2.1. Two-scale momentum theory

The central equation of the two-scale momentum theory (Nishino & Dunstan Reference Nishino and Dunstan2020; Nishino & Smyth Reference Nishino and Smyth2026) is the non-dimensional farm momentum (NDFM) equation

(2.1) \begin{equation} C_{\!T}^* \frac {\lambda }{C_{\!f0}} \beta ^2 + \beta ^{\gamma } = M, \end{equation}

where $C_T^*$ is the internal farm thrust coefficient, $\lambda$ is the array density, $C_{\!f0}$ is the surface friction coefficient of the ABL, $\beta$ is the wind-speed reduction factor, $\gamma$ is the friction exponent and $M$ is the momentum availability factor. The exact definitions of these parameters will be given in the summary of the NDFM derivation described below. A typical usage of the equation is to solve for $\beta$ , given information or models of the remaining parameters, which can then be used to calculate the farm power efficiency.

The NDFM equation is derived through a control volume (CV) analysis of the streamwise mean momentum equation

(2.2) \begin{equation} \frac {\partial \rho U_1}{\partial t} = -\frac {\partial \rho U_1 U_{\!j}}{\partial x_{\!j}} -\frac {\partial\! P}{\partial x_1} + \rho f_c U_2 + \frac {\partial \tau _{1j}}{\partial x_{\!j}} + f_1 \end{equation}

for a volume, $V_{{cv}} = \textit{LWH}_{{cv}} = S_{{cv}} H_{{cv}}$ , containing the wind farm. The Reynolds stress tensor is defined as $\tau _{ij} \equiv -\rho \overline {u_i u_{\!j}}$ in this work. In the original derivation by Nishino & Dunstan (Reference Nishino and Dunstan2020), the height of the CV was chosen to be similar to the boundary-layer (BL) height, $H_{{cv}} \approx \delta _{\textit{ABL}}$ , whereas in later works (e.g. Kirby et al. Reference Kirby, Dunstan and Nishino2023) it has been chosen as $H_{{cv}} = H_{\!F}$ (where typically $H_{\!F} = 2.5 z_{h}$ is chosen with $z_h$ being the turbine hub height, which is also adopted in this work; the sensitivity to $H_{\!F}$ is discussed in Appendix A) to model the effects of turbulent entrainment above the wind farm more explicitly than implicitly. For rectangular farm layouts (both aligned and staggered), the length and width of the CV are typically chosen as the number of turbines times inter-spacing, i.e. $L = N_x s_x$ and $W = N_y s_y$ , where $N_i$ and $s_i$ are the number of turbines and inter-spacing in the $i$ th direction. The NDFM equation will be valid for any choice of CV as long as it encloses the wind farm, but choosing a CV with a size similar to the wind farm is preferred for modelling purposes.

Denoting the volume average over the CV with square brackets (i.e. $ [ \boldsymbol{\cdot }] \equiv V_{{cv}}^{-1} \int _{V_{{cv}}} \boldsymbol{\cdot }\text{d}V$ ) and using Gauss’s theorem for the advection and turbulent stress terms gives

(2.3) \begin{align} \left [\frac { \partial \rho U_1}{\partial t} \right ] &= -\frac {1}{V_{{cv}}}\int _{\varOmega _{{cv}}} \rho U_1 U_{\!j} \text{d}A_{\!j} - \left [ \frac {\partial\! P}{\partial x_1} - \rho f_c U_2 \right ] + \frac {1}{V_{{cv}}} \int _{\varOmega _{{cv}} \setminus \varOmega _w} \tau _{1j} \text{d}A_{\!j} - \frac {S_{{cv}} \tau _w }{V_{{cv}}} - \frac {T}{V_{{cv}}}, \end{align}

where $\varOmega _{{cv}}$ is the surface of the CV, $\tau _{w} \equiv S_{{cv}}^{-1} \int _{\varOmega _w} \tau _{13} \text{d}A$ is the surface-averaged streamwise component of the wall shear stress ( $\varOmega _w$ is the wall surface of the CV) and $T \equiv - \int _{V_{{cv}}} f_1 \text{d}V$ is the integrated body force stemming from the total turbine thrust forces. Multiplying (2.3) through with the CV and rearranging gives

(2.4) \begin{align} T + S_{{cv}} \tau _{w} &= \underbrace {-\int _{\varOmega _{{cv}}} \rho U_1 U_{\!j} \text{d}A_{\!j}}_{X_{{adv}}} \underbrace {- V_{{cv}} \left [ \frac {\partial\! P}{\partial x_1} \right ]}_{X_{\textit{PGF}}} + \underbrace { V_{{cv}} \left [\rho f_c U_2 \right ]}_{X_{\textit{Cor}}} + \underbrace {\int _{\varOmega _{{cv}} \setminus \varOmega _w} \tau _{1j} \text{d}A_{\!j}}_{X_{\textit{turb}}} \underbrace {- V_{{cv}} \left [\frac { \partial \rho U_1}{\partial t} \right ]}_{X_{\textit{uns}}}, \end{align}

where the two terms on the left-hand side are the momentum sinks from the turbine and surface friction drag, respectively, and the terms on the right-hand side are the momentum balancing terms from advection, pressure-gradient forcing, Coriolis, turbulence and unsteadiness. A similar CV analysis can be made for a case without turbines (which has $T = 0$ and is denoted with a zero subscript), and this can be used to normalise (2.4), i.e.

(2.5) \begin{align} \frac {T + S_{{cv}} \tau _{w}}{S_{{cv}} \tau _{w0}} &= \frac {X_{{adv}} + X_{\textit{PGF}} + X_{\textit{Cor}} + X_{\textit{turb}} + X_{\textit{uns}}}{X_{\textit{adv0}} + X_{\textit{PGF0}} + X_{\textit{Cor0}} + X_{\textit{turb0}} + X_{\textit{uns0}}}. \end{align}

The left-hand side can be rewritten as

(2.6) \begin{equation} \frac {T + S_{{cv}} \tau _{w} }{S_{{cv}} \tau _{w0} } = C_{\!T}^* \frac {\lambda }{C_{\!f0}} \beta ^2 + \beta ^{\gamma } \end{equation}

from the definitions $C_T^* \equiv T/(({1}/{2}) \rho U_{\!F}^2 N_t A_d)$ , $U_{\!F} \equiv [U_1]$ , $\lambda \equiv N_t A_d/S_{{cv}}$ , $C_{\!f0} \equiv \tau _{w0} / (({1}/{2}) \rho U_{F0}^2)$ , $\beta \equiv U_{\!F}/U_{F0}$ and $\gamma = \textrm {log}_{\beta } ( {\tau _{w}}/{\tau _{w0}} )$ ; hence, (2.5) is, in fact, the NDFM equation (2.1), and the momentum availability factor is

