4.1. Precursor simulations

Profiles for the horizontally averaged wind speed, wind direction, temperature and turbulence intensity from the precursor simulations are shown in figure 3. Turbulence intensity is defined as $TI = ((2/3) \langle \bar {k}^B \rangle _{xy})^{1/2} / U$ , where $\bar {k}$ is the time-averaged resolved turbulence kinetic energy, $\langle \cdot \rangle _{xy}$ denotes horizontal averaging and the superscript $B$ denotes the base flow without turbines. The horizontally averaged wind speed magnitude is given by $U = (\langle \bar {u}^B \rangle _{xy}^2 + \langle \bar {v}^B\rangle _{xy}^2)^{1/2}$ . In the uniform inflow simulations, shown in figure 3( $a$ – $d$ ), inflow profiles are independent of the Rossby number. This is due to the absence of velocity shear in uniform inflow and a domain with slip-walls. Note that the vertical domain extent is much larger than the profiles shown in figure 3, which are zoomed in to the rotor area.

Figure 3.

Horizontally and time-averaged inflow profile characteristics under ( $a$ – $d$ ) uniform, ( $e$ – $h$ ) truly neutral (TNBL) and ( $i$ – $l$ ) conventionally neutral (CNBL) conditions. The turbine hub height $z_h$ is shown by the dash-dotted line, and the top and bottom extents of the rotor are shown by the dotted lines. Note that a prognostic equation for the potential temperature is not solved for the uniform inflow conditions.

Adding a bottom wall introduces velocity shear and thus vertical variation in the inflow profiles as a function of the Rossby number for the ABL simulations, shown in figure 3( $e$ – $l$ ). In the TNBL, the wind speed magnitude $U$ increases to a maximum of $U_{max } \approx 1.02G$ at the low-level jet. The non-dimensional friction velocity $u_*/G$ shows a weak inverse dependence on the Rossby number, and a summary of inflow and ABL properties is given in table 1. The height of the ABL depends on the Rossby number by the Rossby–Montgomery equilibrium height (Rossby & Montgomery Reference Rossby and Montgomery1935; Zilitinkevich et al. Reference Zilitinkevich, Esau and Baklanov2007). In table 1, we compute the ABL height $h$ as the vertical height where the turbulent fluxes reach 5% of their surface value (Zilitinkevich et al. Reference Zilitinkevich, Esau and Baklanov2007). We note that $h$ changes considerably across the range of Rossby numbers simulated and we keep $z_h$ constant rather than fixing the dimensionless ratio $z_h/h$ . Additional simulations with constant $z_h/h$ are provided in Appendix E to control this dimensionless parameter, and the results show the same qualitative response in the wake dynamics dependent on $Ro$ as for fixed $z_h = 150\,\text {m}$ . The magnitude of wind direction shear at the surface monotonically increases with increasing relative Coriolis strength (decreasing $Ro$ ). However, wind direction profiles are increasingly nonlinear in the rotor area as the TNBL height decreases with decreasing Rossby number. The change in time-mean wind direction from the bottom to the top of the rotor is defined as $\Delta \psi = \psi (z_h+D/2) - \psi (z_h-D/2)$ for the time-averaged wind direction $\psi (z) = \arctan (\langle \bar {v}^B \rangle _{xy}/ \langle \bar {u}^B \rangle _{xy})$ . Note that profiles of $\bar {v}^B(z)$ look very similar to profiles of $\psi (z)$ . Overall, due to the presence of the bottom boundary, the TNBL inflow is significantly more complex than the uniform inflow simulations.

Table 1.

Inflow properties of the LES experiments in this study. Simulations of the TNBL and CNBL use a surface roughness $z_0 = 10^{-4}\,\textrm {m}$ and Coriolis parameter $f_c = 1.03\times 10^{-4}\,\textrm {rad s}^{-1}$ . In the CNBL, the Zilitinkevich number is $N/f_c = 55.5$ , where $N$ is the Brunt–Väisälä frequency.

In the CNBL, the boundary layer growth is suppressed relative to the TNBL due to the presence of the stable free atmosphere. The low-level jet is stronger in the CNBL than in the TNBL and the maximum wind speed reaches $1.05G$ . The non-dimensional friction velocity in the CNBL decreases with increasing Rossby number, which is consistent with simulations from Liu et al. (Reference Liu, Gadde and Stevens2021). The buoyancy length scale $L_b = u_*/N$ , where $N=\sqrt {g\Gamma /\theta _0}$ is the Brunt–Väisälä frequency, varies between 12.8 m at $Ro = 100$ and 100 m for $Ro = 1000$ , which is larger than the vertical grid resolution $\Delta _z = 12\,\textrm {m}$ , allowing buoyancy effects to be resolved. Similar to the TNBL, there is a non-monotonic trend in the wind direction profiles across the turbine rotor $\Delta \psi$ . That is, the change in wind direction across the rotor does not monotonically increase with increasing relative Coriolis forcing strength. Wind direction shear across the rotor of the turbine $\Delta \psi$ spans a larger range of values for the same parametric sweep of $Ro$ in the CNBL than in the TNBL (see table 1).

Differences between the TNBL and CNBL development are a result of the stable free atmosphere in conventionally neutral conditions. Profiles of horizontally and time-averaged horizontal shear stress and buoyancy flux are shown in figure 4. The presence of negative buoyancy flux $\overline {w'\theta '}$ , shown in figure 4( $f$ ), reduces shear stresses $\bar {\tau }_{xz} = \overline {u'w'} + \bar {\tau }^d_{13}$ and $\bar {\tau }_{yz} = \overline {v'w'} + \bar {\tau }^d_{23}$ in the CNBL. Therefore, the ABL height $h$ is lower in the CNBL than in the TNBL for the same Rossby number, and the wind shear (speed and direction) in the CNBL is enhanced relative to the TNBL. A significant consequence of the negative buoyancy flux is in the low-Rossby-number simulations ( $Ro \leq 167$ ), where the CNBL is sufficiently shallow that the turbine, which reaches up to 270 m at the rotor tip, impinges on the free atmosphere. As will be discussed in §§ 4.4 and 4.5, the interaction between the rotor and the free atmosphere induces gravity waves, which affect the wake dynamics.

