text:
\section{Introduction}
\label{sec:introduction}
The wind energy market is strongly growing with a record in new installations in 2015 \cite{WWEAMarch2016, WWEAonline} and increasing importance expected over the next years, see also \cite{WindEnergyScenarios2030}. Wind energy is a fluctuating resource that is converted into electrical power in a nonlinear fashion (see e.g. \cite{Burton2011}). These characteristics constitute challenges with respect to power quality and grid stability, cf.~\cite{MilanPeinkePRL2013}, which raise significant interest in predicting short-term power output fluctuations in wind farms. Understanding and predicting the spatio-temporal structure of velocity fluctuations in wind farms which is a main source of the power output fluctuations, is a crucial step towards this goal.

The atmospheric conditions of wind approaching a wind farm can be described by basic atmospheric boundary layer (ABL) theory. For a recent overview, we refer to \cite{StevensMeneveau2017Annu}.
Wind speed fluctuations in the ABL occur on different time scales, ranging from annual or seasonal variations, down to minutes and seconds \cite{Burton2011, vanderHoven}.
The short-term fluctuations on scales smaller than around, say, ten minutes are usually referred to as turbulent fluctuations \cite{Burton2011}.
An introduction to the characteristics of atmospheric wind including typical turbulence intensities and naturally occurring extreme wind gusts (cf.~also \cite{BoettcherBarthPeinke2007}), as well as other factors such as geographical (onshore, offshore), landscape and climate influences is given in \cite{Burton2011}.
The evolution of wind velocities in the wind farm itself is influenced by the aerodynamics of the turbines (see e.g.~\cite{HansenButterfieldAnnRevFM1993}), the design of the whole wind farm, e.g.~in terms of turbine spacing \cite{MeyersMeneveauWindEnergy2012, StevensGaymeMeneveauWindEnergy2016}, and by the influence of turbines onto each other, primarily mediated by turbine wakes \cite{CrespoHernandezFrandsenWindEnergy1999, VermeerSorensenCrespo2003, KeaneAguirreorque2016, FletcherBrownWindEnergy2010, SamoraniHandbook}. Capturing the full complexity of turbulent flows in wind farms at all scales simultaneously is  currently computationally out of reach, emphasizing the need for simplified, physics-based models.

In the present work, we derive such a model for the space-time correlation of the turbulent short-term fluctuations based on concepts from classical turbulence theory. Specifically, it is based on the Kraichnan-Tennekes random sweeping hypothesis with additional mean flow. Focusing on its simplicity, it depends on a minimal set of parameters and is set up to be applied both in the ABL without turbines, or in the wind turbine array boundary layer. With respect to wind energy applications, it provides a starting point for further developments, such as predictive models of the temporal power output fluctuations, or in the area of wind turbine control \cite{Bossanyi2013}.

Velocity space-time correlations along with modeling approaches have been widely investigated (see \cite{Favre1965} for a classical and \cite{HeJinYangAnnuRev2017,Wallace2014} for recent reviews) and have also found application to turbulent ABLs. Thus, applying similar approaches to the modeling of wind farms appears worth exploring, which is the scope of the present paper.

Considering turbulence with mean flow, velocity fluctuations are primarily advected with the mean velocity, which forms the basis of Taylor's frozen eddy hypothesis \cite{Taylor}. Due to the frozen-in assumption, however, these fluctuations do not decorrelate in time along the mean flow. In the random sweeping hypothesis, as introduced by Kraichnan \cite{Kraichnan} and Tennekes \cite{Tennekes}, it is assumed that small-scale velocity perturbations are advected randomly by large-scale velocity fluctuations. This random advection is the primary source of temporal decorrelation. Both Taylor's and the Kraichnan-Tennekes hypotheses have been thoroughly investigated with respect to their range of validity, e.g.~in \cite{Lin1953,Lumley1965, WuGengYao2017, KatulBanerjee2016}. Due to their simplicity, they have laid the foundation for a range of models for space-time correlations both in real space \cite{HeZhang2006,ZhaoHe2009} as well as in spectral space \cite{HuntBuellWray1987,WilczekNarita2012}. In the spectral formulation of space-time correlations in terms of wavenumber-frequency spectra, the random sweeping hypothesis can be applied to derive the wavenumber-frequency spectrum as a product of the wavenumber spectrum and a frequency distribution, whose properties are fixed by the decorrelation hypothesis \cite{WilczekNarita2012}. In a real space formulation, as adopted in this paper, this corresponds to a convolution of the spatial correlation function with a temporal decorrelation kernel, consistent with the classical  Kovasznay-Corrsin conjecture \cite{Corrsin, Favre1965, Phillips2000}. Recently, the random sweeping model with additional mean flow advection has been extended to include the physics of ABLs to establish a spectral description of ABL space-time correlations \cite{WilczekStevensMeneveau2015}.

In the context of ABLs and wind farms, a variant of space-time correlations, so-called coherence functions, are widely used which contain space-frequency information. Modeling approaches usually combine Taylor's frozen eddy hypothesis with a model for the spectrum of wind speed fluctuations, such as the Kaimal spectrum \cite{Kaimal1972} or the von K\'arm\'an spectrum \cite{vonKarman1948}, see \cite{Burton2011}. As an example, in the Mann model of turbulence \cite{Mann1994, Mann1998}, two-point spatial information is obtained on the basis of one-point spectra with the help of Taylor's frozen eddy hypothesis. Instead of using Taylor's frozen eddy hypothesis, de Mar\'{e} and Mann \cite{MareMann2016} combine the Mann model mentioned above with a model by Kristensen \cite{Kristensen1979} to model space-time correlations in high-Reynolds number flow. Kristensen's longitudinal coherence is determined by the temporal decay of eddies and their transverse motion. Other coherence models which also avoid the use of Taylor's frozen eddy hypothesis can be found in literature, e.g.~\cite{SoerensenHansenRosas2002, Bossanyi2013}.

In the interest of a broader overview, we also mention stochastic modeling approaches which are used to predict velocity increment statistics in wind farms \cite{MeliusTutkunCal2014n1, MeliusTutkunCal2014n2,MilanPeinkePRL2013}, and wind power output fluctuations, e.g.~\cite{GottschallPeinke2007, GottschallPeinke2008, AnahuaBarthPeinke2008}. A description of wind power output fluctuations influenced by atmospheric turbulence can be found in \cite{Bandi2017}.

The model for space-time correlations in wind farms based on the random sweeping hypothesis derived in this paper essentially represents a real space formulation of the previous spectral model for ABL introduced in \cite{WilczekStevensMeneveau2015}, generalized here to include finite time correlations of the random sweeping velocity. Given that the flow through a wind farm is strongly inhomogeneous due to the presence of the wind turbines, a real-space formulation appears favorable as it allows us to discern different flow regions such as along and between streamwise lines of turbines. The finite-time correlation of the random advection velocity is crucial for capturing the decorrelation trends at large times more accurately.

The remainder of this paper is structured as follows. In section \ref{sec:Theory}, the mathematical derivation of the model in real space is presented. In a next step, our model is compared to data obtained from large eddy simulation (LES) of turbulent flow in a wind farm. In section \ref{sec:LES}, we briefly describe the LES tool and simulated flow conditions. The comparison of the model to the numerical data is given in section \ref{sec:Comparison}. The principal aim of the analysis and comparison is to establish whether the space-time correlation model in terms of a convolution of the space correlation and a temporal decorrelation kernel can predict observations made from the LES wind farm data. Further details are discussed in an outlook, section \ref{sec:Outlook}.

\section{Space-time correlations from an advection model}
\label{sec:Theory}
In the following, we provide a derivation for the space-time correlations of streamwise velocity fluctuations $u_1'$ in a plane at hub height in wind farms. It is based on the Kraichnan-Tennekes random sweeping hypothesis \cite{Kraichnan, Tennekes}; we assume that small-scale velocity fluctuations are advected with a large-scale random velocity $\boldsymbol v=(v_1, v_2)$, which we, for simplicity, restrict to the horizontal plane. The indices $1$ and $2$ refer to the streamwise and spanwise direction, respectively. Additionally, we take into account an advection with a mean velocity $\boldsymbol U = (U,0)$ in streamwise direction, corresponding to the mean wind in an ABL. For now, we neglect possible differences between the mean velocity and effective convection velocity, a topic of considerable interest in its own right, see e.g.~\cite{Wills1964,kimhussain1993, delalamojimenez2009} for turbulent shear flows. The simple dynamical model discussed in the following serves to motivate the analytical structure of the space-time correlations. We then conjecture that, in essence, the same model holds for the arguably more complex flow in the wind farm and compare the results to LES.

