1. Introduction
Wall modeling, an approach to modeling near-surface turbulence, is a crucial component in large-eddy simulations (LESs) of wall-bounded flows at high Reynolds numbers (Piomelli and Balaras 2002; Bose and Park 2018; Yang and Griffin 2021), including flows in the atmospheric boundary layer (Deardorff 1980; Moeng 1984; Bou-Zeid et al. 2005; Salesky and Anderson 2018). Despite its importance, there is currently no one-size-fits-all approach for all types of flow (Slotnick et al. 2014; Wang et al. 2023). Traditional wall models are mostly derived in flows that are quasi steady, horizontally homogeneous in surface and flow properties, and horizontally dominated (i.e., the horizontal components of the near-surface velocity are much larger than the vertical component), which are not necessarily suitable for unsteady and locally convective flows.
One of the most widely used wall models to account for hydrostatic stability is based on the Monin–Obukhov similarity theory (MOST; Monin and Obukhov 1954), which assumes a horizontally homogeneous, quasi-steady, and predominantly horizontal flow. In a neutral boundary layer, MOST reduces to the law of the wall (Prandtl 1933). Applying the law of the wall for wall modeling is equivalent to the drag coefficient approach that uses a constant roughness length and first-level vertical grid spacing (e.g., Roberts et al. 2020). However, the limitations of MOST have been well known (Stiperski and Calaf 2023). For example, MOST fails to account for horizontal wind variances (Wyngaard and Coté 1974), wall normal velocity variances, and temperature variances (Mahrt 1999). Additionally, its overly idealized assumptions call into question its broad applicability, for example, in Rayleigh–Bénard convection (RBC) and other natural convection flows.
In spite of the long-standing limitations of MOST, without a more reliable alternative, atmospheric models have used MOST for evaluating surface fluxes, even for the flow over complex terrain (e.g., Schumanndlr 1990) and intense convection (e.g., Schenkman et al. 2014; Markowski and Bryan 2016; Roberts et al. 2020), which clearly violate the MOST assumptions. Numerical simulations have shown that the vertical velocity induced by horizontal convergence (implying a horizontally heterogeneous velocity field) is sensitive to the use of wall models (Wang et al. 2020, 2023). Without relying on the MOST, the nonequilibrium wall models have also been developed in the engineering community (Balaras et al. 1996; Cabot and Moin 2000; Wang and Moin 2002; Park and Moin 2014), but currently, their applications have only been explored in shear-dominated boundary layers, and therefore further discussion of these approaches is not pursued here.
RBC is an example of a locally and horizontally heterogeneous phenomenon that violates the MOST assumptions. RBC is characterized by local warm updrafts and cold downdrafts, where the local near-surface vertical velocity can be on the same order of magnitude as the horizontal velocity. Previous studies have revealed the deviation from MOST in the updraft and downdraft of convection (e.g., Fodor et al. 2019), but MOST is still often used in the LES of RBC to model near-surface turbulence while accounting for near-surface stability (e.g., Salesky and Anderson 2018; Thomas et al. 2019; Yang et al. 2022).
Efforts to construct wall models using neural networks (NNs) began as early as Milano and Koumoutsakos (2002). Over the last decade, significant advances in machine learning (ML) and high-performance computing have provided greater opportunities to develop innovative wall modeling approaches based on NNs. For instance, LES wall models based on NNs have been constructed for canonical boundary layers (e.g., Yang et al. 2019; Bae and Koumoutsakos 2022; Vadrot et al. 2023), periodic hills (Zhou et al. 2021), rotating channels (Huang et al. 2019; Huang and Yang 2021), and irregular surfaces on aircrafts (Lozano-Durán and Bae 2020, 2023). To the best of our knowledge, however, LES wall models for RBCs using an NN-based approach have yet to be explored. Therefore, the objective of this work is to employ NNs to develop wall models as alternatives to the MOST-based models for LES of RBC and to gain a deeper understanding of the mechanisms governing the surface shear stress and heat flux.
