Wind farm-induced atmospheric gravity waves have been the subject of recent research as they can impact wind farm performance. Pressure variations associated with gravity waves can contribute to the global blockage effect and wind farm wake recovery. Therefore, accurate numerical simulation of flow fields, where wind-farm-induced gravity waves may be produced, is important. Three main considerations in such simulations are the overall domain size, the use of Rayleigh damping near domain boundaries to dampen gravity waves, and advection damping at the inlet to prevent spurious oscillations. Often these considerations are treated ad hoc rather than systematically. This work aims to test and extend the systematic modelling of internal gravity waves proposed in a preliminary investigation to modelling of both internal and trapped gravity waves. The preliminary study identifies the length scales to set the domain and damping layer sizes and the time scale to configure the Rayleigh damping coefficient but under linearly stratified conditions. Large eddy simulations of flow through a wind farm canopy are performed under conventionally neutral boundary layer (CNBL) conditions to test the validity of proposed setups for CNBL conditions. Background atmospheric parameters, such as Froude number (Fr), inversion height (Hi), and inversion layer Froude number (Fri) control most of the atmospheric gravity wave characteristics. We validated for CBNL conditions that the effective wavelengths of the internal gravity waves are the correct length scale to configure the domain size and damping layer thickness. Likewise, the optimum damping coefficient to dampen the internal gravity waves relates to the free atmosphere's buoyancy frequency or buoyant perturbations' time scale. We infer that the damping coefficient in the inversion layer may relate to the inversion buoyancy frequency to effectively dampen the trapped gravity waves. Moreover, the advection damping length is linked to the horizontal wavelength of the trapped gravity waves in the inversion layer to prevent spurious waves at the inlet by retaining wave energy accumulation.

Wind farms, particularly offshore clusters, are becoming larger than ever before. Besides influencing the surface wind flow and the inflow for downstream wind farms, large wind farms can trigger atmospheric gravity waves in the inversion layer and the free atmosphere aloft. Wind farm-induced gravity waves can cause adverse pressure gradients upstream of the wind farm, that contribute to the global blockage effect, and can induce favorable pressure gradients above and downstream of the wind farm that enhance wake recovery. Numerical modeling is a powerful means of studying these wind farm-induced atmospheric gravity waves, but it comes with the challenge of handling spurious reflections of these waves from domain boundaries. Typically, approaches which employ radiation boundary conditions and forcing zones are used to avoid these reflections. However, the simulation setup of these approaches heavily relies on ad-hoc processes. For instance, the widely used Rayleigh damping method requires ad-hoc tuning to produce a setup which may be only produce satisfactory results for a particular case. To provide more systematic guidance on setting up realistic simulations of atmospheric gravity waves, we conduct an LES study of flow over a 2D hill and through a wind farm canopy that explores the optimum domain size and damping layer setup depending on the fundamental parameters which determine the flow characteristics.

In this work, we only consider linearly stratified conditions (i.e., no inversion layer), thereby focusing on internal gravity waves in the free atmosphere and their reflections from the domain boundaries. This type of flow is governed by a single Froude number, which dictates most of the internal wave properties, such as wavelength, amplitude, and direction. This in turn will dictate the optimum domain size and Rayleigh damping layer setup. We find the effective horizontal and vertical wavelengths, (the representative wavelengths of the entire wave spectrum), to be the appropriate length scales to size the domain and damping layer thickness, and the optimal Rayleigh damping coefficient scales with the Brunt–Väisälä frequency.

Considering Froude numbers seen in wind farm applications, we propose recommendations to limit the reflections to less than 10 % of the total upwards propagating wave energy. Typically, damping is done at the top boundary, but given the non-periodic lateral boundary conditions of practical wind farm simulation domains, we find that damping the inflow-outflow boundaries is of equal importance to the top boundary. The Brunt–Väisälä frequency-normalized damping coefficient should be between 1 and 10. The damping layer thickness should be at least one effective vertical wavelength; damping layers exceeding 1.5 times the vertical wavelength are found to be unnecessary. The domain length and height should accommodate at least one effective horizontal and vertical wavelength, respectively. Moreover, Rayleigh damping does not damp the waves completely, and the non-damped energy might accumulate over the simulation time.

The interaction of large wind farm clusters with the thermally stratified atmosphere has emerged as an important physical process that impacts the productivity of wind farms. Under stable conditions, this interaction triggers atmospheric gravity waves (AGWs) in the free atmosphere due to the vertical displacement of the atmospheric boundary layer (ABL) by the wind farm. AGWs induce horizontal pressure gradients within the ABL that alter the wind speed distribution within the farm, influencing both wind farm power generation and wake development. Additional factors, such as the growth of an internal boundary layer originating from the wind farm entrance and increased turbulence due to the wind turbines, further contribute to wake evolution. Recent studies have highlighted the considerable computational cost associated with simulating gravity wave effects within large-eddy simulations (LESs), as a significant portion of the free atmosphere must be resolved due to the large vertical spatial scales involved. Additionally, specialized boundary conditions are required to prevent wave reflections from contaminating the solution. In this study, we introduce a novel methodology to model the effects of AGWs without extending the LES computational domain into the free atmosphere. The proposed approach addresses the wave reflection problem inherently, eliminating the need for these specialized boundary conditions. We utilize the recently developed multi-scale coupled (MSC) model of Stipa et al. (2024b) to estimate the vertical ABL displacement triggered by the wind farm, and we apply the deformation to the domain of an LES that extends only to the inversion layer. The accuracy in predicting the AGW-induced pressure gradients is equivalent to the MSC model. The AGW modeling technique is verified for two distinct free-atmosphere stability conditions by comparing it to the traditional approach in which AGWs are fully resolved using a domain that extends several kilometers into the free atmosphere. The proposed approach accurately captures AGW effects on the row-averaged thrust and power distribution of wind farms while demanding 12.7 % of the computational resources needed for traditional methods. Moreover, when conventionally neutral boundary layers are studied, there is no longer a need to solve the potential temperature equation, as stability is neutral within the boundary layer. The developed approach is subsequently used to compare the global blockage and pressure disturbances obtained from the simulated cases against a solution characterized by zero boundary layer displacement, which represents the limiting case of very strong stratification above the boundary layer. This approximation, sometimes referred to as the “rigid lid”, is compared against the full AGW solution using the MSC model. This is done for different values of inversion strength and free atmosphere lapse rate, evaluating the ability of the “rigid lid” to predict blockage, wake effects, and overall wind farm performance.