# Departure from Flux-Gradient Relation in the Planetary Boundary Layer
## 1 Introduction  
Numerical weather prediction models, such as the Weather Research and Forecast (WRF) mesoscale model [1], are widely used to simulate the atmosphere's dynamics. WRF outputs are continuously applied for different purposes and in wind energy, in particular, are the common source of wind stampologies at present. A number of atmospheric processes cannot be resolved at the mesoscale resolution, so we have to rely on model parameterizations to characterize turbulence within the planetary boundary layer (PBL). Hence, it is important to evaluate the ability of PBL parameterizations to reproduce the atmospheric processes they are intended for. Some limitations in mesoscale models are inherent from parameterizations of the PBL. These models use a turbulence closure of the vertical transfer of momentum, which is responsible for well-known and long-standing biases related to enhanced diffusion and underestimation of the turning of the wind, especially under stably stratified conditions [2], as well as strongly baroclinic boundary layers [3]. Therefore, accurate observations of the wind and turbulence measures across the PBL, e.g., using remote sensing technologies, are valuable for the evaluation of PBL parameterizations commonly used in mesoscale models. Ground-based Doppler wind lidars measure the wind speed with an accuracy comparable to traditional anemometry [4, 5] and are able to reach heights of up to 2 km, depending on the aerosol concentration in the atmosphere. Measuring all heights simultaneously from a single instrument, e.g., with a pulsed lidar, removes uncertainties in wind-turningeffects analyses when compared to the use of multiple instruments mounted on a meteorological mast. Such long-range profiling lidars can also be used to estimate second-order moments. [PERSON] et al. [4] presented results with regards to the degree of turbulence attenuation originated from spatial averaging within the lidar measurement volume, while [PERSON] and [PERSON] [6] provided an overview of methods used to derive turbulence from lidar measurements. Lidar measurements, therefore, provide a unique opportunity to evaluate the mesoscale models' ability to reproduce turbulence characteristics and to improve the suite of parametrizations that such models offer. A number of PBL parameterizations in the WRF model are based on a flux-gradient relation, in which the eddy fluxes are related to the mean vertical gradients of velocity  
\[\langle u^{\prime}w^{\prime}\rangle=-K_{m}\frac{\partial U}{\partial z}\quad \text{and} \tag{1}\]  
\[\langle v^{\prime}w^{\prime}\rangle=-K_{m}\frac{\partial V}{\partial z}, \tag{2}\]  
where \(U\) and \(V\), aligned with the geographical coordinates, represent the two horizontal mean wind components, \(u^{\prime}\) and \(v^{\prime}\) are fluctuations around their means, and \(\langle\rangle\) represent an ensemble average. The eddy diffusivity or momentum exchange coefficient \(K_{m}\) can be computed differently in PBL parameterizations. As the influence of the Coriolis force tends to increase in the atmosphere, a misalignment between the stress vector \((-\langle u^{\prime}w^{\prime}\rangle,-\langle v^{\prime}w^{\prime}\rangle)\) and the mean wind vector \((U,V)\) is expected. Further, the flux-gradient relation might break down as a result of a misalignment between the stress vector and the vertical gradient of the mean wind vector \((\partial U/\partial z,\partial V/\partial z)\). Figure 1 illustrates these angles, hereafter referred to as \(\alpha\) and \(\beta\), respectively, and defined as  
\[\alpha=\text{atan2}(U,V)-\text{atan2}(-\langle u^{\prime}w^{\prime}\rangle,- \langle v^{\prime}w^{\prime}\rangle)\quad\text{and} \tag{3}\]  
\[\beta=\text{atan2}\Big{(}\frac{\partial U}{\partial z},\frac{\partial V}{ \partial z}\Big{)}-\text{atan2}(-\langle u^{\prime}w^{\prime}\rangle,-\langle v ^{\prime}w^{\prime}\rangle), \tag{4}\]  
where \(\text{atan2}(x,y)\) is defined such that it provides the angle of the vector relative to the \(x\)-axis.