text:
\section{Introduction}

Transitional patterns in plane shear flows arise naturally from uniform turbulence at sufficiently low Reynolds number. These patterns feature a selected orientation of around $\simeq 24^\circ$ when they emerge \citep[]{prigent2003long, tsukahara2005dns, ShimizuPRF2019, kashyap2020flow}.
When the Reynolds number is further reduced,
these spatio-temporally intermittent structures display important features of non-equilibrium phase transitions; both experimental and numerical studies have demonstrated their membership in
the directed percolation universality class in the case of plane Couette flow \citep[]{lemoult2016directed, chantry_universal,klotz2022phase}.

Oblique patterns consist of turbulent regions or bands alternating with (quasi-) laminar gaps. An inherent feature of the coexistence of these two phases in planar shear flows is the large-scale flow along the laminar-turbulent interface. This along-band flow
has been observed in both experimental and numerical configurations \citep[]{coles1966progress, barkley2007mean, duguet2013oblique,
couliou2015large,
tuckerman2020patterns,  klotz2021experimental}, and
can be seen as a consequence of the breaking of spanwise symmetry and incompressibility \citep[]{duguet2013oblique}.

Transitional turbulence presents a separation of scales: flow along the laminar-turbulent interface paves
the large scales, while the streaks and the rolls governed by the self-sustaining process of turbulence \citep[]{hamilton1995regeneration, waleffe1997self} are the basic ingredients of the small-scale flow.
In channel flow, the streak spacing is commonly found to be around $\lambda_z^+ \approx 100$ \citep[]{kim1987turbulence},
whereas it is found to be larger ($\lambda_z^+ \approx 132$) in plane Couette flow at low-enough Reynolds number \citep[]{komminaho1996very, jimenez1998largest, tsukahara2006dns}.
In contrast, the wavelength of the large-scale patterns is much larger than that of the rolls and streaks, e.g.\ with a ratio on the order of 20 in patterned plane Couette flow.
This scale separation is visible in the spectral analysis presented by several authors. We mention \citet[]{tsukahara2005dns} in channel flow, \citet[]{tuckerman2011patterns, duguet2013oblique} in Couette flow and \citet[]{ishida2017turbulent} in annular pipe flow.
However, the exact contribution of the rolls and streaks in energising the large-scale patterns has never been thoroughly investigated.

In pipe flow, the energy distribution within turbulent structures was measured in the classic experiments of Wygnanski {\em et al.} \citep{wygnanski1973transition, wygnanski1975transition} and later in numerical simulations by \citet[]{song2017speed}. For localised turbulent structures known as puffs, turbulent production $P$ at the upstream side of a puff is larger than turbulent dissipation $\epsilon$, whereas at the downstream side, dissipation dominates production, as it does throughout regions of quasi-laminar flow in general. No local balance between $P$ and $\epsilon$ is found within the puff. In contrast, in expanding or retracting turbulent zones, known as slugs, the flow in the turbulent core is locally in equilibrium, with production balancing dissipation ($P \approx \epsilon$). Theoretical efforts to model turbulent-laminar structures in pipe flow are based on these properties of the turbulent production and dissipation \citep{barkley2011a,barkley2016theoretical}.

Spectral energy budgets have been extensively used to quantify energy transfers and interactions between mean flow and turbulent kinetic energy (TKE) in high Reynolds number wall-bounded flows.
This approach dates from \citet[]{lumley1964spectral}, who conjectured that energy is transferred from small to large scales in shear flows as distance from the wall increases.
This concept of inverse energy transfer was later investigated by \citet[]{domaradzki1994energy, bolotnov2010spectral, lee2015direct, mizuno2016spectra, cho2018scale, lee2019spectral, kawata2021scale} (and references therein).
However, it is only recently that the spectral energy budget has been computed at low $Re_\tau$, in particular by \citet{symon2021energy} in a turbulent channel of minimal size at $Re_\tau =180$ and in an exact coherent state of channel flow \citep[]{park_graham_2015}.
Currently, there is a lack of understanding of the spectral distribution of energy in transitional wall-bounded turbulence, especially regarding the role of energy transfers and triad interactions in the emergence of the large-scale flow.

This article is devoted to the relationship between the inhomogeneous mean flow and turbulent fluctuations in transitional plane Couette flow. These are investigated through the computation of both physical (\S \ref{sec:phys_balance}) and spectral  (\S \ref{sec:spec_balance}) energy balances
in the regime where patterns emerge from uniform turbulence.
We will survey various energy transfers as a function of $Re$ in \S \ref{sec:bal_Re}. In \S \ref{sec:y_spec}, turbulent production and non-linear transfers are analysed at various wall-normal positions. The energy processes reported in this article will be further investigated as a function of the pattern wavelength in \citet[Part 2]{gome2}, where we will discuss their role in selecting the pattern wavelength.

