Two-Scale Momentum Theory

• Two-Scale Momentum Theory is a momentum-conservation framework that separates wind farm aerodynamics into turbine-scale and atmospheric-scale interactions.
• It employs a nondimensional parameter (λ/Cf0) to balance turbine loading and wake effects with atmospheric drag, highlighting trade-offs between efficiency and power density.
• The theory has been extended to finite-size and time-dependent scenarios, facilitating coupled CFD/LES simulations for advanced wind farm optimization.

Searching arXiv for papers on two-scale momentum theory, especially wind-farm literature and closely related formulations. In the wind-energy literature, Two-Scale Momentum Theory denotes a momentum-conservation framework for large wind farms that separates the aerodynamics into two coupled levels: an internal turbine/array scale, where turbine drag, wake interaction, and surface friction are determined, and an external farm/atmospheric scale, where the atmospheric boundary layer determines how much momentum is available to the farm as a whole. The theory originated as a model for ideal very large wind farms, then expanded to finite-size farms, time-dependent atmospheric forcing, and explicit momentum-availability closures. Its central role is to replace a fixed-upstream-speed view of farm performance with a coupled problem in which turbine loading, farm-scale wind-speed reduction, and atmospheric response must be solved consistently (Nishino, 2016, Nishino et al., 2019).

1. Origins in the analysis of very large wind farms

The original formulation was proposed for very large, horizontally periodic wind farms embedded in a fully developed atmospheric boundary layer driven by a constant streamwise pressure gradient, with Coriolis effects neglected (Nishino, 2016). Its main physical claim was that the performance of an ideal very large wind farm is not governed solely by isolated-turbine actuator-disc theory. Instead, the farm interacts with the atmospheric boundary layer through a farm-scale momentum balance, while each turbine still obeys an actuator-disc-like local thrust relation.

In that formulation, the central control parameter is the nondimensional ratio

$$\frac{\lambda}{C_{f0}}, \qquad \lambda=\frac{A}{S},$$

where $A$ is rotor swept area per turbine, $S$ is the horizontal area allocated per turbine, and $C_{f0}$ is the natural friction coefficient of the undisturbed atmospheric boundary layer before farm construction (Nishino, 2016). This ratio measures the effective turbine loading of the site. The theory predicts that when $\lambda/C_{f0}\to 0$, the farm behaves like a collection of isolated turbines and the classical Betz limit is recovered, with

$$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$

The dense-farm limit is qualitatively different. As $\lambda/C_{f0}$ increases, the maximum power coefficient of each turbine decreases, while the normalized farm power density increases asymptotically to an upper limit (Nishino, 2016). For $\gamma=2$, the asymptotic values reported are

$$n_{\max}\to 0.3849, \qquad \beta_{\mathrm{opt}}\to 0.5774, \qquad a_{\mathrm{opt}}\to 1.$$

This established the basic design trade-off that remains central to the later literature: dense farms sacrifice per-turbine efficiency in exchange for higher power per unit site area.

2. Core variables, coupling equation, and power relations

The theory uses a small set of bulk variables to couple the two scales. The farm-scale velocity reduction is written either as

$B=\frac{U_F}{U_{F0}}$

or

$A$0

depending on the paper, where $A$1 is the farm-layer-averaged velocity after construction and $A$2 is its no-farm counterpart (Nishino, 2018, Kirby et al., 2022). At turbine scale, the local velocity reduction is written as

$A$3

with $A$4 the rotor-averaged velocity (Nishino, 2018).

In the time-dependent generalized formulation, the central coupling condition is written as

$A$5

where $A$6 is the local/internal thrust coefficient, $A$7 is array density, $A$8 parameterizes how bottom friction changes with farm-scale velocity reduction, and $A$9 is the momentum availability factor supplied by the external atmospheric problem (Nishino et al., 2019). In the non-dimensional farm momentum formulation used later, the equivalent expression is

$S$0

with $S$1 the farm-averaged internal thrust coefficient and $S$2 the precursor friction coefficient (Kirby et al., 2022, Baungaard et al., 2 Feb 2026).

The power relations follow the same two-scale logic. In the generalized finite-farm model, the turbine power coefficient is

$S$3

so power depends jointly on local turbine loading and farm-scale flow reduction (Nishino, 2018). In the offshore finite/infinite-farm analysis, the same dependence is written as

$S$4

and for ideal actuator discs,

$S$5

The literature therefore treats $S$6 or $S$7 as the single coupling variable linking the atmospheric momentum supply to turbine-scale extraction (Nishino et al., 2019, Kirby et al., 2022).

A related modelling choice is the definition of the nominal farm layer. In the finite/infinite offshore LES study, the farm-layer height is taken as

$S$8

with the note that $S$9 may vary between about $C_{f0}$0 and $C_{f0}$1 as long as the same value is used consistently in both internal and external problems (Kirby et al., 2022).