\subsection{Advection model}
Assuming that advection with $\boldsymbol U$ and $\boldsymbol v$ gives the dominant contribution to capture the temporal decorrelation of velocity fluctuations, we arrive at the simple advection equation for the streamwise small-scale velocity component:
\begin{equation}\label{eq:advection}
  \partial_t u_1' + (\boldsymbol U+ \boldsymbol v) \cdot \nabla u_1' = 0 \, .
\end{equation}
To include the random character of the large-scale advection, we assume that $\boldsymbol v$ is spatially constant with a Gaussian ensemble distribution and decorrelates exponentially in time:
\begin{equation}
 \langle v_1(t)v_1(t+\tau) \rangle = \langle v_1^2 \rangle \exp\left[-   |\tau| /T_1 \right]\, ,
 \label{eq:exp_decay_rsvel}
\end{equation}
for the streamwise random-sweeping velocity $v_1$, and analogously for the spanwise random-sweeping velocity $v_2$. Here, $\langle \dots \rangle$ denotes ensemble-averaging. For practical purposes in the application to wind farm LES data, the ensemble averages are replaced by averages over space and time which is further discussed in the corresponding sections. The assumption of an exponentially decaying correlation function allows us to introduce decorrelation time scales $T_1$ and $T_2$, respectively. This assumption is tested below with our LES data.
In the following, we explicitly compute the space-time correlations of the streamwise velocity
\begin{equation}
 R_{11}(\boldsymbol r,\tau)  =  \left\langle u_1'(\boldsymbol x+\boldsymbol r,t+\tau) \, u_1'(\boldsymbol x,t) \right\rangle
 \label{eq:R_equation1}
\end{equation}
in the framework of the advection model, where $\boldsymbol r = (r_1, r_2)$ is the distance vector in the plane at hub height and $\tau$ is the time lag. The solution of equation \eqref{eq:advection} corresponds to a shifted initial condition such that
\begin{equation}
  u_1'(\boldsymbol x+\boldsymbol r,t+\tau) = u_1'\left(\boldsymbol x+\boldsymbol r - \boldsymbol U\tau - \int_t^{t+ \tau } \!\!\!\!\! \boldsymbol v(s) \, \mathrm{d}s,t \right) \, .
  \label{eq:u_shifted}
\end{equation}
Under the assumption of homogeneity and statistical stationarity, there is no explicit $\boldsymbol x$- and $t$-dependence on the left hand side of equation \eqref{eq:R_equation1}. Using equation \eqref{eq:u_shifted}, the space-time correlations can be obtained according to
\begin{align}
 R_{11}(\boldsymbol r,\tau)  &= \left\langle u_1'\left(\boldsymbol x + \boldsymbol r - \boldsymbol U\tau - \int_t^{t+ \tau} \!\!\!\!\! \boldsymbol v(s) \, \mathrm{d}s,t \right) \, u_1'(\boldsymbol x,t) \right\rangle \label{eq:model_space_time_corr1} \\
		&=  \int\mathrm{d}\boldsymbol a \left\langle u_1'\left(\boldsymbol x+\boldsymbol r-\boldsymbol a,t \right) \, u_1'(\boldsymbol x,t) \, \delta\left(\boldsymbol a-\boldsymbol U\tau- \int_t^{t+ \tau } \!\!\!\!\! \boldsymbol v(s) \, \mathrm{d}s \right) \right\rangle \label{eq:model_space_time_corr2} \\
		&=  \int\mathrm{d}\boldsymbol a \left\langle u_1'\left(\boldsymbol x+\boldsymbol r-\boldsymbol a,t \right) \, u_1'(\boldsymbol x,t) \right\rangle\left\langle \delta\left(\boldsymbol a-\boldsymbol U\tau- \int_t^{t+ \tau } \!\!\!\!\! \boldsymbol v(s) \, \mathrm{d}s \right) \right\rangle \, . \label{eq:model_space_time_corr3}
\end{align}
The average here is taken over both, small-scale and large-scale fluctuations. To factorize the averages in the last step, we have assumed statistical independence of the large-scale random sweeping velocity $\boldsymbol v$ at all times in the interval $[t, t+\tau]$ and the streamwise velocity fluctuations $u_1'$. A similar assumption has been made by Corrsin in the context of the relation between Eulerian and Lagrangian velocity correlations \cite{Corrsin , Phillips2000}. As a result of this factorization, the first average in the integrand is simply the instantaneous correlation function evaluated at $\boldsymbol r-\boldsymbol a$ in the plane, i.e.~$R_{11}(\boldsymbol r-\boldsymbol a)$. The averaged delta function represents the decorrelation kernel which determines the temporal decorrelation of the velocity fluctuations. Under the Gaussian assumption for the large-scale velocity, it can be explicitly evaluated yielding
\begin{equation}
  \left\langle \delta\left(\boldsymbol a-\boldsymbol U\tau- \int_t^{t+ \tau } \!\!\!\!\! \boldsymbol v(s) \, \mathrm{d}s \right) \right\rangle = \frac{1}{2\pi \sigma_1(\tau)\sigma_2(\tau)}\exp\left[ -\frac{(a_1-U\tau)^2}{2\sigma_1(\tau)^2} -\frac{a_2^2}{2\sigma_2(\tau)^2} \right]\, ,
  \label{eq:decorrelation_kernel}
\end{equation}
where
\begin{align}
  \sigma_1(\tau)^2 &= \int_t^{t+\tau}\!\!\!\mathrm{d}s \int_t^{t+ \tau }\!\!\!\mathrm{d}s'  \left\langle v_1(s) v_1(s') \right\rangle = 2\langle v_1^2 \rangle T_1^2 \left[ \frac{ |\tau| }{T_1} - \left( 1-\exp\left[ -\frac{ |\tau | }{T_1} \right] \right) \right]\, , \label{eq:sigma_1} \\
    \sigma_2(\tau)^2 &= \int_t^{t+ \tau }\!\!\!\mathrm{d}s \int_t^{t+ \tau }\!\!\!\mathrm{d}s'  \left\langle v_2(s) v_2(s') \right\rangle = 2\langle v_2^2 \rangle T_2^2 \left[ \frac{ | \tau | }{T_2} - \left( 1-\exp\left[ -\frac{ | \tau |}{T_2} \right] \right) \right] \, . \label{eq:sigma_2}
\end{align}
The effect of the Gaussian decorrelation kernel on the space-time correlations can be interpreted straightforwardly: the mean velocity advection induces a shift of the spatial correlation function, whereas the random advection blurs the spatial correlation by mixing a range of spatial scales. The functional forms of $\sigma_1$ and $\sigma_2$ are implied by the exponential decay of the large-scale velocities. While these formulas are already quite involved, the essential point here is that $\sigma_1(\tau)^2 \approx \langle v_1^2 \rangle \tau^2$ for $ |\tau |  \ll T_1$ and $\sigma_1(\tau)^2 \approx 2\langle v_1^2 \rangle T_1  |\tau |$ for $| \tau |   \gg T_1$, i.e.~the random displacement exhibits a ballistic behavior for short times and then transitions to a diffusive long-time behavior. This behavior, in fact, is not limited to exponentially decaying correlation functions but can be realized with a much larger class of correlation functions.

In the limit of $|\tau| \rightarrow 0$, the decorrelation kernel becomes a delta function; as expected, the space-time correlation function reduces to the spatial correlation only, $R_{11}(\boldsymbol r,0) = R_{11}(\boldsymbol r)$. For large time increments, $|\tau| \rightarrow \infty$, the decorrelation kernel becomes very broad and shallow, which leads to a vanishing correlation in time.

\subsection{Reduction to a one-dimensional model}
\label{subsec:Reduction}
Because we aim for an even simpler, one-dimensional model for the streamwise velocity correlation in the streamwise direction $r_1$, we make the assumption that the spatial correlation function in the plane can be factorized according to $R_{11}(\boldsymbol r) = R_{11}(r_1) \exp\left[-r_2^2/\left(\frac{4}{\pi} L_2^2\right) \right]$, i.e.~the transverse decorrelation of the streamwise velocity is modeled as a Gaussian with a transverse integral length scale $L_2$. Throughout the paper, we use the convention that functions such as $R_{11}(...)$ only contain those parameters as arguments which are not a priori set to zero. The factoring assumption, which is robust with respect to the functional form of the transverse decorrelation, allows to explicitly evaluate the transverse contribution of the convolution integral. As a result, we obtain
\begin{equation}\label{eq:spacetimecorr}
R_{11}(r_1,\tau) = \int\!\mathrm{d}a_1 \, R_{11}(r_1-a_1) \frac{\exp\left[ -\frac{(a_1-U\tau)^2}{2\sigma_1(\tau)^2} \right]}{\sqrt{2\pi \sigma_1(\tau)^2 \left[ 1 + \sigma_2(\tau)^2/\left(\frac{2}{\pi} L_2^2\right)\right]}}  \, .
\end{equation}
The space-time correlation is given by a convolution of the instantaneous correlation function with a Gaussian temporal decorrelation kernel, which is parameterized by the mean velocity and the mean square random sweeping velocities. Interestingly, the width of the decorrelation kernel is given by the effect of streamwise random sweeping $\sigma_1$, whereas its amplitude depends on both, streamwise and spanwise random sweeping. Furthermore, $\sigma_2(\tau)$ and $L_2$ only appear in combination, $\sigma_2(\tau)^2/\left(\frac{2}{\pi} L_2^2\right)$, and not independently of each other.

The derivation of our model resembles parts of the derivation of the wavenumber-frequency model for ABLs \cite{WilczekStevensMeneveau2015}, however, it is further extended by incorporating the temporal decorrelation of the random sweeping. Our resulting model also shows similarities to the Kovasznay-Corrsin convolution as presented in \cite{Phillips2000} for homogeneous turbulence advecting in streamwise direction.

To sum up, the model depends upon the following parameters: the advection velocity $U$, the second moment of the streamwise and spanwise random sweeping velocities $\langle v_1^2 \rangle$, $\langle v_2^2 \rangle$, the correlation time scales of the random sweeping velocities $T_1$, $T_2$, the transverse integral length scale of the streamwise velocity fluctuations $L_2$ and the space correlation of the streamwise velocity fluctuations $R_{11}(r_1)$. By a comparison to LES, we explore the potential of the model presented here to describe space-time correlations of the streamwise velocity fluctuations in the hub height plane in wind farms. In the following, we discuss how the free parameters can be determined from LES. As an alternative, these parameters could also be prescribed by further modeling assumptions, which is beyond the scope of the present paper.

\section{Large eddy simulations and parameters}
\label{sec:LES}
The applicability of the model in equation \eqref{eq:spacetimecorr} for describing space-time velocity correlations in wind farms is investigated by LES data.
The goal is to compare the velocity space-time correlations predicted by the model to velocity space-time correlation data from the LES.
\subsection{Description of LES}
In the wind farm LES, we solve the filtered incompressible Navier-Stokes equations without buoyancy, system rotation or other effects to model a neutral pressure-driven ABL flow over a rough surface \cite{CalafMeneveauMeyers2010,StevensMeneveauRenSus2014}. In horizontal directions, a pseudo-spectral discretization with periodic boundary conditions is adopted and second-order centered finite differences are used in the vertical direction. The subgrid stresses are modeled using a scale-dependent Lagrangian dynamic approach in conjunction with the Smagorinsky model and a sharp spectral cutoff test-filter \cite{bou05}. The wall stress at the ground is captured by a standard rough-wall model \cite{moe84} using velocities test-filtered at twice the grid scale \cite{bou05}. For the top boundary we use a zero-vertical-velocity and zero-shear-stress boundary condition. Therefore, the modeled flow effectively corresponds to a `half-channel flow' with an impermeable centerline boundary. Time-stepping is performed with a second-order Adams-Bashforth scheme. Here, we consider the idealized case of a fully developed wind turbine array boundary layer, modeled by a periodic array of wind turbines. The individual wind turbines are modeled as actuator disks  \cite{jim07,CalafMeneveauMeyers2010,ste14}. Their back-reaction on the flow is modeled by a thrust force added to the momentum equation. For a more complete account on wind farm modeling by means of LES, we refer the reader to \cite{meh14,StevensMeneveau2017Annu}.