\section{Numerical setup}
Plane Couette Flow is driven by two parallel rigid plates moving at opposite velocities $\pm U_\text{wall}$. Lengths are nondimensionalised by the half-gap $h$ between the plates, velocities by $U_\text{wall}$, and time by $h/U_\text{wall}$. The Reynolds number is defined to be $Re= U_\text{wall}h/\nu$.
We will require one last dimensional quantity, the horizontal mean shear at the walls, which we denote by $U^\prime_\text{wall}$.
We will use non-dimensional variables throughout.
We use the pseudospectral parallel code {\tt Channelflow} \citep{channelflow} to simulate the incompressible Navier-Stokes equations
\begin{subequations}
    \begin{align}
    \frac{\partial \utot}{\partial t}
    + \left(\utot \cdot \nabla\right) \utot
    &= -\nabla p  + \frac{1}{Re} \nabla^2 \utot\\
    \nabla \cdot \utot &= 0
    \end{align}
    \label{eq:NS}
\end{subequations}
 in a domain which is periodic in the $x$ and $z$ directions.

Since the bands are found to be oriented obliquely with respect to the streamwise direction, we use a periodic numerical domain which is tilted with respect to the streamwise direction of the flow, shown as the oblique rectangle in figure \ref{fig:domain}. This choice was introduced by
\citet{barkley2005computational}
and has become common in studying turbulent bands \citep[e.g.,][]{ reetz2019invariant, paranjape2020oblique,tuckerman2020patterns}.
The $x$ direction is chosen to be aligned with a typical turbulent band and the $z$ direction to be orthogonal to the band.  The relationship between streamwise-spanwise coordinates and tilted band-oriented $(x,z)$ coordinates is:
\begin{subequations}
\label{tilted}
\begin{align}
\mathbf{e}_\text{strm} &= \quad\cos{\theta} \, \mathbf{e}_x + \sin{\theta} \, \mathbf{e}_z \\
\mathbf{e}_\text{span} &= -\sin{\theta } \, \mathbf{e}_x + \cos{\theta} \, \mathbf{e}_z \quad
\end{align}
\end{subequations}
The usual wall-normal coordinate is denoted by $y$ and the corresponding velocity by $v$.
The field visualised in figure \ref{fig:domain} (black box) is obtained by concatenating four times a field resulting from a simulation in $L_{\text{strm}}=200$, $L_{\text{span}}=100$.

The tilted box effectively reduces the dimensionality of the system by discarding large-scale variations along the short $x$ direction. This direction is considered homogeneous over large scales because it is only determined by small turbulent scales, and because the band is assumed to be infinite in $x$.
The main underlying assumption is the
angle of the pattern. In large non-tilted domains, plane Couette flow
shows two statistical orientations that equilibrate
\citep{prigent2002large, duguet2010formation, klotz2022phase}
whereas only one orientation is permitted by our tilted box.

In our simulations, we fix the angle $\theta = 24^{\circ}$, the $x$ domain length $L_x=10$, the $x$ resolution $\Delta x = L_x/N_x=10/120$, and $z$ resolution $\Delta z = L_z/N_z = 0.08$, (similar to that used by \citet[]{tsukahara2006dns, barkley2007mean}). We will make extensive use of two numerical domains, with different domain sizes $L_z$, shown in figure \ref{fig:domain}.

\begin{enumerate}[leftmargin=*,labelindent=8mm,labelsep=3mm, itemindent=0mm, label=(\arabic*)]
    \item \textbf{Minimal Band Units}, shown as the red box in figure \ref{fig:domain}, which can accommodate a single turbulent band and associated quasi-laminar gap. This effectively restricts the flow to a perfectly periodic turbulent-laminar pattern of wavelength $\lambda=L_z$. The size $L_z$ governing the periodicity of the pattern and can be modified. $L_z$ is fixed to $L_z=40$, which is an approximation of the natural spacing of bands observed experimentally and numerically. The effect of size $L_z$ will be investigated in \citet[Part 2]{gome2}.
    \item \textbf{Long Slender Boxes}, which have a large $L_z$ direction that allows for a large number of gaps and bands in the system. The blue box in figure \ref{fig:domain} is an example of such a domain size with $L_z = 240$, but a larger size of $L_z=800$ will be used throughout this article.
\end{enumerate}

\begin{figure}
    \centering
\includegraphics[width=0.8\columnwidth]{tiltedbox.pdf}
\caption{Visualisation of the numerically simulated flow at $Re=360$ and of our numerical domains.  Colors show $y$ velocity at $y=0$ (blue: $-0.2$, white: 0, red: 0.2)  in a domain of size $L_{\text{strm}}=400$, $L_{\text{span}}=200$. Red and blue boxes show a Minimal Band Unit and a Long Slender Box, respectively. }
    \label{fig:domain}
\end{figure}