3. Finite-size and time-dependent generalizations

The first major generalization addressed a limitation of the original idealized theory: the assumption that the pressure gradient driving the flow over the farm is unchanged by farm construction. For real large but finite-size wind farms, the farm itself can modify the pressure field between its upstream and downstream sides. The generalized model therefore replaces the original fixed-forcing assumption with the farm-scale balance

$C_{f0}$2

where $C_{f0}$3 is the horizontally averaged surface shear stress after farm construction, $C_{f0}$4 is turbine thrust, $C_{f0}$5 is surface area per turbine, and $C_{f0}$6 is the pressure drop after construction (Nishino, 2018).

The additional pressure response is represented empirically as

$C_{f0}$7

where $C_{f0}$8 is an environment-dependent parameter (Nishino, 2018). The paper states that $C_{f0}$9 depends on farm size relative to atmospheric-system size, terrain or offshore environment, and atmospheric conditions and stability. This extension makes the theory applicable to wind farms that are large but not as large as the atmospheric system driving the flow.

A second generalization recast the framework as a time-dependent external problem coupled to a quasi-steady internal problem (Nishino et al., 2019). The external problem is an atmospheric-boundary-layer-scale problem that accounts for pressure gradients, acceleration, Coriolis effects, and farm-scale atmospheric interaction; the internal problem partitions the total bottom resistance into turbine drag and bottom shear stress. The paper emphasizes that this formulation is model-agnostic: an external numerical weather prediction model can be coupled to an internal CFD model through the same momentum equation. It also explicitly frames wind-farm blockage as a time-dependent atmospheric response problem rather than a purely statistical correction.

4. Momentum availability and atmospheric closure

In the more recent formulation, farm-scale atmospheric physics are condensed into the momentum availability factor $\lambda/C_{f0}\to 0$0, defined from a control-volume streamwise momentum budget (Baungaard et al., 2 Feb 2026). The volume-averaged balance is written as

$\lambda/C_{f0}\to 0$1

where the right-hand side contains momentum supplied by advection, pressure-gradient forcing, Coriolis forcing, turbulent transport, and unsteadiness. Comparing the same control volume with and without turbines gives

$\lambda/C_{f0}\to 0$2

and

$\lambda/C_{f0}\to 0$3

This closure program is associated with the analytic model of Kirby, Dunstan, and Nishino, validated and extended using LES data drawn from the database of Lanzilao et al. (Baungaard et al., 2 Feb 2026). The paper compares a full analytic model $\lambda/C_{f0}\to 0$4, a simplified model $\lambda/C_{f0}\to 0$5, a linearized form

$\lambda/C_{f0}\to 0$6

and an older empirical linear form

$\lambda/C_{f0}\to 0$7

Six LES cases were analyzed: three with precursor atmospheric boundary-layer heights $\lambda/C_{f0}\to 0$8, and three with different turbine layouts. The standard farm was a fixed 1.6 GW configuration with 160 turbines; the half-farm and double-spacing cases used 80 and 40 turbines, respectively. All cases used $\lambda/C_{f0}\to 0$9, $$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$0, $$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$1, actuator-disk turbines with $$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$2, $$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$3, and $$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$4 (Baungaard et al., 2 Feb 2026).

The principal validation result is that the full model $$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$5 matches the exact LES-based value $$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$6 very well, typically within about $$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$7 and often better, whereas the simpler $$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$8 and $$C_{P,\max} \to \frac{16}{27}, \qquad a_{\mathrm{opt}}\to \frac13.$$9 increasingly overpredict momentum availability as ABL height increases (Baungaard et al., 2 Feb 2026). The paper attributes this to the ABL Rossby number,

$\lambda/C_{f0}$0

and proposes the Rossby-corrected linear extension

$\lambda/C_{f0}$1

A key physical conclusion is that advection, pressure-gradient forcing, and turbulent entrainment are all of similar importance, while Coriolis and unsteadiness are small for the considered cases. A common misconception is therefore to treat $\lambda/C_{f0}$2 as a single-process correction; the LES budget indicates that the successful closure balances several comparably important mechanisms.

5. Wake effects, blockage, and simulation-based validation

Two-Scale Momentum Theory was explicitly developed to bridge the gap between conventional wake models and farm-scale atmospheric models. The offshore LES study of 50 infinitely large wind farms states that conventional wake models capture the internal scale but not the external scale, whereas top-down models capture the external scale but not the internal scale; the two-scale framework couples them through $\lambda/C_{f0}$3, $\lambda/C_{f0}$4, and $\lambda/C_{f0}$5 (Kirby et al., 2022).

That study used periodic LES of 50 infinitely large offshore farms with actuator discs, spanning $\lambda/C_{f0}$6 and $\lambda/C_{f0}$7, chosen by a maximin design in $\lambda/C_{f0}$8-space (Kirby et al., 2022). For the offshore cases, the parameters were $\lambda/C_{f0}$9, $\gamma=2$0, $\gamma=2$1, and $\gamma=2$2. The reported outcome is that the analytical model predicts the impact of array density very well for all 50 farms and can therefore be used as an approximate upper limit to farm performance.

