# Wind Turbine Wakes Under Different Atmospheric Conditions: Bridging Internal Dynamics and Meandering with Taylor's Diffusion Theory

Emeline Noël, Erwan Jézéquel

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Emeline Noël, Erwan Jézéquel. Wind Turbine Wakes Under Different Atmospheric Conditions: Bridging Internal Dynamics and Meandering with Taylor's Diffusion Theory. Journal of Physics: Conference Series, 2025, 3016, pp.012044. 10.1088/1742-6596/3016/1/012044. hal-05154769

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# Wind Turbine Wakes Under Different Atmospheric Conditions: Bridging Internal Dynamics and Meandering with Taylor's Diffusion Theory

To cite this article: Emeline Noël and Erwan Jézéquel 2025 *J. Phys.: Conf. Ser.* **3016** 012044

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# Wind Turbine Wakes Under Different Atmospheric Conditions: Bridging Internal Dynamics and Meandering with Taylor's Diffusion Theory

Emeline Noël and Erwan Jézéquel

¹IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France

E-mail: emeline.noel@ifpen.fr

**Abstract.** Atmospheric conditions influence wind turbine wake behavior, which in turn affects power production and structural loads. Two major aspects of this impact are wake meandering and wake expansion. Taylor's diffusion theory, originally developed for particle dispersion, has been used to describe the dispersion of the wake structure center, although it is not a particle. The present study proposes a framework that bridges Taylor's diffusion theory from wake particles to the wake structure center by describing the wake structure as a surface-averaged entity of particle positions. Thus, the applicability of Taylor's diffusion theory to describe wake diffusion processes, such as wake meandering and wake expansion, is established. Additionally, an extended version of the theory is proposed to more accurately represent wake expansion. Large Eddy Simulations (LES) are conducted using the Meso-NH solver for three atmospheric stability conditions: stable, neutral, and unstable. To validate theoretical predictions, two distinct wake tracking methodologies are employed: Eulerian and Lagrangian. The results demonstrate that wake meandering is driven by the Eulerian turbulent time scale of the upstream flow, while wake expansion is governed by the Lagrangian turbulent time scale, along with internal dynamics described by mixing layer theory. The significance of this study lies in its conceptualisation of wake diffusion processes as a unifying theoretical framework, emphasising the necessity of understanding the turbulent structure of the upstream flows to enhance the predictive capabilities of wind turbine wakes across a variety of atmospheric conditions.

## 1 Introduction

Atmospheric conditions play a critical role in the performance of wind farms, affecting both power production [1, 2] and structural loads on wind turbines [3]. These effects are primarily due to changes in the incoming airflow [4], which alter the wake behavior of wind turbines in two key aspects: wake meandering [5, 6] and wake expansion [7]. Specifically, wake meandering refers to the lateral and vertical movement of the wake downstream of the turbine, while wake expansion describes the gradual widening of the wake as it moves further from the turbine. These wake characteristics directly influence the rate of wake recovery [8], and therefore the efficiency of downstream turbines and overall wind farm productivity.

The pioneering work of Larsen et al. [5] has significantly influenced the conceptualization of wind turbine wake meandering. In their study, wake meandering is viewed as the result of large-scale atmospheric turbulence influencing the trajectory of the wake, treated as an object or structure. This led to the development of the Dynamical Wake Meandering model, which provides a semi-steady representation of the wake. In contrast, steady-state representations of the wake have been developed through analytical models, grounded in the assumption that the statistical mean wake velocity deficit exhibits self-similarity [9]. These models typically consist of a shape function and a scaling factor [10, 11]. The shape function depends on a characteristic wake width, which

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evolves with the downstream turbine coordinate, reflecting the wake's expansion. However, this parameter is often calibrated through simulations, and deriving a physical model for it remains challenging. Some authors [12, 13, 14] have derived a physical relationship for wake expansion with downstream distance based on Taylor's diffusion theory [15], which is closely related to plume theory. Ali et al. [16] build on this approach by deriving an analytical solution for wake expansion based on a disk source formulation, offering a closed-form representation of the velocity deficit evolution. Alternatively, Braunbehrens and Segalini [17], in their statistical wake meandering model, determined the probability density function of the wake structure center using Taylor's diffusion theory, which is related to wake meandering. This raises a fundamental question: does Taylor's diffusion theory primarily describes wake expansion, wake meandering, or both? While these phenomena are linked to turbulent dynamics, it remains uncertain whether they arise from the same underlying process or distinct turbulent mechanisms.