For training and evaluation purposes, we perform direct numerical simulations (DNSs), which resolve all scales of motion and do not require a wall model, and then filter the DNS results to resolutions that are relevant to LES. We employ a feedforward neural network (FNN) to relate the near-wall flow properties to the surface shear stress and heat flux. We also use the same DNS results to evaluate the accuracy of the MOST-based model for the wall fluxes. Compared to the previous ML-wall-model studies, the complexity of the flow considered in this paper is the involvement of thermal stratification and heat transfer. It is not entirely clear what off-wall information should be used when modeling the wall shear stress and wall heat flux. The absence of such knowledge puts us at risk of overfitting or underfitting. To determine the most relevant input features, we evaluate the importance of each input using the Shapley additive explanations values (SHAP value; Shapley 1953; Štrumbelj and Kononenko 2014), which, combined with deep learning, is a form of interpretable ML. To better understand the SHAP values and gain the physical insight, we also analyze the velocity and temperature profiles within the filter grid cells conditionally selected according to the low and high ranges of the input variables.
One challenge for ML algorithms is to apply the trained model beyond the scope of its training data. Yang et al. (2019) demonstrated that the NN including physical information has the capability to model the boundary layer at a Reynolds number higher than that used to train the NN. On the contrary, an NN trained without incorporating any physical information struggles to extrapolate. For RBC, however, there is no well-known relationship for the near-surface flow quantities like that between the velocity and wall-normal distance in a canonical boundary layer. The NN ability to extrapolate can be improved by using dimensionless parameters within the local and near-surface regions, which allows NN to account for relationships observed in large-scale circulations, such as power laws among the Nusselt number, Reynolds number, and Rayleigh number (Ahlers et al. 2009). As we do not invoke the governing equation in our training, we refer to our approach as the “physics-aware” approach to distinguish it from the physics-informed NN approach (Cai et al. 2021), where one penalizes the solution based on the governing equation. Specifically, the NNs here are called physics-aware feedforward neural networks (PAFNNs).
In this paper, we first evaluate the performance of the NN models (referring to both FNN and PAFNN models hereafter) and the importance of the features using a priori tests. Specifically, we will use DNS-filtered data as input to the models and compare their output surface shear stress and heat flux with the DNS-filtered ones. Later, to evaluate their applicability, we will implement the NN models in LES for the a posteriori tests. To assess the extrapolation capability of the NN models, an additional set of DNS and LES runs with more intense RBC is performed.
The rest of the paper is organized as follows: Section 2 introduces the methods, which cover the DNS configuration, the filtering approach to mimic the LES data, the MOST wall model, the deep learning configuration, the SHAP value, and the a posteriori tests. Section 3 presents the results. Finally, section 4 provides the conclusions. For the readers’ reference, we place the flow statistics from the interior of the domain in appendix A and more details of the training process in appendix B.
2. Methods
The training data used in this work are filtered from DNS. Therefore, we first present the DNS model description, simulation setup, and the filtering approach used to obtain data relevant to the LES grid resolutions. Next, we introduce the traditional MOST and several newly trained NN models, explaining how the filtered data are used as inputs. To understand the importance of the inputs for each FNN wall model, we introduce the SHAP value. Finally, we describe how we implement the NN models in LES for a posteriori tests in both the training and the extrapolation scenarios.
a. DNS
The DNS is performed using the System for Atmospheric Modeling (SAM), which solves dimensionalized equations for velocity and moist static energy (see details in Khairoutdinov and Randall 2003, appendix A). The velocity components are solved on a staggered Arakawa C grid and advected using a second-order central scheme. The scalar is solved at the center of each grid cell and advected using a multidimensional positive-definite advection transport algorithm (Smolarkiewicz and Grabowski 1990). Pressure is diagnosed by solving a Poisson equation. Moisture is not introduced in the simulations of this study.
SAM was originally designed for atmospheric LES but has recently been adapted to simulate cloud chambers, which have a scale similar to the case in this study (Thomas et al. 2019, 2023; Yang et al. 2022; Wang et al. 2024). For the DNS used in this study, the anelastic approximation is replaced by the Boussinesq approximation. Additionally, the eddy viscosity is substituted with the kinematic viscosity (υ = μ/ρ, where μ = 1.717 × 10−5 Pa s is the dynamic viscosity of air, and ρ = 1.2096 kg m−3 is the density of air), and the wall model is replaced by an explicit no-penetration and no-slip boundary condition. The Prandtl number is 0.72. The simulation setup largely follows the DNS performed by Chandrakar et al. (2022): a 1 m3 cube domain with a cold top surface of 282 K and a warm bottom surface of 294 K (indicated as ΔT = 12 K hereafter), yielding a Rayleigh number of ∼109 (Table 1). Different from Chandrakar et al. (2022), sidewalls are removed in this work, and the doubly periodic lateral boundary conditions are applied for simplicity. The horizontal size of the domain results in a domain aspect ratio of 1, which may not be ideal for studying the structure of RBC (Grötzbach 1983; Stevens et al. 2018), but it is sufficient for capturing the characteristics of near-surface turbulence within RBC.