Figure 4.

Profiles of horizontally and time-averaged fluxes in the ( $a$ – $c$ ) TNBL and ( $d$ – $f$ ) CNBL showing horizontal shear stress in the ( $a{,}d$ ) streamwise and ( $b{,}e$ ) lateral directions, normalised by the friction velocity $u_*$ , as well as the ( $c{,}f$ ) buoyancy flux.

4.2. Qualitative time-averaged wake behaviour

In this section, we examine the time-averaged wake velocity fields. Cross-sections of the hub height velocity deficit field $\overline {\Delta u}_i = \bar {u}_i - \langle \bar {u}_i^B \rangle _{xy}$ in the streamwise direction ( $i=1$ ) are shown in figure 5, where $\langle \bar {u}_i^B \rangle _{xy}$ is the horizontally and time-averaged precursor flow. The precursor flow is the base flow without wakes. In uniform inflow, wake recovery does not begin until very far downstream ( $x/D \gtrsim 12$ ) due to the lack of free stream turbulence (Olivares-Espinosa et al. Reference Olivares-Espinosa, Breton, Masson and Dufresne2014; Howland et al. Reference Howland, Bossuyt, Martínez-Tossas, Meyers and Meneveau2016). Wake deflection is anti-clockwise as viewed from above (deflection is in the $+y$ direction in the chosen coordinate system). The amount of wake deflection increases with increasing relative Coriolis forcing strength (i.e. decreasing $Ro$ ).

Figure 5.

Hub height wind turbine wakes visualised as the streamwise velocity deficit with respect to the inflow, viewed from above, for varying Rossby numbers in ( $a$ , $b$ , $c$ ) uniform inflow, ( $d$ , $e$ , $f$ ) TNBL inflow and ( $g$ , $h$ , $i$ ) CNBL inflow. The dashed white line indicates $y=0$ while pink dash-dotted lines show the wake centreline.

For the TNBL and CNBL inflows, wakes recover due to free stream turbulence. The wake recovery in all simulations is qualitatively only minimally affected by the parametric effects of $Ro$ , despite the large variation in turbulence intensity, which is dependent on $Ro$ , across the rotor plane. Wake deflection is also increasingly anti-clockwise as the relative strength of Coriolis forcing increases, which is particularly noticeable in the CNBL simulations at low Rossby numbers. A quantitative investigation of the wake dynamics in the ABL flows is given in §§ 4.4 and 4.5.

Figure 6.

Wake cross-sections at $x/D = 8$ visualised as the streamwise velocity deficit with respect to the inflow for varying Rossby numbers in ( $a$ , $b$ , $c$ ) uniform inflow, ( $d$ , $e$ , $f$ ) TNBL inflow and ( $g$ , $h$ , $i$ ) CNBL inflow. The turbine location is given by the white circle centred around the origin and the wake centroid position is given by the pink $+$ .

Vertical slices through the $yz$ -plane are shown in figure 6 at a distance $x = 8D$ downwind of the rotor. By $x=8D$ , variations in the relative Coriolis forcing strength can be observed in the uniform inflow wakes, with lower Rossby numbers deflecting farther in the $+y$ direction. In the TNBL, wakes are skewed, which is a result of the inflow wind direction shear (Magnusson & Smedman Reference Magnusson and Smedman1994; Abkar & Porté-Agel Reference Abkar and Porté-Agel2016). The difference in wake skewing between Rossby numbers is small, owing to the small spread in wind direction change over the rotor $\Delta \psi$ for the TNBL simulations.

In the CNBL, the decreasing boundary layer height as the Rossby number decreases strongly affects the wake shape. The large differences in wake shape between the CNBL simulations result from the changing inflow properties. Because the inflow wind direction is increasingly nonlinear with decreasing $Ro$ , wake velocity slices assume complex shapes that are not well approximated by axisymmetric or elliptical (skewed) Gaussian wakes (Abkar & Porté-Agel Reference Abkar and Porté-Agel2016). Additionally, the wind speed shear across the rotor is increasingly complex as the ABL height decreases and the rotor interacts directly with the low-level jet. Finally, the ambient turbulence intensity varies substantially across the rotor, particularly for $Ro = 125$ and $Ro = 100$ , which is distinct from the TNBL simulations.

4.3. Uniform inflow results

The uniform inflow simulations represent the canonical flow building block for understanding the dynamics of turbine wakes in the presence of Coriolis forces. Specifically, the uniform inflow environment separates Coriolis forcing from wind shear. Although the presence of uniform (i.e. gradient-free and non-turbulent) inflow cannot exist in the ABL due to the presence of the ground, it is helpful to study as a basis for parsing the leading-order dynamics governing wake structure and evolution in more complex inflows. The main limitation of the uniform inflow numerical set-up is the unphysical delay in the onset of a fully turbulent far wake, due to the absence of inflow wind shear or turbulence (Olivares-Espinosa et al. Reference Olivares-Espinosa, Breton, Masson and Dufresne2014; Howland et al. Reference Howland, Bossuyt, Martínez-Tossas, Meyers and Meneveau2016), as shown in figure 5. Therefore, we will not attempt to draw conclusions from the streamwise momentum budgets regarding wake recovery using the uniform inflow set-up. Nonetheless, we will show that the dynamics of the uniform inflow problem are complex and instructive for parsing mechanisms of lateral momentum transport which affect wake evolution in the presence of Coriolis forces.

We begin by investigating the wake deflection due to the Coriolis force in uniform inflow. Here, we choose to measure the wake deflection $y_c(x)$ as the centroid of the streamtube cross-section in the $yz$ -plane. The wake deflection increases monotonically with increasing relative Coriolis forcing strength (decreasing $Ro$ ) and follows a parabolic trajectory, as shown in figure 7( $a$ ). By scaling the wake deflection $y_c$ by the Rossby number, shown in figure 7( $b$ ), we see that the wake deflection collapses on to one curve until $x/D\approx 12$ . The collapse worsens for $x/D \gt 12$ due to wake breakdown and turbulence. The collapse in the scaled wake deflection allows us to focus on understanding the wake dynamics at one Rossby number with the expectation that the relevant physical transport mechanisms will scale with the Rossby number.

Figure 7.