Understanding the dynamics of wake meandering and expansion is not only fundamental for optimizing turbine performance but also for the strategic placement of turbines within a wind farm to minimize wake losses and improve overall farm productivity. Optimizing these processes can help maximize the energy output of a wind farm while minimizing the adverse effects of wake losses on downstream turbines

The objective of this study is to investigate the turbulent mechanisms governing wake meandering and wake expansion, assessing whether they arise from the same diffusion process or distinct physical phenomena. To this end, Large Eddy Simulations (LES) of the IEA 15 MW wind turbine wake have been conducted under three atmospheric stability conditions: stable, neutral, and unstable. Employing both Eulerian and Lagrangian wake tracking methods enables analysis of the wake's evolution from different perspectives. The Eulerian method provides statistical insights into wake meandering by tracking the displacement of the wake's center, revealing how the wake structure shifts laterally and vertically over time. In contrast, the second method follows individual fluid parcels, capturing their trajectories and providing insights into wake expansion. This dual approach enables a more comprehensive understanding of both wake meandering and wake expansion, linking these phenomena to the underlying turbulent mechanisms across varying atmospheric conditions.

The manuscript is structured as follows: Section 2 details the LES solver, numerical configurations, and the two distinct wake tracking methodologies: Eulerian and Lagrangian. Subsequently, Section 3 presents a comprehensive analysis of the results obtained from both wake tracking approaches, contextualizing them within the theoretical framework of Taylor diffusion theory and its applicability. The study concludes with Section 4, which synthesizes the key findings and their implications.

## 2 Methods

### 2.1 Meso-NH Solver

LES were performed using the Meso-NH solver, an open-source, non-hydrostatic mesoscale atmospheric model developed through the collaborative efforts of the Laboratoire d'Aérologie and the Centre National de Recherches Météorologiques [18]. Actuator methods have been introduced within this framework to effectively model wind turbines [19]. The solver is based on anelastic approximation equations [20]. The model uses a turbulence parameterization based on a mixing length closure scheme [21]. This is a 1.5-order closure model, which includes a prognostic equation for turbulent kinetic energy. The equations are discretized with a staggered Arakawa C-grid for accurate spatial representation. Time integration uses a fourth-order Runge-Kutta scheme. Momentum advection uses a fourth-order scheme, while scalar advection uses the piecewise parabolic method, which handles sharp gradients well. The actuator line method [22] and the actuator disk with rotation method (ADR) [23] for wind turbine modeling have been validated against the New Mexico wind tunnel experiments, demonstrating their effectiveness in simulating wind turbine aerodynamics and wake dynamics. The model's ability to represent wind turbine interactions with various stratified atmospheric boundary layers has also been confirmed through comparisons with in situ measurements [24].

### 2.2 Large Eddy simulations setup

The simulations employ a grid nesting strategy [25] with three nested meshes. The finest mesh, designated M3, exhibits a horizontal resolution of 4 m and ensures 60 grid cells per rotor diameter. This resolution is deliberately more stringent than the sensitivity thresholds reported by Joulin [19]. The streamwise extent of M3 spans 25 rotor diameters, allowing for adequate wake development beyond 10 rotor diameters downstream and incorporating a scale-up region of 12 rotor diameters. The intermediate mesh, designated M2, has a horizontal resolution of 8 m, which ensures a refinement ratio of 2 relative to M3. This conservative refinement ratio is chosen to mitigate discontinuities in the resolved turbulence spectrum across nested domain interfaces. In order to enhance computational efficiency and facilitate parallel domain decomposition, the width of M2 is set equal to its length. The coarsest mesh, designated as M1, serves as the precursor domain and is coupled to M2 using a one-way nesting approach. The domain M1 encompasses an area twice the size of M2 and has a horizontal resolution that is five times coarser (40 m). This configuration enables M1 to generate realistic atmospheric inflow conditions while
<figure><img src="image_7.png" /><figcaption>Figure 1: Three-domain nested simulation setup with progressively refined spatial resolution (M1 → M2 → M3). The symbol ⇔ represents the two-way coupling between the nested meshes, while the symbol → represents the one-way coupling.</figcaption></figure>