(2.7) \begin{equation} M \equiv \frac {X_{\!F}}{X_{\!F0}} = \frac {X_{{adv}} + X_{\textit{PGF}} + X_{\textit{Cor}} + X_{\textit{turb}} + X_{\textit{uns}}}{X_{\textit{adv0}} + X_{\textit{PGF0}} + X_{\textit{Cor0}} + X_{\textit{turb0}} + X_{\textit{uns0}}} . \end{equation}

The contributions of the various physical processes to $M$ can conveniently be split into separate parts by rewriting (2.7) as

(2.8) \begin{align} M &= 1 + \frac {X_{\!F} - X_{\!F0}}{X_{\!F0}} \\ &= 1 + \underbrace { \frac {X_{{adv}} - X_{\textit{adv0}}}{X_{\!F0}}}_{\Delta M_{{adv}}} + \underbrace { \frac {X_{\textit{PGF}} - X_{\textit{PGF0}}}{X_{\!F0}}}_{\Delta M_{\textit{PGF}}} + \underbrace { \frac {X_{\textit{Cor}} - X_{\textit{Cor0}}}{X_{\!F0}}}_{\Delta M_{\textit{Cor}}} + \underbrace { \frac {X_{\textit{turb}} - X_{\textit{turb0}}}{X_{\!F0}}}_{\Delta M_{\textit{turb}}} + \underbrace { \frac {X_{\textit{uns}} - X_{\textit{uns0}}}{X_{\!F0}}}_{\Delta M_{\textit{uns}}}, \nonumber \end{align}

where one can note that $X_{\!F0} = S_{{cv}} \tau _{w0}$ . To solve the NDFM equation, one must model $M$ or each individual subpart of it.

2.2. Analytical momentum availability models

The simplest model for $M$ is to assume that all $\Delta M$ contributions in (2.8) are negligibly small, hence,

(2.9) \begin{equation} M_{\textit{constant}} = 1, \end{equation}

which corresponds to assuming that the momentum available to the wind farm is fixed, i.e. there is no ABL response to the farm.

A linear model for $M$ was proposed by Nishino & Dunstan (Reference Nishino and Dunstan2020), i.e.

(2.10) \begin{equation} M_{\textit{lin}} = 1 + \zeta (1-\beta ), \end{equation}

and further studied by Patel et al. (Reference Patel, Dunstan and Nishino2021) through NWP simulations with a wind farm modelled as an increased surface roughness patch. They concluded that the linear relationship appeared to satisfactorily fit their data with a momentum response parameter $\zeta \sim \mathcal{O}(10^1)$ depending on the atmospheric conditions. Using a similar methodology, Kirby et al. (Reference Kirby, Nishino and Dunstan2022) noted that $\zeta$ strongly depends on the wind farm size with smaller $\zeta$ values being observed for larger wind farms.

A more elaborate model of $M$ was developed by Kirby et al. (Reference Kirby, Dunstan and Nishino2023) who considered each term in (2.8) and used various physical arguments to construct submodels of these, which will be discussed in detail in § 4. Their final combined model,

(2.11) \begin{equation} M_{\textit{KDN1}} = \frac {1 + \frac {H_{\!F}}{\textit{LC}_{\!f0}} \left ( 1 - \beta ^2 \right ) - \frac { \tau _{t0}}{\tau _{w0}}}{\beta \left (1 -\frac {\tau _{t0}}{\tau _{w0}}\right )}, \end{equation}

was successfully used in conjunction with the NDFM equation to predict the wind farm efficiency for various wind farm LES cases.

Equation (2.11) requires $({\tau _{t0}}/{\tau _{w0}})$ , which is often unknown and unavailable as input for prediction of wind farm performance. Kirby et al. (Reference Kirby, Dunstan and Nishino2023) thus suggested to model this ratio as $({\tau _{t0}}/{\tau _{w0}}) = 1 - H_{\!F}/h_0$ , which stems from assuming that the (undisturbed) ABL streamwise shear stress is linear from the ground to the ABL height, $h_0$ . With this model, one obtains a second simpler version,

(2.12) \begin{equation} M_{\textit{KDN}2} = \frac {1 + \frac {h_0}{\textit{LC}_{\!f0}} \left ( 1 - \beta ^2 \right ) }{\beta }, \end{equation}

which, for example, has been used by Nishino & Smyth (Reference Nishino and Smyth2026).

Some further linearisations were given in the appendix of Kirby et al. (Reference Kirby, Dunstan and Nishino2023) to produce a third version, i.e.

(2.13) \begin{equation} M_{\textit{KDN}3} = 1 + \underbrace {\left (1.18 + 2.18 \frac {h_0}{\textit{LC}_{\!f0}} \right )}_{\zeta _{\textit{KDN}3}} (1 - \beta ), \end{equation}

which is in a linear form similar to (2.10). The model correctly captures that the momentum response factor, $\zeta _{\textit{KDN}3}$ , should decrease with increasing farm size, as discussed earlier, through its dependence on the farm length $L$ . Various quasi-one-dimensional (quasi-1D) assumptions were used to obtain (2.11), which is why the farm size is represented by $L$ rather than the farm surface area $S_{{cv}}$ . A generalisation of $M_{\textit{KDN}3}$ to flows with a non-zero inverse ABL Rossby number, $\textit{Ro}_{h0}^{-1} \equiv f_c h_0/G$ , named $M_{\textit{BNK}}$ , will be developed in § 5.

An overview of the various momentum availability models is given in figure 1, where they are, roughly, categorised according to their accuracy, generality, complexity and the amount of required input. In the next section, these models will be tested using LES data.

Figure 1.

Hierarchy of momentum availability models. We propose a new model, $M_{\textit{BNK}}$ , in this paper; see § 5. The arrows indicate assumptions to move from one model to another.

4.1. General observations relevant for the two-scale momentum theory

Some integral quantities of the LES relevant for the two-scale momentum theory are shown in tables 1 and 2. The BL height $h_0$ and concavity $p_0$ have been estimated through a fit of the normalised total shear stress to $|\tau |_0/|\tau |_{w0} = (1 - z/h_0)^{p_0}$ , where $|\tau | \equiv \sqrt {\tau _{13}^2 + \tau _{23}^2}$ and the ‘0’ subscript denotes the undisturbed ABL. A discussion about the shape of the ABL shear stress profiles and its relevance for the two-scale momentum theory will be given in § 5. The values of the friction coefficient $C_{\!f0}$ are within the typical expected range for offshore wind farms.

Table 1.

Integral quantities for the LES precursor simulations. The quantities are the ABL height $h_0$ , concavity of the total shear stress profile $p_0$ , normalised mean streamwise velocity in the CV without turbines $U_{F0}/G$ , friction coefficient $C_{\!f0}$ and the ratio of streamwise shear stress at the top of the CV to streamwise wall shear stress $\tau _{t0}/\tau _{w0}$ .

Table 1. Long description

A table comparing integral quantities for LES precursor simulations. The table has four rows and six columns. The column headers are h0 (m), p0, UF0/G, CF0, and τf0/τw0. The row labels are H300, H500, and H1000. Row 1: h0 (m), 360; p0, 1.18; UF0/G, 0.95; CF0, 0.00181; τf0/τw0, 0.099. Row 2: h0 (m), 555; p0, 1.29; UF0/G, 0.94; CF0, 0.00185; τf0/τw0, 0.324. Row 3: h0 (m), 1100; p0, 1.94; UF0/G, 0.92; CF0, 0.00187; τf0/τw0, 0.417.

