with the four PBL closing parameterizations were analyzed, comparing them with the ERA5 data for five different variables, in three different geographic ranges, and using three different error calculations, totaling 720 sets of results. In this section, a brief overview of the simulated maps for each variable will be exposed. The scenarios where Taylor’s PBL parameterization stood out will be addressed, first separated by season and simulation variable, then separated by geographic range.

5.1. The Simulated Maps

The two sets of data, the one simulated by the BAM model (in this section, always using the Taylor PBL parameterization) and the ERA5 reanalysis data, can be plotted on the terrestrial globe and visually compared. An example is the Cloud Cover map (Figure 2), where one can clearly see the similarities between the 12-h forecast simulations performed by the model and the ERA5 reanalysis data. However, there is an overestimation of the Cloud Cover field, which may be related to greater stability of the nocturnal boundary layer simulated by the BAM model. In other words, the wind intensity in the boundary layer (1000 to 850 mb) must be weak and produce little mechanical turbulence, favoring the process of saturation and cloud formation.

The distinct selection of color palettes for each variable helps to highlight the differences in the amounts predicted by the model. Consequently, some discrepancies between the predicted variables and the reanalysis data variables may present abrupt color transitions. The thermal radiation field emerging at the top of the atmosphere simulated by the model after 72 h of the initial condition, seen in Figure 3, exhibits intensity differences, particularly within the convective systems of the tropical and subtropical regions when comparing the simulated field in Figure 3a and the reanalysis data in Figure 3. Despite these minor differences in intensity, the BAM model accurately reproduced the spatial pattern of thermal radiation emanating from the top of the atmosphere, as well as the spatial distribution of convective systems.

Figure 4 illustrates the PBL height field map after 3 days of simulation, where it is verified that the model consistently simulates the convective boundary layer during the day (Europe and Africa). However, at night (Oceania, Asia, and the Americas), the model has difficulty simulating the height of the boundary layer in relation to the ERA5 reanalysis data. Analyzing the general aspects of Figure 4a,b, it appears that the height of the simulated PBL is generally underestimated compared to the ERA5 data. However, the correct position of the maximum and minimum height of the PBL simulated by the BAM model is observed in comparison with the reanalysis data.

The map for the temperature field at 2 m in Figure 5 shows 72 h of forecast. In general, the magnitude of temperature is similar in most of the terrestrial globe. However, over the Tropical Atlantic, Tropical Pacific, and Indian Oceans, there is a temperature difference of up to a few degrees. This temperature pattern at 2 m indicates that the sea surface temperature used in the simulations is colder in relation to the ERA5 reanalysis data. Thus, as a consequence of the colder sea surface temperature used in the integration of the model, some meteorological systems may be affected, such as tropical convection, atmospheric saturation, the cloud cover field, and the interaction of radiation with the atmosphere. In some regions of South America, the temperature at 2 m is also overestimated by the BAM model.

The most interesting result for a weather forecast model is the simulated precipitation. In Figure 6a, the precipitation field after 72 h of forecast is illustrated alongside the ERA5 data in Figure 6b, where it is verified that practically all the precipitating systems in the terrestrial globe are simulated consistently by the BAM model. However, the precipitation intensity of convective systems is underestimated by the BAM model. This may be related to the spatial and vertical resolution of the model, as well as aspects of the variables discussed earlier.

5.2. Dry Season

The simulation for the dry season obtained the best results for the Taylor PBL parameterization. The largest number of variables and the most expressive advantages are discussed in the following sections.

5.2.1. Time Mean Temperature at 2 m

The average 2-m temperature is not the most accurately simulated variable when using the Taylor parameterization, except in the Amazon region. In most scenarios, the Bretherton–Park parameterization provides better performance in simulating near-surface temperature. However, in the Amazon, the Taylor scheme performs better in both seasons, particularly during the first week, as illustrated in Figure 7. In this figure, the shorter peaks—indicating smaller errors (closer to 0 K)—belong to the Taylor parameterization, represented by the black line.