<table><caption>Table 1: Key specifications of the IEA 15 MW</caption><thead><tr><td>Parameter</td><td>IEA 15 MW</td></tr></thead><tbody><tr><td>Rotor diameter, D (m)</td><td>240</td></tr><tr><td>Hub height, H<sub>hub</sub> (m)</td><td>150</td></tr><tr><td>Rated wind speed (m/s)</td><td>10.59</td></tr><tr><td>Maximum rotor speed (rpm)</td><td>7.56</td></tr><tr><td>Minimal rotor speed (rpm)</td><td>5</td></tr></tbody></table>

delegating turbulence downscaling to the finer-resolution meshes, M2 and M3. To ensure effective downscaling between M1 and M2, a buffer zone of 2 km is introduced. This distance is derived from empirical findings, which suggest that each additional resolved scale in the finer mesh requires approximately 10 upstream observations for accurate representation. Given the fivefold increase in resolved scales from M1 to M2, a buffer length equivalent to 50 times the initial M1 grid spacing (40 m) is necessary, leading to the 2 km buffer specification. The same vertical resolution is used for all meshes, starting with a constant resolution of 4 m from the ground to 350 m. Above this, a stretching technique is applied with a 5% ratio, leading to a maximum cell size of 40 m, extending up to the top of the domain at 1.5 km. Figure 1 shows the three embedded domains along with their spatial extent. The simulations are conducted within the idealized framework of Meso-NH, with flow driven by a geostrophic wind forcing and Coriolis forces. Horizontal periodic boundary conditions are applied, while a rigid lid with a Rayleigh damping layer is used at the top boundary starting at 1.3 km. For the bottom boundary, the velocity at the first grid point is prescribed using the Monin-Obukhov similarity theory, and a specified heat flux is applied in the potential temperature equation. The initial velocity field is uniform, set to the value of the prescribed geostrophic wind. The initial potential temperature is initialized as a constant value of 290 K from the ground to 600 m, followed by a temperature inversion of 20 K/km up to 800 m. Above this, the free atmosphere exhibits a stable stratification of 6 K/km. Wind turbine power production is regulated through adjustments to the rotor's rotational speed and blade pitch, in accordance with the implementation of the ROSCO controller [26] within the Meso-NH framework. The wind turbine is modeled using an ADR approach. The ADR model employs 60 radial elements and 200 azimuthal elements to represent the turbine's rotor disk.

Three atmospheric conditions (neutral, stable, unstable) were simulated. The simulation process was initiated with a 20-hour neutral atmospheric boundary layer computation utilizing mesh M1. Convergence of the mean wind velocity at hub height was observed after approximately 15 hours of simulation time. These converged results were then interpolated to meshes M2 and M3 for a 30-minute nested simulation without a wind turbine, followed by a 30-minute nested simulation with a wind turbine. A wind turbine was positioned at (9.5 km, 10 km); key specifications [27] are in Table 1. For stable and unstable cases, surface fluxes of -9W/m² and 90W/m², respectively, were introduced on mesh M1 following the convergence of the neutral simulation. These fluxes were applied over a 5-hour period. For each case, a one-hour interval exhibiting minimal fluctuations (approximately 0.5 m/s for velocity and 0.5° for wind direction) was selected from this initial simulation. This selected time period was then re-simulated with the addition of meshes M2 and M3. The nested simulation ran for 30 minutes without a wind turbine, followed by 30 minutes with a wind turbine present, mirroring the approach used for the neutral case. The simulations share common parameters, including a surface roughness of z₀ = 50 mm, geographical coordinates of 33.3° latitude and -119.5° longitude, and geostrophic wind components (UG, VG) = (11.04, -4.81) m/s. The stability parameter ζ was evaluated at the first grid point above the surface, with values of 0 (neutral), -0.023 (weakly unstable), and 0.014 (weakly stable). For analysis, high-frequency data was sampled every 1.2 seconds on mesh M3 during the final 20 minutes of the simulation.