The Ra and resulting Nu from the DNS runs.
The DNS has 512 grid points in each direction (similar to those applied in Chandrakar et al. 2022), yielding a homogeneous and constant grid spacing of ∼2 mm. The time step is adjustable to account for the Courant–Friedrichs–Lewy criterion. The quasi-steady state of mean surface heat flux is reached in roughly 1 min (Fig. A1), and the result at 1–10 min is used for training. Although a quasi-steady state of domain-averaged quantities is reached, the locations of updraft and downdraft shift horizontally, which can represent the flow’s horizontal heterogeneity and unsteadiness. An illustration of the resulting temperature field, turbulent coherent structures, and energy spectrum is shown in Fig. 1. Because the focus of this research is on the wall model rather than the structure of RBC, we leave the statistics of domainwide flow quantities in appendix A and focus on the near-surface area in the main context hereafter.
The definition of boundary layer depth in this study is complicated by the slightly negatively tilted temperature profile observed within the domain’s interior (see Figs. A2a,b), which is a consequence of the limited horizontal domain with periodic boundaries. If we define the boundary layer by temperature gradient (∂T/∂z; i.e., as ∂T/∂z is 99% close to the centerline value, where z is the distance from the surface), there are 12 grid points in the boundary layer, which meets the requirement of roughly 10 points for this Ra (Stevens et al. 2010). Table 1 shows that the resulting Nusselt numbers align with experimental data from Niemela et al. (2000) and Chavanne et al. (2001), as well as the DNS from Stevens et al. (2010) (see the intercomparison of Nu Ra−1/3 in Stevens et al. 2010, Fig. 1a).
For an a posteriori test, a separate DNS run is carried out with a 276-K cold top surface and a 300-K warm bottom surface (indicated as ΔT = 24 K hereafter). The grid resolution follows the DNS with ΔT = 12 K. If we define the boundary layer by ∂T/∂z as above, the boundary layer will contain 11 grid points. Concerning the limitation of computational resource, this run is performed for a physical time of 4 min, but the data are enough for analysis (Fig. A1). A comparison of the energy spectra of two DNS runs is presented in Fig. 1b. It should be noted that this second DNS run is not used for training the NN models but instead is employed to evaluate the extrapolation ability of the NN models after they are implemented in LES.
Table 1 demonstrates that the two DNS runs yield similar values of Nu Ra−1/3, and the values are consistent with the experiments and DNS results presented by Stevens et al. (2010), Fig. 1a. However, Stevens et al. (2010) show that a broader Ra range reveals a decrease in Nu Ra−1/3 with increasing Ra, implying that the true scaling power of Ra for Nu is less than 1/3 [see Grossmann and Lohse (2000, Table 1) for the scaling power in various experiments].
b. Data preparation
For parameterization development and evaluation, the DNS output is mapped to the LES grid (allowing the overlapping of filter grid cells to increase the amount of data) by applying a top-hat filter (i.e., DNS results are averaged over each LES grid cell). The filter length is chosen to be consistent with the grid spacing used in recent LES studies of cloud chambers, ranging from 3.125 to 6.25 cm (Thomas et al. 2019, 2023; Yang et al. 2022; Wang et al. 2024). Note that these filter lengths are within the small-scale part of the inertial subrange, as shown by the red dashed and dotted lines in Fig. 1b, confirming the appropriateness of these grid spacings for LES.