Throughout the paper, $x_1$ and $x_2$ denote the streamwise and spanwise directions, respectively, and $x_3$ denotes the vertical direction. The simulation is performed on a $512 \times 256 \times 128$ grid discretizing a domain of size $L_{x_1}/H \times L_{x_2}/H \times L_{x_3}/H = 4\pi \times 2\pi \times 1$, where $H$ is the height of the ABL, i.e.~the domain height. In this domain, $16 \times 16$ turbines are placed with a turbine diameter $D = 0.1 \, H$ in an aligned configuration. The streamwise spacing of the turbines $S_{x_1}$ is $7.85\, D = 0.785\, H$, the spanwise spacing $S_{x_2}$ is $3.93 \, D =0.393 \, H$. The simulation is conducted with a time step of $6 \times 10^{-5}\, H/u^*$, where $u^*$ is the friction velocity associated with the driving pressure gradient. The data are collected after the statistically stationary state is reached. The velocity field at hub height ($x_3=0.1\, H$) is stored every ten time steps at $8192$ snapshots, thus, over a total duration of about $4.9 \, H/u^*$. This corresponds to about $3.5$ flow through times at hub height, or about $55$ inter-turbine travel times (i.e.~the time for traveling from one turbine to the next).
\begin{figure}
\centering
  \includegraphics[trim =  1.0cm 0.0cm 0.5cm 0.0cm, clip, width=0.48\textwidth]
 {figure1.pdf}
\centering
  \includegraphics[trim =  1.0cm 0.0cm 0.5cm 0.0cm, clip, width=0.48\textwidth]
 {figure2.pdf}
\caption{Left: Streamwise velocity component normalized by the friction velocity $u^*$ at one instant in time. $16 \times 16$ turbines are located in the domain indicated by the black lines. Right: Temporally averaged streamwise velocity component normalized by $u^*$. The individual turbines clearly introduce inhomogeneities in the flow field, especially along the lines of the turbines. One can also observe high-speed streaks between the turbines when averaging over time. All plots in this paper have been created with matplotlib \cite{Hunter2007}.
}
\label{fig:figure1_2}
\end{figure}
 \begin{figure}
  \centering
   \includegraphics[trim =1.0cm 0.0cm 0.5cm 0.0cm, clip, width=0.48\textwidth]
  {figure3.pdf}
\centering
\includegraphics[trim =  1.0cm 0.0cm 0.5cm 0.0cm, clip, width=0.48\textwidth] {figure4.pdf}
\caption{Left: Temporally averaged second moment of the streamwise velocity fluctuations as given in equation \eqref{eq:time_average}, normalized by $u^{*2}$. The field is highly inhomogeneous. Right: Turbulence intensity of streamwise components, $\mathrm{TI} = \sqrt{\langle u_1'^2 \rangle_t (\boldsymbol x)}/\langle u_1 \rangle_t(\boldsymbol x)$. The highest intensity can be found at the edges of the turbine wakes.}
\label{fig:figure3_4}
\end{figure}
To give an impression of the flow, the left panel in figure \ref{fig:figure1_2} shows a visualization of the streamwise velocity. One can see that higher-velocity flow streaks meander between wind turbine columns. In the present paper, columns refer to turbines aligned in the streamwise direction, whereas rows refer to neighboring turbines in spanwise direction.
The right panel of figure \ref{fig:figure1_2} shows the temporal mean velocity. Even after time averaging there remain spatial large scale inhomogeneous streaks, visible as elongated regions. It is well known that such features are very slow to converge in this type of flow \cite{MuntersMeneveauMeyers2016}. Here, we average over these in space as appropriate, i.e.~when we investigate the applicability of the model between lines of turbines, we only average over all lines in the middle between the turbines in order to minimize the effect of these streaks.
Furthermore, pronounced wakes behind the wind turbines are clearly visible.

For the subsequent analysis, we decompose the total streamwise velocity into its temporal mean and the remaining fluctuations
\begin{equation}
u_1(\boldsymbol x,t) = \langle u_1 \rangle_t(\boldsymbol x) + u_1'(\boldsymbol x,t)  \label{eq:decomposition} \, .
\end{equation}
The $\langle ... \rangle_t$-brackets denote an average over time. As a result, the temporal mean velocity is a function of the position in the plane. The mean square velocity fluctuations are obtained analogously:
\begin{equation}
\langle u_1'^2\rangle_t(\boldsymbol x) = \langle u_1^2 \rangle_t(\boldsymbol x)   -  \langle u_1 \rangle_t^2 (\boldsymbol x) \, .
\label{eq:time_average}
\end{equation}

The mean square velocity fluctuations are shown in the left panel of figure \ref{fig:figure3_4}.
As can be seen, the inhomogeneities introduced by the wind farm also carry over to the statistical quantities and have to be taken into account when computing statistical averages.
As expected, the temporal mean velocity is higher between the lines of wind turbines than along the lines, cf.~right panel in figure \ref{fig:figure1_2}. There, the velocity deficit behind the wind turbines is clearly visible.

As for the fluctuations, they appear lower between the turbine columns. The highest velocity fluctuations are generated at the edges of the wind turbine wakes. The turbulence intensity, defined as the relation of the root mean square fluctuations and the temporal mean velocity, is shown in the right panel of figure \ref{fig:figure3_4}. The highest turbulence intensity can be found at the edges of the turbine wakes accompanying the regions of the highest velocity fluctuations. Furthermore, it is visible that along the lines of the turbines, the highest intensity is located in the region behind the turbines within half the distance to the following turbines. This is followed by a less intense region in front of the following turbine.

\subsection{Determination of parameters and spatial correlations from LES}
\label{subsec:Determination from LES}

In the following, we compare the model space-time correlation to the space-time correlation obtained directly from the LES data. To this end, we compute the wavenumber-frequency spectrum from the LES data and determine the velocity space-time correlation by an inverse Fourier transform. To evaluate the model \eqref{eq:spacetimecorr}, we also compute the spatial correlation function. Based on the LES data, we determine values for the mean velocity $U$, the streamwise random sweeping velocity fluctuations $\langle v_1^2 \rangle$, the corresponding integral time scale $T_1$, as well as values for the spanwise random sweeping fluctuations $\langle v_2^2 \rangle$ and $T_2$, and the transverse integral length scale $L_2$.
In the following, we discuss how to obtain these parameters from the LES data.

At this stage it is worth recalling that the derivation of the advection model contains a number of simplifying assumptions.
To derive the analytical model, scale separation between the spatially constant random sweeping advection velocity and the advected velocity fluctuations has been assumed. It has been discussed by Kraichnan \cite{Kraichnan}, that even with spatially slowly varying large-scale random sweeping fluctuations, the random sweeping hypothesis as discussed here remains valid. To make it more realistic, we have additionally included a large-scale flow with a finite correlation time in our model. In the application of the model to LES data, however, we do not distinguish anymore between small- and large-scale fluctuations, but rather apply the model to the overall velocity fluctuations.
For the model, we furthermore restrict ourselves to advection in the plane at hub height; any vertical transport of streamwise velocity fluctuations is therefore neglected. Also, effects like large-scale shear which have been discussed, e.g.~in the context of the elliptic \cite{HeZhang2006} and the Mann \cite{Mann1998} models, are currently not taken into account. Additionally, we have assumed a statistically homogeneous flow throughout the derivation in section \ref{sec:Theory}, which is not met in the wind farm application: due to the equidistantly spaced wind turbines, the continuous shift invariance of statistical quantities implied by homogeneity reduces to a discrete one. For averages in spanwise direction, we take these spatial inhomogeneities into account and distinguish two cases in the following: Firstly, we investigate the space-time correlations along the wind turbine columns. The description of velocity correlations along these lines is the primary purpose of the application of the model. In a second case, the lines between the wind turbine columns are investigated for a comparison.
As a consequence, averages to compute the reference space-time correlations from the LES and the space correlation to feed the model space-time correlation are taken over all streamwise positions and times, whereas the spanwise average is restricted to the corresponding subset of lines. Also the correlation times $T_1$ and $T_2$ are determined for the two cases separately. The distinction in two cases is not done for $U$, $\langle v_1^2 \rangle$, $\langle v_2^2 \rangle$, and $L_2$ as discussed below. In the following, averaging brackets such as $\langle ... \rangle_{\boldsymbol x t}$ denote an average over all times and the whole $x_1, x_2$-plane. A subscript $\tilde{\boldsymbol x}$ at the $\langle ... \rangle_{ \tilde{\boldsymbol x} t}$-brackets denotes an average over all $x_1$ in streamwise direction and a spanwise average over $\tilde{x}_2$ referring to the reduced subset of lines.

Figure \ref{fig:figure5_6} shows the longitudinal and transverse spatial velocity correlations. The longitudinal correlations for the two cases are shown in the left plot. For a comparison, the right panel of figure \ref{fig:figure5_6} shows the transverse space correlation which is used for the computation of the integral length scale $L_2$ below.
\begin{figure}
 \centering
    \includegraphics[trim =  0.5cm 0.5cm  2.5cm 1.8cm, clip, width=0.49\textwidth]{figure5.pdf}
    \includegraphics[trim =  0.5cm 0.5cm  2.5cm 1.8cm, clip, width=0.49\textwidth] {figure6.pdf}
    \caption{
    Left: Comparison of the longitudinal normalized streamwise velocity space correlations obtained from LES data for the two cases, $R_{11}(r_1 ) = \langle u_1'(x_1 + r_1,x_2 ,t) u_1'(x_1,x_2,t) \rangle_{\tilde{\boldsymbol x} t}$. The decay of the correlation is higher on the lines of the turbines.
    Right: transverse normalized spatial correlation of the streamwise velocity fluctuation,
$R_{11}(r_2 ) = \langle u_1'(x_1,x_2+r_2,t) u_1'(x_1,x_2,t) \rangle_{\boldsymbol x t}$. The average here is taken over the whole $x_2$-range. The integral over the positively correlated range of the blue dashed line (LES) is used to determine $L_2$, cf.~equation \ref{eq:L2_integral}. The transverse velocity correlation is modeled as a Gaussian (red solid line) in the modeling process in section \ref{subsec:Reduction}.}
    \label{fig:figure5_6}
  \end{figure}

As for the mean velocity $U$, we use the overall mean velocity, i.e.~the temporally and spatially averaged streamwise velocity $\langle u_1 \rangle_{\boldsymbol x t} = 7.88\, u^*$ for both cases. However, as a comparison, the mean velocity along the lines of the turbines is $\langle u_1 \rangle_{\tilde{\boldsymbol x} t} = 6.51\, u^*$, while it is $\langle u_1 \rangle_{\tilde{\boldsymbol x} t} = 8.59\, u^*$ over the subset of lines between the turbines.

The second moment of the streamwise and spanwise random sweeping fluctuations are obtained from the LES straightforwardly by computing the variance of the respective velocity components, i.e.~$ \langle v_1^2 \rangle = \langle u_1^2 \rangle_{\boldsymbol x t}-\langle u_1 \rangle_{\boldsymbol x t}^2 =  3.60\, u^{*2}$, and $ \langle v_2^2 \rangle = \langle u_2^2 \rangle_{\boldsymbol x t}=  1.48\, u^{*2}$ (note that there is no imposed mean flow in the spanwise direction). The way $\langle v_1^2 \rangle$ is computed does not yield the same result as a spatial average of equation \eqref{eq:time_average}, i.e.~$\langle u_1'^2 \rangle_{\boldsymbol x t}= 2.68\, u^{*2}$.
For $\langle v_1 \rangle$ we do not distinguish the two cases because the highest turbulence intensity is at the edges of the wakes behind the turbine, cf.~figure \ref{fig:figure3_4} which is neither exactly on the lines of the turbines nor exactly between the turbines. For the spanwise random sweeping fluctuations we also use the overall mean square spanwise fluctuations from LES.