## 2.3 Pollutant wake tracking

Several methods have been proposed to define the time series of the wake center's coordinates [28], but they require careful, and somewhat heavy post-process in order to avoid outliers that can skew the results [6]. Alternatively, it
is here proposed to do the wake tracking using a passive scalar, similarly to a pollutant. A scalar of concentration $s_*$ is added in the code. Its conservation equation is reduced to an advection term, and a source that is defined to be equal to 1 in the rotor disk area and 0 outside. The concentration of $s_*$ does not appear in any other equation of the system, hence the designation as passive scalar. The position of the wake structure center ($y_c, z_c$) at a time $t$ and position $x$ is then defined as the coordinates of the center of gravity of $s_*$ at $t$ and $x$. This method is compared in Figure 2 to the ConstantFlux method [28] for a wake in an unstable atmospheric boundary layer (where the wake is difficult to track), from the calibration dataset of [29]. The pollutants method shows very good results, with the same overall trajectory (both in the near and far wake regions), but much less outliers in the far wake. Since the ConstantFlux is arguably one of the best algorithm available to this day [6], we consider the present "pollutant" algorithm as a reliable wake tracking method.

<figure><img src="image_8.png" /><figcaption>Figure 2: Coordinates ($y_c, z_c$) of the wake center 2 and 6 diameters behind the wind turbine computed with the ConstantFlux algorithm [28] and the pollutant method.</figcaption></figure>

## 2.4 Tracers wake tracking

The utilization of high-frequency data enables the tracking of wake behaviour using tracers. In the primary post-processing step, tracers are stochastically injected at the turbine rotor disk location. The trajectories of these tracers evolve temporally using the high-frequency data within a Lagrangian transport framework, facilitating detailed tracking of individual particle movements within the wake. At 1.2-second intervals, the position of each emitted particle is recorded and stored in a trajectory file. In the subsequent post-processing phase, the trajectory file is analyzed to compute the displacements, $\Delta y$ and $\Delta z$, between the initial and current positions of the particles at selected downstream locations. Figure 3 provides visualization of tracers particles in wind turbine wake.

<figure><img src="image_9.png" /><figcaption>Figure 3: Visualization of tracer particles in wind turbine wake simulations.</figcaption></figure>
# 3 Results and Discussion

## 3.1 Inflow characteristics and turbine performance metrics

The analysis focuses on the final 10 minutes of each simulation case. The upstream flow is characterized by calculating the mean velocity and turbulence intensity at a location 4D upstream of the wind turbine, within a disk of 2D centered around the hub height. The statistical characteristics of the upstream flow as well as turbine performance metrics are presented in Table 2. As the simulation procedure for stable and unstable flows is initiated from the neutral case, we observe that the mean velocity profile changes in response to variations in atmospheric conditions. Under stable conditions, as turbulent mixing near the ground decreases, the mean velocity near the surface tends to increase. Conversely, under unstable conditions, enhanced turbulent mixing near the ground leads to a reduction in mean velocity near the surface. As expected, the stable case exhibits a lower overall turbulence intensity. However, the streamwise turbulence intensity remains relatively high, while the lateral and vertical components of turbulence intensity are significantly reduced due to the suppression of vertical mixing. In contrast, the unstable case shows a similar overall turbulence intensity to the neutral case, but with a lower streamwise component, as enhanced vertical mixing reduces the streamwise velocity fluctuations. It is expected that atmospheric stability influences the redistribution of turbulence intensity, as discussed by [7].