The inputs to the MOST and NN models include the distance from the walls (h, which is half of the filter length) and two filtered variables: magnitude of horizontal wind speed (
For training the PAFNN models, a choice needs to be made about whether to invoke global (as in Zhou et al. 2021) or only local quantities for input. Because a wall model is intended to be applicable to any geometry and flow type, using only local variables is highly preferred (Yang et al. 2017).
c. The Monin–Obukhov similarity model
d. The neural network models
The training of the NN models is performed using Keras (Chollet et al. 2015) and TensorFlow (Abadi et al. 2015) in Python. The input layer consists of standardized and filtered variables from DNS data and a bias. The output layer contains the standardized values of
Several LES wall models are trained for comparison purposes, as summarized in Table 2. The first model (FNN_basic) is trained using the basic inputs as those for the MOST model to examine whether the traditional wall model could be improved with the same inputs. The next case (FNN_+w) is trained with the addition of
Introduction of the FNN cases.
To construct PAFNN models, three additional cases are developed using nondimensional variables, as detailed in section 2(b). The variables are represented on a logarithmic scale, except for Frw and cosθ, which can take negative values. The use of logarithmic scales is based on previous research that suggested power-law relationships between the surface heat flux (scaled as Nuh) and the surface shear stress (scaled as CD) with the Reynolds number and Rayleigh number (Grossmann and Lohse 2000, 2002, 2011; Stevens et al. 2011, 2013), at least for sufficiently large Reynolds numbers and Rayleigh numbers. Invoking the logarithmic scaling simplifies the regression task in the asymptotic regime. In addition, to extrapolate outside the training conditions, the behavior of the target variable at infinity must conform to the behavior of the activation function at infinity, as per the extrapolation theorem in Bin et al. (2022). By employing the leaky ReLU activation function, the above is only true if we invoke the logarithmic scale. In section 3e, we will evaluate PAFNN performance on extrapolated input parameters.
e. The Shapley additive explanations value
In addition to comparing different NN cases, the importance of each feature in the NN models is further evaluated by the SHAP values (Shapley 1953). We refer interested readers to Štrumbelj and Kononenko (2014) for the detailed equations. The primary concept of the SHAP value is to quantify how a variation in an input feature contributes to a variation in the output. A SHAP value is assigned to each feature for a particular event, and the distribution of SHAP values provides a measure of the overall importance and correlation (positive or negative) of each feature on a specific output.
The SHAP values and related figures in this work are produced mainly by the SHAP package for Python (Lundberg and Lee 2017). Although the SHAP values are standardized when directly computed from the input and output for NN, those presented in this work are scaled back by the output standard deviation to better quantify their contribution.
f. A posteriori test
To perform a posteriori tests, we implement the trained wall models in SAM LES, which uses the same solver and configuration as SAM DNS with a few modifications. The grid spacing is coarsened to 3.125 cm, and we use an anelastic approximation (as originally adopted in SAM LES for cloud chambers; Thomas et al. 2019, 2023; Wang et al. 2024), a turbulent-kinetic-energy (TKE) subgrid-scale (SGS) model (Deardorff 1980), and either a MOST or NN wall model. The upper boundary adopts the same wall model as the lower surface but with reverse values of
While the evaluations of LES of some flows are based on systematic behavior at certain locations (e.g., the flow over hills; Zhou et al. 2021), this is not the case for RBC in this work. The locations of updraft and downdraft plumes shift with time, making it challenging to compare the LES and DNS results at specific locations. Instead, we compare the results based on the mean and distribution of local
3. Results
The snapshot of the DNS run reveals a turbulent convective layer with updraft and downdraft plumes (Fig. 1a). The concentration of turbulent vortex tubes is higher along these plumes. Near the lower surface, the turbulent coherent structure is characterized by sparse vortex tubes below the downdraft plume and abundant vortex tubes being drawn into the updraft plume. This structure differs from regularly organized hairpin vortices in a horizontally homogeneous boundary layer, implying that MOST’s assumption of horizontal statistical homogeneity is not satisfied in this case.