We furthermore introduce the friction Reynolds number:
\begin{align}
Re_\tau = \frac{ u_\tau h}{\nu}, ~~~~\text{with } {u_\tau^2} = \nu
U^\prime_\text{wall}
= \frac{U_{\rm wall}^2}{Re}\left<
\frac{\partial{u}_{\rm strm}}{\partial y}(y=\pm 1)
\right>_{x,z,t}
\label{eq:Retaudef}
\end{align}
Note that $Re_\tau = \sqrt{Re}$ in the laminar state.
Later we will need the wall-normal coordinate in wall units $y^+=(1-y)Re_\tau$, and similarly for other quantities.
In a Minimal Band Unit at a transitional Reynolds number, the turbulence may be uniform or patterned during different time periods, i.e. it is temporally as well as spatially intermittent.
For this reason, for each value of $Re$, we take the time average in \eqref{eq:Retaudef} over a period during which the flow retains qualitatively the same state.
This yields two slightly different values, $Re_\tau^u$ for a uniform state and $Re_\tau^p$  for a patterned state, as presented in table \ref{tab:Re_tau} for $L_z=40$. In later nondimensionalisations, we will use either $Re_\tau^u$ or $Re_\tau^p$, as appropriate for the flow state.

This procedure does not take into account the local variability of the wall shear stress due to spatial intermittency; for this, we would need to omit $z$-averaging in \eqref{eq:Retaudef} to produce $z$-dependent values of $Re_\tau$; see \citet[]{kashyap2020flow} for a thorough analysis of fluctuations of $Re_\tau$ within and outside of turbulent bands.

\begin{table}
  \begin{center}
\def~{\hphantom{0}}
  \begin{tabular}{cccccccccc}
      $Re$ & 400 & 420 & 440 &460 & 480 & 500 & 550 & 600 & 1000\\

      \hline
      $Re_\tau^p$ & 29.68 & 31.09 & 32.82 &34.61 & 35.90&37.33 & - & - & -\\
      $Re_\tau^u$ & 30.65 & 32.24 & 33.69 & 35.08 &36.42 & 37.67 &40.66 & 43.62 & 66.42 \\

     \end{tabular}
  \caption{Values of $Re_\tau$ for various values of $Re$ in a Minimal Band Unit of size $L_z=40$.  $Re_\tau^p$ and $Re_\tau^u$ are obtained by averaging over the patterned or uniform state, respectively. For $Re>500$, the patterned state does not occur. }
  \label{tab:Re_tau}
  \end{center}
\end{table}

\begin{figure}
    \centering
\subfloat[]{  \includegraphics[width=0.5\columnwidth]{probes_q_R440_5000_or.pdf}  \label{fig:probes_Lz800_R440}}
\subfloat[]{  \includegraphics[width=0.5\columnwidth]{probes_q_R380_or_5000.pdf} \label{fig:probes_Lz800_R380}} \\
\vspace{-1em}
\subfloat[]{  \includegraphics[width=\columnwidth]{probes_MBU_1.pdf} \label{fig:probes_MBU_R430}}
\caption{Spatio-temporal  visualisations of the emergence of gaps and patterns in a Long Slender Box with $L_z=800$, for (a) $Re=440$ and (b) $380$. Flow at $t=0$ is initiated from uniform turbulence at $Re=500$. Colors show local cross-flow energy $(v^2 + u_{\text{span}}^2)/2$ at $x=L_x/2$, $y=0$ (white: 0, dark orange: 0.02). (c)
Intermittent alternation between uniform and patterned turbulence at $Re=430$ in a Minimal Band Unit of $L_z=40$.
}
\label{fig:probes_Lz800}
\end{figure}

\label{sec:balance}
\section{Spectra in different configurations}
\label{sec:spectra_box_size}