( $a$ ) Streamtube centroid position $y_c(x)$ as a function of downstream distance $x$ in uniform inflow varying the Rossby number. ( $b$ ) Scaling by the Rossby number collapses all uniform inflow simulations onto one curve.

The wake deflection is the integrated form of the lateral momentum budget. Therefore, in uniform inflow, the lateral momentum budgets should also collapse if scaled by the Rossby number. Individual terms $M_y$ in the streamtube-averaged lateral momentum balance (3.3) are shown in figure 8 as a function of streamwise position downwind of the turbine. The individual terms, when scaled by the Rossby number, are of the order of unity and collapse together. All of the forcing terms are nearly constant in $x$ outside of the wake expansion region $x/D \lesssim 1$ until wake recovery begins at $x/D\approx 12$ . Therefore, the uniform inflow centroid position follows a parabolic trajectory due to constant acceleration (mean advection). When wake recovery begins, the wake velocity deficit $(G_1 - \bar {u})$ decreases, which decreases the strength of the Coriolis term and, by extension, the lateral advection of the wake. The Coriolis forcing term is the largest in magnitude and positive in sign (in the northern hemisphere), which results in a positive lateral acceleration and therefore a net positive lateral advection of the wake region. However, the presence of a lateral pressure gradient opposes the direct Coriolis forcing term. The result of the pressure gradient is a net acceleration which is only approximately 25 % as strong as the direct Coriolis forcing term. Note that as the advection term is non-negligible, the wake flow is not in geostrophic balance.

Figure 8.

All terms in the lateral momentum balance for the uniform inflow simulations, streamtube-averaged and plotted as a function of streamwise distance. The streamtube-averaged quantities are scaled by the Rossby number $Ro$ , showing similarity in the flow.

Figure 9.

( $a$ ) Streamlines of in-plane velocities $\bar {v}$ and $\bar {w}$ at $Ro = 100$ and $x = 10D$ . Contours of vorticity are superimposed, showing a counter-rotating vortex pair (CVP) with positive streamwise vorticity above the hub plane and negative vorticity below. ( $b$ ) Streamwise vorticity budget terms $M_\omega$ scaled by $Ro$ for the case $Ro = 100$ , integrated for $z \gt z_h$ . Streamwise vorticity advection (dashed blue) is balanced by Coriolis production (purple). Buoyancy torque (term C) is omitted as the uniform inflow simulations do not solve the potential temperature equation.

We seek to investigate the physical mechanism that results in a non-zero lateral pressure gradient. The streamtube averaged lateral pressure gradient in the absence of Coriolis forcing is equal to zero. To investigate how Coriolis effects give rise to the lateral pressure gradients, we visualise streamlines of the in-plane velocity components $\bar {v}, \bar {w}$ at a cross-section $x/D = 10$ in figure 9( $a$ ). Streamlines of the in-plane velocity components at $Ro = 100$ show the presence of a counter-rotating vortex pair (CVP), which is the source of the wake deflection. The formation of the CVP, which is visualised by the formation of streamwise vorticity $\omega _1 = \omega _x$ , can be understood through the vorticity budget equation. We take the curl of (3.1) to solve for the Reynolds-averaged vorticity equation. This yields

(4.1) \begin{equation} \underbrace {\bar {u}_j \frac {\partial \bar \omega _{i}}{\partial x_j} }_{\text {A}} = \underbrace { \bar {\omega }_j \frac {\partial \bar {u}_i}{\partial x_j} }_{\text {B}} \underbrace { + \frac {\varepsilon _{ij3}}{Fr^2 \theta _0} \frac {\partial \bar {\theta }}{\partial x_j} }_{\text {C}} \underbrace { - \varepsilon _{ijk} \frac {\partial }{\partial x_j} \left ( \frac {\partial \bar {\tau }_{km}}{\partial x_m} + \frac {\partial \overline {u^{\prime}_k u^{\prime}_m}}{\partial x_m} \right ) }_{\text {D}} \underbrace { + \frac {1}{Ro} \frac {\partial \bar {u}_i}{\partial x_3} }_{\text {E}}, \end{equation}

where the bracketed terms are: (A) advection by the mean flow; (B) vortex stretching; (C) buoyancy torque; (D) torque from subgrid and Reynolds stresses and (E) Coriolis. The streamwise vorticity causes the CVP formation in uniform inflow. The integrated streamwise vorticity budget ( $i=1$ ) in figure 9( $b$ ) shows that the transfer of vorticity due to the Coriolis force is the largest contributor to the vorticity budget (4.1). The Coriolis term is primarily balanced by the mean advection of vorticity, where terms are again of order unity after being scaled by $Ro$ . This balance can be written as

(4.2) \begin{equation} \bar {u}_j \frac {\partial \bar {\omega }_x}{\partial x_j} \approx \bar {u}\frac {\partial \bar {\omega }_x}{\partial x} \approx \frac {1}{Ro} \frac {\partial \bar {u}}{\partial z}. \end{equation}

In the vorticity balance, the role of the Coriolis force is to introduce planetary vorticity into the wake dynamics. The presence of the wake induces vertical velocity gradients ( $\partial \bar {u}/\partial z$ ) such that positive streamwise vorticity is generated above hub height (in the northern hemisphere), and vice versa below hub height.

Lateral velocities induced by a CVP are also the mechanism for wake deflection in yaw-misaligned turbines (Howland et al. Reference Howland, Bossuyt, Martínez-Tossas, Meyers and Meneveau2016; Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016). Wakes trailing yawed turbines are a widely studied case in recent literature (Meyers et al. Reference Meyers, Bottasso, Dykes, Fleming, Gebraad, Giebel, Göçmen and van-Wingerden2022), and therefore, we make an analogy between the flow physics from Coriolis effects and yawed turbine wakes here. Traditionally, modelling the deflection of yaw-misaligned wind turbine wakes with momentum-based approaches has yielded overpredictions of the wake deflection because the lateral pressure gradient forces on the streamtube, which are non-negligible, are neglected for model simplicity (Jiménez et al. Reference Jiménez, Crespo and Migoya2010; Shapiro et al. Reference Shapiro, Gayme and Meneveau2018; Heck et al. Reference Heck, Johlas and Howland2023). For wakes in uniform inflow with Coriolis forcing, the presence of a CVP also induces a non-negligible lateral pressure gradient force. The role of the pressure gradient forcing, as is the case in yaw-misaligned wind turbine wakes, is to enforce mass conservation (Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016). Neglecting the pressure gradient forcing in (3.3) leads to an order-of-magnitude overprediction in the wake deflection (not shown).