5.2.2. Planetary Boundary Layer Height

The PBL height was better simulated by the Holtslag–Boville [4] and Bretherton–Park [19] parameterizations, and Taylor parameterization placed second in all scenarios (see

Table 3. BAM TQ126 RMSE Global averages for each simulation variable, where: H-B: Holtslag–Boville; B-P: Bretherton–Park; M-Y: Mellor–Yamada. The bold value is the smallest error in each column, compared to the ERA5 reanalysis data.

September (dry season for the Brazilian Amazon)
Method	timeMeanTemperatureAt2m	planetaryBoundaryLayerHeight	outgoingLongwaveAtTop	cloudCover	totalPrecipitation
H-B	3.43 °K	416 m	35.46 Wm−2	38.22%	12.52 kg (m2; day)−1
B-P	3.19 °K	601 m	35.19 Wm−2	39.91%	12.49 kg (m2; day)−1
Taylor	3.80 °K	494 m	34.40 Wm−2	37.70%	11.99 kg (m2; day)−1
M-Y	3.61 °K	535 m	35.71 Wm−2	39.05%	12.43 kg (m2; day)−1
January (wet season for the Brazilian Amazon)
Method	timeMeanTemperatureAt2m	planetaryBoundaryLayerHeight	outgoingLongwaveAtTop	cloudCover	totalPrecipitation
H-B	4.50 °K	385 m	34.72 Wm−2	37.68%	12.76 kg (m2; day)−1
B-P	4.08 °K	569 m	34.58 Wm−2	39.39%	12.88 kg (m2; day)−1
Taylor	4.67 °K	469 m	34.05 Wm−2	37.53%	12.32 kg(m2; day)−1
M-Y	4.64 °K	471 m	35.12 Wm−2	38.39%	12.75 kg (m2; day)−1

,

Table 4. BAM TQ126 RMSE Full South America averages for each simulation variable, where: H-B: Holtslag–Boville; B-P: Bretherton–Park; M-Y: Mellor–Yamada. The bold value is the smallest error in each column, compared to the ERA5 reanalysis data.

dry season
Method	timeMeanTemperatureAt2m	planetaryBoundaryLayerHeight	outgoingLongwaveAtTop	cloudCover	totalPrecipitation
H-B	3.02 °K	422 m	37.42 Wm−2	38.98%	13.23 kg (m2; day)−1
B-P	2.83 °K	586 m	37.64 Wm−2	41.02%	13.19 kg (m2; day)−1
Taylor	3.40 °K	493 m	36.60 Wm−2	38.08%	12.22 kg (m2; day)−1
M-Y	3.23 °K	533 m	38.85 Wm−2	39.90%	12.79 kg (m2; day)−1
wet season
Method	timeMeanTemperatureAt2m	planetaryBoundaryLayerHeight	outgoingLongwaveAtTop	cloudCover	totalPrecipitation
H-B	2.90 °K	418 m	40.33 Wm−2	37.10%	15.54 kg (m2; day)−1
B-P	2.68 °K	567 m	39.79 Wm−2	37.95%	15.28 kg (m2; day)−1
Taylor	3.33 °K	484 m	39.95 Wm−2	37.42%	14.71 kg (m2; day)−1
M-Y	3.13 °K	491 m	40.63 Wm−2	37.56%	14.62 kg (m2; day)−1

and

Table 5. BAM TQ126 RMSE Amazon averages for each simulation variable, whre: H-B: Holtslag–Boville; B-P: Bretherton–Park; M-Y: Mellor–Yamada. The bold value is the smallest error in each column, compared to the ERA5 reanalysis data.

dry season
Method	timeMeanTemperatureAt2m	planetaryBoundaryLayerHeight	outgoingLongwaveAtTop	cloudCover	totalPrecipitation
H-B	3.03 °K	500 m	37.27 Wm−2	40.91%	13.80 kg (m2; day)−1
B-P	3.53 °K	383 m	37.44 Wm−2	42.51%	14.51 kg (m2; day)−1
Taylor	2.52 °K	408 m	35.23 Wm−2	38.44%	12.57 kg (m2; day)−1
M-Y	3.82 °K	742 m	39.37 Wm−2	40.45%	12.81 kg (m2; day)−1
wet season
Method	timeMeanTemperatureAt2m	planetaryBoundaryLayerHeight	outgoingLongwaveAtTop	cloudCover	totalPrecipitation
H-B	2.10 °K	336 m	60.84 Wm−2	19.39%	22.88 kg (m2; day)−1
B-P	2.18 °K	273 m	60.67 Wm−2	19.16%	22.69 kg (m2; day)−1
Taylor	2.04 °K	287 m	60.47 Wm−2	19.60%	21.42 kg (m2; day)−1
M-Y	2.27 °K	365 m	61.99 Wm−2	20.07%	21.02 kg (m2; day)−1

). During the dry season, the results from Taylor are almost indistinguishable from Bretherton–Park (Figure 8).