We carried out simulations in a Long Slender Box of size $L_z=800$ for various $Re$, with the uniform state at $Re=500$ as an initial condition. These simulations are shown via the spatio-temporal diagrams of figure~\ref{fig:probes_Lz800} for $Re=440$ and for $Re=380$.
With decreasing $Re$, the flow shows intermittent gaps (white spots on the figure) that emerge from the turbulent field at seemingly random locations.
A \emph{gap} is defined as a weakened turbulent structure, or a quasi-laminar zone, surrounded by turbulent flow. A gap is the opposite of a \emph{band}, which is a turbulent core surrounded by quasi-laminar flow. In plane Couette flow, bands are observed at $Re\in[300, 440]$ \citep{prigent2003long, barkley2007mean,duguet2010formation,shi}.
Gaps and bands self-organize into \emph{patterns} as $Re$ is decreased. This is the situation observed in a Long Slender Box in figure~\ref{fig:probes_Lz800_R380} ($Re=380$), where a regular alternation of gaps and turbulent bands is visible.
In a Minimal Band Unit, the system is constrained and the distinction between gaps and patterns is lost.
While the system cannot exhibit the spatial intermittency seen in figure \ref{fig:probes_Lz800_R440}, temporal intermittency is possible and is seen as alternations between uniform turbulence and patterns, as illustrated in figure \ref{fig:probes_MBU_R430} at $Re = 430$.
\citet[Part 2]{gome2} investigate extensively the emergence of gap and patterns out of turbulent flow.

We define the total physical energy and total spectral energy of the flow as:
$$
E(y,z) \equiv  \frac{1}{2} \overline{ \utot \cdot \utot} ~~~
\text{and} ~~~
\Etot(y,k_z) \equiv  \frac{1}{2} \overline{ \uhat^{*} \cdot \uhat},  $$
where $\overline{(.)}$ denotes the time and $x$ average and the Fourier transform is taken in the band-orthogonal direction $z$:
\begin{align}\uhat(x, y, k_z) = \frac{1}{L_z}\int_{0}^{L_z}  \boldsymbol{u}(x,y,z) e^{- i k_z z } \text{d}z.
\label{eq:fft}
\end{align}

\begin{figure}
    \centering
\subfloat[Long Slender Box]{ \includegraphics[width=0.5\columnwidth]{Spec_Lz800_Re.pdf}
\label{fig:spec_Lz800_Re}}
\subfloat[Minimal Band Unit at $Re=430$]{ \includegraphics[width=0.5\columnwidth]{spec_MBU.pdf}
\label{fig:spec_MBU}} \\
\subfloat[Patterned mean flow in a Minimal Band Unit]{ \includegraphics[width=\columnwidth]{mf_u_p.pdf}
\label{fig:mf_u_P}}
\caption{(a) Total energy spectra in a large domain $L_z=800$ (black line) at $y=0$, for different $Re$. The spectra are averaged irrespective of the nature of the state (patterned or uniform).
(b) Total energy spectra (continuous lines) in a Minimal Band Unit of size $L_z=40$. The spectra are individually computed in the uniform turbulence (red) and patterned states (blue), at $Re=430$.
(c) Visualisation of the mean flow: colors show the deviation of the along-band velocity from the laminar base flow $\overline{U}-U_b$ (top) and the pressure $\overline{P}$ (bottom). Streamlines of the mean flow perpendicular to the bands are shown as grey curves.
}
\label{fig:spec}
\end{figure}

Figure~\ref{fig:spec_Lz800_Re} shows $\Etot(y=0, k_z)$ for simulations in a Long Slender Box at different values of $Re$. Here, the average is carried out over a long period of time ($t\in [100, 5000])$.
The total energy spectra show two prominent energy-containing scales: one at small wavenumbers (around $k_z=0.15$, i.e.\ $\lambda_z \simeq 42$) corresponding to the
alternation of turbulent bands and quasi-laminar gaps, and a second one at large wavenumbers ($k_z \simeq 1.41$, $\lambda_z\simeq 4.45$), which we will denote $\ksmax$. This small wavelength corresponds to a spanwise spacing of $\lambda_{\text{span}} = 2\pi\cos{\theta}/\ksmax  = 4.06$, which is approximately the idealised periodicity of pairs of streaks and rolls in Couette flow \citep{waleffe1997self}, with individual rolls occupying the height $L_y=2$ of the shear layer.
In wall units, this peak corresponds to $\lambda_{\text{span}}^+ = 130$ at $Re=430$ ($Re_\tau = 31.9$). This is not far from the streak spacing of $\lambda_{\text{span}}^+=136$ measured by \citet[]{komminaho1996very} in plane Couette flow at $Re_\tau = 52$.
For $k_z>\ksmax$, the energy decreases with $k_z$ up to the resolution scale.
The scale separation between the large-scale gaps and bands and the small-scale streaks and rolls was already observed in the transitional regime by many authors \citep[]{tsukahara2005dns,tuckerman2011patterns, ishida2016transitional}.