<table><caption>Table 2: Inflow characteristics and turbine performance metrics for each simulated atmospheric condition.</caption><thead><tr><th></th><th>Stable case</th><th>Neutral case</th><th>Unstable case</th></tr></thead><tbody><tr><td>(I<sub>u</sub>, I<sub>v</sub>, I<sub>w</sub>) (%)</td><td>(6, 3.4, 1.9)</td><td>(7.2, 6.8, 6.1)</td><td>(5.8, 6.2, 6.7)</td></tr><tr><td>TI (%)</td><td>4.1</td><td>6.7</td><td>6.2</td></tr><tr><td>U<sub>∞</sub> (m/s)</td><td>11.2</td><td>10.2</td><td>9.65</td></tr><tr><td>C<sub>T</sub> (-)</td><td>0.68</td><td>0.71</td><td>0.79</td></tr><tr><td>C<sub>P</sub> (-)</td><td>0.46</td><td>0.49</td><td>0.53</td></tr></tbody></table>

## 3.2 Wake structure center dispersion

The last 10 minutes of high-frequency data are employed to compute the wake structure center position at each time step for various downstream locations. This post-processing is conducted for each atmospheric condition case, following the methodology described in section 2.3. Figure 4 illustrates a kernel density estimation used to visualize the probability density function of the wake structure center positions. Figure 4a for the stable case shows minimal dispersion in the wake structure center position, which can be attributed to the low vertical and lateral turbulence intensities. In contrast, Figure 4c for the unstable case displays a significantly higher dispersion in the wake structure center position. Interestingly, the lateral and vertical turbulence intensities in the unstable case are similar to those observed in the neutral case, where the wake structure center dispersion is much smaller. This suggests that additional flow characteristics must be considered to fully explain the observed behavior.

Braunbehrens and Segalini [17] build upon the heuristic conjecture proposed by Larsen et al. [5], which suggests that the wake behaves as a passive scalar, transported by large-scale atmospheric eddies. Based on this idea, they propose that the movement of the wake structure center follows Taylor's diffusion theory. The standard deviation of the wake structure center position can thus be expressed as:

$$ \sigma_{c_i}(T) = \sigma_i \sqrt{2A_i T - 2A_i^2 (1 - e^{-T/A_i})} \quad (1) $$

where *i* represents the direction, *σ*<sub>*i*</sub> is the standard deviation of the velocity in this direction, *A*<sub>*i*</sub> is the integral time scale, and *T* is the travel time. In their work, the integral time scales are estimated based on general observations of the atmospheric boundary layer. Although the study yields valuable insights, certain aspects of the methodology and results warrant further scrutiny. Notably, the wake structure center position in Larsen et al.'s work [5] is not directly associated with a particle but rather represents the center of the wake as an object. Therefore, it is not immediately clear that the movement of this center would follow the same diffusion behavior as a particle subject to turbulent diffusion, as described by Taylor's diffusion theory.
<figure><img src="image_10.png" /><figcaption>Figure 4: Kernel density estimation of the probability density function of the wake structure center positions at various downstream locations. Red markers indicate the center positions for each time step.</figcaption></figure>

To address this apparent inconsistency, we propose a distinction between the Lagrangian and Eulerian perspectives of wake particle movement. This approach aims to reconcile the treatment of the wake structure center as a single entity with the collective behavior of individual particles within the wake. We define the ensemble of particle positions that constitute the wake area $A(t)$ as $\{X_p(t)|p \in A(t)\}$, where $X_p(t) = (x_p(t), y_p(t), z_p(t))$ represents the position of each particle $p$ in $A(t)$ at time $t$. At $t=0$, the wake area is the rotor disk. The mean position of wake particles, the wake structure center, is defined as the surface average over the wake area, given by:

$$ \langle \mathbf{X}_p(t) \rangle = \frac{1}{|A(t)|} \int_{A(t)} \mathbf{X}_p(t) dA \qquad (2) $$

where $|A(t)|$ is the total area of the wake at time $t$. We apply the Reynolds transport theorem to equation 2 as follows:

$$ \frac{D}{Dt} \langle \mathbf{X}_p(t) \rangle = \frac{1}{|A(t)|} \int_{A(t)} \frac{\partial \mathbf{X}_p(t)}{\partial t} dA + \frac{1}{|A(t)|} \int_{\partial A(t)} (\mathbf{v}_b \cdot \mathbf{n}) \mathbf{X}_p(t) dl \qquad (3) $$

