Current turbulence parameterizations in numerical weather prediction models at the mesoscale assume a local equilibrium between production and dissipation of turbulence. As this assumption does not hold at fine horizontal resolutions, improved ways to represent turbulent kinetic energy (TKE) dissipation rate (

This work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.

While turbulence is an essential quantity that regulates many phenomena in the atmospheric boundary layer

Current boundary layer parameterizations of

The inaccuracy of the mesoscale model representation of

Several studies have documented the variability of

This knowledge on the variability of TKE dissipation rate provided by observations lays the foundation to explore innovative ways to improve the model representation of

Here, we train and test different machine-learning algorithms to predict

The Perdigão field campaign

At Perdigão, 184 sonic anemometers were mounted on 48 meteorological towers, which provided an unprecedented density of instruments in such a limited domain (Fig.

Map of the Perdigão valley showing the location and height of the 48 meteorological towers whose data are used in this study. Digital elevation model data are courtesy of the US Geological Survey.

The height of the towers ranged from 2 to 100 m, with the sonic anemometers mounted at various levels on each tower, as detailed in Table

To classify atmospheric stability, we calculate the Obukhov length

Heights where sonic anemometers were mounted on the meteorological towers at the Perdigão field campaign.

Histogram of the heights a.g.l. of the 184 sonic anemometers considered in this analysis.

TKE dissipation rate from the sonic anemometers on the meteorological towers is calculated from the second-order structure function

To account for the uncertainty in the calculation of

As additional quality controls, to exclude tower wake effects, data have been discarded when the recorded wind direction was within

Before testing the performance of machine-learning algorithms in predicting TKE dissipation rates, we first assess the current accuracy of the parameterization of

To evaluate the accuracy of the MYNN parameterization of

Scatter plot showing the comparison between observed and MYNN-parameterized

The TKE dissipation rate predicted by the MYNN parameterization shows, on average, a large positive bias compared to the observed values, with a mean bias of

Different atmospheric stability conditions give different biases. Figure

Scatter plot showing the comparison between observed and MYNN-parameterized

Stable cases show the largest bias (mean of

Different heights also impact the accuracy of the parameterization of

Bias in the MYNN-parameterized

As shown in Fig.

To test the power of machine learning to improve the numerical representation of the TKE dissipation rate, we consider three learning algorithms in this study: multivariate linear regression, multivariate third-order polynomial regression, and random forest. Given the proof-of-concept nature of this analysis in proving the capabilities of machine learning to improve numerical model parameterizations, we defer an exhaustive comparison of different machine-learning models to a future study and only consider relatively simple algorithms in the present work. The learning algorithms are trained and tested to predict the logarithm of

For the purpose of machine-learning algorithms, the data set has to be divided into three subsets: training, validation, and testing sets

Before testing the models, however, it is important to avoid overfitting by setting the values of hyperparameters. Each learning algorithm has specific model-specific hyperparameters that need to be considered, as will be specified in the description of each algorithm. To test different combinations of hyperparameters and determine the best set, we use cross validation with randomized search, with 20 parameter sets sampled for each learning algorithm. For each set of hyperparameters, the RMSE between the actual and predicted

Cross-validation approach used to evaluate the performance of the machine-learning models considered in this study.

In the following paragraphs, we describe the main characteristics of the three machine-learning algorithms used in our study. A more detailed description can be found in machine-learning textbooks

To check whether simple learning algorithms can improve the current numerical parameterization of

To avoid training a model that overfits the data, regularization techniques need to be implemented, so that the learning model is constrained: the fewer degrees of freedom the model has, the harder it will be for it to overfit the data. We use ridge regression

Multivariate polynomial regression can easily be achieved by adding powers of each input feature as new features. The regression algorithm is then trained as a linear model on this extended set of features. For a third-order polynomial regression, the model becomes

Random forests (

As an ensemble of decision trees, a random forest trains them on different random subsets of the training set. Once all the predictors are trained, the ensemble (i.e., the random forest) can make a prediction for a new instance by taking the average of all the predictions from the single trees. In addition, random forests introduce some extra randomness when growing trees: instead of looking for the feature that, when split, reduces the overall error the most when splitting a node, a random forest searches for the best feature among a random subset of features.

Decision trees make very few assumptions about the training data. As such, if unconstrained, they will adapt their structure to the training data, fitting them closely, and most likely overfitting them, without then being able to provide accurate predictions on new data. To avoid overfitting, regularization can be achieved by setting various hyperparameters that insert limits to the structure of the trees used to create the random forests. Table

Hyperparameters considered for the random forest algorithm.