The PBL height was better simulated by Holtslag and Park parameterizations in all scenarios. The Amazon basin region was where Taylor parameterization received its best approximations compared to the ERA5 reanalysis data (see Table 2, Table 3 and Table 4).

5.2.3. Outgoing Longwave at Top

The thermal radiation emitted into space at the top of the atmosphere simulated with the BAM model has a smaller error compared to the ERA5 reanalysis using Taylor PBL parameterization. The other PBL parameterizations present similar scenarios but with greater error. Figure 9 shows a lower significant error for the Taylor simulation in all three geographic regions: Global (GBL), over the territory of “Full” South America (FSA), and Amazon (AMZ). In Figure 10, the time series of thermal radiation emitted into space at the top of the atmosphere for each PBL method are plotted. The simulation obtained with the Taylor parameterization has the values at the bottom during the two simulated weeks, keeping the smallest error.

5.2.4. Cloud Cover

In Figure 11 and Figure 12, cloud cover was best simulated during the dry season using Taylor PBL parameterization. The three geographic regions (GBL, FSA, and AMZ) in Figure 11 indicate that the Taylor PBL parameterization has the smallest cloud cover error in comparison to other boundary layer schemes for the entire planet. Figure 12 is the cloud cover forecast time series, where is verified after the first week that Taylor parameterization presents a smaller error in relation to the other boundary layer schemes. Determining the influence of the boundary layer on the cloud cover field is very complex, as cloud formation depends on the interaction of many physical processes.

5.2.5. Total Precipitation

Precipitation is generally well represented by the Taylor parameterization, especially in the dry season simulation. The difference in relation to the other methods is small, but it is indeed relevant given the importance of the variable. Figure 13 illustrates the time series of the global MRE, where the results with the smallest error were obtained by the simulation using Taylor PBL parameterization.

5.2.6. Comparative Discussion

Across the dry season experiments, the Taylor PBL parameterization showed a recurring advantage for variables that integrate column processes, like Outgoing Longwave Radiation at the Top of the Atmosphere, Cloud Cover, and to a lesser extent, Total Precipitation, while moving behind Bretherton–Park in near-surface temperature and Holtslag–Boville/Bretherton–Park in PBL height. Taken together, these patterns suggest that Taylor’s turbulence formulation may be mixing moisture and heat vertically in a way that supports a more realistic vertical distribution of fluxes yet does not translate into equally accurate local surface coupling. The Amazon region stands out: Taylor reduces 2-m temperature errors there relative to the other schemes, which could reflect stronger land–atmosphere feedbacks over deep vegetation. Conversely, where stable or transitional regimes dominate (nighttime, higher latitudes), schemes with more robust treatment of stability better diagnose PBL depth, which likely helps their surface temperature skill outside the Amazon. Overall, the dry-season results indicate that Taylor’s strengths emerge most in convectively active, moisture-rich environments, particularly when vertically integrated metrics are evaluated.

5.3. Wet Season

The wet season maintained Taylor PBL parameterization best results in relation to the dry season consistently; however, the results with the other PBL parameterization schemes were more similar to Taylor PBL parameterization. In the global and Amazonian scenarios, Taylor PBL parameterization presented the most relevant results, while for South America in general, no improvements were obtained.

5.3.1. Time Mean Temperature at 2 m

During the wet season, the temperature at 2 m was best simulated in the Amazon region by the BAM model using Taylor PBL parameterization (Table 5). It can be verified in Figure 14 that the results in the first simulated week, where the temperatures have their smallest variations, are in the Taylor results. However, the results are not as clear as they were for the dry season.

