Recent research suggests that atmospheric gravity waves can affect offshore wind-farm performance. A fast wind-farm boundary layer model has been proposed to simulate the effects of these gravity waves on wind-farm operation by
In recent years, it has been well documented that wind farms form a blockage to the flow in and around them
Previous work on the interaction between gravity waves and wind farms has assumed the free atmosphere to be uniformly stratified, with a constant background wind
Currently, the TLM can only describe uniformly stratified free atmospheres, which places a strong restriction on the atmospheric conditions that can be represented. This work adapts the TLM for flow profiles that vary with altitude and studies how these variations change the interaction between the ABL flow and gravity waves. A common approach in gravity-wave theory is to use a piecewise representation of the upper atmosphere, where the profiles of the stratification and the wind speed are split up in a discrete number of layers
It is well known that variations in the atmospheric state can cause wave reflection, which might lead to internal gravity-wave resonance
The remainder of this paper is organized as follows. Section
Wind farms form a blockage to the ABL flowing through and around them, thereby pushing the inversion layer, and the free atmosphere above, upwards. These displacements can trigger gravity waves, which may influence the ABL flow by inducing pressure gradients. The first part of this section discusses these induced pressures, while the second part gives an overview of how the TLM models ABL flow and how it incorporates gravity-wave effects.
The vertical displacement
The second type of waves propagates vertically through the free atmosphere above the capping inversion if it is stably stratified, as is usually the case. The free atmosphere perceives
The TLM is based on earlier work by
The TLM improves on the model by Smith in several ways, the most important of which is to divide the ABL in two separate layers. The resulting lower and an upper layer are denoted by subscripts 1 and 2, respectively.
The two layers are separated by a pliant surface, similar to the interface between the ABL and the free atmosphere. The wind-farm forcing terms are added in the momentum equations for the lower layer while only affecting the upper layer through interaction through the pliant surface. For this reason, the lower layer is also called the wind-farm layer. The resulting approximation of the ABL is visualized in Fig.
Schematic representation of the three-layer model. Figure from
In Eqs. (
The turbines are represented individually using an actuator disk model. To incorporate their interactions, the TLM is coupled with a wake model, which in this work is a Gaussian wake model coupled with linear superposition of velocity deficits
In reality, the free atmosphere is not uniform, and the stratification strength and wind speed can strongly depend on altitude. This will of course impact internal gravity-wave propagation through the atmosphere and thus the pressure feedback of these waves in the ABL. Currently, the TLM does not incorporate this as the simplified version of the internal wave equation on which Eq. (
The internal gravity-wave equation in vertically non-uniform atmospheres with continuous background velocities is
The relation between
To evaluate the expression for the stratification coefficients derived in the previous section, the Helmholtz equation for vertically non-uniform atmospheres has to be solved. This is no longer trivial as the vertical wavenumber now varies with altitude. In earlier studies, one of the main approaches to solving the Helmholtz equation has been the so-called piecewise or multilayer methods
The basic principle of piecewise methods is to represent the atmosphere as a discrete number of sublayers
The main advantage of this method is that realistic wave patterns can be obtained with a relatively small number of sublayers. Within a sublayer, only 2 degrees of freedom have to be determined. Another advantage compared to other methods such as Wentzel–Kramers–Brillouin (WKB) theory is that piecewise methods can account for wave reflection, although they cannot incorporate weakly non-linear effects
As the number of sublayers has to be limited for computational reasons, not all of the atmosphere can be approximated. An appropriate height
The values of the coefficients
Above the highest sublayer, the atmosphere is assumed to be uniformly stratified. This results in the same situation as discussed in Sect.
Combined with the radiation condition applied at height
If there are inversion layers in the free atmosphere, these can be modeled by discontinuities in
As the wind can vary, the situation can arise where
To verify the implementation of the piecewise-constant method, it was compared to a second-order finite-difference (FD) code with a central difference scheme on various continuously varying background velocities and buoyancy frequencies. The piecewise method consistently outperformed the FD code, achieving second-order convergence as expected through the proof in Appendix
We also reproduced results from
Idealized upper-atmosphere velocity
An overview of the different upper-atmospheric flow profiles used for verification. The upper atmosphere set up by
Mountain wave drag on a small ridge with a two-layer Brunt–Väisäla frequency profile and constant background wind
By using the piecewise-constant method developed in Sect.