We distinguish the two cases for the determination of the correlation time scales of the random sweeping velocities. The model time scales $T_1$ and $T_2$ are determined from the temporal correlation of the LES streamwise and spanwise velocity fluctuations with respect to the subset of lines in the respective cases. The integration stops at the first zero crossing:
\begin{equation}
\label{eq:T1_T2}
 T_1  = \frac{1}{\langle u_1'^2\rangle_{\tilde{\boldsymbol x} t}} \int\limits_{0}^{\tilde{t}} \mathrm{d}\tau \, \langle u_1'(\boldsymbol x , t+\tau) u_1'(\boldsymbol x,t)\rangle_{\tilde{\boldsymbol x} t}  \, , \ \
 T_2   = \frac{1}{\langle u_2^2\rangle_{\tilde{\boldsymbol x} t}} \int\limits_{0}^{\tilde{t}} \mathrm{d}\tau \, \langle u_2(\boldsymbol x , t+\tau) u_2(\boldsymbol x,t)\rangle_{\tilde{\boldsymbol x} t}  \, .
\end{equation}
The $\tilde{t}$ indicates the first zero-crossing of the correlation so that for the integral time scales only the positively correlated range is counted.
As a result, the model integral time scales $T_1$ and $T_2$ on the lines of turbines (case 1) are $T_1 = 0.033\, H/u^*$ and $T_2 = 0.016\, H/u^*$. The time scales between the lines of the turbines (case 2) result in $T_1 = 0.054\, H/u^*$ and $T_{2} = 0.021  \, H/u^*$.
The dashed blue lines in figures \ref{fig:figure7_8} and \ref{fig:figure9_10} are the normalized streamwise and the spanwise temporal correlation from simulation data. The solid red lines are exponentials with the above given integral time scales which compare favorably to the LES data.
 \begin{figure}
 \centering
    \includegraphics[trim =  0.5cm 0.5cm  2.5cm 1.8cm, clip, width=0.49\textwidth]{figure7.pdf}
    \includegraphics[trim = 0.5cm 0.5cm  2.5cm 1.8cm, clip, width=0.49\textwidth]{figure8.pdf}
    \caption{
    Comparison of the normalized temporal velocity correlations on the turbine columns from LES data (blue dashed line) to the exponential decorrelation in equation \eqref{eq:exp_decay_rsvel}, (red solid line). Left: normalized streamwise temporal velocity correlation $R_{11}( \tau)= \langle u_1'(\boldsymbol x ,t + \tau) u_1'(\boldsymbol x,t) \rangle_{\tilde{\boldsymbol x} t}$ from LES data and exponential decorrelation with the same streamwise integral time scale as given through the LES.
    Right: normalized spanwise temporal velocity correlation $R_{22}(\tau)= \langle u_2(\boldsymbol x ,t + \tau) u_2(\boldsymbol x,t) \rangle_{\tilde{\boldsymbol x} t}$ from LES data and exponential decorrelation with the same spanwise integral time scale as given through the LES.}
    \label{fig:figure7_8}
  \end{figure}
  \begin{figure}
    \includegraphics[trim =  0.5cm 0.8cm 3.5cm 1.8cm, clip, width=0.49\textwidth]{figure9.pdf}
    \includegraphics[trim =  0.5cm 0.8cm 3.5cm 1.8cm, clip, width=0.49\textwidth]{figure10.pdf}
    \caption{Comparison of the normalized temporal velocity correlations between the turbine columns from LES data (blue dashed line) to the exponential decorrelation in equation \eqref{eq:exp_decay_rsvel}, (red solid line). Left: normalized streamwise temporal velocity correlation $R_{11}(\tau)$ from LES data and exponential decorrelation with the same streamwise integral time scale as given through the LES.
    Right: normalized spanwise temporal velocity correlation $R_{22}(\tau)$ from LES data and exponential decorrelation with the same spanwise integral time scale as given through the LES.}
    \label{fig:figure9_10}
  \end{figure}
As a reminder, we also use the LES data for obtaining the streamwise space correlation $R_{11}(r_1)$ for the model. The transverse correlation $R_{11}(r_2)$ is modeled as a Gaussian in section \ref{subsec:Reduction} which requires the determination of the transverse integral length scale $L_2$.
The transverse integral length scale $L_2$ of the streamwise velocity is taken over the whole spanwise direction, without a distinction of the two cases, by
\begin{equation}
  L_2 =  \frac{1}{\langle u_1'^2\rangle_{\boldsymbol x t}}\int\limits_{0}^{\tilde{l}} \mathrm{d}r_2 \, \langle u_1'(x_1, x_2 +r_2, t) u_1'(x_1,x_2,t)\rangle_{\boldsymbol x t}\, .
  \label{eq:L2_integral}
\end{equation}
As above, the $\tilde{l}$ indicates the first zero-crossing of the correlation so that also for the integral length scale only the positively correlated range is taken into account. The right-hand plot of figure \ref{fig:figure5_6} shows the transverse spatial correlation of the streamwise velocity component from the LES in comparison to the Gaussian model. The choice of the Gaussian with the integral length scale $L_2$ as computed by \eqref{eq:L2_integral} shows good agreement with the LES data.

A summary of the free parameters is shown in table \ref{tab:parameters}. In fact, the free parameters from the case along the lines of turbines and between differ essentially only in the computation of the space correlation data along the respective subset of lines and the integral time scales $T_1$ and $T_2$. With the possibility to set the free parameters, the characteristics of the LES space-time correlation can be captured, e.g.~the choice of the spanwise random sweeping variance and the integral length scale influences the amplitude of the time kernel, whereby the streamwise random sweeping influences the amplitude and the width.

\begin{table}
\begin{center}
\begin{tabular}{l| p{3cm} |p{3cm}  }
   Parameter evaluated from LES & {Case 1: along\newline the lines of turbines} & Case 2: between\newline the lines of the turbines \\
  \hline
 $\phantom{\langle} U = \langle u_1 \rangle_{\boldsymbol x t}$  & \multicolumn{2}{c}{ $ 7.88\, u^{*\phantom{2}}$ } \\
$ \langle v_1^2 \rangle =   \langle u_1^2 \rangle_{\boldsymbol x t} -\langle u_1 \rangle^2_{\boldsymbol x t} $  & \multicolumn{2}{c}{$3.60 \, u^{*2}$ } \\
$ \langle v_2^2 \rangle =   \langle u_2^2 \rangle_{\boldsymbol x t}  $  & \multicolumn{2}{c}{$1.48 \, u^{*2}$ } \\
$ \phantom{\langle} L_2  $  &  \multicolumn{2}{c}{$0.09 \, H\phantom{^{*2}}$}  	 \\
 \hline
$\phantom{\langle} T_1  $     & $0.033 \, H/u^*$ &  $0.054 \, H/u^*$ \\
$\phantom{\langle} T_2 $   & $0.016 \, H/u^*$ &  $0.021  \, H/u^*$\\
 \end{tabular}\\
  \end{center}
    \caption{Free parameters for the model for the two cases; $U$, $\langle v_1^2\rangle$,  $\langle v_2^2\rangle$, and $L_2$ are the same for both cases whereas the integral time scales change. Furthermore, the space correlation $R_{11}(r_1)$ which is used as input for the model differs for the two cases.
    }
  \label{tab:parameters}
  \end{table}

\section{Comparison of the space-time correlation model with LES data}
\label{sec:Comparison}

For the comparison of our model space-time correlations to LES, we consider velocity correlations both on wind turbines and between turbine columns in the following sections \ref{subsec:on turbines} and \ref{subsec:between turbines} separately. In section \ref{subsec:discussion}, the results are discussed and the two cases are compared.

\subsection{Space-time correlations on wind turbine columns}
\label{subsec:on turbines}
In the first case, all spanwise averages for the LES space-time correlation $R_{11}(r_1, \tau)$ and for the space correlation $R_{11}(r_1)$ are taken with respect to the reduced subset of $\tilde{x}_2$-coordinates along the lines of turbines, i.e.~the spanwise average over 16 lines separated by the spanwise turbine spacing.
The parameters $U$, $\langle v_1^2 \rangle$, $\langle v_2^2\rangle$, $T_1$, $T_2$, and $L_2$ are chosen according to case 1 in table \ref{tab:parameters}.
\begin{figure}
  \centering
    \includegraphics[trim = 0cm 0cm 0cm 0cm, clip,  width=0.49\textwidth]{figure11}
    \includegraphics[trim = 0.0cm 0cm 0cm 0cm, clip,  width=0.49\textwidth]{figure12}
    \caption{Comparison of space-time correlations $R_{11}(r_1, \tau)/\langle u_1'^2 \rangle_{\tilde{\boldsymbol x} t}$ on the turbine columns from LES data to the model space-time correlations.
    Left: normalized space-time correlation of the streamwise velocity component from LES data on lines of wind turbines. White isocontours are shown for the interval $[0.0, 1.0]$ in $0.1$-steps. The slope of the LES space-time correlation originates from the choice of $U$ which is used to non-dimensionalize $\tau$, a slightly larger $U$ would increase the slope towards 45 degrees. Right: normalized model space-time correlation with $R_{11}(r_1)$ from LES with parameters $U$, $\langle v_1^2 \rangle$, $\langle v_2^2 \rangle$, $T_1$, $T_2$, $L_2$ as given in table \ref{tab:parameters} (case 1).
    The kink in the isocontour lines in the right-hand side plot results from the analytical form of the decorrelation kernel.}
    \label{fig:figure11_12}
  \end{figure}

The contour plots of the LES space-time correlation and the model space-time correlation are shown in figure \ref{fig:figure11_12} which allow for a qualitative comparison.
The left panel of figure \ref{fig:figure11_12} shows the space-time correlation evaluated from the LES data. The streamwise velocity fluctuations decorrelate both in space as well as in time. The characteristic tilt of the space-time correlation is clearly given by the mean flow advection. The highly correlated region (red color) is small, the main decorrelation takes place within one turbine spacing, $|r_1|/H \leq 0.785$ and one inter-turbine travel time (i.e.~the time for traveling from one turbine to the next, here with the mean velocity $U = 7.88 \, u^*$), $|\tau| U/H \leq 0.785$.
In this iso-correlation contour plot, we find wiggles caused by the periodic modulation of the flow due to the presence of wind turbines. The right panel of figure \ref{fig:figure11_12} shows the model space-time correlation where the free parameters have been obtained from the LES which clearly shows the same characteristics as the space-time correlations from LES in the left panel. The wiggles are not captured by the model.

For a more quantitative comparison, figure \ref{fig:figure13_14} shows cuts through the contour plots in figure \ref{fig:figure11_12}. The cuts through the contour plots confirm a very good agreement with the LES data, demonstrating that our model for the space-time correlations captures the temporal decorrelation trends well. The temporal increments in the left panel of figure \ref{fig:figure13_14} are given in terms of the inter-turbine travel times. The spatial increments in the right panel are related to the turbine distance. The results are comparable also for other increment sizes than the ones chosen in the right panel of figure \ref{fig:figure13_14}, as can be confirmed through figure \ref{fig:figure11_12}.
  \begin{figure}
  \centering
    \includegraphics[trim = 0.5cm 0.5cm 2.5cm 1.8cm, clip,  width=0.49\textwidth]
    {figure13.pdf}
  \centering
    \includegraphics[trim = 0.5cm 0.5cm 2.5cm 1.8cm, clip,  width=0.49\textwidth]{figure14}
\caption{Horizontal and vertical cuts through the contour plots \ref{fig:figure11_12} for the comparison of space-time correlations on the turbine columns from LES to model space-time correlations. Dashed lines correspond to the LES data, solid lines to the model space-time correlation with $R_{11}(r_1)/\langle u_1'^2 \rangle_{\tilde{\boldsymbol x} t}$ from LES, and $ U$, $\langle v_1^2 \rangle$, $\langle v_2^2 \rangle$, $T_1$, $T_2 $, $L_2$ as given in case 1 in table \ref{tab:parameters}. Left: normalized $R_{11}(r_1, \tau)$ at fixed instants in time as a function of $r_1$. The solid orange curve in the left panel is per definition the normalized space correlation $R_{11}(r_1)/\langle u_1'^2 \rangle_{\tilde{\boldsymbol x} t}$ from LES data. Right: normalized $R_{11}(r_1, \tau)$ at fixed spatial increments. $S_x=0.785\, H$ is the turbine spacing.
The dashed orange curve from the LES in the right panel for $r_1/H = 0$ is the same as the dashed blue line in the left panel in figure \ref{fig:figure7_8}. Although not exactly the same, the orange solid model curve for $r_1/H = 0$ is in good approximation the corresponding exponential in the left panel of figure \ref{fig:figure7_8}.}
\label{fig:figure13_14}
\end{figure}

The  value of the overall mean velocity $U = 7.88 \, u^*$ (cf.~table \ref{tab:parameters}) matches well with the moving peaks.
The amplitude and width of the curves corresponding to full multiple turbine distances or full multiple inter-turbine travel times show a visibly better agreement between the model and the LES.
Furthermore, the amplitude of the LES correlation does not seem to change considerably when increasing the spatial increment from half increments of turbine distances to full integer multiples of turbine distances.
This effect causes the wiggles in the LES contour plot \ref{fig:figure11_12} are not directly visible in the cuts in figure \ref{fig:figure13_14}. They can be observed indirectly by the fact that the amplitude of the curves stay constant over several increment sizes.
This may be related to the fact that the main decorrelation takes place within half the turbine distance to the downstream turbine. The turbulence intensity is considerably higher in the first half distance and then decreases immediately in front of the next turbine, cf.~right panel in figure \ref{fig:figure3_4}.