The results of the a priori tests in sections 3a to 3c demonstrate the inadequacy of MOST in modeling local surface shear stress and heat flux, the better results obtained by the NN models, and the gradual improvement of the NN models with the inclusion of additional input features. Later in sections 3d and 3e, we present the a posteriori tests and evaluate the ability of the PAFNN models to extrapolate.
a. A priori test: The Monin–Obukhov similarity model
Figure 2 presents a comparison between
Low values of the coefficient of determination for
b. A priori test: Feedforward neural network trained with primitive flow quantities
Figures 3a and 3b present the results for FNN_basic. For a better comparison, the color-bar scales remain the same for all the similar two-dimensional histograms (Figs. 2–9). Compared to the MOST results shown in Fig. 2, the R2 values for
With the addition of
Figure 5 shows that the results can be further improved considering the influence of the pressure gradient via two input variables,
To better understand how each input variable affects
Figure 6 reveals the deviation of the mean |U| and T profiles simulated by DNS from those suggested by MOST (comparing the red solid line to the blue line in each panel). Figure 6a shows that high
c. A priori test: Physics-aware feedforward neural network
The performance of the NN can be improved by incorporating information about the physical system, e.g., by forming a reduced set of input variables using dimensional analysis (Gunaratnam et al. 2003). Nondimensionalizing FNN_basic inputs using Eqs. (4) and (5) reduces the number of input features from three to two in the PAFNN_basic case (2). However, as shown in Figs. 7a and 7b, the R2 of PAFNN_basic is inferior to FNN_basic with much narrower ranges of modeled
After considering the influence of vertical velocity, PAFNN_+w significantly improves the results (cf. Fig. 8 to Fig. 7). Specifically, R2 for
Accounting for the influence of pressure gradient force further improves the R2 values for both
In summary, when comparing the PAFNN with the FNN models, only PAFNN_basic shows noticeably poorer performance than FNN_basic. To ensure a fairer comparison between the FNN and PAFNN models, their results are evaluated against the number of features (Fig. 10, which is also known as the elbow method; Thorndike 1953). We see that 1 − R2 is reduced significantly as the number of input features increases from 2 to 3, and then declines slowly as the number of features increases further. This suggests that including more than three input features increases the risk of overfitting, especially for
d. A posteriori baseline tests
In this section, we subject the NN models to a posteriori tests by implementing these models in SAM, performing LES, and comparing the simulated distributions of
The distribution of
For the physics-aware models, PAFNN_basic overestimates the probability for
Regarding the surface heat flux, Figs. 11b and 11d show that the LES with the MOST model (green lines) predicts a negatively skewed distribution compared to a positively skewed PDF from DNS and overestimates the mean
The performance of the wall models heavily depends on the quality of the input variables. During the NN training and in the a priori testing, these variables are taken from a reference DNS and are the same for all wall models, while in the a posteriori (interactive) testing the inputs, provided now by LES, are affected by the employed wall models. Figure 12 shows the primitive flow quantities on the first LES grid level that contribute to the input layer of the NN models. The distributions of
Overall, including the input from vertical velocity into the NN wall models improves the peak and smoothness of the considered PDFs. In terms of the mean value, FNN_+w yields the best result for
e. A posteriori extrapolation tests
In the final set of tests in this study, we investigate the performance of the NN wall models when they operate on input variables outside the ranges for which they were trained. In these extrapolation tests, we conduct simulations of more unstable RBC by doubling the temperature difference between the bottom and top boundaries (from 12 to 24 K), while keeping the rest of the model setup unchanged.
First, we note that the MOST model cannot accurately capture either the mean or the PDF shape of
Figure 14 shows the near-surface primitive flow quantities for examining the input features. Here, we focus only on the distribution obtained by the PAFNN models compared to that in the DNS (represented by the red and black lines in Fig. 14, respectively). Similar to the training case as shown in Fig. 12,
In summary, although comparing the different LES runs with DNS is complicated by the difference in the flow quantities in each run, we can conclude that PAFNN models possess the capability to extrapolate, at least within the RBC regime considered in this study, and including vertical velocity captures more details. Although including pressure gradient force does not improve the mean surface shear stress and heat flux, the smoothness of PDF is improved, and the local surface shear stress is reproduced more accurately according to the a priori tests. In a pressure-driven flow, including pressure gradient force may be crucial.