Given the large variability of

Wind speed (WS).

The logarithm of TKE. This quantity is calculated as

The logarithm of friction velocity of

The log-modulus transformation

The standard deviation SD

The mean vegetation height

Example of an upwind terrain elevation sector with a 1 km radius centered on the location of one of the meteorological towers at Perdigão.

Scatter plot showing the comparison, performed on the testing set, between observed and machine-learning-predicted

The distribution of the input features and of

While we acknowledge that the input features are not fully uncorrelated, we found that including all these features provides a better predictive power for the learning algorithms, despite negatively affecting the computational requirements of the training phase. The application of principal component analysis can help reduce the number of dimensions in the input features while preserving the predictive power of each, but it is beyond the scope of the current work.

To evaluate the prediction performance of the three machine-learning algorithms we considered, we use, for each method, a scatter plot showing the comparison between observed and machine-learning-predicted

Performance of the machine-learning algorithms trained and tested at Perdigão, measured in terms of RMSE and MAE between the logarithm of observed and MYNN-parameterized

Scatter plot showing the comparison, performed on the testing set, between observed and machine-learning-predicted

The predictions from all the considered learning algorithms do not show a significant mean bias, as found in the MYNN representation of

Table

Given the large gap in the performance of the MYNN parameterization of

The random forest for unstable conditions provides, on average, more accurate predictions (RMSE of 0.37, MAE of 0.28) compared to the algorithm used for stable cases (RMSE of 0.44, MAE of 0.33), thus confirming the complexity in modeling atmospheric turbulence in quiescent conditions. However, when the error metrics are compared to those of the MYNN parameterization, the random forest for stable conditions provides the largest relative improvement, with a 50 % reduction in MAE, while for unstable conditions the reduction is of 20 %.

Not only do machine-learning techniques provide accuracy improvements to represent atmospheric turbulence, but additional insights on the physical interpretation of the results can – and should – be achieved. In particular, random forests allow for an assessment of the relative importance of the input features used to predict (the logarithm of)

Partial dependence plots for the input features used in the analysis. Distributions of the considered features are shown in the background.

Feature importance classification as derived from the random forest.

The feature importance results are affected by the correlation between some of the input features used in the models. We find how the logarithm of turbulence kinetic energy is the preferred feature for tree splitting, with the largest importance (47 %) in reducing the prediction error for

Finally, to assess the dependence of TKE dissipation rate on the individual features considered in this study, Fig.

The strong relationship between

Despite turbulence being a fundamental quantity for the development of multiple phenomena in the atmospheric boundary layer, the current representations of TKE dissipation rate (

The results of this study show how machine learning can provide new ways to successfully represent TKE dissipation rate from a set of atmospheric and topographic parameters. Even simple models such as a multivariate linear regression can provide an improved representation of

Multiple opportunities exist to extend the work presented here. In the future, additional learning algorithms, such as support vector machines and extremely randomized trees, should be considered. Deep learning methods, such as recurrent neural networks, and specifically long- to short-term memory, which are well suited for time-series-based problems, could also be considered to obtain a more complete overview of the capabilities of machine-learning techniques for improving numerical representations of

High-resolution data from sonic anemometers on the meteorological towers

The supplement related to this article is available online at:

NB and JKL designed the analysis. NB analyzed the data from the sonic anemometers and applied the machine-learning figures, in close consultation with JKL and MO. NB wrote the paper, with significant contributions from JKL and MO.

The authors declare that they have no conflict of interest.

The views expressed in the article do not necessarily represent the views of the DOE or the US Government.

We thank the residents of Alvaiade and Vale do Cobrão for their essential hospitality and support throughout the field campaign. In particular, we are grateful for the human and logistic support Felicity Townsend provided to our research group in the field. We thank Jose Laginha Palma for providing the vegetation data used in the analysis. We thank Ivana Stiperski for her exceptionally thoughtful review of our discussion paper, which greatly improved the quality of this work. Support to Nicola Bodini and Julie K. Lundquist is provided by the National Science Foundation, under the CAREER program AGS-1554055 and the award AGS-1565498. This work utilized the RMACC Summit supercomputer, which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder, and Colorado State University. The Summit supercomputer is a joint effort of the University of Colorado Boulder and Colorado State University.

Funding provided by the US Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office. This research has been supported by the National Science Foundation (grant no. AGS-1554055) and the National Science Foundation (grant no. AGS-1565498).

This paper was edited by Lutz Gross and reviewed by Ivana Stiperski and two anonymous referees.