The spectrum varies with Reynolds number, but mostly at large scales (low $k_z$):
the large-scale peak is barely visible at $Re=500$ and grows in intensity with decreasing $Re$, becoming dominant for $Re<440$. Meanwhile, the small-scale spectrum is only very weakly affected by the change in $Re$.

We now turn to the Minimal Band Unit, which has exactly the periodicity of a single wavelength of the pattern.
The flow in this configuration does not have localised gaps like those which appear in figure \ref{fig:probes_Lz800_R440}.
The system is instead
fluctuates between patterned and uniform states as seen in figure \ref{fig:probes_MBU_R430}, and each of the two states can be distinguished and consequently analysed separately. In particular, we can take means for patterned and uniform states independently.
For this reason, the remainder of this article will focus on the Minimal Band Unit with a fixed length of $L_z=40$.

The total energy spectrum in a Minimal Band Unit
at $Re=430$ is presented in figure~\ref{fig:spec_MBU}. Contrary to figure~\ref{fig:spec_Lz800_Re}, where unconditional averaging mixes uniform turbulence and localised gaps in the spectrum, here we have conditionally computed the spectrum for the patterned state (blue line) and the uniform state (red line) separately.
As expected, the spectrum for the uniform state lacks the peak at the pattern scale.
The energy of the streak-roll structures $\Etot(\ksmax)$ is higher in the uniform case than in the patterned case. This hints at a redistribution of the energy from small scales (near $\ksmax$) to large scales ($\ll \ksmax$) when the flow changes from uniform to patterned turbulence. For $k_z > 2$, both spectra appear to collapse, suggesting that the small-scale turbulent cascade is the same in both cases.

We now decompose the flow into a mean and fluctuation: $\utot = \ubar + \uprime$,
where the mean flow $\ubar(y,z)$ is computed from an $(x,t)$ average over long time intervals in either the patterned or the uniform state in the Minimal Band Unit.
The mean flow in this configuration was studied by \citet{barkley2007mean}. The mean flow
$\ubar = (U(y,z), V(y,z), W(y,z))$ is visualised on figure \ref{fig:mf_u_P}, by showing $U-U_b$ and $\overline{P}$ (colors) and plotting the streamlines of $(V,W)$ (grey lines).
(Note that figure~\ref{fig:mf_u_P} corrects the erroneous pressure displayed in \citet[figure 5]{barkley2007mean}.)
The flow is centered around the quasi-laminar region, and the total in-plane velocity $(V,W)$ shows a circulation around this region of the flow. $U-U_b$ shows two centro-symmetrically related zones of flow parallel to the band, localised in the upper layer (blue zone) and in the bottom layer (red zone).

The mean flow $\ubar$ can also be decomposed into Fourier modes:
\vspace*{-0.2cm}
\begin{equation}
    \label{eq:trig}
    \ubar(y, z) = \ubar_0(y) + 2\mathcal{R} \left(\ubar_1(y) e^{2\pi i z/L_z} \right) + \ubar_{>1} (y,z)\\[-0.2cm]
\end{equation}
where $\mathcal{R}$ denotes real part, $\ubar_0 \equiv \widehat{\ubar}(y, k_z=0)= (U_0(y), 0 , W_0(y))$ is the $z$-independent (uniform) component of the mean flow, $\ubar_1 = \widehat{\ubar}(y, k_z=2\pi/L_z)$ is the Fourier coefficient corresponding to wavelength $\lambda_z = L_z$, and $\ubar_{>1} \equiv \sum_{k_z>2\pi/L_z} \widehat{\ubar}(y, k_z) $ is the remainder of the decomposition.
(To lighten the notation, we omit the hats on $\ubar$ when subscripts $0$, $1$, or $>1$ are used to indicate the corresponding Fourier coefficients.)
Most of the mean-flow energy lies in the uniform mode $\ubar_0$, with a few percent in the trigonometric component $\ubar_1$. The energy in the remaining terms ($\ubar_{>1}$)
is at least two orders of magnitude lower than that of $\ubar_1$ \citep[]{barkley2007mean}.