4. Conclusions
Many traditional LES wall models rely on MOST, which is derived for a shear-dominated flow and may not be well suited for RBC. One important flow quantity in RBC is the near-surface vertical velocity, which is not considered by MOST. To explore alternative wall models, we perform DNS to train the NN (including the FNN and PAFNN) models with various input features for a better modeling of surface shear stress and heat flux. To understand the influence of the input flow quantities, the SHAP values and the conditional average of the |U| and T profiles are investigated. For the a priori study, we use the DNS-filtered data as inputs for the MOST and NN models, and then compare the models’ outputs with the DNS-filtered data. For the a posteriori tests, we implement the NN models in LES and evaluate the distribution of surface shear stress and heat flux against the DNS results. For the a posteriori extrapolation tests, additional DNS and LES runs are performed with a doubled temperature difference between the bottom and top boundaries.
A priori tests show that, in the same scenario (temperature difference between the bottom and top boundaries) as in the training process, the NN models are able to capture the local surface shear stress and heat flux. Incorporating vertical velocity into the NN models enhances both the modeled surface shear stress and heat flux. Meanwhile, the inclusion of the horizontal pressure gradient force yields a noticeable improvement in surface shear stress but only marginally enhances the heat flux. An analysis of the input features shows that a strong horizontal velocity magnitude is positively correlated with surface shear stress (analogous to the MOST model) but is negatively correlated with heat flux (contrary to the MOST model). A pronounced vertical temperature difference is positively correlated with both surface shear stress and heat flux. Negative near-surface vertical velocities, or downdrafts, drive the high-|U| and low-T air toward the surface. This enhances the vertical wind shear and temperature gradient near the surface, subsequently amplifying both surface shear stress and heat flux. A significant horizontal pressure gradient force and its alignment with the horizontal velocity both augment the near-surface vertical gradient of |U|, thereby increasing surface shear stress. However, they exert only a minimal impact on T and, consequently, the heat flux. When trained with nondimensional parameters as data and with more than three input features, the PAFNN models perform comparably well to the FNN models in the scenarios of the training process. Finally, all NN models exhibit shortcomings when modeling the lower ranges of surface shear stress and heat flux.
In the a posteriori tests simulating the same RBC scenario used for the training, LES runs utilizing the newly implemented NN models can achieve reasonable mean values of surface shear stress and heat flux. Nevertheless, when including only the same input as the MOST model, the NN models exhibit unsmooth distributions near the peaks, revealing the limitation owing to insufficient input features. The details in the distributions of surface shear stress and heat flux are improved after including the influence of vertical velocity. Including the influence of pressure gradient force results in the smoothest distribution of PDF, although the mean surface shear stress and heat flux are underestimated. However, the input features provided in LES are different from DNS, possibly because of a greater dissipation from the SGS models and advection schemes. The quality of the input features can affect the performance of the wall models. For example, the limitations of the NN models in modeling the lower ranges can be worsened by the overestimated near-surface temperature difference in LES, and the lack of improvement after including pressure gradient force may result from the underestimated pressure gradient force in LES. Last, the a posteriori tests with a more intense RBC scenario show that the FNN models are incapable of extrapolating, whereas the PAFNN models demonstrate the ability to do so.
This work contributes to advancing the understanding of near-surface turbulence and applying ML techniques to the development of NN-based wall models. Our analysis of feature importance highlights the negative correlation of the near-surface vertical velocity with surface shear stress and heat flux. The vertical velocity’s influence should exist and be considered in the wall model of any flow where vertical velocity is nonnegligible. Furthermore, although vertical velocity and near-surface temperature difference are not completely independent in this study, they can be independent in flows such as a tornado, where vertical velocity is driven by the strong perturbation pressure gradient force (e.g., Wang et al. 2020, 2023). In terms of application, the PAFNN models have successfully extrapolated to a more unstable flow, suggesting their potential for use in other atmospheric conditions.
From the presented results, several directions for follow-up research can be identified. To enhance the NN-based wall models, a more systematic exploration of hyperparameters for training is recommended, and testing different filter types for DNS data, such as the Gaussian filter, is worth considering. Additionally, it remains to be explored whether the means and standard deviations derived from the training data are suitable for various flows. Regarding limitations, the NN models do not currently account for surface roughness like the MOST-based model. Also, the current PAFNN models are not yet applicable to a stable boundary layer, because the Rayleigh number is restricted to positive values when used in logarithmic scale as an input. For application to atmospheric flows, the temperature difference may need to be parameterized for different hydrostatic regimes like MOST, and further experiments are required to confirm its reliability. Finally, for atmospheric applications, the proposed approach needs to be expanded to include the development and testing of NN representations for fluxes of moisture and other scalars.
