The disturbed atmospheric pressure near a wind farm arises from the turbine drag forces in combination with vertical confinement associated with atmospheric stability. These pressure gradients slow the wind upstream, deflect the air laterally, weaken the flow deceleration over the farm, and modify the farm wake recovery. Here, we describe the airflow and pressure disturbance near a wind farm under typical stability conditions and, alternatively, with the simplifying assumption of a rigid lid. The rigid lid case clarifies the cause of the pressure disturbance and its close relationship to wind farm drag.

The key to understanding the rigid lid model is the proof that the pressure field

The construction of offshore wind farms may significantly help our society transition to renewable energy, but the wind slowing by these farms may ultimately limit their potential for electric power generation (Ahktar et al., 2022). This issue has an extensive literature, reviewed recently by Stevens and Meneveau (2017), Archer et al. (2018), Porte-Agel et al. (2020), Pryor et al. (2020), and Fischereit et al. (2021). An integral part of the wind slowing by turbine drag is the creation of a local pressure field. This pressure disturbance was initially neglected (Jensen, 1983) but has been recently estimated in connection with gravity wave (GW) generation (Smith, 2010, 2022; Wu and Porté-Agel, 2017; Allaerts and Meyers, 2018, 2019). In a stably stratified atmosphere, the lifting of the air caused by farm drag creates gravity waves aloft whose pressure field acts back on the lower atmosphere.

This pressure field modifies the airflow in ways that the direct action of turbine drag cannot. First, it can decelerate the flow before it reaches the first row of turbines, so-called “blockage” (Bleeg et al., 2018) in wind farm terminology or “blocking” in mountain meteorology. Second, it can deflect the air to the left and right. Third, over the farm, it can fight back against the turbine drag, helping to keep the wind flow strong. Finally, it slows the downwind recovery of the wake.

The pressure field near a wind farm is analogous in some respects to that for a single turbine. The airflow approaching a turbine disk begins to decelerate upwind due to an adverse pressure gradient, and its corresponding axial induction factor reduces the turbine efficiency to the Betz limit (Hanson, 2000). According to Gribben and Hawkes (2019), the local non-hydrostatic pressure disturbance decays inversely as the square of the distance upstream. The farm-generated hydrostatic pressure disturbance may be more far-reaching.

In discussing the cause of the pressure field, we shall exercise caution as the cause may be model dependent. In compressible subsonic aerodynamics, acoustic waves play a role in creating the pressure field. In stratified flow, gravity waves play a role. In presumed non-divergent flow, the pressure field is usually determined diagnostically as the cause is hidden from view. The pressure field exists simply to keep the flow non-divergent.

In this paper, we compare the wind farm pressure field in the realistic gravity wave (GW) model with the idealized rigid lid (RL) model. The rigid lid approximation retains some of the features of the atmospheric problem but allows us to derive simple theorems and closed-form solutions that clarify the cause, nature, and impact of the pressure field.

We begin by recalling the governing equations for the two-layer model of Smith (2010) and describe the rigid lid (RL) limit. Second, we derive approximate closed-form expressions for the far-field and near-field pressure. Third, we discuss the cause of the pressure field and its role in the wind farm disturbance. Finally, we consider using pressure measurements to estimate total farm drag and the use of the RL model in industrial applications.

Our method for computing the response to wind farm drag forces uses a two-layer stratified hydrostatic gravity wave (GW) model solved with fast Fourier transforms (FFTs). This model consists of a lower turbine layer from which momentum is removed by specified drag forces and a Rayleigh restoring force that decays the farm wake (Smith, 2010, 2022). An overlying density-stratified layer responds to vertical displacement and creates a hydrostatic pressure field

The GW model discussed herein uses the hydrostatic assumption and thus does not take into account the pressure field associated with vertical fluid acceleration. Pressure fields in this model are generated only by density anomalies aloft. If an airflow streamline approaching a wind farm curves sharply upwards, a region of non-hydrostatic high pressure will be generated below it. These effects are easily incorporated in the linearized FFT modeling framework, but we do not do that here. In mountain wave theory, for example, such effects are usually neglected for horizontal scales greater than 1 km. Non-hydrostatic effects are certainly important on the scale of an individual turbine but less so on the farm scale where hydrostatic effects are expected to dominate. Some wind farm models, such as that of Gribben and Hawkes (2019), include the non-hydrostatic effect but neglect the hydrostatic effect.

We first ran the two-layer GW model with the realistic parameters shown in Table 1. The model's two stability parameters are the reduced gravity

Parameters of the reference gravity wave (GW) model.

Wind farm disturbance properties as stability is increased towards the rigid lid limit. The value

To investigate the influence of atmospheric stability (Eq. 4), we ran the GW model several more times – first with the two stability parameters

We then increased each stability parameter (Eq. 4) from zero towards a large value (Table 2). The vertical displacement of the fluid decreased towards zero and the pressure perturbations increased from zero. Other model output values changed only slightly. The maximum wind speed deficit decreased slightly from 0.445 to 0.323

One striking aspect of Table 2 is that the

The planform patterns of the gravity wave (GW) and rigid lid (RL) solutions are compared in Figs. 1 and 2. The wind speed deficit patterns (Figs. 1a and 2a) show the wake caused by the farm drag but also show the influence of the pressure fields. Both show upstream deceleration, which is stronger in the RL case, and lateral regions of accelerated flow downwind of the farm. The wind speed deficit patterns over the farm are different also due to pressure forces acting on the flow. The pressure fields (Figs. 1b and 2b) show an upwind maxima and downwind minima of approximately similar magnitude. The RL case, however, has these two extrema shifted upwind, and the whole field is exactly anti-symmetric with respect to the upwind–downwind direction.