We further investigate how this influences the interaction between wind farms and gravity waves. To this end, Sect.
To determine the effects of vertical non-uniformity on the pressure feedback of internal gravity waves, the stratification coefficients are calculated for the upper atmosphere used by
The stratification coefficients for a uniform atmosphere with
While in a uniform atmosphere the wind farm can only trigger waves with an upwards group velocity, changes in the stratification and wind speed can cause waves to reflect. This allows both up- and downgoing waves to propagate throughout the atmosphere. As up- and downgoing internal gravity waves pass through each other, they interfere, potentially causing resonance. The resulting large wave amplitudes can drastically affect the pressure feedback of the waves
Figure
We investigate how variations in the wind and stability change the effects wind-farm operation has on the ABL flow by revising an example case used by
To analyze how the changes in the stratification coefficients impact the flows around wind farms, a flow case discussed earlier by
The example case is set up to have a
The example case is set up as follows. The mean flow in the two layers of the ABL is based on the analytical boundary layer model of
Table
Flow parameters of the flow case based on the one used by
Wind-farm configuration of the reference flow cases, as analyzed by
The results of the uniform and non-uniform simulations are shown in Fig.
Planform view of the inversion layer displacement
We further analyze the lee waves that appear in Fig.
The left-hand side of Eq. (
In order to apply this theory to the case from Sect.
While the above analysis explains how vertical variations in the atmospheric profile change the interaction between internal gravity waves and the ABL flow, it does not offer insight on how to predict the resulting impact on wind-farm performance. It is not clear what parameters could describe this. Extensive research has been done on internal gravity-wave resonance, with most of it focusing on flow around topographies. However, this has to be used with caution as the overall flow is then analyzed in the context of mountain wave drag, where the height displacement is given by the shape of the terrain
To determine the impact of varying wind speeds and stability on wind-farm energy production, we follow the approach of
The simulations were performed on a 2000 by 2000 grid, on a 1000
The atmospheric profiles of potential temperature
The maximum capping inversion displacement
From the 8746 cases, we only use those where the atmosphere is statically stable at every altitude in the free atmosphere. Additionally, the cases without capping inversion, (cf. earlier discussion following Eq.
Average perturbations over all the analyzed flow cases for both uniform and non-uniform upper atmospheres.
The difference between the maximum capping inversion displacements in the non-uniform and uniform simulations as normalized by the ABL height
Figure
The goal of this study was to extend the applicability of a wind-farm gravity-wave model to vertically non-uniform free atmospheres. This was done by changing the expressions for the stratification coefficients
The effects of the variations in background wind and stability were studied by analyzing how free-atmospheric wave reflection influences the wave pressure feedback, the ABL flow, and overall wind-farm performance. Firstly, the stratification coefficients for the idealized atmosphere used by
Finally, the extended TLM was used to simulate 1 year of operation of the Belgian–Dutch wind-farm cluster, repeating a similar analysis by
The results of this study show that vertical atmospheric non-uniformity could play a major role in the interaction between wind farms and gravity waves. This suggests that variations with altitude of the free atmosphere's wind and stability should be taken into account when simulating wind-farm operation in specific atmospheric conditions and may be important for the optimization of turbine control in the future (see, e.g.,
Adding more sublayers leads to a better approximation of the actual profiles of atmospheric variables. This would then also result in a better approximation of
It is notable that the above derivation is not limited to piecewise-constant methods but is valid for general piecewise methods. Therefore, as long as
The code used for the simulations and the raw data of the simulation results can be provided by contacting the corresponding author. An open-source version of the code is planned to be released by the end of the year. The code used for the simulations is written in Python.
KD, LL, NvL, and JM jointly derived how to apply the piecewise method to the TLM and set up the verification and simulation studies. KD performed code implementations and carried out the simulations. SJ provided the optimization tool used to determine the altitude of the tropopause. KD, LL, SJ, NvL, and JM jointly wrote the manuscript.
At least one of the (co-)authors is a member of the editorial board of
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The computational resources and services in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation Flanders (FWO) and the Flemish Government department EWI. The work of
This research has been supported by the Energy Transition Fund of the Belgian Federal Public Service for Economy, SMEs, and Energy (FOD Economie, K.M.O., Middenstand en Energie).
This paper was edited by Andrea Hahmann and reviewed by two anonymous referees.