In summary, the net effect of the Coriolis force on wakes in uniform inflow is an anti-clockwise deflection that increases with increasing relative Coriolis forcing strength. In the lateral momentum budget, the Coriolis term is the largest in magnitude but partially cancelled by a lateral pressure gradient force in the wake. Wake deflection is caused by the formation of a CVP due to the transfer of planetary vorticity into the streamwise direction via the wake-added shear. The CVP is responsible for creating $\bar {v} \gt 0$ inside the wake (in the northern hemisphere), which deflects the wake anti-clockwise and gives rise to the non-negligible lateral pressure gradient force. Neglecting the pressure gradient forcing in a momentum-based modelling approach results in an overprediction in the wake deflection.

4.4. Coriolis effects on streamwise momentum in the ABL

This section focuses on wake recovery in TNBL and CNBL conditions. The streamtube-averaged streamwise velocity deficit is shown as a function of streamwise distance in figure 10. In the uniform inflow cases, shown as dotted lines in figure 10, the streamtube-averaged velocity deficit is constant after the pressure recovery in the near-wake of the turbine. Additionally, the uniform inflow wake recovery does not depend on the strength of Coriolis forcing. Overall, in uniform inflow, we observe very weak parametric dependence of relative Coriolis forcing strength on wake recovery.

Figure 10.

Streamtube-averaged streamwise velocity deficit as a function of streamwise coordinate $x/D$ in $(a)$ TNBL inflow and $(b)$ CNBL inflow. Uniform inflow wakes are shown with dotted lines.

In the TNBL, the relative strength of Coriolis forcing weakly affects the wake recovery. Due to wind shear in the ABL inflow, the turbine thrust in the TNBL is lower than in uniform inflow. As a result, the wake velocity deficit magnitude is lower in the TNBL than in uniform inflow. In the far wake, the strongest relative Coriolis forcing experiences the fastest wake recovery, and the wake recovery rate generally decreases as $Ro$ increases. We emphasise that the relative differences in wake strength between Rossby numbers are modest, and we further elaborate on the statistical significance of the differences in wake recovery for varying $Ro$ in § 4.6.

Trends in wake velocity as a function of $Ro$ are less clear in the CNBL. This is due to the suppressed ABL height, which results in complex inflow conditions across the rotor area, as noted in § 4.2. For example, the wind speed profile changes significantly as the boundary layer height decreases and the low-level jet interacts with the rotor. This affects the rotor thrust, which increases monotonically with decreasing CNBL height. As a result, the maximum velocity deficit, which occurs around $x \approx 2.5D$ , changes in magnitude with the Rossby number. Further, the turbulence intensity also varies more in the CNBL than in the TNBL as a function of the Rossby number. In contrast with the TNBL wakes, wake recovery rates in the CNBL do not follow a clear trend with changing Rossby numbers. To parse the dynamics of the wake recovery, we analyse the streamwise momentum budget in the wake.

Figure 11.

Streamtube-averaged streamwise momentum budget terms $M_x$ for wakes in the ( $a$ , $b$ ) TNBL and ( $c$ , $d$ ) CNBL. The mean advection is the sum of all forcing terms.

Wake recovery is described by the streamwise momentum budget. The streamtube-averaged streamwise momentum budget is shown in figure 11 for wakes in the TNBL and CNBL. For $x \lesssim 3D$ , the modified pressure forcing is large due to the pressure recovery from the forcing at the rotor. Beyond this region, turbulent fluxes (divergence of Reynolds stresses) replenish momentum into the wake, comprising the leading-order dynamics.

In the TNBL, the wake recovery due to the turbulent flux of momentum accounts for nearly all (95 %–99 %) of the wake recovery. Effects of direct Coriolis forcing, subgrid stresses and pressure gradients are negligible in the streamwise momentum budget. In other words, the dependence of the streamwise momentum budget on relative Coriolis forcing strength is primarily through changes in the ABL structure, which depends on the Rossby number (see figure 3). The Reynolds stress divergence in the wake, and therefore wake recovery rate, increases as the relative strength of Coriolis forcing increases. For all Rossby numbers, the Reynolds stress divergence reaches a maximum value near $x = 8D$ . The marginal increase in wake recovery rate with decreasing Rossby number shown in figure 10( $a$ ) is enabled by the increasing turbulent flux in the wake.

We observe two notable differences between the TNBL and CNBL streamwise momentum budgets. One notable difference is that the pressure gradient force deviates from the asymptotic recovery expected from classical momentum theory for $Ro \leq 167$ . Instead, oscillations are observed in the pressure gradient term. These oscillations stem from gravity waves induced by the free-standing turbine, which interact directly with the capping inversion when $z_{tip} \gtrsim h$ , where $z_{tip} = z_h + 0.5D$ is the rotor tip height. Gravity wave excitation from large wind farms has been studied extensively in recent literature (e.g. Allaerts & Meyers Reference Allaerts and Meyers2017; Sanchez Gomez et al. Reference Gomez, Miguel, Julie, Mirocha and Arthur2023; Lanzilao & Meyers Reference Lanzilao and Meyers2024). The presence of the turbine perturbs the capping inversion upward, exciting gravity waves that non-locally redistribute kinetic energy through pressure oscillations. While gravity waves are not the main focus of this study, the pressure gradient forcing due to gravity waves is dynamically significant to the streamwise and lateral momentum budgets (see § 4.5). Sensitivity tests at low Rossby numbers with a doubled vertical domain height $L_z$ and adjusted RDL strength show that our numerical set-up does not significantly affect the wake dynamics or results (see figure 20 in Appendix A.5). We suggest future work investigating the interactions between the rotor, wakes and the free atmosphere. Finally, we note that because gravity wave effects are inherently non-local, additional surrounding turbines may change the development of gravity waves and should be studied separately.