Table 2.

Integral quantities for the LES wind farm simulations. The quantities are the wind-speed reduction factor $\beta$ , the local wind-speed reduction factor at the start of the farm $\beta _{\textit{local}}(0)$ , the local wind-speed reduction factor at the end of the farm $\beta _{\textit{local}}(L)$ , the friction exponent $\gamma$ , the internal thrust coefficient $C_T^*$ and the normalised perturbation pressure jump over the farm $\Delta P^*/(\rho G^2)$ .

Table 2. Long description

A table comparing integral quantities for wind farm simulations. The table has three rows and seven columns. The columns are labeled Beta, Beta local at start, Beta local at end, Gamma, C T star, and Delta P star over rho G squared. The rows are labeled H 300, H 500, and H 1000. Row 1: Beta, 0.71; Beta local at start, 0.85; Beta local at end, 0.70; Gamma, 1.26; C T star, 1.04; Delta P star over rho G squared, -0.178. Row 2: Beta, 0.74; Beta local at start, 0.88; Beta local at end, 0.74; Gamma, 1.13; C T star, 1.03; Delta P star over rho G squared, -0.180. Row 3: Beta, 0.76; Beta local at start, 0.94; Beta local at end, 0.71; Gamma, 1.05; C T star, 1.03; Delta P star over rho G squared, -0.123.

In earlier studies, it has generally been assumed that the surface friction exponent, $\gamma$ , is around 1.7–1.8 (e.g. Kirby et al. Reference Kirby, Nishino and Dunstan2022), but the current LES, see table 2, show that it can be significantly smaller, i.e. around 1.0–1.3. A smaller value of $\gamma$ means that the wall friction contribution, i.e. $\beta ^\gamma = \tau _w/\tau _{w0}$ , in the NDFM equation (2.1), becomes larger (because $0 \lt \beta \lt 1$ ). Nevertheless, as shown in figure 4, the farm thrust term, i.e. $C_{\!T}^*({\lambda }/{C_{\!f0}}) \beta ^2 = T/(S_{{cv}} \tau _{w0})$ , is still the dominant momentum sink on the left-hand side of the NDFM equation for all cases. This is an important observation because it implies that the modelling of $\gamma$ is not critical to the prediction of $\beta$ . One can therefore, in many cases, assume that $\gamma = 2$ to simplify the NDFM equation without incurring any significant error, even if the true value of $\gamma$ is closer to one. On the other hand, the modelling of the internal thrust coefficient, $C_T^*$ , is more important for obtaining good predictions of $\beta$ with the NDFM equation. Regarding $C_T^*$ , table 2 shows that it is nearly constant for all three simulations, which is consistent with the fundamental two-scale assumption in the sense that it is mainly an ‘internal’ parameter of the wind farm (i.e. depends mainly on turbine layout and turbine operating conditions) and, therefore, is not expected to be sensitive to the ABL height, which is an ‘external’ parameter (Kirby et al. Reference Kirby, Nishino, Lanzilao, Dunstan and Meyers2025). The modelling of $C_T^*$ was investigated by Kirby et al. (Reference Kirby, Nishino and Dunstan2022) and Nishino & Smyth (Reference Nishino and Smyth2026) and will not be further discussed in this paper.

Figure 4.

Momentum sinks in the NDFM equation, i.e. left-hand side of (2.1) and (2.6). For completeness, the momentum sinks in the aligned, double-spacing and half-farm cases are also included, although these cases will not be further investigated in this section.

The development of the local wind-speed reduction factor $\beta _{\textit{local}} \equiv (H_{\!F} {\textit{WU}}_{F0})^{-1} \int _0^{H_{\!F}} \int _0^{\textit{W}} \,U_{1}\text{d}y \,\text{d}z$ and the perturbation pressure $P^* \equiv P - P_\infty$ (where $P_\infty$ is the background pressure, which is linearly decreasing in the horizontal direction according to the driving pressure gradient force (PGF), $\partial\! P_\infty /\partial x_i$ , of the simulation) are shown in figure 5. The small ‘high-frequency’ fluctuations in the farm region are from the individual turbines, whereas the larger fluctuations, which are also present in farm wakes (Lanzilao & Meyers Reference Lanzilao and Meyers2025), are from farm-induced gravity waves. Gravity waves are also clearly visible in figure 2 and have been studied in detail by Lanzilao & Meyers (Reference Lanzilao and Meyers2024) among others. They are generally stronger for lower ABL heights and it is also seen in figure 5(b) that a lower ABL height induces a larger pressure perturbation drop, $\Delta P^*$ , over the farm region, which is favourable for farm power production. The pressure perturbation, $P^*$ , should go to the same constant value, $P_{\textit{inlet}}^*$ , far upstream and downstream of the farm, but it does not despite the fact that a large domain was used for the LES, which suggests that an even longer domain should ideally have been used. An error can be expected from this in the PGF term of the momentum availability factor, but this is not expected to change the main conclusions of the following analysis.

Figure 5.

(a) Local wind-speed reduction factor, and (b) normalised perturbation pressure (subtracted by inlet value) spanwise averaged over the width $y=[y_{\textit{centre}}-W/2,y_{\textit{centre}}+W/2]$ and height $z=[0,H_{\!F}]$ of the CV. The CV region enclosing the wind farm is marked in grey.

Figure 6.

(a) Contributions from CV surfaces to the advection term for LES (full) and the model (dashed). A positive sign means advection of $U_1$ into the CV. Note that the north and south contributions are almost identical. (b) Total advection contribution and error of the model.

4.2. Advection

Streamwise momentum is advected in and out of the CV, and its contribution, $\Delta M_{{adv}}$ , to the momentum availability factor is defined in (2.8). For our current cases, the expression for the advection part simplifies to

(4.1) \begin{align} \Delta M_{{adv}} &= \frac {X_{{adv}}}{X_{\!F0}} = \frac {-\int _{\varOmega _{{cv}}} \rho U_1 U_{\!j} \text{d}A_{\!j}}{X_{\!F0}}, \end{align}

because $X_{\textit{adv}0} = 0$ due to the horizontal homogeneity of the precursor simulations (the front/rear and north/south advection cancel due to periodicity and the top/bottom advection are zero because $U_3 = 0$ ). There is also no flow through the bottom surface in the main simulation, hence, only the front/rear, north/south and top surfaces of the CV contribute to the integral, which are calculated directly from the LES flow field data and displayed in figure 6(a). In the advection submodel of Kirby et al. (Reference Kirby, Dunstan and Nishino2023) it was assumed that the north/south contributions were zero, but as shown in figure 6(a), both of these are in fact negative for the current cases (meaning there is advection out of the CV at these surfaces) due to the flow through the farm diverting horizontally spanwise. However, in comparison with the other contributions, the north/south contributions are indeed small and it is therefore a reasonable approximation to neglect them. The front/rear contributions used in their submodel are exact, as they can be rewritten as