\begin{figure}
    \centering
\includegraphics[width=0.9\columnwidth]{meanflow.pdf}
\caption{mean-flow decomposition in two Fourier modes $\ubar_0$ and $\ubar_1$ \eqref{eq:trig}, visualised in the planes $y=\pm0.47$. Colors show turbulent kinetic energy $K(y,z)$: turbulent and laminar zones are respectively in blue and white.
}
\label{fig:mf_trig}
\subfloat{ \includegraphics[width=0.9\columnwidth]{scales.pdf}}
\caption{Illustration of the most-relevant scales in transitional patterns. The upper part depicts the mean flow: $\ubar_0 $ is the uniform ($z$-independent) mean shear, illustrated by the mean streamwise velocity profile, while $\ubar_1$ corresponds to scale $\lambda_z \sim 40$ and is dominated by the flow along the laminar-turbulent interfaces as illustrated in the streamwise-spanwise plane. The lower part depicts the fluctuations: scale $\lambda_z \sim 40$ is illustrated by the periodic presence and absence of fluctuating cross-flow velocity. The scale $\kLS$ is that below which fluctuations receive energy by non-linear interactions (via $\Transnls$, eq. \eqref{eq:tke_bal_spec}), and $\ksmax$ is the scale of rolls and streaks, illustrated in the spanwise-$y$ plane.
}
\label{fig:scales}
\end{figure}

\newpage

The decomposition of $\ubar$ into $\ubar_0$ and $\ubar_1$ is illustrated in figure \ref{fig:mf_trig}. The mean flow and the turbulent kinetic energy $K(y,z) \equiv  \frac{1}{2} \overline{\uprime \cdot \uprime}$ are visualised at planes $y=\pm0.47$.
The most relevant scales involved in the mean flow and the fluctuations are illustrated in figure \ref{fig:scales}. Mode $\ubar_0$ has a S-shape profile in $y$ with small spanwise component.
Mode $\ubar_1$ contains the large-scale flow along laminar-turbulent interfaces.

\section{Physical balance in a Minimal Band Unit}
\label{sec:phys_balance}
\nopagebreak
Before turning to the energy balance in spectral space, we first consider the traditional turbulent energy decomposition in the physical-space representation \citep{pope2000turbulent}, as carried out in transitional pipe flow by \cite{wygnanski1973transition} and \cite{song2017speed} and in bent pipe flow by \cite{rinaldi2019vanishing}.
We write the balance equation for the turbulent kinetic energy (TKE), $K(y,z)$, in the physical representation:
\begin{equation}
    \label{eq:tke_bal_phys}
    \frac{\partial K}{\partial t} + \ubar \cdot \nabla K = P - \epsilon  + \Transnl + \Transp + \Transv
\end{equation}
where the production term, dissipation term, and rate of strain are:
\begin{equation}
    \label{eq:P_eps}
    P \equiv - \overline{u_i^\prime u_j^\prime} \frac{\partial \overline{u_i}}{\partial x_j} \text{,} ~~~~~
    \epsilon \equiv \frac{2}{Re} \overline{s_{ij}^\prime s_{ij}^\prime}
    \text{,} ~~~~~
s_{ij}^\prime \equiv \frac{1}{2}\left( \frac{\partial u_i^\prime}{ \partial x_j} + \frac{\partial u_j^\prime}{\partial x_i}  \right).
\end{equation}
Subscripts $i$ and $j$ range over $\{1,2,3\}$ (or equivalently $\{x,y,z\}$) and we use the Einstein summation convention. The transfer terms read:
\begin{align}
    \label{eq:T}
    \Transnl \equiv -\frac{1}{2} \frac{\partial}{\partial x_i}\overline{u_i^\prime u_j^\prime u_j^\prime}, ~~~~~
    \Transp \equiv - \frac{\partial}{\partial x_i} \overline{u_i^\prime p^\prime } , ~~~~~
    \Transv \equiv \frac{2}{Re} \frac{\partial}{\partial x_i}\overline{u_j^\prime s_{ij}^\prime}
\end{align}
which account respectively for non-linear interactions, work by pressure and viscous diffusion.  We also introduce the total transfer $T\equiv \Transnl + \Transp + \Transv$. This TKE balance is accompanied by the energy balance of the mean flow, $\Emean = \frac{1}{2}\ubar \cdot \ubar = E - K$ \citep[eq. 5.131]{pope2000turbulent}:
\begin{equation}
    \label{eq:mean_bal_phys}
    \frac{\partial \Emean}{\partial t} + \ubar \cdot \nabla \Emean = - \Prodmp - \Dissipmp  + \Transnlm + \Transpm + \Transvm
\end{equation}
where
\begin{equation}
    \label{eq:epsm_Tm}
    \Dissipmp \equiv \frac{2}{Re} \overline{s}_{ij} \overline{s}_{ij}, \qquad
    \overline{s}_{ij} \equiv
\frac{1}{2}
\left(\frac{\partial \overline{u}_i}{\partial x_j} + \frac{\partial \overline{u}_j}{\partial x_i} \right)
    \end{equation}
    and
\begin{equation}
    \Transnlm \equiv -\frac{\partial}{\partial x_i} \overline{u}_j \overline{u_i^\prime u_j^\prime}, ~~~~~
    \Transpm \equiv   - \frac{\partial}{\partial x_i}\overline{u}_i \overline{p} ~~~~ \text{and} ~~~~
    \Transvm \equiv  \frac{2}{Re}\frac{\partial}{\partial x_i} \overline{u}_j   \overline{s}_{ij}
\end{equation}