where $\mathbf{v}_b$ is the velocity on the boundary $\partial A(t)$ and $\mathbf{n}$ is the outward-pointing normal vector to $\partial A(t)$.
The equation 3 can be rewritten to express the relationship between Lagrangian and Eulerian mean velocities as follows:

$$ \langle \mathbf{U}_p \rangle^L = \langle \mathbf{U}_p \rangle^E + \frac{1}{|A(t)|} \int_{(t)} (\mathbf{v}_b \cdot \mathbf{n}) \mathbf{X}_p(t) dl \quad (4) $$

where $\langle \mathbf{U}_p \rangle^L$ is the Lagrangian mean velocity of the wake and $\langle \mathbf{U}_p \rangle^E$ is the Eulerian mean velocity of the wake, the last term is the Stokes drift velocity [30]. The hypothesis that the wake behaves as a passive scalar suggests that the decomposition of equation 4 holds for any group of fluid parcels. It can thus be concluded that the Eulerian velocity field associated with these fluid parcels would exhibit a spectral structure analogous to that observed for the Eulerian mean velocity of the wake. The Eulerian integral time scale [31] can be expressed as:

$$ T_i^E = \int_0^\infty \frac{u_i^E(t_0)u_i^E(t_0 + \tau)}{u_i^{E2}(t_0)} d\tau \quad (5) $$

where $u_i^E$ is the fluid Eulerian velocity in the i direction for an Eulerian point, and the $\tau$ denotes temporal averaging. Accordingly, the wake structure center would comply with Taylor's diffusivity theory by employing the Eulerian integral time scale in the following manner:

$$ \sigma_{E_i}(T) = \sigma_i \sqrt{2T_i^E T - 2T_i^E T_i^E \left(1 - e^{-T/T_i^E}\right)} \quad (6) $$

The Eulerian integral time scale, as defined in equation 5, is determined from the simulation results by identifying the time at which the mean of the autocorrelation function first decays to zero. For this calculation, the mean autocorrelation values are obtained from 10 probes evenly distributed within a grid located at a distance of 4D upstream of the rotor. Table 3 provides a summary of the Eulerian integral time scales for each case. To estimate the travel time T to a fixed downstream location, we use the relation for convective velocity in a mixing layer $U_c = 0.5(U_\infty + U_{wake})$, as described by [32]. The wake velocity is estimated as the mean of the Eulerian velocity of the wake and the free stream flow velocity, accounting for the influence of free-flow parcels incorporated into the wake. Therefore, the convective velocity of the wake can be expressed as follows:

$$ U_c(x_D) = 0.75U_\infty + 0.25\langle U_p \rangle^E(x_D) \quad (7) $$

where $\langle U_p \rangle^E(x_D)$ is estimated using the tracer trajectory data described in section 2.4 with $x_D = (x - x_{Hub})/D$. The travel time T to reach a downstream location x can be defined as $(x - x_{Hub})/U_c(x_D)$, where $x_{Hub}$ is the hub location and $U_c(x_D)$ is the convective velocity at the normalized downstream distance $x_D$. Following the approach [13], we propose a one-dimensional reduction of the lateral and vertical standard deviations of the rotor-normalized center displacement using the geometric mean. This is defined as $\sigma_{center}^{LES}(x_D) = \sqrt{\sigma_y^{LES}(x_D)\sigma_z^{LES}(x_D)}$. From a modeling perspective, the combination of Taylor diffusion standard deviations is contingent upon the interdependence of flow characteristics. In scenarios with dependent flow characteristics, such as the neutral and stable cases where streamwise turbulent intensity predominates, a geometric mean approach is employed. Conversely, for independent flow characteristics, a summation of Gaussian variances provides a more appropriate methodological framework. Figure 5 presents a comparison between the wake structure center dispersion calculated from the simulation results and the predictions obtained from Taylor's diffusion theory, using the Eulerian integral time scale and normalised with the rotor diameter. The confidence interval is computed using the Chi-squared distribution. The strong agreement observed between the theoretical predictions and the numerical results suggests that Taylor's diffusion theory, when applied with the Eulerian integral time scale, provides a good representation of the wake structure center dispersion. This dispersion can, in fact, be interpreted as the wake meandering phenomenon.