(4.2) \begin{equation} - \frac {1}{X_{\!F0}} \int _{\varOmega _{\textit{front}}} \rho U_1 U_{\!j} \text{d}A_{\!j} = \frac {2 \beta _{\textit{local}}(0)^2 H_{\!F}}{\textit{LC}_{\!f0}}, \end{equation}

(4.3) \begin{equation} - \frac {1}{X_{\!F0}} \int _{\varOmega _{\textit{rear}}} \rho U_1 U_{\!j} \text{d}A_{\!j} = - \frac {2 \beta _{\textit{local}}(L)^2 H_{\!F} }{\textit{LC}_{\!f0}} . \end{equation}

The sign change on the former is because the vectors $U_{\!j}$ and $\text{d}A_{\!j}$ have opposite directions on the front face. Finally, for the top surface Kirby et al. (Reference Kirby, Dunstan and Nishino2023) used a simple quasi-1D mass flux analysis to estimate its contribution as

(4.4) \begin{align} - \frac {1}{X_{\!F0}} \int _{\varOmega _{\textit{top}}} \rho U_1 U_{\!j} \text{d}A_{\!j} &= \frac {H_{\!F} \left ( \beta _{\textit{local}}(L)^2 - \beta _{\textit{local}}(0)^2 \right )}{\textit{LC}_{\!f0}} . \end{align}

In all cases, $\beta _{\textit{local}}(L) \lt \beta _{\textit{local}}(0)$ , hence, the model always predicts advection out of the top surface (i.e. a negative contribution). The LES data in figure 6(a) however show that there is advection into the CV (i.e. a positive contribution) for the lowest ABL height case, H300, which can occur due to the strong gravity waves present in this case. The model for the top surface advection (4.4) overpredicts the advection out of the top surface in all cases, which in effect compensates for the neglected advection out of the north/south surfaces. The overall advection submodel of Kirby et al. (Reference Kirby, Dunstan and Nishino2023) is the sum of (4.2)–(4.4), i.e.

(4.5) \begin{equation} \Delta M_{{adv}} = \frac { H_{\!F}}{\textit{LC}_{\!f0}} \big ( \beta _{\textit{local}}(0)^2 - \beta _{\textit{local}}(L)^2 \big), \end{equation}

and figure 6(b) shows it is within approximately $\pm 10\,\%$ of the LES data. As it depends on $\beta _{\textit{local}}(0)$ and $\beta _{\textit{local}}(L)$ , which are in general unknown, it is by itself not a very practical analytical model and it will therefore be further simplified in § 4.4.

4.3. Pressure gradient forcing

The CNBLs used in the current cases are driven by a horizontal pressure gradient forcing, $(\partial\! P_\infty /\partial x_1,\partial\! P_\infty /\partial x_2)$ , which is constant throughout the domain, i.e. it is a barotropic ABL, and, as shown in figure 5(b), a perturbation PGF, $\partial\! P^*/\partial x_1$ , develops in the CV region due to the presence of the wind farm (recall that $P^* \equiv P - P_\infty$ is defined as the pressure perturbation from the linearly decreasing background pressure $P_\infty$ that drives the flow). The contribution from the pressure gradient to the momentum availability is defined in (2.8) and the expression for the PGF part can be rewritten with the perturbation pressure as

(4.6) \begin{align} \Delta M_{\textit{PGF}} &= -\frac {V_{{cv}}}{X_{\!F0}} \left [ \frac {\partial\! P^*}{\partial x_1}\right ] = - \frac {H_{\!F}}{L \tau _{w0}} \underbrace {\left ( P_{\textit{rear}}^* - P_{\textit{front}}^* \right )}_{\equiv \Delta P^*}, \end{align}

where $P_{\textit{rear}}^*$ and $P_{\textit{front}}^*$ are the rear and front surface-averaged values, respectively. Equation (4.6) is exact and was the starting point for Kirby et al. (Reference Kirby, Dunstan and Nishino2023), who used Bernoulli’s equation to estimate the front perturbation pressure as

(4.7) \begin{equation} P_{\textit{front}}^* - P_{\textit{inlet}}^* = \frac {1}{2} \rho U_{F0}^2 \big ( 1 - \beta _{\textit{local}}(0)^2\big ), \end{equation}

where $P_{\textit{inlet}}^*$ is the far upstream value. The second assumption of their PGF model is that the rear perturbation pressure is simply the negative value of the above, i.e.

(4.8) \begin{equation} P_{\textit{rear}}^* - P_{\textit{inlet}}^* = -\big(P_{\textit{front}}^* - P_{\textit{inlet}}^* \big), \end{equation}

which corresponds to assuming that $P^* - P_{\textit{inlet}}^*$ is anti-symmetric around the centre of the farm. Combining (4.8) with (4.6) and (4.7) gives the final submodel of Kirby et al. (Reference Kirby, Dunstan and Nishino2023):

(4.9) \begin{equation} \Delta M_{\textit{PGF}} = 2 \frac {H_{\!F}}{\textit{LC}_{\!f0}} \big (1 - \beta _{\textit{local}}(0)^2 \big ). \end{equation}

Figure 7.

(a) Contributions from the front and rear surfaces to the PGF term for LES (full) and the model (dashed). (b) Total PGF contribution and error of the model.

The two assumptions, i.e. (4.7) and (4.8), are checked with the LES data in figure 7(a), which shows that the simple Bernoulli estimate fairly well captures the trend of the front pressure, whereas the rear assumption is less consistent. Regarding the latter, it can indeed be observed in figure 5(b) that the perturbation pressure is not perfectly anti-symmetric around the centre of the farm. The H500 case is closest to being anti-symmetric, which is consistent with the fact that this is the case with the best rear prediction. As discussed at the end of § 4.1, the LES domain length is possibly too short to simulate the pressure field around the farm accurately, although it is not clear how large the impact of the LES domain length is and if a longer inlet fetch would improve the agreement between the LES and model.

The difference between the two dashed lines (or full lines) in figure 7(a) is the pressure perturbation jump, $\Delta P^*$ , over the farm, which per (4.6) determines the PGF contribution to $M$ . It can be seen that the pressure jump is overestimated by the model for the H300 and H500 cases, which is why $\Delta M_{\textit{PGF}}$ is overestimated for these two cases; see figure 7(b). Nevertheless, the model correctly captures that the PGF contribution tends to decrease with increasing ABL height, which is expected because the ABL top acts as a ‘lid’ on the flow and thereby affects the pressure gradient within the farm (a taller ABL height gives a smaller enhancement of the favourable wind farm pressure gradient). The effect of the ABL height on the wind farm pressure field has also been studied in detail by Lanzilao & Meyers (Reference Lanzilao and Meyers2024).

Figure 8.