A second notable difference between the TNBL and CNBL streamwise momentum budgets is that the inflow turbulence intensity at the rotor hub height increases more with increasing $Ro$ in the CNBL than in the TNBL. As a result, the turbulence flux term in figure 10( $d$ ) monotonically increases with increasing Rossby number for $x \lesssim 5D$ . The peak Reynolds stress divergence still generally increases with increasing relative Coriolis forcing (decreasing $Ro$ ), as was observed in the TNBL. The exception is for the cases where stable thermal stratification in the free atmosphere suppresses turbulence and therefore wake recovery due to the shallow CNBL height ( $Ro = 125$ and $Ro = 100$ ).

To examine the parametric dependence of the turbulent momentum flux on the Rossby number, we take a further look at the Reynolds stress divergence term. In figure 12, we break out the wake recovery due to the Reynolds stress divergence into lateral ( $M_{x,2}$ ) and vertical ( $M_{x,3}$ ) turbulent momentum fluxes. The streamwise Reynolds stress component is not shown because it is small compared with the in-plane components and integrates to nearly zero inside the streamtube for $x \in [0, 15]D$ .

Figure 12.

( $a$ ) Ratio of vertical to lateral turbulent momentum flux into the wake streamtube. ( $b$ – $j$ ) Profiles in the $yz$ -plane at $x = 8D$ of gradients of turbulent momentum flux, where reds indicate acceleration (wake recovery) and blues indicate deceleration. The streamtube boundary is outlined in black.

Both the magnitude and structure of turbulent fluxes change as a function of $Ro$ . In figure 12( $a$ ), the ratio of vertical to lateral turbulent momentum flux $M_{x,3}/M_{x,2}$ , integrated over the entire streamtube from the rotor to $x = 15D$ , are shown as a function of $Ro$ . In the present simulations of a single turbine wake in the TNBL, the vertical momentum flux contribution is between 55 % and 80 % of the lateral momentum flux contribution to the wake recovery. While the ratio of vertical to lateral momentum flux changes modestly with $Ro$ in the TNBL, the dependence of $M_{x,3}/M_{x,2}$ on $Ro$ in the CNBL is more pronounced. In the CNBL, as the relative Coriolis forcing strength increases and ABL height decreases, wake recovery is increasingly driven by vertical momentum fluxes. We note that while the ratio of turbulent fluxes is dependent on ABL properties and Rossby number, as shown here, it is also dependent on lateral turbine spacing if neighbouring turbines are present (Calaf et al. Reference Calaf, Meneveau and Meyers2010; van der Laan et al. Reference van der, Paul, Baungaard and Kelly2023). Therefore, the dependence of wake recovery on Rossby number may differ for turbine wakes in wind farms compared with the wakes of free-standing turbines studied here.

We visualise turbulent momentum flux contributions to the wake recovery in figure 12( $b$ – $j$ ). Contours of the mean streamwise advection, lateral Reynolds stress gradient and vertical Reynolds stress gradient are shown in a $yz$ -plane at a distance $x=8D$ downwind for three atmospheric conditions. In all cases, the streamtube boundary corresponds closely to the boundary between net Reynolds stress convergence (acceleration) and divergence (deceleration). That is, the streamtube-averaged quantities robustly select the wake region which is accelerating and recovering lost momentum. At $Ro = 500$ , the TNBL and CNBL wake recovery is very similar, as shown by figure 12( $a$ ), so only the contours of the TNBL wake are shown. Between $Ro = 500$ and $Ro = 100$ in the TNBL and CNBL, the magnitude of the mean advection within the streamtube increases. In the TNBL, changes in the Rossby number only weakly affect the wake shape and the ABL structure, so the individual turbulent flux contributions are similar between $Ro = 100$ and $Ro = 500$ . In the CNBL, wakes experience strong wind direction shear as the ABL height decreases with decreasing $Ro$ . This strongly skews the wake and causes more vertical momentum entrainment than lateral entrainment of streamwise momentum. To summarise, while differences in overall wake recovery are small between Rossby numbers, wake recovery mechanisms change substantially as the ABL height decreases.

4.5. Coriolis effects on lateral momentum

As noted in the introduction, there are two distinct coupled effects from Coriolis forces that alter the structure of wakes in the ABL. First, the wake skews due to wind direction shear, where different vertical levels $z$ are subjected to different inflow wind angles. This skewed wake is visualised in figure 6( $d$ – $i$ ), and the effect has been explored in previous literature (Abkar & Porté-Agel Reference Abkar and Porté-Agel2016; Churchfield & Sirnivas Reference Churchfield and Sirnivas2018) and modelled with wake modelling approaches (Abkar et al. Reference Abkar, Sørensen and Porté-Agel2018; Martínez-Tossas et al. Reference Martínez-Tossas, King, Quon, Bay, Mudafort, Hamilton, Howland and Fleming2021). In addition to the lateral wake advection induced by Coriolis forces through wind direction shear, Coriolis forces also affect wake dynamics, primarily through lateral momentum equations. Wake deflection is caused by an imbalance of lateral forcing terms in the wake as a whole. Previous literature has mostly explored the advective mechanisms that cause wake skewing. In contrast, we focus on Coriolis effects on wake dynamics, which primarily affect lateral deflection of the wake. The extent to which wake skewing or wake deflection are significant to farm design will depend on the layout, control strategy and ABL conditions of a wind farm. Future work should compare these different effects in a case study.

Wake deflection, which is shown in figure 5, is influenced by the relative strength of Coriolis forces. Here, we measure the wake deflection by computing the centroid of the streamtube position, consistent with the analysis in § 4.3. The wake centroid evolution $y_c(x)$ is shown in figure 13( $a{,}b$ ) for varying Rossby numbers with each inflow type. Because the hub height wind direction oscillates after the wind angle controller is turned off at the beginning of the concurrent simulations, the wind direction drifts up to $1^\circ$ throughout the averaging time of the primary simulation. The advection due to the time-mean wind direction at hub height is removed by subtracting $x\tan (\psi _{hub}) \approx \psi _{hub} x$ from the streamtube centroid $y_c(x)$ , where $\psi _{hub}$ is given in table 1.

Figure 13.