The same paper also proposes a decomposition of losses into turbine-scale and farm-scale components: $\gamma=2$3 with total loss

$\gamma=2$4

For the large offshore wind farms studied, farm-scale losses are typically more than twice as large as turbine-scale losses, with

$\gamma=2$5

(Kirby et al., 2022). The same study further states that wake and blockage effects are not independent: wake-induced changes in $\gamma=2$6 alter total farm drag, which changes the atmospheric response and therefore the equilibrium farm-layer wind speed.

The finite-size generalization was validated separately with 3D incompressible RANS simulations of a $\gamma=2$7 staggered array of 625 actuator discs, with spacing $\gamma=2$8 in both streamwise and spanwise directions and disc diameter $\gamma=2$9, representing a farm of about $$n_{\max}\to 0.3849, \qquad \beta_{\mathrm{opt}}\to 0.5774, \qquad a_{\mathrm{opt}}\to 1.$$0 (Nishino, 2018). Two roughness lengths were used, $$n_{\max}\to 0.3849, \qquad \beta_{\mathrm{opt}}\to 0.5774, \qquad a_{\mathrm{opt}}\to 1.$$1 and $$n_{\max}\to 0.3849, \qquad \beta_{\mathrm{opt}}\to 0.5774, \qquad a_{\mathrm{opt}}\to 1.$$2, across eight operating conditions defined by four local-drag settings $$n_{\max}\to 0.3849, \qquad \beta_{\mathrm{opt}}\to 0.5774, \qquad a_{\mathrm{opt}}\to 1.$$3 in each roughness environment. The generalized theory matched the CFD results well for both onshore/high-roughness and offshore/low-roughness cases; example fitted values were $$n_{\max}\to 0.3849, \qquad \beta_{\mathrm{opt}}\to 0.5774, \qquad a_{\mathrm{opt}}\to 1.$$4 for onshore and $$n_{\max}\to 0.3849, \qquad \beta_{\mathrm{opt}}\to 0.5774, \qquad a_{\mathrm{opt}}\to 1.$$5 for offshore (Nishino, 2018). Row-by-row variation was captured qualitatively, with higher power in the first rows and lower power downstream, while the farm-averaged power remained well described by the theory.

6. Applications, scope, and limitations

The theory is designed as a coupling architecture rather than a single closure. In the time-dependent formulation, the external problem may be supplied by a numerical weather prediction model and the internal problem by CFD, LES, RANS, or an engineering wake model, with the farm wind-speed reduction factor serving as the coupling variable (Nishino et al., 2019). In the finite-size pressure-response formulation, the theory is explicitly proposed as a component that could be embedded in a regional-scale atmospheric model to predict average farm power effectively (Nishino, 2018).

A second application is coupled turbine/farm optimization. The finite-size generalization states that the framework may be combined with blade element momentum theory for coupled wind turbine/farm optimisation, so that blade-level rotor design and farm-scale pressure-feedback effects are handled within one model chain (Nishino, 2018). This suggests a design methodology in which rotor loading is not optimized against a fixed free stream, but against a farm-mediated atmospheric state.

The scope conditions are also clear. The rigorous derivations assume identical turbines, regular layout, flat terrain or sea, fully developed flow away from edges, horizontally uniform undisturbed wind across the farm, and a farm layer much thinner than the full atmospheric boundary layer (Nishino et al., 2019). The same paper explains that approximate extension to real farms is possible by defining a farm-average $$n_{\max}\to 0.3849, \qquad \beta_{\mathrm{opt}}\to 0.5774, \qquad a_{\mathrm{opt}}\to 1.$$6, using a larger control volume enclosing the whole farm, and replacing local quantities by farm averages, but at the cost of a more complicated external momentum-availability factor.

Several limitations follow directly from the validation literature. The theory does not prescribe wind direction, vertical profiles, or turbulence structure except through the chosen external and internal models (Nishino et al., 2019). The simple linear response form for $$n_{\max}\to 0.3849, \qquad \beta_{\mathrm{opt}}\to 0.5774, \qquad a_{\mathrm{opt}}\to 1.$$7 is useful, but later LES evidence shows that linearized closures increasingly overpredict momentum availability for increasing ABL heights unless rotational effects are included through a Rossby-sensitive correction (Baungaard et al., 2 Feb 2026). Likewise, wake mitigation alone cannot be interpreted as a sufficient design principle for large farms, because the offshore LES results indicate that farm-scale slowdown is the larger contributor to power loss under the studied conditions (Kirby et al., 2022).

Taken together, these developments define Two-Scale Momentum Theory as a general wind-farm framework in which atmospheric momentum supply, farm-scale wind-speed reduction, turbine loading, and surface friction are solved as a coupled system. Its distinctive contribution is to make farm performance a problem of multiscale momentum partition rather than a superposition of isolated wakes.