\subsection{Space-time correlations between wind turbine columns}
\label{subsec:between turbines}
In the second case, $R_{11}(r_1, \tau)$ and $R_{11}(r_1)$ from LES are computed with respect to the 16 lines between the turbines, separated by the spanwise turbine spacing. The values for $U$, $\langle v_1^2 \rangle$, $\langle v_2^2 \rangle$, $T_1$, $T_2$, and $L_2$ are chosen according to case 2 in table \ref{tab:parameters}.

The LES space-time correlation from between the turbines is shown in the iso-correlation contour plot in the left panel of figure \ref{fig:figure15_16}. The right panel of figure \ref{fig:figure15_16} shows a contour plot of the model space-time correlation \eqref{eq:spacetimecorr}. The model space-time correlation exhibits a less extended high correlated (red) area than the LES space-time correlation. For a more quantitative comparison of the model and the LES space-time correlation, figure \ref{fig:figure17_18} shows cuts through the contour plot \ref{fig:figure15_16} for various choices of $r_1$ and $\tau$. The less extended red area of the model carries over to considerably faster decreasing amplitudes of the model curves in comparison to the LES curves in figure \ref{fig:figure17_18}.
\begin{figure}
  \centering
    \includegraphics[trim = 0cm 0cm 0cm 0cm, clip,  width=0.49\textwidth]{figure15}
    \includegraphics[trim = 0.0cm 0cm 0cm 0cm, clip,  width=0.49\textwidth]{figure16}
    \caption{Comparison of space-time correlations $R_{11}(r_1, \tau)/\langle u_1'^2 \rangle_{\tilde{\boldsymbol x} t}$ between the turbine columns from LES data to the model space-time correlations. Left: normalized space-time correlation of the streamwise velocity component from LES data. White isocontours are shown for the interval $[0.0, 1.0]$ in $0.1$-steps. The slope of the LES space-time correlation can be increased by increasing $U$. Right: normalized model space-time correlation with $R_{11}(r_1)$ from LES and parameters $U$, $\langle v_1^2 \rangle$, $\langle v_2^2 \rangle $, $T_1$, $T_2$, $L_2$ according to table \ref{tab:parameters}, case 2. As in the previous case along the turbine lines, the kink in the isocontour lines of the model plot results from the decorrelation kernel.
    }
    \label{fig:figure15_16}
  \end{figure}

  \begin{figure}
  \centering
    \includegraphics[trim = 0.5cm 0.5cm 2.5cm 1.8cm, clip,  width=0.48\textwidth]{figure17}
    \includegraphics[trim = 0.5cm 0.5cm 2.5cm 1.8cm, clip,  width=0.48\textwidth]{figure18}
   \caption{Horizontal and vertical cuts through the contour plots \ref{fig:figure15_16} for the comparison of  normalized space-time correlations between the turbine columns from LES to the model space-time correlations.
   Dashed lines correspond to the LES data, solid lines to the normalized model space-time correlation with $R_{11}(r_1)/\langle u_1'^2 \rangle_{\tilde{\boldsymbol x} t}$ from LES, and the parameters $ U$, $\langle v_1^2 \rangle$, $\langle v_2^2 \rangle $, $T_1$, $T_2$, $L_2 $ from table \ref{tab:parameters}, case 2. Left: normalized $R_{11}(r_1, \tau)$ at fixed instants in time as a function of $r_1$. The solid orange curve in the left panel is per definition the normalized space correlation $R_{11}(r_1)/\langle u_1'^2 \rangle_{\tilde{\boldsymbol x} t}$ from LES data. Right: normalized $R_{11}(r_1, \tau)$ at fixed spatial increments. $S_x=0.785\, H$ is the turbine spacing. The dashed orange curve from the LES in the right panel for $r_1/H = 0$ is the same as the dashed blue line in the left panel in figure \ref{fig:figure9_10}. As in case 1, the solid orange model model curve approximates the corresponding exponential in figure \ref{fig:figure9_10} well.}
    \label{fig:figure17_18}
  \end{figure}

Figure \ref{fig:figure17_18} shows that, for small temporal and spatial increments, a value of $U = 7.88\, u^*$ seems to fit reasonably in terms of the moving peak of the model space-time correlation. For larger increments, however, a higher value for $U$ approaching the temporal and spatial mean velocity over the lines between the turbines seems more appropriate. The left panel plot of figure \ref{fig:figure17_18} shows very good agreement for small temporal increments (blue curve). Comparably good results are found for the spatial increments up to one turbine distance, $0.785\, H$, in the right panel of figure \ref{fig:figure17_18}.
For larger temporal and spatial increments, the model still captures the characteristics of the space-time correlations qualitatively, i.e.~in terms of amplitude, the broadening and the peak shift, however, quantitatively there are discrepancies observable.

To test whether these quantitative discrepancies can be in principle remedied, we evaluate the model for between the turbines with a different set of parameters, only for illustrative purposes. It turns out that by slightly adjusting the mean velocity and the second moment of the spanwise random sweeping, excellent results can be obtained for the lines between the turbines also for large temporal and spatial increments. Figure \ref{fig:figure19_20} shows a comparison to LES for the model space-time correlation with a mean velocity adjusted by a factor $1.14$ to $U=9.0 \, u^*$ and a spanwise variance adjusted by a factor $0.34$ to $ \langle v_2^2 \rangle=0.5\, u^{*2} $.
In contrast to the streamwise random sweeping, the spanwise random sweeping has influence only on the amplitude but not the width of the model curve. Therefore, we may conclude that the functional form of the proposed model is very satisfactory while the main discrepancies with data are due to deficiencies in prescribing the model parameters.
 \begin{figure}
  \centering
    \includegraphics[trim = 0.5cm 0.5cm 2.5cm 1.8cm, clip,  width=0.48\textwidth]{figure19}
    \includegraphics[trim = 0.5cm 0.5cm 2.5cm 1.8cm, clip,  width=0.48\textwidth]{figure20}
   \caption{
   Comparison of model to LES data for normalized space-time correlations between the turbine columns. It turns out that with the subsequently described set of parameters excellent agreement can be obtained. Dashed lines correspond to the LES data, solid lines to the normalized model space-time correlation $R_{11}(r_1,\tau)/\langle u_1'^2 \rangle_{\tilde{\boldsymbol x} t}$ where the values for $ U  = 9.0\, u^*$ and $\langle v_2^2 \rangle =  0.5\, u^{*2}$ are adjusted for a better accordance with the LES space-time correlation. As above, $R_{11}(r_1)/\langle u_1'^2 \rangle_{\tilde{\boldsymbol x} t}$ is evaluated from LES for the lines between the turbines, and the other parameters $\langle v_1^2 \rangle $, $T_1 $, $T_2$, and $L_2 $ are chosen according to case 2 in table \ref{tab:parameters}. Left: normalized $R_{11}(r_1, \tau)$ at fixed instants in time as a function of $r_1$. Right: normalized $R_{11}(r_1, \tau)$ at fixed spatial increments. $S_x=0.785\, H$ is the turbine spacing.}
    \label{fig:figure19_20}
  \end{figure}

\subsection{Discussion}
\label{subsec:discussion}
The comparison of the model \eqref{eq:spacetimecorr} to LES data shows good qualitative agreement in the sense that it captures the main characteristics of spatio temporal velocity correlations in a wind farm, including the spatial and temporal decorrelation and the characteristic tilt in the contour plots generated by an advection velocity. The decorrelation on the lines of turbines is higher than between which is visible in the comparison of the extension of the highly correlated (red) area in the contour plots \ref{fig:figure11_12} and \ref{fig:figure15_16}. The same observation follows from the cuts through the contour plots which show that between the turbine columns, velocity correlations decay slower in space and time, cf.~figure \ref{fig:figure13_14} and \ref{fig:figure17_18}. This is qualitatively reflected by the model.

Considering the fact that all free parameters are obtained from the LES (table \ref{tab:parameters}), the model performs overall well also in terms of quantitative results.
The mean velocity over the respective subsets of lines along the turbines is considerably smaller than between. Still, the choice of the overall mean velocity in our model yields reasonable results on the turbine lines when compared to the LES data.
One possible explanation could be that velocity fluctuations are not primarily advected with the mean velocity at hub height, but might be entrained from slightly larger heights with a correspondingly higher mean velocity, or influenced by shear effects as suggested in \cite{HeZhang2006}.

A similar effect became visible in the case between the turbines which in comparison to the LES fits much better for an increased value for the mean velocity. Increasing the value for the mean velocity with respect to the inherent physics of the problem, and at the same time adjusting (by a significant amount) the second moment of the spanwise random sweeping, gives almost perfect overlaps of the curves for between the turbines, cf.~figure \ref{fig:figure19_20}.
Figure \ref{fig:figure17_18} shows that the width of the model space-time correlation between the turbines fits well, however the amplitude is decreasing too fast.
A decreased $\sigma_1$ increases the amplitude and decreases the width of the distribution, a decreased $\sigma_2$ only increases the amplitude and has no influence on the width. This observation suggests that the values chosen for $T_1$ and $\langle v_1^2 \rangle$ are reasonable, whereas a decreased spanwise random sweeping fluctuation $\langle v_2^2 \rangle$ serves to increase the amplitude, as done for figure \ref{fig:figure19_20}.
In fact, a similar effect would be obtained by decreasing the correlation time of the random sweeping velocity $T_2$.

Smaller $T_1$ and $T_2$ result in smaller $\sigma_1$ and $\sigma_2$, respectively shorter correlated random sweeping velocities $v_1$ and $v_2$ in equation \eqref{eq:exp_decay_rsvel}. Furthermore, smaller $\sigma_1$ and $\sigma_2$ increase the correlation of $u'_1$ in the model \eqref{eq:spacetimecorr} due to their influence on the amplitude and the width of the decorrelation kernel.
Effectively, this observation gives rise to further discussion with respect to the choice of values for $T_1$ and $T_2$ in the model.
On the one hand, the integral time scales obtained from the LES on the turbine lines are smaller than between the turbine lines, cf.~table \ref{tab:parameters}. This is reasonable considering an expected higher temporal decorrelation of velocity fluctuations on the turbine lines.
On the other hand, that means that $\sigma_1$ and $\sigma_2$ on the turbine lines are smaller than between the turbine lines. Consequently, with the current choice of values for $T_1$ and $T_2$, the influence of the temporal decorrelation kernel in the model space-time correlation is smaller on the turbine lines than between. However, we leave the discussion of obtaining alternative values for $T_1$ and $T_2$ in the model open for future work.