(a) Test of the advection pressure (AP) approximation (red lines) and the underlying assumption (i) (4.13) (blue lines). (b) Combination of the advection and PGF terms of the momentum availability factor. The error percentage indicated is between the model and AP approximation bars.

4.4. The AP approximation

Combining the advection and PGF models ((4.5) and (4.9)) gives

(4.10) \begin{align} \Delta M_{{adv}} + \Delta M_{\textit{PGF}} &= \frac {H_{\!F}}{\textit{LC}_{\!f0}} \big ( - \beta _{\textit{local}}(0)^2 - \beta _{\textit{local}}(L)^2 + 2 \big ), \end{align}

which can be simplified to

(4.11) \begin{align} \Delta M_{\textrm {adv}} + \Delta M_{\textrm {PGF}} &= \frac {H_{\!F}}{\textit{LC}_{\!f0}} \big ( 1 - \beta ^2 \big ), \end{align}

resulting from the ‘AP approximation’,

(4.12) \begin{equation} \beta _{\textit{local}}(0)^2 + \beta _{\textit{local}}(L)^2 \approx 1 + \beta ^2 . \end{equation}

This approximation was proposed by Kirby et al. (Reference Kirby, Dunstan and Nishino2023) by assuming that the decrease of $\beta _{\textit{local}}$ from upstream (where it has a value of 1) to the front of the farm is equal to the increase from the farm-averaged value (i.e. $\beta$ ) to $\beta _{\textit{local}}$ at the end of the farm, i.e.

(4.13) \begin{equation} 1 - \beta _{\textit{local}}(0) \approx \beta _{\textit{local}}(L) - \beta, \quad \textrm {assumption (i)}, \end{equation}

which when squared and rearranged becomes

(4.14) \begin{align} 1 + \beta ^2 &\approx \beta _{\textit{local}}(L)^2 + \beta _{\textit{local}}(0)^2 + 2 \left ( \beta _{\textit{local}}(L) \beta _{\textit{local}}(0) - \beta \right ) . \end{align}

This equation simplifies to (4.12) when the second assumption is invoked:

(4.15) \begin{align} 2 \left ( \beta _{\textit{local}}(L) \beta _{\textit{local}}(0) - \beta \right ) \approx 0, \quad \textrm {assumption (ii)} . \end{align}

The data from the LES in figure 8(a) show an absolute error of around 0.1–0.2 for assumption (i) (4.13). The error from this first assumption means that (4.14) is not exactly correct, which together with the error from the second assumption leads to the fact that the AP approximation gives an overestimation of $\beta _{\textit{local}}(L)^2 + \beta _{\textit{local}}(0)^2$ by around 10–25 %. In contrast, Kirby et al. (Reference Kirby, Dunstan and Nishino2023) only found an overprediction of around 10 % for the four LES cases they analysed. The overprediction of the AP approximation means that the sum of the advection and PGF terms gets underpredicted by 30–35 %, see figure 8(b), because $\beta _{\textit{local}}(0)^2 + \beta _{\textit{local}}(L)^2$ appears with a negative sign in (4.10). These results highlight the difficulty in accurately predicting the advection and PGF terms without using the local flow information (such as $\beta _{\textit{local}}$ ) as input.

4.5. Coriolis

It was assumed by Kirby et al. (Reference Kirby, Dunstan and Nishino2023) that the contribution from the Coriolis force to the momentum availability factor could be neglected, i.e.

(4.16) \begin{equation} \Delta M_{\textit{Cor}} = 0 . \end{equation}

To verify the correctness of this, we evaluate its contribution directly from the LES data using its definition (2.8), which is

(4.17) \begin{align} \Delta M_{\textit{Cor}} &= \frac {V_{{cv}} \rho f_c}{X_{\!F0}} \left ( \left [U_2\right ] - \left [U_2\right ]_0\right ) = \frac {2 }{C_{\!f0} \textit{Ro}_{F0}} \left ( \frac {\left [U_2\right ] - \left [U_2\right ]_0 }{U_{F0}}\right ) \!, \end{align}

where $\textit{Ro}_{F0} \equiv U_{F0}/(f_c H_{\!F})$ is a Rossby number based on the CV height and no-turbine farm velocity. All three simulations have $ {2 }/({C_{\!f0} \textit{Ro}_{F0}}) \approx 4$ (which can be evaluated from the values in table 1), but the change of CV-averaged spanwise velocity from the precursor to main simulation is small, $ (( { [U_2 ] - [U_2 ]_0 )}/{U_{F0}} ) = \mathcal{O}(0.01)$ , and therefore $\Delta M_{\textit{Cor}}$ is only around $0.05$ , see table 3. This is much smaller than the contributions from the advection, PGF and turbulence terms, and the Coriolis term in the momentum equation thus contributes to less than $1 \,\%$ of the total $M$ . It is therefore a good assumption to neglect the direct Coriolis contribution. However, as shown in § 5, the Coriolis effect still has an important indirect influence on the momentum availability through shear stress veering, which cannot be neglected.

Table 3.

The LES and model predictions of the Coriolis contribution.

Table 3. Long description

A table comparing Coriolis contribution predictions across different simulations. The table has four columns and three rows. The columns are labeled as follows: 2/Cf0RoF0, [Uz]−[U2]0/Uf0, ΔMCor (LES), ΔMCor (model), and ΔMCor/M [%](LES). The rows are labeled as H300, H500, and H1000. Row 1: 2/Cf0RoF0, 3.94; [Uz]−[U2]0/Uf0, 0.02; ΔMCor (LES), 0.08; ΔMCor (model), 0.00; ΔMCor/M [%](LES), 0.8. Row 2: 2/Cf0RoF0, 3.93; [Uz]−[U2]0/Uf0, 0.01; ΔMCor (LES), 0.06; ΔMCor (model), 0.00; ΔMCor/M [%](LES), 0.5. Row 3: 2/Cf0RoF0, 3.97; [Uz]−[U2]0/Uf0, 0.01; ΔMCor (LES), 0.04; ΔMCor (model), 0.00; ΔMCor/M [%](LES), 0.4.

4.6. Unsteadiness

The effect of (farm-scale) unsteadiness on the momentum availability was assumed to be negligible in the model of Kirby et al. (Reference Kirby, Dunstan and Nishino2023), i.e.

(4.18) \begin{align} \Delta M_{\textit{uns}} = 0. \end{align}

To validate this, the time history of the flow is necessary, which is not available in the LES database of Lanzilao & Meyers (Reference Lanzilao and Meyers2024). As an alternative, we instead consider the time-averaged and horizontally averaged streamwise momentum budgets to implicitly give an assessment of the stationarity of the flow in the present LES.