## 3.3 Wake particles dispersion

The particle dynamics within the wake are tracked using the method detailed in section 2.4, employing 100 tracers. We propose a one-dimensional reduction of lateral and vertical particle displacements, defined by the geometric mean:

$$ \Delta^{LES} = \frac{\sqrt{|\Delta_y^{LES}| |\Delta_z^{LES}|}}{D} \quad (8) $$

The standard deviation of particle displacement is then derived from all tracers' displacements at the defined streamwise position as:
<figure><img src="image_11.png" /><figcaption>Figure 5: Wake structure center dispersion normalized by the rotor diameter, compared with Taylor's diffusion theory using the Eulerian integral time scale. The grey region represents the 95% confidence interval.</figcaption></figure>

<table><caption>Table 3: Eulerian integral time scales.</caption><thead><tr><th></th><th>Stable</th><th>Neutral</th><th>Unstable</th></tr></thead><tbody><tr><th>(T<sub>v</sub><sup>E</sup>, T<sub>w</sub><sup>E</sup>) (s)</th><td>(20, 7.3)</td><td>(5.1, 2.8)</td><td>(11, 11)</td></tr></tbody></table>

$$ \sigma_{\text{tracers}}^{\text{LES}}(x_D) = \sqrt{E[(\Delta^{\text{LES}})^2] - (E[\Delta^{\text{LES}}])^2} \qquad (9) $$

As noted by Vahidi and Porté-Agel [13], the wake and the free-stream form a mixing layer. Consequently, the particles constituting the wake are subject to a dynamic internal contribution, arising from turbulence enhancement in the mixing layer. This internal dynamic contribution exerts an influence on the particles' transport. This can be modeled through an additional deviation term, as follows, with S the spreading parameter fixed to 0.043 [13]:

$$ \sigma_{ML} = \frac{2S(U_{\infty}T - (x - (x_{Hub} + D)))}{D} \qquad (10) $$

Thus, the overall dispersion of tracers can be modeled as the superposition of the Lagrangian inflow contribution and the internal dynamic contribution, representing an extension of Taylor's diffusion theory for wakes:

$$ \sigma_{\text{tracers}}^{\text{LES}} = \sigma_L + \sigma_{ML} \qquad (11) $$

where $\sigma_L$ is defined by equation 6, but with the Lagrangian integral time scale $T_L$. To estimate $T_L$, we observed that the integral time scale ratio, using Lagrangian and Eulerian definition of Taylor's diffusivity, can be expressed as:

$$ \beta = \frac{T_L}{T_E} \approx \left( \frac{\sigma_L}{\sigma_E} \right)^2 \approx \left( \frac{\sigma_{\text{tracers}}^{\text{LES}} - \sigma_{ML}}{\sigma_{\text{center}}^{\text{LES}}} \right)^2 \quad \text{where the approximation holds as } T \gg T_L \text{ and } T \gg T_E \qquad (12) $$

Figure 6a presents the estimated ratio of Lagrangian to Eulerian integral time scales, denoted as $\beta$, as a function of downstream distance for the different atmospheric stability conditions. The results show that $\beta$ converges rapidly, reaching an asymptotic value at approximately 4D downstream for both neutral and unstable atmospheric conditions. However, the stable case exhibits a notably different behavior, failing to achieve clear convergence within the observed domain. The observed $\beta$ ratios fall within the common range reported in literature [31], and the increase in $\beta$ for shear-dominated flows is consistent with theoretical predictions. Interestingly, the unstable case exhibits a higher $\beta$ ratio compared to the neutral case, despite similar turbulent intensities (lateral and vertical) and mean velocities. This suggests that atmospheric stability plays a significant role in determining the $\beta$ ratio, beyond what can be explained by turbulence intensity alone. It is important to note that our approach estimates the Lagrangian particle standard deviation by subtracting the mixing layer contribution from the tracer values, representing the internal dynamic of the particles. The discrepancy observed in the stable case may be attributed to limitations in the mixing layer model, which may not fully capture the behaviour of the mixing layer in stably stratified flows.
<figure><img src="image_12.png" /><figcaption>(a) Estimation of the ratio between Lagrangian and Eulerian integral time scale for the simulated cases</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(b) Wake tracers dispersions for the three simulated atmospheric conditions, normalized by the rotor diameter, compared with predictions from an extended Taylor's diffusion theory incorporating both Lagrangian integral time scales and mixing layer contributions. The grey region represents the 95% confidence interval.</figcaption></figure>