Zoom of the disturbance caused by a 7 km by 7 km wind farm from the realistic gravity wave (GW) model:

Zoom of the disturbance caused by a 7 km by 7 km wind farm from the idealized rigid lid (RL) case:

We can understand the rigid lid (RL) solution more fully by noting that the pressure field

To illustrate the harmonic property of

Laplacian of the pressure field with units

Recall that a harmonic function has no local maxima or minima and therefore only takes on values that are between the boundary values. As

In non-divergent flow, the role of pressure is to maintain the non-divergent property of the flow. As the turbine force field

This interpretation is supported by noting that pressure is insensitive to the Rayleigh restoring force coefficient

Centerline properties of the farm disturbance including the farm mask, wind speed deficit (

The two pressure fields, GW and RL, are compared along the centerline in Fig. 4a and b. Both transects have an upwind maximum and downwind minimum. The GW pressure field (Fig. 4a) is smoother with a maximum over the farm and a smaller minimum in the near wake. In the rigid lid case (Fig. 4b), the pressure maximum and minimum points are equal in magnitude and shifted upstream slightly to the farm edges. In both cases, the air decelerates as it approaches the farm under the adverse pressure gradient. The linearized Bernoulli equation derived from Eq. (1),

A key feature of the rigid lid solution is the linear pressure field over the farm, so we define

Using values from Tables 1 and 2, the non-dimensional force ratio is

Equation (8) is the Poisson equation where the scalar

In the present computation (Table 1), the drag force is

Green's function method with two delta functions (Eqs. 12 and 13) can also be used to find the pressure field near the farm center. The result is

In the previous section, we used Green's function solution to Eq. (8) to derive the far-field pressure dipole Eq. (14). We now re-derive this formula using a physical volume-conservation argument. When the farm drag slows the flow, it creates a volume flow deficit (

If the mean flow

As the RL source function expression Eq. (19) provides good estimates of the far-field pressure, we can use it to estimate airflow blockage and deflection. For upstream blocking, the wind disturbance will decay inversely with distance upwind. At the front edge of the farm, we evaluate Eq. (19) to give

The upwind pressure field deflects the airflow to the left and right. The maximum lateral speed is located near the farm lateral edge at

In addition to the conceptual value of the rigid lid (RL) model emphasized herein, it could also be used in industrial or engineering models of wind farm disturbance. Any quasi-analytic model or computational fluid dynamics (CFD) model could utilize the RL assumption to simplify the computation. This is an easy way to incorporate the effects of atmospheric stability. Our results confirm this logic but only in a qualitative way. We have shown that the RL model overidealizes the pressure dipole and shifts the pressure field slightly upwind. Worse still would be the assumption that RL models will not have a pressure field because they do not support gravity waves. In fact, the rigid lid assumption requires that a pressure field be generated from the leading and trailing edge of the farm where the turbine drag vector field is divergent. Any properly designed RL model would have a dipole pressure field very similar to that described in Eqs. (14) and (22).

The direct link between farm drag and far-field pressure dipole (Eqs. 14 and 24) in the RL case allows us to determine total farm drag with a pair of pressure measurements. If pressure sensors are located a distance

In the reference GW case (Fig. 1b), the upstream and downstream pressure values are

When turbine drag in a wind farm slows the wind, the lowest layer must thicken to conserve mass and push the higher layers upwards. The influence of this lifting depends on the atmospheric static stability. With no stratification, this upward displacement will not generate a hydrostatic pressure disturbance.

When moderate stable stratification is present, the upward displacement will create pressure anomalies that act on the turbine layer. The computation of the pressure field typically requires the use of a gravity wave (GW) model. When the stratification is very strong, the GW solutions approach the rigid lid (RL) limit where little or no vertical displacement occurs. In this situation, we can compute the pressure field directly from the non-divergent assumption, without having to consider gravity waves. A pressure field dipole is then created to prevent flow divergence at the front and back edge of the wind farm where the turbine drag is divergent. The rigid lid approximation allows closed-form expressions that deepen our understanding of the wind farm pressure disturbance.

Surprisingly, the GW and RL solutions are qualitatively similar. Both have an upwind–downwind, high–low pressure difference. Pressure forces act to smooth out the deceleration of the wind by the farm. They reduce the deceleration over the farm with a favorable pressure gradient and add deceleration zones upwind and downwind with an adverse pressure gradient. They also produce small areas of deflected and accelerated airflow to the left and right of the farm.

In the real atmosphere, the inversion strength is only about

We propose two applications for the RL solutions. First, they provide an approximate way to compute total farm drag from upwind and downwind pressure measurements. Second, they may apply directly to industrial wind farm models that use a rigid lid to reduce computational time and complexity.

The MATLAB code used in this paper is available from the author.

There are no experimental data in this paper. The data for plots and tables come from MATLAB code.

The author has declared that there are no competing interests.

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I appreciate useful conversations with Brian Gribben, Graham Hawkes, Neil Adams, Xiaoli Guo Larsen, Jake Badger, Jana Fischereit, and Idar Barstad. Insightful comments came from reviewers Dries Allaerts and James Bleeg.

This paper was edited by Johan Meyers and reviewed by Dries Allaerts and James Bleeg.