To sum up, as already mentioned in the introduction, the temporal decorrelation of the random sweeping velocity has a major influence on the results and we conclude that a temporal decorrelating random sweeping, as incorporated in our model, is necessary to capture the decorrelation more accurately.

\section{Summary and Outlook}
\label{sec:Outlook}

In this paper, we have presented a model for velocity space-time correlations in turbulent flow along with its application to streamwise velocity fluctuations in fully developed wind turbine array boundary layers. The model is based on the classical Kraichnan-Tennekes random sweeping hypothesis with additional mean flow, and takes into account finite correlation times of the large-scale random sweeping velocity.

Under the given assumptions, the streamwise space-time correlation is obtained as a convolution of the space correlation and a decorrelation kernel, consistent with the classical Kovaznay-Corrsin conjecture. The properties of the analytical decorrelation kernel are fixed by the modeling assumptions, such that the spatial correlation function along with simple statistical characteristics of the velocity field (mean fluctuations, integral time and length scales) suffice to make a prediction for the full space-time correlation.

We then presented detailed comparisons of the model to LES results of a periodic wind farm flow. Taking space-time correlations along the wind turbine columns and between the wind turbine columns as two typical benchmark cases, we find good qualitative agreement given the simplicity of the modeling approach. In this context, it appears worth mentioning that the needed model parameters have been obtained directly from the LES data, such that no free fitting parameters are contained in the formulation. Therefore, the model can be easily adapted to different simulation scenarios and field measurements, which presents one direction of future research.

Regarding limitations of the model, quantitative differences between the model and the LES space-time correlations were observed, especially for the correlations between the turbine columns. These shortcomings are presumably rooted in some oversimplied modeling assumptions as well as the small number of parameters. However, we demonstrated excellent agreement by adjusting the mean velocity and the variance of the spanwise random sweeping. This suggests the potential of the model to capture the space-time correlations quantitatively. One interesting question for future research is therefore how additional physical effects such as vertical energy entrainment or shear affect  the model and its parameters. While vertical energy entrainment may help to justify a higher mean convection velocity, effects like shear can give rise to increased mean velocity fluctuations \cite{HeZhang2006, ZhaoHe2009}. Aiming towards a fully analytical model, it will be interesting to also obtain the spatial correlation function, which is currently obtained from the LES data, from additional theoretical arguments. Such an approach, for example, has been taken in the derivation of the wavenumber-frequency spectrum model with a model wavenumber spectrum \cite{WilczekStevensMeneveau2015}.

Considering power output fluctuations as a next step, both LES results such as \cite{StevensMeneveauRenSus2014} as well as wind tunnel measurements of micro wind farms \cite{BossuytHowlandMeneveauMeyersTorque2016, BossuytMeneveauMeyers2017} may serve as important reference points for future tests of our model.

\acks Laura J.~Lukassen and Michael Wilczek are supported by the Max Planck Society.
Richard J.A.M.~Stevens is funded in part by the Shell-NWO/FOM-initiative Computational sciences for energy research of Shell and Chemical Sciences, Earth and Live Sciences, Physical Sciences, FOM and STW. This work was carried out on the national e-infrastructure of SURFsara, a subsidiary of SURF cooperation, the collaborative ICT organization for Dutch education and research. Charles Meneveau was supported by NSF (grant OISE-1243482, the WINDINSPIRE project).

redpajama_set_name: RedPajamaArXiv


Tsalanang ECD – Learning is fun!
The team at Ripple Reading, together with Sakata Seed Southern Africa, are supporting Teacher Dorah with a dedicated Teacher (Thembi) for the upgraded Grade R class and programme so that young children build a good foundation from which their academic school career can develop. The programme aims to improve the young learner's vocabulary as well as their experience and understanding of words and the English language. Drama plays an important role in building confidence and helping with the learning experience.
Julia an experienced teacher at Trinity Private College is assisting and mentoring Thembi, the new teacher for the Gr R class.
Elize is in full swing using Drama to develop language skills. Teacher Samantha, who is currently being trained through the Heron Bridge Teacher Resource Centre, with the help of Sakata, as well as Teacher Thembi, join in the fun!
The Principal of Tsalanang ECD, Teacher Dorah, gets up close with the learners to share the experience!


This Cemetery is located 2.5 miles from Proctor on Huff Ridge. It is on the right side of the road and up on a knoll. It is overgrown. It was verified by Thomas N. Fankhouser and Bill Cunningham on 6 Dec 1998.


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Grand Forks' Center for Exploration receives major financial boost as children's museum, destination park
Operation Exploration committee receives $100,000, Tourism Planning Grant from the state's Department of Commerce
A conceptual drawing of the Grand Forks Center for Exploration (Contributed image)
By Pamela Knudson
An idea that has been floating around the community for about 30 years has taken a big step forward with a grant from the North Dakota Department of Commerce.
The Grand Forks Center for Exploration, otherwise referred to as a children's museum, has received a $100,000 Tourism Planning Grant to support professional planning for the facility, which is being developed to enhance the lives of area residents but also to draw visitors from out of state.
The Grand Forks Center for Exploration is one of five recipients of this award, selected from a field of 18 applications. It was the only project in the Red River Valley to receive this type of grant this year. The awards, ranging from $100,000 to $400,000, totaled $1 million.
The funds will be used to develop a comprehensive plan needed to build the Center for Exploration, described as a destination children's museum that provides STEAM – science, technology, engineering, arts and mathematics – learning activities for people of all ages and abilities.
The building will serve as a connection point for the region with exhibits featuring agriculture, wind energy, technology and an event space for gatherings.
The project, still in the conceptual phase of the fundraising process, is part of a larger visionary campaign called "Operation Exploration," said Jill Nelson, director of operations and community relations with the Grand Forks Park District and member of the Operation Exploration Campaign board, which is guiding the progress of the project.
Through philanthropy, naming opportunities and grant funding, the Operation Exploration Campaign committee, along with the Grand Forks Park District, is working on plans for the Center for Exploration children's museum and destination park, Nelson said.
The center will offer "an interactive learning experience by highlighting regional industries and resources with multiple permanent and rotating exhibits, while the year-round destination park will provide an imaginative and inclusive play experience that will be free for all to enjoy," she said.
The outdoor exhibits could include vegetable gardens, with the produce being used inside the facility as the basis for "play-based learning – and how it goes from field to table" said Dean Opp, retired director of the Grand Forks school system's Summer Performing Arts program.
Similar types of practical learning could be applied to exhibits focused on wind energy and aerospace, he said.
Operation Exploration committee members estimate that the entire project is valued at $30 million, including about $20 million for the facility, which may occupy 22,000 to 25,000 square feet.
Members plan to continue to seek support for an endowment for the center and seek city, county and state awards as well.
The Center for Exploration would be sustained with donor membership and admission will be charged. Renting space for events, receptions and meetings will help to generate income, Nelson said.
The committee is hopeful the project could start in early 2023.
The proposed location for the exploration center is not yet being made public, because details of an agreement to attain the property are not finalized, committee members said in a recent interview with the Herald.
'Great idea'
When first approached with the idea of a children's museum, Opp said he recalled thinking, "this isn't just an interesting idea, but a great idea."
It surfaced at a time when, because of the pandemic, people were "looking for something positive," Opp said, noting that Mayor Brandon Bochenski's interest in developing Grand Forks helped to fuel enthusiasm.
Opp also noted the importance of inclusivity in developing the center, both in terms of accessibility for all and the types of exhibits that will be displayed. The notion of inclusivity extends to people who are dealing with Alzheimer's, with committee members thinking of ways to ensure their comfort, or the elderly who appreciate chairs with arms to help them more easily rise from being seated.
About the committee members who are involved in the project, Opp said, "It's amazing to see the vision they have – they're thinking big. It's impactful."
Receiving the state Department of Commerce grant gives the project not only a big boost, but "it adds a lot of validity to what we're doing," he said.
"The passion of everyone on this committee is inspiring," Nelson said, noting that they've been meeting and brainstorming with potential donors. "It's a bigger-than-life goal. And we've been met with an overwhelming response from the community."
Sally Miskavige, vice president of the Operation Exploration Campaign, said the idea of the children's museum has been around for decades, as several education and other community leaders discussed a project of this type.
"Grand Forks is such a wonderful place to live. I raised my family here; we have wonderful teachers in Grand Forks," Miskavige said.
"We want to continue to invest in the community, and keep it a great place to live."
'Destination project'
The 2021 Tourism Planning Grant, appropriated by the Legislature, is intended to support professional planning for tourism developers who have a destination project that would have the ability to attract visitors from outside North Dakota and retain them for at least four to six hours – and preferably an overnight stay in or near the community.
"North Dakota has ample opportunities to expand tourism experiences that enhance the quality of life for residents and attract visitors," said Sara Otte Coleman, state tourism and marketing director. "We expect the plans developed will provide insight for future attractions that will be a driving force in boosting our economy by creating more entertainment and recreation attractions."
Said Miskavige: "We are excited and thankful to be awarded a 2021 North Dakota Tourism Planning Grant. This grant provides our board with the assistance needed to help launch this amazing project beyond the conceptual phase."
With construction costs raised through the Operation Exploration capital campaign, the committee plans to build a destination playground adjacent to the Center for Exploration, Nelson said. With the commitment of the Grand Forks Park District to own and operate the park, it will be free to residents and visitors, making this an attractive amenity for tourism in Grand Forks.
"Unlike standard neighborhood parks, destination playgrounds draw in hundreds of visitors and residents daily, due to their unique themes and large play structures," Nelson said.
Julie Rygg, executive director of Visit Greater Grand Forks, said, "The Center for Exploration will assist with visitation and new resident efforts not only for Greater Grand Forks, but all of North Dakota. This is a wonderful opportunity for our community."
Speaking for the Grand Forks Parks District, Nelson said, "We are honored to be part of this innovative project. This regional attraction will greatly benefit Greater Grand Forks while demonstrating our community's investment in our local industries, our region and our future generations."
More information about the Tourism Planning Grant can be found at www.ndgov.link/Tourism-Planning-Grant .
For more information about the Operation Exploration campaign, or, if interested, how to get involved in the project, contact the board at info@centerforexploration.com .
A conceptual drawing of the proposed Grand Forks Center for Exploration and destination park (Contributed image)
Related Topics: NONPROFITS
Letter: Comparing the Electoral College vote with the will of the people
Mayville Public Library raises $21,000 for repairs on historic building
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My student's friend came to me for some lessons on grade 8 aural test. She was a quick learner, and had quite a good ear, so she did well even just for a few hours of this rather difficult aural training (especially the harmonic progression part). Then she asked for my advice on her grade 8 pieces. So I listened to her playing, and was horrified to find she play this first piece incredibly fast. I asked her why she took such a quick tempo. She replied that her teacher told her it was a dance piece so it should be played as fast as possible (apparently Prestissimo was not even fast enough!).
The piece is a Baroque sarabande. Indeed it is a dance piece, but a slow one! In Baroque era, dance pieces were popular, and they were grouped as sets of dance suites. For instance, in the keyboard works of J.S. Bach, there are French Suites and English Suites. Each suite consists of several numbers of dance pieces, e.g. Allemande, Courante, Sarabande, Gigue etc. Under the specified name there is a set style and tempo, that means "Gigue" is definitely a quick dance, while "Sarabande" is a slow one. If the girl played that exam piece at such a fast speed, she would for sure fail. It was fortunate she could change the way she had been playing for a year in a few days, and eventually she was awarded grade 8 proficiency with merit. I could tell she could have gotten distinction should she study with a well-informed teacher.
There was a student who played a wind instrument needed my help for aural training and piano accompaniment for her graded exam. She seemed to feel pretty confident in her preparation. We started our practice with the first piece; however I found she was not that well prepared: the tempo and pulse were unsteady. I thought okay at least this song could be fixed in a short period of time (the exam was due in ten days!). As we proceeded to work on the second piece, I realized the way she played one repeated rhythmic pattern was just plain wrong. It was a regular two-quaver rhythm, but in this particular song, the composer indicated in the score that it was a "swing" rhythm, that means the first quaver should be played double the length as the second one, and therefore, we could view the whole rhythmic pattern as a triplet, so that the first note was a crotchet and the second note a quaver. Or in simple language, the first note should be played longer than the second one.
It was easy to recognize the mistake, but difficult to fix it. When a student had worked on something wrong for long time, it would take much longer time and much more effort to correct it and re-learn something right. I did not want to hurt the student's feeling and confidence, while at the same time I needed to tell her to change almost everything. It was not a smooth path to be taken.
I did not want to venture into fixing the third piece since it was a solo piece without piano accompaniment. But since the student's parent asked for my advice, I listened to it. I was filled with headache: it was totally under preparation! There was no sense of direction, the notes were crumbled together, the tempo and rhythm were off, there was not much of dynamic contrast, etc. Fortunately, it was not a long piece of music. Still, I worked with the student phrase by phrase, identifying the repeated patterns and sequences she had never recognized before. It was like learning a new piece from scratch.
Here came the best part, or the worst case scenario: scales. The student simply had no idea which ones she should be practising on and therefore the whole time during her practice she had been working on the wrong ones. That was not something I could fix in two days before the exam. The only thing I could do was simply to tell her which ones she needed for the exam and work on them a bit with her in lesson so that she could go practise on her own at home.
Be positive and you will get positive results. Be negative and you might never get there.
In my studio policy I suggest beginners and students of first year's learning keep a daily routine of 20-30 minutes' practice. I feel that I have to further elaborate on such guideline.
This guideline indicates a "general" schedule. For really young beginners (4-5 years' old), they should not be pushed to practise 20-30 minutes every day. During the first few months of piano training, young children should probably practise about 10-15 minutes which can be extended to 20 minutes only when necessary and when the children themselves want to play more. I suggest my students' parents that it would be more effective when they break daily practice into 2 sessions of 10 minutes and keep each session short and precise. For instance, children can practise 10 minutes after school, having a little rest and small high-protein snack for energy boost. Then they can go on with their daily routine.
Another 10-minute session can occur after dinner. If the students get everything done in the first 10-minute session or there is really no more time, there is no need to make them practise another 10 minutes. But students should work on different music or aspects of music in the two sessions or else they would feel bored and obliged to practise, wondering why they have to repeat same practise twice.
Remember, this is only applied to beginner students of ages 4-5. From age 6 onwards children should be able to sit still and practise for at least 15 minutes a day or in one session.