In order to emphasise the derivation of
\eqref{eq:tke_bal_phys} and
\eqref{eq:mean_bal_phys} from the Navier-Stokes equations, we have retained temporal derivatives, even though these equations described $t$ and $x$ averaged quantities. Averaging in time is justified by the fact that turbulent-laminar banded patterns are statistically steady in plane Couette flow. There is, in fact, some slight motion of the band position. To gain in precision, we position the pattern at each time based on the phase of the $z$-trigonometric Fourier coefficient of the along-band flow at the mid-plane: $z_{\text{loc}}(t) = - \phi(t) L_z/2\pi$, where $\phi(t) = \arg \left<\widehat{u}(x, 0, 2\pi/L_z,t) \right>_x$. Temporal averages are computed with this phase alignment and we consider $\partial K/\partial t = 0$ and $\partial \overline{E}/\partial t = 0$. The results in this section are all presented in a frame centered around the quasi-laminar zone.

In figure \ref{fig:schema_flux} we represent the streamwise mean flow with arrows and the turbulent kinetic energy $K(y,z)$ by colors. The center of the turbulent region is at $z\pm20$, while \emph{overhang regions} \citep{lundbladh1991direct, duguet2013oblique} are located around $z\pm10$, where the along-band large-scale flow is strongest (see figure \ref{fig:mf_u_P}).
Figures \ref{fig:budget} and \ref{fig:mean_bal_phys}
display the energy budgets of equations \eqref{eq:tke_bal_phys} and \eqref{eq:mean_bal_phys}.
To better relate these results to those from pipe flow, we integrate the energy budgets over the upper half of the domain, where the $z$ component of the mean flow is from left to right. We use the same symbols $P$, $\epsilon$, etc.\ to denote these half-height averages.
(The lower half can be obtained from the upper half by symmetry and should be compared to pipe flow with the opposite streamwise direction.) All quantities depend strongly on $z$ and it is this dependence on which we will focus.

Figure~\ref{fig:budget} shows the TKE budget.
The energy balance is dominated by production and dissipation. Unsurprisingly, production is minimal in the quasi-laminar region where the fluctuations, and hence the Reynolds stresses, are small. The regions where production is larger than and smaller than dissipation are indicated in the figure. There are approximately, but not exactly, centered on peaks and troughs in production.
This local disequilibrium between production and dissipation is accounted for by the transfer terms: for $z \in [-10, 0]$ the advective transfer $\ubar \cdot \nabla K$ is negative and of larger intensity than the fluctuation transfer terms $T$, so the net transfer is negative.
This is compensated for
by a positive transfer that exists
in most of the turbulent domain,
from $z\simeq 6.5$ to $z\simeq-13$. The spatial flux of energy goes from the turbulent core to the quasi-laminar zone, as schematically indicated by the grey arrows in figure \ref{fig:budget}.
These results are consistent with those in a band in plane Poiseuille flow \citep[Fig. 5]{brethouwer2012turbulent} and in a puff in pipe flow \citep[]{song2017speed}: when entering the turbulent region from upstream to downstream, $P>\epsilon$ first, and then $P<\epsilon$, which signifies a spatial flux of energy from upstream to downstream. (In the upper half of our Couette domain, increasing $z$ corresponds to going downstream in a pipe.)

\begin{figure}
    \centering
\subfloat[]{ \includegraphics[width=\columnwidth]{schema_flux.pdf}
\label{fig:schema_flux}}\\
\vspace{-0.9em}
\subfloat[]{ \includegraphics[width=0.5\columnwidth]{budget_L40_R400.pdf}
\label{fig:budget}}~
\subfloat[]{ \includegraphics[width=0.5\columnwidth]{mean_bal_phys_L40_R400.pdf} \label{fig:mean_bal_phys}}
\label{fig:mean_bal_phys_all}
\caption{(a) Mean-flow profiles (streamwise velocity) and TKE,  $K(y,z)$ (white: 0, blue: 0.08), in a turbulent-laminar pattern centered around the laminar gap at $z=0$.
(b) TKE and (c) mean-flow energy budgets for $Re=400$.
Each term is integrated over the upper half of the domain, $y\in [0, 1]$, where advection by the mean flow is towards the right.
}
\end{figure}