5.3.2. Planetary Boundary Layer Height

The height of the PBL was not best simulated by the Taylor parameterization in most regions, but in the Amazon region in the rainy season, it moved a little closer to the reanalysis data again. In this region, the Taylor parameterization obtained the second lowest error, with Bretherton–Park parameterization in first place, as shown in Figure 15 and Table 5.

5.3.3. Outgoing Longwave at Top

When the Taylor PBL parameterization is employed in BAM simulations, the thermal radiation at the top of the atmosphere exhibits the lowest error compared to ERA5 reanalysis data. Figure 16 presents the mean difference between the model output and ERA5, highlighting the relatively small error associated with the Taylor parameterization in comparison to the other boundary layer schemes implemented in the BAM model.

5.3.4. Total Precipitation

The precipitation simulated by the model with Taylor PBL parameterization in the rainy season did not approach the ERA5 reanalysis data as it did in the dry season in South America and in the Amazon, but maintained its global performance, as seen in Figure 17. The MRE demonstrates that parameterization of Taylor is better by the end of the second week.

5.3.5. Comparative Discussion

During the experiments of the wet season, the performance gaps between the PBL schemes narrowed but the basic structure of the differences persisted. Taylor generally retained its global advantage in top-of-atmosphere longwave flux and remained competitive for cloud cover. However, these good results, specifically its skill in representing outgoing longwave radiation and cloud cover, became less pronounced when averaging over all of South America, where a large variety of ecosystems and complex surface characteristics challenge the method. By comparison, in the Amazon, Taylor again produced low differences for 2-m temperature and Outgoing Longwave at the Top, suggesting that its representation of turbulence remains effective in very humid, vegetation-dominated conditions even under active convection. Park continued to perform well for PBL height, indicating that its stability functions and mixing length behavior may better track shallow and transitional boundary layers.

5.4. Global Results

The global average RMSE values for the five variables, presented in Table 3, were consistent across both seasons. These results synthesize the individual analyses discussed previously and indicate good performance of the Taylor parameterization in simulating Outgoing Longwave Radiation at the Top of the Atmosphere, Cloud Cover, and Total Precipitation at the end of the two-week BAM model simulations. The Taylor parameterization ranked second in estimating PBL height, while it exhibited the poorest performance for near-surface (2 m) mean temperature.

5.5. Full South America Results

The South America scenario brought some difficulties for the presented method to reproduce the reanalysis data, especially in the wet season. The overall loss of performance in the wet season is visible in Table 4 alongside the consistent good performance in the dry season, repeating similar results to those previously seen in the global simulations. Taylor’s parameterization holds the second place in the wet season; together with the good dry season performance, it is still a considerable option.

5.6. Amazon Region Results

The Amazon region achieved the closest results to the ERA5 reanalysis data when simulated using Taylor’s PBL parameterization. Table 5 shows results for the dry season; good performance is still visible alongside a new good result in the Mean Temperature at 2 m (in both seasons), only observed in the Amazon results. The method gathered some good results even in the wet season, differently than the Full South America experiment, proving its utility in weather forecasts or future studies about the climatology in the Amazon region.

Comparative Discussion

When comparing the three spatial domains (Global, Full South America, and Amazon) across both seasons, a pattern might emerge in how each PBL scheme behaves. At the global scale, Taylor consistently exhibits the lowest errors for column-integrated variables such as Outgoing Longwave at the Top and Cloud Cover, reinforcing the idea that its turbulence formulation promotes coherent vertical structures. Over South America, however, Taylor’s advantage is less systematic, particularly during the wet season. In contrast, the Amazon domain consistently promotes Taylor’s relative strengths, yielding improvements in both Outgoing Longwave at the Top and near-surface temperature. Conversely, Holtslag–Boville and Bretherton–Park tend to perform better in diagnosing PBL height and stabilizing surface temperature under more stable or transitional conditions, an effect more visible outside the Amazon and in dry-season regimes.