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Home / Stories/Articles / Featured / 10 Most Important Battles in American History and Why: Part IV—the Final Two.
10 Most Important Battles in American History and Why: Part IV—the Final Two.
By Mr. Twisted
By RU Twisted
Well, here we are. The final two. I'd like to say something profound that would be the poetically written version of a drum roll, but instead I'll save the hoopla and get right to it, as you've probably already guessed what we're covering by this point.
The Battle of Gettysburg
On July 1, 1863, Union and Confederate forces collided in the town of Gettysburg, Pennsylvania, which had an estimated population of about 2,000 people. The battle lasted through July 3 and is considered by most to be the seminal conflict in the Civil War.
Confederate General Robert E. Lee, after his victory at Chancellorsville, led his Army of Northern Virginia on a second invasion of the north with the intent on taking the fighting away from an already war-ravaged South. Major General George Meade, three days into his command of the Union Army, had moved his forces into Pennsylvania with the intent of being in between the Confederate Army and Washington, D.C.
For three days, elements of both armies met at places like Oak Hill, Seminary Ridge, Barlow's Knoll, Devil's Den, Little Round Top, the Wheatfield, and many others in what would come to be known as not only a turning point in the war but also a focal point of American history. Casualties on both sides were extreme but the Confederate forces suffered absolutely devastating losses and would never be the same again.
Why it's important: I'm not sure that the significance of the Battle of Gettysburg can be overstated. Between the first and third of July, Union and Confederate forces are estimated to have lost between 45,000 and 51,000 in killed, wounded, or missing soldiers. Of the 120 general officers present, 9 were either killed or mortally wounded during the battle. 63 medals of honor were given to Union Soldiers alone for their actions. Days 1, 2, and 3 of the battle by themselves rank in the top 15 of the bloodiest battles of the Civil War.
But let's be clear; the Battle of Gettysburg is important for much larger reasons than just statistics. Its significance goes well beyond mere numbers of casualties.
Gettysburg was without question a major turning point in the war that did more to shape our country than probably any other. After July 3, 1863, Confederate forces would be permanently on the defensive in a conflict that is still largely misunderstood by a great deal of our population. The inability of Confederate forces to mount significant offense into the North proved their lack of material and personnel resources. Conversely, it showed both sides that the Union Army had resources to spare and was willing to do so in any number of ways.
The Battle of Gettysburg marked the beginning of the end of an idea that dates back to the Articles of Confederation and the Constitution itself. The concepts of states' rights, of the right to secede, and what federalism meant were all radically changed during and after the Civil War, and the events at Gettysburg signified that change. As many historians have pointed out, the Civil War was a demarcation point when we went from "these" United States to "the" United States. Gettysburg was the same point within the war itself and, as a result, has become one of the most significant battles in our history.
Operation Overlord, aka The Battle of Normandy
On June 6, 1944, after months of planning and high-level deceptions thrown at the Germans, Allied forces invaded Normandy, France on what has come to be one of the most easily recognized names in military history—D-Day. Over 150,000 troops in 5,000 ships and 1,200 planes crossed into German-occupied France under the leadership of GEN Dwight D. Eisenhower, who had been named Commander of the Supreme Headquarters Allied Expeditionary Force (SHAEF—proving that awkward nomenclature didn't begin in the 1980s). It was an operation so large and taken so seriously that the live-fire rehearsal, Exercise Tiger, cost the lives of 946 American Servicemen.
American, Canadian, British, and French forces inserted via water and air in military offensive that was unparalleled in size and scope—especially when factoring in this new and totally kick-ass methodology known as "airborne!" (I may, for the record, be somewhat biased). By the end of the operation in late August (Operation Neptune was the Normandy assault; Overlord was the name for the entire operation), over two million Allied troops were in France. Between 6 June and the end of August, American forces suffered over 20,000 killed and over 90,000 wounded (total Allied casualties topped 200,000).
Why it's important: In the context of the European theater of World War II, nothing was more significant than Operation Overlord. From a strategic point of view, the level of planning and deception that went into setting the stage for the Normandy invasion was without comparison. From a tactical perspective it was not only successful, but also provided numerous lessons for future leaders and students of military theory. Consider the fact that, despite the tremendous level of preparation as well as the experience of the leadership involved, it was almost as if they planned for every contingency on the first day and neglected to make provision for what would meet them on the second day and following. The bocage—or thorny hedgerows—proved to be a major stumbling block that forced some of the heaviest and bloodiest fighting and seems to have been almost overlooked completely in the planning stage.
Yet it remains, perhaps for that reason, as one of the greatest examples of American fighting forces overcoming obstacles in the history of our nation. It is due to this that the Normandy invasion's importance transcends military strategic significance.
No other battle or series of battles features so prominently in our minds when considering the topic of American exceptionalism. When we think of our nation's heroes fighting against tremendous adversity in horrible conditions, chances are pretty good that we reflect on the opening scene of Saving Private Ryan, The Big Red One, or shots from one of several other D-Day themed movies. And while that could indicate that our ideas of American greatness are somewhat skewed by Hollywood, the simple fact is that those movies resonated with us in the first place because this was such an extraordinary event.
D-Day, and Operation Overlord as a whole, marked a truly significant moment in American history. It wasn't just one of the biggest military operations ever conducted by our forces; it marked the beginning of the end of the largest war the world had ever seen. The end of WWII ushered in a period of American greatness that, although possibly mythologized by popular media, still gave birth to huge numbers of people that resulted in me, you, and an explosion of technological advancement unlike any the world has ever seen. The invasion of Normandy signals the introduction of that era.
Well, there you have it—the ten most important battles in our country's history. I'm sure there are several that deserved to be on this list that didn't make it, but I'm also sure that the ones I included are pretty damn important and worthy of study before being dismissed.
America, for better or worse, is what it is because of a lot of bloodshed. The very least we can do is understand how important those events were and why.
Editor's note: click the links here for Part 1, Part 2, and Part 3 of the series.
American historyD-dayfeaturedGettysburg
leftoftheboom
Outstanding list.
You called it correctly in the comments of the last installment.
Whoo Hoo. What do I win?
But seriously it was a good list and I think it would be pretty hard to argue the last two because your write up told the tale pretty well.
I don't know if you every checked out the Green Books, from the Center for Military History but the entire website is pretty awesome for a military history buff.
I was not familiar with the Green Books until you mentioned them and then I looked them up.
And I now have you to thank for an even bigger reading list. Wow.
There is an awesome amount of history there with actual battlefield information in detail.
I personally love the maps but there are definitely years worth of reading there and you can download most of it.
Here is the link for everyone interested. http://www.history.army.mil/ and it is not just the Green Books, under the research tab, you got it all.
The Battle of New Orleans was nowhere on this list. We are now mortal enemies.
I challenge you to a duel*, Sir!
*This is, in my circle, common parlance for "buy you some whiskey."
Why would Hurricane Katrina make the list?
Baaaaaaaaaaaahahahahahahahaha.
That is pretty funny!
Really enjoying these article's, a note though at the same time that you mention the significance of Gettysburg you should combine in, similar to the Manila, San Juan article, the fall of Vicksburg as well. Partly being that Lincoln actually recieved word of it fall to Grant, and Meade's victory within hours of each other. It helped cement Lincoln position as president for now in an unpopular civil war he had able commanders in both the Eastern and Western theaters and a hell of a lot of good PR for the election that was upcoming later in the year. I could go on, don't get to use my history degree often doing commercial roofing, but then my post would be as long as the article. God bless, love the articles, the clothing line, and the site, keep up the good work.
Mean Tweets and PTSD: Facts and Fakers
By Havoc 13 Americans love labels, and we can't wait to show ours off. We look to distinguish ourselves...