Acknowledgments.
This research is supported by the U.S. Department of Energy Office of Science Atmospheric System Research (ASR) project at PNNL. PNNL is operated for the Department of Energy by Battelle Memorial Institute under Contract DE-AC05-76 RL01830. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract DE-AC02-05CH11231 using NERSC Awards BER-ERCAP0021333 and BER-ERCAP0024023. Yang acknowledges AFOSR for financial support. Figure 1a is produced using Python PyVista package (Sullivan and Kaszynski 2019). Figure 1b is produced by the code modified from Python wrapper for the John Hopkins University Turbulence Database Cluster library (Li et al. 2008; Perlman et al. 2007). For transparency and following the position statement of the Committee on Publication Ethics (COPE), we acknowledge the assistance of ChatGPT and Bing Chat in examining the possible reasons for errors in the data processing code, generating the first draft of the data processing code, checking spelling, correcting grammar, along with some refinements for the context.
Data availability statement.
The SAM model was kindly provided by Prof. Marat Khairoutdinov and publicly available at http://rossby.msrc.sunysb.edu/∼marat/SAM.html. The output from the SAM simulations, including the DNS and LES runs, and the Python scripts for training NN are stored on NERSC HPSS storage system at https://portal.nersc.gov/archive/home/w/wang1202/www/Wang2023JAS. TensorFlow is an open source available at https://www.tensorflow.org/. Keras is also an open source available at https://keras.io/. The SHAP value package for Python is available at https://github.com/slundberg/shap.
APPENDIX A
Flow Statistics
As introduced in section 2a, the focus of this study is on near-surface turbulence. The domain aspect ratio with periodic lateral boundaries is not ideal for examining the interior structure of RBC. Keeping this in mind, the flow statistics of the DNS runs and LES runs with two different wall models (MOST and PAFNN_+w+∇p) are provided for the readers’ information.
Figure A1 displays the time series of both the DNS and LES runs. For the LES runs, only two representative wall models are chosen to maintain clarity in the figures. The surface sensible heat flux can represent the steadiness of near-surface turbulence, while the resolved TKE can represent the steadiness of the interior of the domain. In each case, the quasi-steady states are reached in approximately 1 min. The LES runs achieve the quasi-steady states slightly faster than the DNS runs, due to the assumption of complete mixing within the LES grid cells.
Figure A2 depicts horizontally averaged vertical profiles of flow statistics for the two a posteriori tests, with ΔT of 12 and 24 K. Figures A2a and A2b reveal that the temperature profile in the interior of the flow is slightly tilted, complicating the definition of boundary layer depth. The tilt in the temperature profile appears to be related to periodic lateral boundaries, because an additional LES run with no-penetration lateral boundaries shows no tilt (not shown). If the boundary layer is examined by the temperature gradient, before the temperature gradient reaches 99% close to the centerline value, the LES runs consist of only one grid point in the boundary layer, while the DNS runs have 12 and 11 grid points for the ΔT = 12 K and ΔT = 24 K cases, respectively. Figures A2c and A2d present the temperature variance. DNS resolves a much higher temperature variance near the surface. In the LES, the MOST model produces a slightly larger temperature variance than the PAFNN model. Figures A2e and A2f illustrate that the DNS runs contain more than 10 grid points below the near-surface peaks of TKE, whereas in the LES runs the local maxima of TKE are at levels closest to the walls. It is important to note that the resolved TKE in LES should be less than the total TKE represented by DNS. Therefore, LES with the MOST wall model that yields resolved TKE similar to, or greater than, that of DNS is indicative of an overpredicted sensible heat flux (i.e., excessive energy input). This is further confirmed in Figs. A2g and A2h, where the resolved vertical velocity variance in LES with the MOST wall model exceeds that predicted by DNS.
APPENDIX B
Training Details
Using the early stopping criterion, the training for each case halts when the loss stops decreasing for 20 consecutive epochs. As the number of features increases, the loss can reach a lower minimum, and the required number of epochs increases.
Figures B2 and B3 display the distributions of features versus targets used in the training of both the FNN and PAFNN models. Generally, features that are more correlated with targets result in higher feature importance, as seen in Figs. 3–5 and 7–9.
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