5.7. Net Radiation

An interesting result from the simulations was the strong performance of the Taylor parameterization in representing the Outgoing Longwave at the Top of the Atmosphere in most scenarios. According to Coelho et al. [22], a good representation of atmospheric radiation in a climate model is essential for successful climate simulations: the amount of radiation at the top of the atmosphere is a good indicator of the energy balance in the simulation. A further analysis was conducted to compare the net radiation ($NR$) between the simulation results and the reanalysis data across the three geographic regions. The net radiation (difference) is given by the following [22]:

$$\begin{array}{c}\hfill NR=(\mathrm{incident} \mathrm{short} \mathrm{wave} \mathrm{flux}-\mathrm{upward} \mathrm{short} \mathrm{wave} \mathrm{at} \mathrm{the} \mathrm{top})+\\ \hfill (\mathrm{incident} \mathrm{long} \mathrm{wave} \mathrm{flux}-\mathrm{outgoing} \mathrm{long} \mathrm{wave} \mathrm{at} \mathrm{the} \mathrm{top}).\end{array}$$

The incident longwave is considered negligible (there is no significant heat coming from space for this radiation band). So, this variable is set to zero.

However, the net radiation results were less favorable to Taylor’s parameterization, which exhibited the largest global average error in both seasons, particularly during the dry season, as shown in Table 3.

Despite the energy balance,

Table 6. BAM TQ126 net radiation RMSE for each geographic location for the dry season (H-B: Holtslag–Boville, B-P: Brethetron–Park, M-Y: Mellor–Yamada). The bold value is the smallest error in each column, compared to the ERA5 reanalysis data.

	GLB	FSA	AMZ
H-B	73.4597	65.9437	55.5906
B-P	72.7019	65.3901	60.0827
Taylor	90.3283	76.7280	53.7867
M-Y	74.6171	66.9587	55.3842

and

Table 7. BAM TQ126 net radiation RMSE for each geographic location for the wet season (H-B: Holtslag–Boville, B-P: Bretherton–Park, M-Y: Mellor–Yamada). The bold value is the smallest error in each column, compared to the ERA5 reanalysis data.

	GLB	FSA	AMZ
H-B	102.9497	106.5917	118.3482
B-P	103.1809	106.1878	119.0044
Taylor	113.8907	111.7826	116.8788
M-Y	104.1775	106.2570	119.0742

, Taylor still achieved a good score in the Amazon region, consistent with most of its results.

5.8. Experiment Afterwords

Preliminary results were presented with a new turbulence parameterization based on Taylor’s statistical theory. The numerical experiments were not exhaustive. However, they were conducted under two typical dry and wet season conditions for South America. These are important climatic conditions for precipitation, which have a strong impact on society and various economic sectors in Brazil.

Holtslag–Boville’s scheme is a first-order closure method, and its main goal was to include a counter-gradient term for representing turbulent fluxes, where the eddy diffusivity combined length scale with a fitting curve. Another scheme designed for better representation of precipitations was developed by Bretherton–Park, and it also is a first-order closure. The second-order closure is the parameterization presented by Mellor–Yamada. For this second-order representation, many other equations need to be included and this closure can represent some atmospheric movements in the PBL that are not be simulated by first-order schemes. The parameterization based on Taylor’s statistical theory adds some theoretical aspects for the turbulent energy distribution into different space scales (turbulent kinetic energy spectra) with the maximum observed peak frequency, allowing the representation of important features from the atmospheric turbulence.

Taylor’s parameterization yielded better results in some aspects and worse results in others. The results show that no parameterization offers hegemonic advantages for all variables and for all regions of the planet. The results indicate that multi-model prediction (runs with different parameterizations together) has the potential to improve the model’s prediction skill.

5.9. Discussion

This study provides a systematic evaluation of the PBL parameterization based on Taylor’s theory implemented in the BAM model. The results with the new parameterization were confronted with the results achieved using PBL schemes presented by Holtslag–Boville, Bretherton–Park, and Mellor–Yamada when compared to the ERA5 reanalysis data. A set of simulations with the BAM model using two different resolutions (a lower resolution one-week simulation and a higher resolution two-week simulation) in two different seasons (dry and wet), evaluating five different simulation variables in three different geographic domains (global, South America, and Amazon).