The momentum budget for a CNBL precursor simulation is given by

(4.19) \begin{align} \left \langle \frac {\partial U_1}{\partial t}\right \rangle = \left \langle - \frac {1}{\rho } \frac {\partial\! P_\infty }{\partial x_1}\right \rangle + \left \langle f_c U_2\right \rangle + \left \langle \frac {1}{\rho } \frac {\partial {\tau _{13}}}{\partial x_3}\right \rangle \!, \end{align}

where $\langle \boldsymbol{\cdot }\rangle$ is the horizontal-average operator. The advection term is zero and the stress divergence term is simplified due to the periodic boundary conditions applied in the horizontal directions. A subscript $\infty$ is used on the pressure to denote that this is the driving pressure of the ABL. The budgets of the three considered ABLs are shown in figure 9 together with the sum of the terms on the right-hand side of (4.19). In the interior of the ABLs, the sum has a small positive value (meaning that $U$ is increasing in time), but it is nevertheless significantly smaller than the other terms, and we thus conclude that the ABLs are close to being stationary (i.e. ‘quasi-stationary’). Previous studies of the CNBL have indicated that around six non-dimensional time units are sufficient for the flow to reach a quasi-stationary state (Pedersen, Gryning & Kelly Reference Pedersen, Gryning and Kelly2014; Liu et al. Reference Liu, Gadde and Stevens2021a , Reference Liu, Lu and Stevens2024) and Lanzilao & Meyers (Reference Lanzilao and Meyers2024) indeed time-averaged the flow over the non-dimensional time period $f_c t = [8.6, 9.2 ]$ .

Figure 9.

Momentum budget for CNBL precursor simulations. The grey area marks the CV region.

The momentum budget for a wind farm simulation is given by

(4.20) \begin{align} \left \langle \frac {\partial U_1}{\partial t}\right \rangle = \left \langle - \frac {\partial U_1 U_{\!j} }{\partial x_{\!j}}\right \rangle + \left \langle -\frac {1}{\rho } \frac {\partial\! P^* }{\partial x_1}\right \rangle + \left \langle -\frac {1}{\rho } \frac {\partial\! P_\infty }{\partial x_1}\right \rangle + \left \langle f_c U_2\right \rangle + \left \langle \frac {1}{\rho }\frac {\partial \tau _{1j}}{\partial x_{\!j}}\right \rangle + \left \langle \frac {1}{\rho } f_1\right \rangle \!, \end{align}

where the horizontal-average operator $\langle \boldsymbol{\cdot }\rangle$ is over the CV surface area, $S_{{cv}}$ , only. No body force fields are available in the analysed LES dataset, but the body force term in (4.20) can nevertheless still be obtained, because the total farm thrust, $T$ , is known, and because the thrust on each actuator disk is distributed uniformly; hence, similar to the actuator wind farm (AWF) model (van der Laan et al. Reference van der Laan, Baungaard, Meyer Forsting and Réthoré2024), we find that the force is distributed in the vertical direction as

(4.21) \begin{equation} q_{{ver}}(z) = \begin{cases} \frac {4}{\pi D} \sqrt {1 - ([z-z_H]/R)^2} &\quad \textrm {if} \quad |(z-z_H)/R| \leqslant 1, \\[7pt] 0 &\quad \textrm {if} \quad |(z - z_H)/R| \gt 1, \\ \end{cases} \end{equation}

and thus, we have

(4.22) \begin{align} \left \langle \frac {1}{\rho }f_1\right \rangle = q_{{ver}}\frac {T}{S_{{cv}} \rho } . \end{align}

Figure 10 shows the budgets for the three cases and again the sum of the terms is close to zero at all heights, thus signifying quasi-stationarity of the wind farm flows as well. In summary, we conclude that the transient evolution of the streamwise momentum is small in the present LES and that the contribution of unsteadiness to the momentum availability can be neglected for all three cases.

Figure 10.

Momentum budget for wind farm simulations. The grey area marks the CV region.

4.7. BL height

Kirby et al. (Reference Kirby, Dunstan and Nishino2023)’s submodel of the momentum availability contribution from turbulence, to be discussed in § 4.8, depends on the height of the undisturbed ABL $h_0 \equiv \delta (x=-\infty )$ and the farm BL $h \equiv L^{-1} \int _{x_{wf,\textit{start}}}^{x_{wf,\textit{start}}+L} \delta (x) \,\text{d}x$ . There are several ways to define the BL height, $\delta (x)$ , and in this section we define it as the height at which the shear stress, $\tau _{13}$ , is equal to 5 % of its ABL surface value $\tau _{w0}$ (e.g. Kosović & Curry Reference Kosović and Curry2000). Figure 11 shows the spanwise-averaged streamwise shear stress from the three cases and highlights the contour line corresponding to the BL height. A large shear stress region develops in and above the wind farm in all cases due to the large velocity gradient, $\partial U / \partial z$ , induced by the turbines, and this region eventually reaches the capping inversion layer and pushes it upwards until the buoyant destruction from the stable stratification of the free atmosphere is sufficiently large to suppress the turbulence. It is this complex balance that dictates the spatial evolution of $\delta (x)$ .

Figure 11.

Contours of the spanwise-averaged (over the CV width $y=[y_{\textit{centre}}-W/2,y_{\textit{centre}}+W/2]$ ) streamwise shear stress. The line styles used denote the BL height $\delta (x)$ from LES () and the model (). The farm region is highlighted with a grey box and the model error, $(h^{\textit{model}} - h^{\textit{LES}})/h^{\textit{LES}}$ , is shown.

Kirby et al. (Reference Kirby, Dunstan and Nishino2023) proposed a simple model for the spatial development of the BL height from a quasi-1D continuity analysis

(4.23) \begin{equation} \delta (x) = \frac {h_0}{\beta _{\textit{local}}(x)}, \end{equation}

which is shown with dashed lines in figure 11. It can be seen that the model correctly captures an increase of the BL height in the farm region, but that it is less accurate for taller ABLs. In the H1000 case, the BL height first starts to increase after approximately a half farm length, which is in line with the fact that the shear stress region develops at an angle (not directly upwards from the wind farm entrance) and that, for taller ABLs, it thus takes a longer streamwise distance to reach the ABL height. The inability of the model to capture this is due to its quasi-1D assumption and that the model is derived from mass continuity, which implicitly implies that the BL height is directly related to the velocity field, i.e. that $\delta (x)$ follows the ‘separating’ streamline between the ABL and the free atmosphere, rather than the shear stress field.

To obtain a model for the farm BL height, $h$ , Kirby et al. (Reference Kirby, Dunstan and Nishino2023) used (4.23) in the definition of $h$ described in the beginning of the current subsection to obtain

(4.24) \begin{equation} h = \frac {1}{L} \int _{x_{wf,\textit{start}}}^{x_{wf,\textit{start}}+L} \delta (x) \text{d}x = \frac {h_0}{L} \int _{x_{wf,\textit{start}}}^{x_{wf,\textit{start}}+L} \frac {1}{\beta _{\textit{local}}} \text{d}x \approx \frac {h_0}{\beta }, \end{equation}

and assumed that the last integral is equal to $L/\beta$ , which thereby gives their final BL height displacement model

(4.25) \begin{equation} \frac {h_0}{h} = \beta . \end{equation}

The ‘integral assumption’, i.e. the last equality in (4.24), was confirmed by Kirby et al. (Reference Kirby, Dunstan and Nishino2023) with four LES cases from Wu & Porté-Agel (Reference Wu and Porté-Agel2017) to be very good (within 0.5 % of the actual value) and we find the same for the cases studied in this paper. This means that the error of $h^{\textit{model}}$ compared with $h^{\textit{LES}}$ , which, as shown in figure 11, is of the order of 7–17 %, is almost entirely from (4.23). This overprediction of $h$ has a significant influence on the model prediction of $\Delta M_{\textit{turb}}$ , which we will return to in the next section.