<caption>Figure 6: Wake tracers dispersions characterizations</caption>

Figure 6b presents a comparative analysis of wake particle dispersion derived from numerical simulations and theoretical predictions based on the extended Taylor's diffusion theory for wakes, as formulated in equation 11. The results demonstrate excellent agreement for unstable and neutral atmospheric conditions. This concordance suggests that the extended Taylor's diffusion theory for wakes, incorporating both Lagrangian inflow contribution and the internal dynamic contribution, provides a good representation of wake particle dispersion under these conditions. However, the stable case exhibits a notable discrepancy between simulated and predicted dispersion. This divergence can be attributed to the non-convergence of the Lagrangian to Eulerian integral time scale ratio $β$ in stable conditions, as illustrated in Figure 6a. Despite the aforementioned limitation, the agreement between numerical simulations and theoretical predictions of the extended Taylor's diffusion theory demonstrates the model's capability to characterize wake dynamics through particle dispersion. This particle dispersion can be interpreted as a direct manifestation of wake expansion, thereby providing valuable insights into the complex behavior of wind turbine wakes across various atmospheric stability regimes.

## 4 Conclusion

In the literature, Taylor's diffusion theory, initially developed for particle dispersion, has been applied to describe various wake phenomena. It has been employed to model wake meandering, as in the Statistical Wake Meandering model (SWM), and to characterize wake expansion using principles from plume theory. However, applying this theory directly to wake structure modeling, as in SWM—where the wake is treated as a coherent entity rather than a collection of particles—introduces conceptual challenges. To address this, we propose a framework that considers the wake as a surface-averaged entity derived from the positions of its constituent particles. This perspective establishes a relationship between the Lagrangian and Eulerian mean wake velocities. Assuming passive scalar behavior, we demonstrate that the wake entity adheres to the Eulerian turbulent characteristics of the upstream flow. Numerical analyses of the standard deviation of the wake structure center reveal strong agreement with Taylor's theoretical predictions when using the Eulerian framework, thereby validating the model's predictive capabilities. This approach conceptualizes wake meandering as an intrinsic Eulerian component of the broader wake diffusion process. Simultaneously, the behavior of individual particles within the wake—characterized by their displacement standard deviation—can be interpreted as a composite effect of Lagrangian upstream flow dynamics and internal mixing layer dynamics, extending classical Taylor diffusion theory. This necessitates removing the passive scalar assumption at the Lagrangian level to account for internal wake dynamics. The ratio between the standard deviation derived from the corrected Lagrangian description (tracers) and that from the Eulerian description (pollutant) provides insights into the relationship between Lagrangian and Eulerian integral time scales of the upstream flow. Numerical results align well with findings in existing literature, with minor discrepancies observed under stable flow conditions, likely due to limitations in modeling the mixing layer. Overall, numerical results demonstrate good agreement with the extended Taylor diffusion theory, suggesting that wake expansion can be interpreted as a Lagrangian perspective on wake diffusion. To conclude, wake diffusion can be comprehensively described using
Taylor's diffusion theories: while the Eulerian framework elucidates wake meandering, the Lagrangian framework reveals wake expansion. However, incorporating internal dynamic terms is essential for a complete description within the Lagrangian framework. In contrast, these terms are negligible in the Eulerian framework since particle-scale variations are significantly smaller than rotor diameter and thus have minimal impact on wake displacement at an Eulerian scale. The proposed framework offers valuable insights into the impact of the turbulent structure of atmospheric flows on wind turbine wakes.

## Acknowledgements

This project was provided with computer and storage resources by GENCI at TGCC thanks to the grant 2024-A0150114592 on the supercomputer Joliot Curie's the SKL partition.

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