Porcelain Hand Bell, featuring The Reichsadler ("Imperial Eagle") along with German city crests DEUTSCHLAND - GERMANY Gold rings and tip.
DEUTSCHLAND - GERMANY Gold ring and tip.
Click the button below to add the Porcelain Hand Bell - The Imperial Eagle and German city crests DEUTSCHLAND to your wish list.


Highest Quality full Restoration, including Mechanicals. Dual Spare Tires with Stainless Trim, a Gravel Guard for the Grille, a Trunk, Wing Windows, Side Curtains, a Flying Quail Radiator Cap, Hood Mounted Driving Lamps, a "Sportlite" Spotlight, Leather Upholstery, and more.
Total frame off, every nut and bolt restoration, way beyond better-than-new standards. All mechanicals were also done during the restoration. Spectacularly finished in black with red interior, this is one of the finest examples anywhere.
High quality restoration in the early '90s and has been in a noted Seattle area collection for the past decade. Silver exterior over Black leather interior with red piping. Chrome wire knock-off wheels, Lucas driving lamps. Recent servicing including rebuilt brakes, fresh fluids, and Avon tires. Ready to enjoy.
This is a very well preserved older restoration that shows excellent. It is finished in original Starmist White with Blue and White Leather Interior with Exceptional Bright Work. It has rare rear window stabilizer bars along with front fender signal lights which can only be found on very few early MK II's.
Fuel injected 327/360 hp in correct Honduras Maroon. Fully documented, all numbers recorded, pictures, receipts, and ownership documentation. Very rare CA car with crankcase ventilation for emission, windshield washer delete, hard top and original wheel covers. NCRS Regional Top Flight car.
Highest quality Cobra Re-creation by Unique Motorcars completed Nov. 2008. Genuine Ford NASCAR aluminum side oiler 427 - pro-built with 475hp/495lb-ft. 13" Wilwood brakes, all leather interior, black soft top, and more. 72 miles since completion.
Correct Ermine White Exterior ordered from the Factory with an L36 396cid Big Block. Options include bucket seats, center console, AM/FM stereo radio, Powerglide transmission, gauge package, power steering, black vinyl top, Air conditioning, and cruise control. Celebrity owned.
A high quality frame-off restoration. Features an upgraded, Pro-built big block 396cid V8 with upgraded internals making approximately 450hp. Ceramic-coated headers, high performance exhaust, Tremec TKO 5-speed, 12-bolt Posi., much more.
Frame off restoration costing in excess of $65,000. Pro built 350ci, ceramic-coated headers and stainless exhaust, MSD ignition, Edelbrock RPM intake and carb set-up, Muncie 4-speed transmission, 12-bolt Positraction with 3:73 gearing, power front disc Brakes, fully powder coated suspension and frame.
It is Matching Numbers Ford Torino GT and Caringly Restored in Red over Black Dressed with Magnum 500 Wheels Shows 84,682 Miles on the Odometer. With less than 1% of all 1970 Torinos being GT Convertibles, the Marti Report Reveals this GT to be the Only One Built in this Specific Configuration.
Fully restored Pontiac Trans Am, white exterior with a black interior. Original numbers matching 455 Big Block. The engine, transmission, and rear end have all been professionally rebuilt with receipts. Factory options include extremely rare Air Conditioning. Complete Pontiac Historical Society paperwork.
Three owner car from new in Seattle. 43,295 original miles, original Black on Black. Sold with the original owner's manual, stamped maintenance booklet, full ownership history, Porsche Certificate of Authenticity, and maintenance records with a fresh major service.
"The Best of Both Worlds" Pro-built 3.8L twin turbo with two boost settings, one for street use and one for track. Dynoed at 739hp at the wheels on highest setting. Bilstein coil-overs, 14" Brembos, Magnesium BBS wheels, EFI engine management, roll cage, Recaros. Over $90k in receipts; dyno sheets included.
The outrageous 1998 Viper GTS-R commemorated Viper's 1997 FIA GT2 World Championship (the first of five between 1997 and 2002). The example you see here is car #003 of only 100 produced and has had just one enthusiast owner from new.
Conceived as a modern day version of the classic Cobra that would be built using the latest technology. This car, number 190 of 249 built and is fully documented as the very first supercharged and intercooled example produced and was built especially for Louis Kapcsandy, a personal friend of Carroll Shelby.
Only 19,020 original miles on this gorgeous Aston Martin DB7 convertible. Stronsay Silver exterior with special order two-tone all leather interior. 6.0L 12 cylinder, 420hp motor. A pristine one owner Aston Martin that drives new with all services are up to date.


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Where business meets politics: how one network made a president
For a quarter of a century, Fox News has been the jewel in the crown of the Murdoch empire. But with Trump on his way out, what's next?
Rodney Tiffen
(Image: Fox News/YouTube)
Fox News is by far Rupert Murdoch's greatest success of the last quarter-century. It's also been his most successful fusion of commercial profitability with his desire to be a central political player.
Founded in 1996, its chief executive for the first 20 years was Roger Ailes. Murdoch and Ailes was a marriage made in right-wing heaven. They shared extensive contacts in Republican politics, a disdain for mainstream journalism and a combative competitiveness.
Murdoch brought his capital, his financial courage and his sense of business strategy. Ailes brought energy and decisiveness and deep expertise in television and propaganda.
It's not simply that Fox offered a more conservative alternative to other TV networks. Rather, more than any other network its sense of newsworthiness was determined by its immediate political agenda. There was heavy and positive coverage of the heady early days in the Iraq War. When the conflict started to sour, Fox ignored it.
Not going to take this anymore: Fox News anchor and Trump critic quits on-air
Its constant diet of confected outrage fed its audience's sense of resentment. An early directive in this most centralised and hierarchical of news organisations was to "seek out stories that cater to angry, middle-aged white men who listen to talk radio and yell at their television".
In 2002 it overtook CNN as America's top cable news channel, a position it has occupied for most of the time since. It has been one of the most profitable parts of Murdoch's empire.
As Fox succeeded commercially, Ailes' and Murdoch's political ambitions grew. Any sense of professional restraint disappeared. Fox commentators called on-air for viewers to donate to favoured candidates; it promoted rallies to be held by the right-wing Tea Party; it offered commentary positions to prominent Republicans; it actively sought to recruit Republican candidates.
By 2016 the stars were aligned. Donald Trump had appeared frequently on Fox, and Fox had given great publicity to his outrageous "birther" campaign, claiming that Barack Obama had not been born in America. This evidence-free conspiracy was a forerunner of much to come.
By the time of the Republican convention, Trump was Fox's favoured candidate, and his eventual election brought about a relationship between a head of government and a news organisation unprecedented among English-speaking democracies.
Trump was a TV news junkie, in particular a viewer of Fox, and he would often respond in real time. Fox was also Trump's preferred outlet when he wanted to give an interview or make a statement, knowing he would not be questioned but that other media would feel compelled to carry what he said.
For journalists from other organisations, two daily tasks became mandatory every day: check what Trump had tweeted, and monitor Fox News.
Fox was at its political pinnacle, and for the next four years faithfully marched in step with its most important viewer's agendas. But it was also on a slippery slope of professional compromise which by 2020 was provoking more acute dilemmas.
First came the pandemic. Trump's inability to confront COVID-19 — to use the national government's capacity to coordinate and mobilise the states' responses — is one of the greatest leadership failures in living memory.
One side of journalism normalised Trump, the other fed his base. Both were wrong
Instead Trump blamed the "China virus" and the failure of "blue states", promoted quack cures, and held out false hopes that it was all about to disappear. And Fox, like its political patron, developed a herd immunity to evidence.
The same was true during the 2020 election race. Fox campaigned strongly for Trump, but the post-defeat denial of reality brought new tests, as Trump built an edifice of conspiracy without any foundation in evidence. Fox News gave considerable uncritical coverage to Trump's claims about vote dumps. "We saw blatant election-law violations in state after state," said Sean Hannity. Still, the evasion of reality was becoming more difficult and creating strains inside the organisation.
The attacks on the Capitol again caught Fox's commentators off balance. Tucker Carlson defended the rioters as "solid Americans" who are "deeply frustrated"; Hannity said most of them were peaceful. Elsewhere some claimed they were left-wing infiltrators. But majority opinion, even inside the Republican Party, was shifting decisively.
Fox News' dilemmas are just beginning. Already some of its audience has defected to even more outrageous right-wing populist outlets like Newsmax and One America News. It remains to be seen whether these will have deep enough pockets and the editorial stamina to keep on fragmenting Fox's populist constituency.
In 2009, Ailes was in no doubt about Fox's position, immediately casting it in an oppositional role to America's first black president. On day two of Obama's presidency Hannity promised Obama "is not going to succeed". On day three Laura Ingraham declared "our country is less safe today". On day four Glenn Beck said Obama had declared the end to the war on terrorism.
This approach succeeded for Fox in 2009. But after four years of continual confrontation, and of a presidency that thrived on creating chaos, it's less clear whether it's a viable strategy in 2021.
Rod Tiffen is an emeritus professor of government and international relations at Sydney University and the author of Rupert Murdoch: A Reassessment.
For more on how Murdoch helped give us Trump, go here.
Capitol Hill Siege
Murdoch and the making of Trump
Make ours a double
Free world less free as governments fail basic tests of governing
Janine Perrett
Imperial America dies in parts as Washington surrenders to Trump's chaos
Guy Rundle
zut alors
Which raises the question ie: who will Murdoch anoint next?
The US invented religion of the 'Prosperity gospel' Evangelicals is aligned and closely cultivated, controlled and radicalized by the Trump political campaign.
Hundreds of Trump supporters comments I have read and listened to on social media show that they consider Democrats as anti Christ demons, pedophiles and Communists.
In a country where mass school shootings are commonplace, the predictable actions of 4% of sociopaths with guns is hard to ignore.
Does Australia have something to learn from allowing a right wing media promoting and polarising anti welfare, intolerant, religious driven politics.
Dog's Breakfast
"seek out stories that cater to angry, middle-aged white men who listen to talk radio and yell at their television."
Well, I'm, ahem, middle aged and white and very rarely angry, except when I accidentally happen upon one of these tv or radio shows. Then I'm apoplectic.


The Feature Page for the approved proposal narrow is located at Key:narrow.
Sometimes, waterways get narrower for a brief period of time, such as "under" a long-removed bridge. It's useful, when reading a map, to know that this is coming. It's also a handy navigational aid.
This page was last edited on 24 June 2009, at 11:20.


Above: the Long family home Gorge River South of Haast in remote Sth Westland Fiordland NZ.
New Lives in the Wild – Ben Fogle: https://www.radiotimes.com/tv-programme/e/wbsy2/ben-fogle-new-lives-in-the-wild-episode-guide/ Series 1 Episode 3: 'The adventurer travels to the west coast of New Zealand's South Island to meet Robert and Catherine Long and their sons Christan and Robin, who live as hunter-gatherers in an area of wilderness that is a two-day trek from the nearest road. Ben helps Robert hunt for rock lobsters and catch eels, and joins Catherine to climb a waterfall to check for storm damage to the family's gravity-fed water pipe'.