( $a$ , $b$ ) Wake centroid location, defined as the centroid of the streamtube, as a function of streamwise coordinate $x/D$ for varying Rossby numbers in the ( $a$ ) TNBL and ( $b$ ) CNBL. ( $c$ , $d$ ) Wake deflections are scaled by the Rossby number. Uniform inflow wake deflections are overlaid with the dotted lines.

In all inflow conditions (uniform, TNBL and CNBL), the amount of wake deflection is parametrically dependent on the relative strength of Coriolis forcing. The sign (clockwise or anti-clockwise) of the wake deflection is also dependent on the relative strength of Coriolis forcing for wakes in the ABL. This differs from the uniform inflow cases, which only deflect anti-clockwise in the northern hemisphere. Note that as clockwise and anti-clockwise deflections are observed in ABL inflow conditions, the wake deflection does not collapse when scaled by the Rossby number, as shown in figure 13( $c{,} d$ ). Between $Ro = 500$ and $Ro = 250$ , we observe a transition in wake deflection direction in the ABL. For $Ro = 500$ , wakes in ABL inflow are deflected clockwise as viewed from above ( $-y$ direction), while for $Ro \leq 250$ , wakes are deflected anti-clockwise ( $+y$ direction). While the wake deflection will depend on ABL conditions, including surface roughness, subsidence, orography, etc., we expect that the qualitative trends presented here will apply to other neutral ABLs, even if no universal transitional Rossby number applies to all neutral ABLs. Finally, the wake deflection is not bounded above or below by the uniform inflow results. For example, in the CNBL at $Ro = 100$ , the observed wake deflection is several times larger than the wake deflection in uniform inflow for $x/D \lesssim 6$ .

To understand why wakes are deflected increasingly anti-clockwise with decreasing Rossby number (increasing relative Coriolis forcing strength), we examine the lateral momentum budget terms $M_y$ averaged within the streamtube, shown in figure 14. Recall that in the absence of planetary rotation (i.e. $Ro \to \infty$ ), the wake is symmetric across the $y$ -axis, and the streamtube-averaged momentum budget averages to zero. With the introduction of Coriolis forces at finite Rossby numbers, which break this symmetry, terms in the lateral momentum budget become non-zero. The wake Coriolis forcing scales directly with $Ro^{-1}$ and is large in magnitude compared with the other terms in the lateral momentum balance. This is in contrast to the streamwise momentum (§ 4.4) where the turbulent flux constitutes the largest forcing term and the direct Coriolis forcing term is negligible.

Figure 14.

Streamtube-averaged momentum terms in the $y$ -momentum balance for wakes in ( $a$ , $b$ , $c$ ) TNBL inflow and ( $d$ , $e$ , $f$ ) CNBL inflow.

In the TNBL, the mean advection term is of the same order of magnitude as in uniform inflow. The mean advection of the streamtube is increasingly positive as the Rossby number decreases. Because the wake deflection is the integrated form of the mean lateral advection, the increasingly positive advection term results in increasingly anti-clockwise wake deflection. The mean advection is the sum of the wake Coriolis, modified pressure forcing and turbulent flux forcing terms (subgrid forcing is negligible). As shown in figure 14( $a$ – $c$ ), the Coriolis, pressure gradient and Reynolds stress terms are all dynamically active in the wake, but the relative importance of each term varies with streamwise location $x$ .

The wake Coriolis term is larger in magnitude than all other lateral forcing terms in the near-wake region. Similar to the uniform inflow simulations, the direct Coriolis forcing is again partially opposed by a lateral pressure gradient force. However, unlike in uniform inflow, the pressure gradient forcing term begins to decay almost immediately after the initial wake expansion. The decay of lateral pressure gradients in the TNBL occurs in conjunction with the lack of a coherent CVP formation in the TNBL turbine wake. A coherent CVP is not formed in the TNBL, as shown in the colour contours of figure 15( $c$ ), due to atmospheric turbulence. The lateral pressure gradients associated with the CVP formation in uniform inflow are obscured by turbulence in the TNBL wake. The lateral pressure gradient field primarily opposes the Reynolds stress divergence, similar to pressure gradients in turbulent jet flows (Pope Reference Pope2000). The asymmetry in the pressure gradient forcing in figure 15( $d$ ) is due to the Coriolis force. We note that a weak CVP can be observed in streamlines of the velocity deficit $\overline {\Delta v}$ , $\overline {\Delta w}$ in the TNBL wake, seen in figure 15( $c$ ). Even so, the structure of the full lateral velocity field $\bar {v}$ , shown by the colour contours, and pressure gradient field indicate that the effect of the CVP formation does not constitute the primary mechanism of momentum transport.

Figure 15.

Wake cross-sections at $x = 8D$ for ( $a$ , $b$ ) uniform inflow and ( $c$ , $d$ ) TNBL simulations at $Ro = 100$ . ( $a$ , $c$ ) Contours of lateral velocity $\bar {v}$ are shown with streamlines of $\overline {\Delta v}$ and $\overline {\Delta w}$ overlaid. ( $b$ , $d$ ) Lateral pressure gradient fields in the wake. Note the different colourbar magnitudes. The streamtube boundary is shown by the black line.

As the wake recovers, the direct Coriolis forcing term, which is proportional to the velocity deficit, begins to diminish. At the same time, the turbulent entrainment of lateral momentum increases. In the TNBL, we observe that the pressure gradient term responds to the combined Coriolis and pressure gradient forcing. For instance, the pressure gradient forcing flips sign in the wake region when the Reynolds stress divergence exceeds the direct Coriolis forcing. As a result, even in the far wake when the wake begins to recover and the direct Coriolis forcing decays, the wake continues to deflect in the TNBL, as quantified in figure 13. In contrast with the uniform inflow wakes, where the CVP formation drives the lateral pressure gradient forcing, turbulence primarily controls the lateral pressure gradient for wakes in the TNBL.