4.8. Turbulence

The last contributing term to the momentum availability factor is from turbulence, $\Delta M_{\textit{turb}}$ , which entrains mean momentum into the wind farm through enhanced mixing. It is defined in (2.8) and it should be noted that it does not include the wall shear stress. Writing out its terms gives

(4.26) \begin{align} \Delta M_{\textit{turb}} &= \frac { X_{\textit{turb}} - X_{\textit{turb}0}}{X_{\!F0}} = \frac { \int _{\varOmega _{{cv}}\setminus \varOmega _w} \tau _{1j} \text{d}A_{\!j} - \int _{\varOmega _{\textit{top}}} \tau _{1j,0} \text{d}A_{\!j} }{X_{\!F0}}, \end{align}

where only the top surface contribution from the precursor case is present, because the front/rear and south/north terms cancel in the precursor due to horizontal homogeneity. Equation (4.26) is exact, but to obtain a practical model, Kirby et al. (Reference Kirby, Dunstan and Nishino2023) proceeded without the front/rear and south/north contributions from the farm case. As shown in figure 12, this is a good assumption, except for the rear contribution, which is negative (it entrains momentum out of the CV) with a magnitude of around 10 % of the top surface entrainment (which is positive and entrains momentum into the CV). If one nevertheless neglects the rear contribution, the equation simplifies to

(4.27) \begin{equation} \Delta M_{\textit{turb}} = \frac {\tau _t - \tau _{t0} }{\tau _{w0}}, \end{equation}

where we have dropped the subscript ‘13’ and denote the surface-averaged shear stress on the CV top surface as $\tau _{t} \equiv S_{{cv}}^{-1} \int _{\varOmega _{\textit{top}}} \tau _{13} \text{d}A$ and used the fact that $X_{\!F0} = \tau _{w0} S_{{cv}}$ .

Figure 12.

Contributions to $\Delta M_{\textit{turb}}$ evaluated from LES data at the various surfaces of the CV.

To model $\tau _t$ , Kirby et al. (Reference Kirby, Dunstan and Nishino2023) assumed that the streamwise shear stress profiles of the ABL and farm simulations, see figure 13(a), are self-similar above the top-tip height, i.e.

(4.28) \begin{equation} \frac {\tau _0\left (z/h_0\right )}{\tau _{w0}} = \frac {\tau (z/h)}{\tau _{w1}} \quad (\mathrm{for}\,z \gt z_{\textit{hub}} + D/2), \end{equation}

where a second (fictive) wall shear stress is defined as $\tau _{w1} \equiv M \tau _{w0}$ . One can interpret $\tau _{w1}$ as the wall shear stress one would obtain by extrapolating the $\tau (z)$ profile above the top-tip height of the turbines to the ground. Figure 13(b) shows that all profiles do not exactly collapse to a single curve, but that each pair of ABL/farm simulations does, which is the important part because this justifies the use of (4.28).

Figure 13.

Streamwise shear stress profiles of the precursor () and farm region of the wind farm simulation () (a) and test of their self-similarity (b). The fictive wall shear stress $\tau _{w1} \equiv M \tau _{w0}$ is directly obtained through LES data of $M$ and $\tau _{w0}$ . The ABL and farm heights are defined with the 5 % (of $\tau _{w0}$ and $\tau _{w1}$ , respectively) rule.

Equation (4.28) can be rearranged to

(4.29) \begin{align} \tau (z) = \tau _{w1}\frac {\tau _0\left (z \frac {h_0}{h}\right )}{\tau _{w0}}, \end{align}

which is used to obtain the top shear stress

(4.30) \begin{align} \tau _t = \tau _{w1}\frac {\tau _0\left (H_{\!F} \frac {h_0}{h}\right )}{\tau _{w0}} . \end{align}

To use this expression one needs $\tau _0 (H_{\!F} {h_0}/{h} )$ , which Kirby et al. (Reference Kirby, Dunstan and Nishino2023) obtained by assuming a linear profile of $\tau _0(z)$ in the interval $z= [0,H_{\!F} ]$ , i.e.

(4.31) \begin{align} \tau _0(z) = \tau _{w0} + \frac {\tau _{t0} - \tau _{w0}}{H_{\!F}} z \quad \quad (\mathrm{for}\, 0 \lt z \lt H_{\!F}), \end{align}

leading to $\tau _0 (H_{\!F} {h_0}/{h} ) = \tau _{w0} + (\tau _{t0} - \tau _{w0}) ({h_0}/{h})$ . Inserting this expression into (4.30) gives

(4.32) \begin{align} \tau _t &= M \left (\tau _{w0} + (\tau _{t0} - \tau _{w0}) \frac {h_0}{h} \right ) \!, \end{align}

which combined with (4.27) results in Kirby et al. (Reference Kirby, Dunstan and Nishino2023)’s submodel for the turbulence contribution to the momentum availability:

(4.33) \begin{align} \Delta M_{\textit{turb}} = M + M\left (\frac {\tau _{t0}}{\tau _{w0}} - 1\right ) \frac {h_0}{h} - \frac { \tau _{t0}}{\tau _{w0}} . \end{align}

Unlike the other submodels, this expression is implicit ( $\Delta M_{\textit{turb}}$ is inside $M$ on the right-hand side), but, due to its simple form, inserting (4.33) into (2.8) with the other submodels nevertheless leads to an explicit relation for $M$ :

(4.34) \begin{equation} M = \frac {1 + \frac {H_{\!F}}{\textit{LC}_{\!f0}} \left ( 1 - \beta ^2 \right ) - \frac { \tau _{t0}}{\tau _{w0}}}{ \left (1 -\frac {\tau _{t0}}{\tau _{w0}}\right ) \frac {h_0}{h}} . \end{equation}

This expression is equal to the first version of the KDN model, $ M_{\textit{KDN1}}$ (2.11), when the BL height model $h_0/h = \beta$ (4.25) is used.

Figure 14.

(a) The BL height ratio $h_0/h$ from LES ( $h$ based on 5 % of $\tau _{w1}$ , c.f. figure 13) and the model of $h_0/h = \beta$ (4.25). The LES value of $h/h_0$ with $h$ based on 5 % of $\tau _{w0}$ , c.f. figure 11, is also displayed, although it is not the relevant height ratio for the $\Delta M_{\textit{turb}}$ model. (b) Model of turbulence contribution to the momentum availability and its error.