The results shown in this work bring a novelty, implementing Taylor’s parameterization in a global operational meteorological forecast model, with a comparison against other schemes. Some limitations arise from the resolutions used in this study. Even though it is a common practice for a global model to use a coarse resolution for performance purposes, the adopted resolutions of 106 km and 215 km are different from the original 31 km resolution of the ERA5 data. Representative days in two important seasons in South America, summer (wet season) and winter (dry season), were selected because they have an important impact on the Brazilian economy. More than 80% of energy production in Brazil comes from hydroelectric power plants, agriculture is a very strong sector in the Brazilian economy, and intense rainfall and deep dry seasons are important information for the people and civil defense. Of course, including the simulations of the other two seasons and many other years could generate a more complete comparison, but this requires more extensive computer resources.

All the scenarios experimented with were simulated for no longer than two weeks; any behavior after this period is unknown. Another limitation is the use of the reanalysis data for comparison. A comparison with observed data is mandatory for future work.

Nevertheless, this new planetary boundary layer closure scheme implemented in the BAM model, differently from the other methods using a fitting curve to model the vertical turbulence profiles, takes into account physics-based processes, such as the maximum of the observational turbulent kinetic energy spectrum and Kolmogorov’s law of $-5/3$ in the inertial subdomain of the spectrum. It is also important to note that, from a computational point of view, the parameterizations using Taylor’s theory and presented by Holtslag–Boville show better performance, while Sungsu-Park’s approach requires more processing time, and Mellor–Yamada’s scheme requires a processing time between those cited as the best performance and that of Sungsu-Park.

With good results for simulations and its computational performance on the global model with Taylor’s parameterization, it can be used as initial/boundary conditions for regional meteorological models.

Results obtained for the Amazon region recorded the best simulation for precipitation. This is relevant information, considering the importance of the Amazon forest’s evapotranspiration in precipitation [18], and its influence on the weather forecast in neighboring regions. Precipitation forecasting is another issue to be pointed out, due to the relevance of the Amazon region to the southeast part of Brazil [17].

As future work, a counter-gradient [43,44] term derived by using Taylor’s theory will be added to the first-order closure scheme for a better representation of the turbulent fluxes. Taylor’s theory can also be applied to express the mixing length for higher-order schemes [5] for modeling Reynolds tensors.

6. Conclusions

This article brought the first results with the implemented atmospheric turbulence parameterization based on Taylor’s theory for all PBL stability conditions in the BAM global model. The results showed a competitive performance for the turbulence representation in a meteorological global model. It is pertinent to note that simulations carried out in the dry season presented better predictions, on the Outgoing Longwave at the Top, Cloud Cover, and Total Precipitation variables, in all three geographic regions.

The good results obtained in our simulations for both seasons qualify the BAM global model to supply data for regional models for initial and boundary conditions.

The Taylor theory here was applied in a first-order closure scheme. Results using this parameterization show that it is competitive with other parameterizations, including the higher-order one. Higher-order schemes can simulate some movements, as those due to the interactions between the PBL and the higher atmospheric levels (entrainment). Our results bring the evidence that the modeling, coupled with some important turbulent characteristics in the same parameterization (observed spectral maxima and the Kolmogorov spectral law), can yield good results.

At the same time, Taylor’s approach did not excel across all metrics. Schemes such as Holtslag–Boville and Bretherton–Park more often produced lower differences from the ERA5 data in the diagnosis of the PBL height and temperature at 2 m, especially considering the full South America averages and the global wet season results. In many scenarios, as expected, the turbulence schemes produced stable simulations. There is no scheme with better results in different scenarios, for different variables, and over different regions on the planet. Taken together, the results indicate there is no single universally superior closure scheme and instead support a scenario-dependent view.

Considering all the different results analyzed, as part of future research, a multi-model ensemble using all four planetary boundary layer parameterizations described here can be evaluated, attributing a weight to each method according to the results gathered in this work, valuing each region and season with a proper parameterization to maximize the simulation accuracy. One technique for estimating multi-model forecast weights is based on an inverse problem formulation, as applied for precipitation by Santos et al. (2013) [45] and Luz and co-authors (2015) [46]. The counter-gradient term using Taylor’s theory could be added to the Reynolds tensors to represent the entrainment action (interaction between the PBL and higher atmospheric level). Finally, the mixing length formulated by Taylor’s theory (12) could also be used for the master scale in the Mellor–Yamada second-order.