Comparing the TNBL and CNBL simulations in figure 14, we see both similarities and differences in the lateral momentum budgets. For example, the direct Coriolis forcing term, which depends only on wake velocity and Rossby number, is nearly the same in the TNBL and CNBL. This is because the streamtube-averaged wake velocity is very similar between the TNBL and CNBL for all $Ro$ , as shown in figure 10. However, the magnitude of the turbulent flux of lateral momentum into the wake is greater in the CNBL than in the TNBL, in general. The lateral momentum flux in the CNBL is generally larger than in the TNBL because the magnitude of wind direction shear is larger in the CNBL than in the TNBL for the same Rossby number. The most substantial difference between the TNBL and CNBL lateral momentum budgets is the presence of oscillations in the lateral pressure gradients with wavelengths of several turbine diameters or greater, shown in figure 14( $f$ ). As mentioned in § 4.4, the pressure oscillations are induced by gravity waves from the turbine interacting directly with the capping inversion. The wavelength of the oscillations decreases as the geostrophic wind speed decreases, as predicted by linear wave theory ( $\lambda _x = 2\pi G_x/N$ ). Where waves are present ( $Ro \lesssim 167$ ), oscillations due to gravity waves influence the pressure field and therefore the lateral pressure gradient forcing term. In the $Ro = 100$ and $Ro = 125$ simulations, the lateral pressure gradient decays rapidly in the near wake, which drastically increases the net lateral advection and therefore wake deflection in the CNBL inflow compared with the TNBL or uniform inflow. While the dynamics differ between the TNBL and CNBL due to free atmosphere stratification, we emphasise that the qualitative trends in wake deflection (figure 13) are the same: we observe parametrically varying wake deflection which can be clockwise or anti-clockwise, but is increasingly anti-clockwise as relative Coriolis forcing strength increases (Rossby number decreases).

The parametric dependence of the lateral momentum budget terms on the Rossby number has a direct connection to the wake deflection. In particular, because the momentum budget terms represent a forcing (and therefore acceleration) on the flow, integrating the lateral momentum budgets twice in time first yields lateral velocity, then wake deflection. Additionally, using the streamtube-averaged velocity $\langle \bar {u}\rangle$ to advect the flow, time integration can be transformed into spatial integration in the streamwise direction $x$ . A mathematical derivation including the assumptions and transformations that connect the wake deflection to the forcing terms $\bar {f}_y$ in the lateral RANS budget (terms II–V in (3.3)) are given in Appendix D. The final result is

(4.3) \begin{equation} y_c(x) = \sum \underbrace { \int _{x_0}^{x} \frac {1}{\langle \bar {u} \rangle (x')} \int _{x_0}^{x'} \frac {1}{\langle \bar {u} \rangle (x'')} \langle \bar {f}_y(x'', y, z) \rangle \,\textrm{d}x'' \,\textrm{d} x' }_{\text {Individual forcing contributions}} + \underbrace { \int _{x_0}^x \frac {\langle \bar {v} \rangle (x_0)}{\langle \bar {u} \rangle (x')} \,\textrm{d}x' }_{\text {Base Advection}}. \end{equation}

As a result of the integration, the observed streamtube deflection (figure 13) can be reconstructed from the streamtube-averaged momentum budgets, as shown in figure 16( $a$ – $b$ ). Overall, the integrated budgets reconstruct the observed streamtube lateral deflection well, quantitatively capturing the streamtube deflection amount.

Figure 16.

Integrated form of the lateral momentum budgets to reconstruct the streamtube centroid deflection in ( $a$ ) TNBL inflow and ( $b$ ) CNBL inflow. ( $c$ ) Integrated contributions to the net streamtube deflection for each lateral forcing term at $x = 10D$ . The net deflection, $y_c(x=10D)$ , is equal to the sum of deflection contributions from all forcing terms to the right of the vertical dashed line. The linear advection $x\psi _{hub}$ is subtracted from the net deflection and base advection columns.

To extend this analysis, we represent the integrated contribution of each lateral forcing term as an equivalent wake deflection amount at $x = 10D$ , shown in figure 16( $c$ ). Some forces, such as direct Coriolis forcing, cause anti-clockwise ( $+y$ direction) wake deflection in the northern hemisphere. Other forces, such as the turbulent flux of lateral momentum, deflect wakes clockwise. Importantly, the turbulent flux of lateral momentum does not increase in magnitude as fast as the direct Coriolis forcing, resulting in increasingly anti-clockwise wake deflections with decreasing $Ro$ . The lateral pressure gradient force generally opposes the Coriolis force, although in the CNBL, the integrated magnitude of the pressure gradients is strongly affected by gravity waves. For example, at $Ro = 125$ with the integration bounds $x \in [0, 10]D$ , the pressure gradient contribution to the wake deflection is nearly zero (see figure 14 $f$ ). While the net deflection and direct Coriolis forcing contribution monotonically increase with increasing relative Coriolis forcing (decreasing $Ro$ ), some forces, such as the lateral pressure gradients and turbulent fluxes, vary non-monotonically as a result of structural changes in the ABL with decreasing $Ro$ such as decreasing inflow $\textit{TI}$ . Finally, the base advection term represents the nonlinear interaction between the non-zero lateral velocity at the rotor, which is due to the wind direction drift in the ABL and to Coriolis effects in the induction region of the wind turbine (Gadde & Stevens Reference Gadde and Stevens2019), with the wake velocity deficit. As figure 16( $c$ ) shows, multiple lateral forcing terms have a leading-order importance on the wake deflection of a wake in the ABL, and each forcing term has a different parametric dependence on the relative strength of Coriolis forcing.

The integrated lateral momentum budgets link the momentum equations with the observed wake deflections. Due to the transformation from time coordinates to spatial coordinates through the streamwise velocity $\langle \bar {u}(x) \rangle$ , the lateral forcing (acceleration) that occurs in the near-wake has a stronger effect on the wake deflection than forcing in the far-wake. This is because the wake velocity in the near-wake region is slower than in the far-wake and, therefore, the residence time of the near-wake is longer than in the far-wake. Notably, the largest lateral forcing term in the near-wake region is the anti-clockwise Coriolis forcing, shown in figure 14. Assuming a linearised advection velocity, $\langle \bar {u}(x) \rangle \approx u_{hub}$ fails to reconstruct the wake deflection from integrating the momentum budget and persistently underpredicts the wake deflection due to Coriolis effects. In summary, through a streamtube-averaged analysis of the wake momentum budget, we parse the importance of each ABL forcing term on the net wake deflection to explain the parametric variation of wake deflection on relative Coriolis forcing strength.