To test the $\Delta M_{\textit{turb}}$ submodel, as shown in figure 14(b), one first evaluates $M$ from (4.34) and subsequently inserts this into (4.33). Hence, the turbulence contribution indirectly depends on the other submodels and the BL height ratio $h_0/h$ through (4.34). It should be emphasised that the dependence on the farm BL height $h$ originates from the self-similarity assumption (4.28), which as shown in figure 13(b) is based on the height where the streamwise shear stress is equal to 5 % of $\tau _{w1}$ . It thus differs from the farm BL height displayed in figure 11, which was based on 5 % of $\tau _{w0}$ , and this explains why, counter-intuitively, $h_0/h$ can exceed one, e.g. the H1000 case in figure 14(a). As discussed in the previous section, the BL height ratio model (4.25) overpredicts $h/h_0$ (based on $\tau _{w0}$ ) by 7–17 % corresponding to an underprediction of $h_0/h$ of about the same amount. However, when comparing the model with $h_0/h$ based on $\tau _{w1}$ , the underprediction is larger, 19–36 %, which is the main reason for the large overprediction of $\Delta M_{\textit{turb}}$ shown in figure 14(b). Other sources of errors are from the no rear-entrainment, linear shear stress profile across the CV height and self-similarity assumptions, but these are found to be secondary. We find that $\Delta M_{\textit{turb}}$ is also overpredicted for shallow ABLs (see Appendix B) and the ‘half-farm’ case, which underlines that this term is the most complex term of (2.8) to model.

4.9. Summary of the momentum availability contributions

The findings from the previous sections are summarised in figure 15. Each bar has the same total value as the corresponding one from figure 3(a–c), but the current figure reveals the split between the various momentum contributions to $M$ . From this, we conclude that $\Delta M_{{adv}}$ , $\Delta M_{\textit{PGF}}$ and $\Delta M_{\textit{turb}}$ are approximately equally important and that none of these can be neglected. For the $M_{\textit{KDN1}}$ model, the advection and PGF contributions are grouped together due to the AP approximation (4.11). The model correctly captures the fact that the turbulence contribution increases with increasing ABL height (this is because the shear stress at the top of the CV increases), although the magnitude is overpredicted as discussed in the previous subsection. Nevertheless, the turbulence overprediction is consistently compensated by the underprediction of the AP approximation, hence, the final model prediction of $M_{\textit{KDN1}}$ is within $\pm 1$ % of $M_{\textit{exact}}$ , which is excellent considering the complexity of the momentum availability. Despite the good total performance of $M_{\textit{KDN1}}$ , its practical applicability is limited, since it is nonlinear in $\beta$ and a priori requires a value of $\tau _{t0}/\tau _{w0}$ , which is the motivation for the $M_{\textit{KDN}2}$ and $M_{\textit{KDN}3}$ models.

Figure 15.

Split of the momentum availability for the LES data (full) and the $M_{\textit{KDN1}}$ model (crossed).

4.10. The $M_{\textit{KDN}2}$ and $M_{\textit{KDN}3}$ models

The $M_{\textit{KDN}2}$ and $M_{\textit{KDN}3}$ models are derived from the $M_{\textit{KDN1}}$ model as shown in the overview in figure 1. The $M_{\textit{KDN}2}$ model is first derived by assuming a linear shear stress profile, which is then further simplified to $M_{\textit{KDN}3}$ by two linearisations for the relationship between $M$ and $\beta$ , but the order in which these simplifications are applied is arbitrary and results in the same linear model, $M_{\textit{KDN}3}$ . We have selected the current order of simplifications, because the nonlinear $M_{\textit{KDN}2}$ model is a practical model on its own, as, for example, used by Nishino & Smyth (Reference Nishino and Smyth2026).

The assumption required to simplify $M_{\textit{KDN1}}$ (2.11) to $M_{\textit{KDN}2}$ (2.12) is that $\tau _0(z)$ is linear from $z=0$ to $z=h_0$ , which gives

(4.35) \begin{equation} 1 -\frac {\tau _{t0}}{\tau _{w0}} \approx \frac { H_{\!F}}{h_0}. \end{equation}

This assumption is more restrictive compared with (4.31), where $\tau _0(z)$ was only assumed to be linear up to $z=H_{\!F}$ . Also, $h_0$ in this context refers not to the 5 % height but to the full ABL height, i.e. where the streamwise shear stress goes to zero. In the ABL literature (e.g. Berg, Patton & Sullivan Reference Berg, Patton and Sullivan2020; Liu et al. Reference Liu, Gadde and Stevens2021b ), $h_0$ is usually defined from the total shear stress $|\tau |_0 = \sqrt {\tau _{13,0}^2 + \tau _{23,0}^2}$ , which is preferred due to its frame invariance and because it better represents the actual ABL height, and we therefore choose to use this height as $h_0$ (obtained from LES by fitting $|\tau |_0/|\tau |_{w0} = (1 - z/h_0)^{p_0}$ ; see table 1) in the evaluation of the $M_{\textit{KDN}2}$ model and in the rest of the paper. As already presented in figure 3, the LES data show that the $M_{\textit{KDN}2}$ model is significantly less accurate than the $M_{\textit{KDN1}}$ model, especially for taller ABLs, and we propose a model extension to amend this in § 5.

The $M_{\textit{KDN}2}$ model (2.12) can be rearranged to

(4.36) \begin{align} M_{\textit{KDN}2} &= \frac {1}{1 - (1 - \beta )} + \frac { \frac {h_0}{\textit{LC}_{\!f0}} \left ( 2\left ( 1 - \beta \right ) - (1 - \beta )^2 \right ) }{1 - (1 - \beta )}, \end{align}

and simplified to $M_{\textit{KDN}3}$ (2.13) using the linearisations suggested by Kirby et al. (Reference Kirby, Dunstan and Nishino2023):

(4.37) \begin{equation} \frac {1}{1 - (1 - \beta )} \approx 1 + 1.18 (1 - \beta ), \quad \textrm {linearisation (i)}, \end{equation}

(4.38) \begin{equation} \frac { 2\left ( 1 - \beta \right ) - (1 - \beta )^2 }{1 - (1 - \beta )} \approx 2.18 (1 - \beta ), \quad \textrm {linearisation (ii)}. \end{equation}

Figure 16 shows the results of applying these two linearisations. Linearisation (i) is applied to the first term in (4.36), which is approximately one order of magnitude smaller than the second term, hence, this linearisation has little impact on the solution. Adding the second linearisation has a larger effect, which, actually, improves the prediction of $M$ slightly, making it closer to $M_{\textit{KDN1}}$ . Overall, both linearisations, however, have a relatively small effect, compared with the linear shear stress assumption (4.35) used for simplifying $M_{\textit{KDN1}}$ to $M_{\textit{KDN}2}$ , and therefore, we focus on modifying this assumption in the next section.

Figure 16.

Test of the two linearisations (4.37–4.38) used to simplify $M_{\textit{KDN}2}$ to $M_{\textit{KDN}3}$ . For reference, $M_{\textit{KDN1}}$ is also included.