4.6. Contextualising Coriolis-driven wake deflection with wind farm flow control

Atmospheric conditions typically change on time scales of $\mathcal {O}(1\,\textrm {h})$ (Stull Reference Stull1988). A time averaging period of one inertial period $2\pi /f_c$ is used in the ABL simulations presented in this study to ensure that the reported dynamics is not affected by a partial period of an inertial oscillation. This time averaging period is much longer than the response time of the ABL. In other words, the ABL is rarely quasi-stationary for a whole inertial oscillation period. In this section, we compare the variation across Rossby numbers to the inherent variation in turbulence over one-hour averages.

To assign an interval of uncertainty to the reported wake statistics, we compare the full time average over one inertial period with non-overlapping one-hour time averages. The standard deviation of the mean $\bar {\sigma }$ is computed from the standard deviation of the ensemble of one-hour averages. A 2 $\bar {\sigma }$ band around the sample mean wake deflection and wake recovery is shown in figure 17 for two Rossby numbers in TNBL inflow. The advection from the base flow $x\psi _{hub}$ for each 1-hour average is subtracted from the wake deflection, like in figure 13, to focus on the variation in wake dynamics rather than on the variation in wind direction due to inertial oscillations. The Rossby numbers $Ro = 125$ and $Ro = 500$ approximately correspond with the cut-in and rated wind speed for the IEA 15-MW reference turbine.

Figure 17.

Variation in ( $a$ ) wake recovery and ( $b$ ) wake deflection for two Rossby numbers in TNBL inflow, showing a distribution from one-hour time-averaged flow fields.

The one-hour time-averaged metrics highlight the variability of mean winds and turbulence in the ABL within an inertial oscillation. For the streamwise velocity deficit shown in figure 17( $a$ ), the intervals between the two Rossby number values overlap. This indicates that the variability in wake recovery due to stochastic turbulence in the TNBL is greater than the variability in wake recovery due to Coriolis effects. That is, for the range of Rossby numbers studied here, we find no statistically significant impact of Coriolis effects on wake recovery of a free-standing turbine, relative to the internal variability of the ABL, for time scales relevant to the ABL or wind turbine control.

In contrast, the impact of relative Coriolis forcing strength on wake deflection is statistically significant, as shown in figure 17( $b$ ). For the Rossby number values relevant to the IEA 15-MW reference turbine, the difference in wake deflection due to Coriolis forcing exceeds the variance due to ABL turbulence over one-hour time scales. Additionally, the difference in wake deflection between Rossby numbers increases with increasing distance downwind. It is also interesting to note that for relatively weak Coriolis forcing ( $Ro = 500$ ), the $2\bar {\sigma }$ envelope of variability includes zero and anti-clockwise wake deflections. Previous literature has focused on shorter time averaging lengths than presented here. Alongside variation in the ABL conditions, the spread of reported wake deflection may be partially attributed to variability of ABL turbulence over relatively short time scales.

Wake steering, or the intentional deflection of wind turbine wakes induced through yaw-misalignment of the rotor, has garnered attention as one way to increase collective power production of wind turbine arrays (Fleming et al. Reference Fleming2019; Howland et al. Reference Howland, Quesada, Martínez, Larra, Palou, Yadav, Chawla, Sivaram and Dabiri2022). Previous work has focused on the combined effects of Coriolis forcing and wake steering, which both can act to deflect wakes in the ABL. In LES of a five-turbine array at $Ro = 1005$ , Nouri et al. (Reference Nouri, Vasel-Be-Hagh and Archer2020) used wake steering to increase aggregate farm power production. They found that power production gains for positive yaw-misalignment angles increased over negative yaw misalignments when Coriolis forces were present in the inflow (Nouri et al. Reference Nouri, Vasel-Be-Hagh and Archer2020). Here, positive yaw misalignments are anti-clockwise turbine rotations about the vertical axis, which induce clockwise wake deflection. Similarly, Wei et al. (Reference Wei, Wang, Wan and Strijhak2023) recommend positive yaw misalignments for fully waked conditions in the northern hemisphere.

While we focus on wakes of yaw-aligned turbines in this study, we can quantify the magnitude of equivalent yaw misalignment which would equal the wake deflection observed due to Coriolis forcing alone. To predict the wake deflection due to yaw misalignment, we use the model proposed by Shapiro et al. (Reference Shapiro, Gayme and Meneveau2018) and extended by Heck et al. (Reference Heck, Johlas and Howland2023) using a wake spreading rate of $k_w = 0.083$ and the same turbine thrust coefficient $C^{\prime}_T = 1.33$ . We find that for $Ro = 125$ , the equivalent yaw misalignment for the observed wake deflection at $x = 10D$ in the TNBL is approximately $-13^\circ$ , while in the CNBL, it is approximately $-24^\circ$ . The equivalent yaw misalignment angles are negative because, for strong relative Coriolis forcing (low Rossby numbers), we observe that the Coriolis force deflects wind turbine wakes anti-clockwise. Therefore, for the Rossby number ( $Ro = 125$ ) and ABL conditions simulated here, positive yaw misalignment and Coriolis-based wake deflections would be in opposition and would nearly cancel out, while negative yaw misalignment will enhance the inherent Coriolis-based anti-clockwise deflection. Future research should study the interactions of relative Coriolis forcing strength with wake steering control.

Previous studies that investigated Coriolis effects on wind turbine wakes have focused on turbines approximately half the size of modern offshore wind turbines, observing both clockwise and anti-clockwise wake deflections. Studying a rotor twice as large in diameter halves the relevant Rossby numbers for that turbine. For example, the NREL 5-MW turbine operating at rated wind speed sees a Rossby number $Ro \approx 880$ . Our results suggest that in this operating regime, wakes for the NREL 5-MW reference turbine would deflect clockwise (van der Laan & Sørensen Reference van der, Paul and Sørensen2017), with a spread of deflection values (in magnitude and direction) due to turbulence which is dependent on the time averaging interval. However, as turbines become larger, Coriolis forces become more dynamically important, relative to the other terms in the lateral momentum balance. The next generation of turbines with rotors greater than 200 m in diameter will increasingly operate in regimes where the dynamics studied here is relevant, and wakes will be deflected increasingly anti-clockwise for lower Rossby numbers.