We now look at the energy budget of the mean flow, presented in figure~\ref{fig:mean_bal_phys}, again centered around the laminar region and integrated over the upper half of the domain.
Unlike pressure-driven channel or pipe flows, the energy is injected into the plane Couette flow by the imposed motion of the wall, and this is captured by the viscous diffusion term $\Transvm$ in the mean-flow energy equation.
This injection is mostly balanced by dissipation and production which fuels fluctuations by extracting energy from the mean flow). The pressure term $\Transpm$ is very weak and non-linear and advective fluxes are non-uniform. The advective contribution to the mean flow $\ubar\cdot\nabla\Emean$
behaves in the opposite way as the contribution of the Reynolds stress $\Transnlm$
This advection term transfers energy of the mean flow from laminar regions, where the production $\Prodp$ is small, to turbulent regions, where the production is large.

In the case of pipe flow, \citet{song2017speed} reported that the peak in TKE dissipation is shifted downstream from the peak in the production. Our data in plane Couette flow does not support a $z$-shift in the peaks in $\Dissipp$ and $\Prodp$. Interestingly, we observe a considerable shift between the peaks in mean-flow dissipation $\Dissipmp$ at $z=-15$ and production $\Prodmp$ at $z=-20$, as shown in figure~\ref{fig:mean_bal_phys}. Recall that these plots show quantities integrated only over the upper half of the domain.
The shift in mean-flow peaks is consistent with that between mean flow and turbulent kinetic energy due to the overhang region surrounding localised turbulence,
which we illustrated in figure \ref{fig:schema_flux}: the upper-half mean flow is mostly dissipated in the turbulent-to-laminar overhang region, and not in the center of the turbulent band.
In contrast, in the laminar-to-turbulent interface, the mean flow accelerates, as it is energised by an advective transfer from the laminar region.

\section{Spectral decomposition}
\label{sec:spec_balance}

We now analyse the spectral balance of kinetic energy. In shear flows at higher $Re$, this analysis leads to a detailed understanding of the energy sources and transfers within the flow. We refer the reader to \citet{bolotnov2010spectral, lee2015direct, mizuno2016spectra, cho2018scale} for studies at higher $Re_\tau$, and to \citet[]{symon2021energy} for a minimal channel study at $Re_\tau=180$.
In a similar vein, \citet{lee2019spectral} recently computed two-point correlations in channel flow.

\subsection{Notation and governing equations}
We begin by writing the Reynolds-averaged Navier-Stokes equations and the equation for fluctuations from the mean:

\begin{align}
    \label{eq:RANS}
    \frac{\partial \overline{u}_j}{\partial t} + \overline{u}_i \frac{\partial \overline{u}_j}{\partial x_i}
    +  \frac{\partial}{\partial x_i} \overline{ u_i^\prime u_j^\prime  }
    &=   - \frac{\partial \overline{p}}{\partial x_j}
    + \frac{2}{Re} \frac{\partial \overline{s}_{ij}}{\partial x_i}
\end{align}

\begin{align}
    \label{eq:devNS}
    \frac{\partial u_j^\prime}{\partial t} + \overline{u}_i \frac{\partial u_j^\prime}{\partial x_i}
    +  u_i^\prime \frac{\partial u_j^\prime}{\partial x_i}
    &= - u^\prime_i \frac{\partial \overline{u}_j}{\partial x_i}
    + \frac{\partial}{\partial x_i}  \overline{ u_i^\prime u_j^\prime }
        - \frac{\partial p'}{\partial x_j}
    + \frac{2}{Re} \frac{\partial s_{ij}^\prime}{\partial x_i}
\end{align}
By taking the $z$ Fourier transform of (\ref{eq:devNS}) and multiplying by $\ujphatc$, followed by averaging over $x$ and $t$, we obtain a balance equation for the spectral kinetic energy
$\tke(y,k_z) \equiv  \frac{1}{2} \overline{ \uprimehat^{*} \cdot \uprimehat } $:

\begin{align}
    \underbrace{ \vphantom{\mathcal{R}\left\{ \overline{\ujphat \widehat{u^\prime_i \frac{\partial \overline{u}_j}{\partial x_i}} } \right\}}
    \frac{\partial \tke(y, k_z)}{\partial t}}_0
&+ \underbrace{\mathcal{R}\left\{ \overline{\ujphatc  \widehat{ \overline{u}_i\frac{\partial u_j^\prime}{\partial x_i} }} \right\} }_{-\Advs(y,k_z)}
    =   \underbrace{- \mathcal{R}\left\{ \overline{\ujphatc \widehat{u^\prime_i \frac{\partial \overline{u}_j}{\partial x_i}} } \right\}}_{\Prods(y, k_z)} ~
   - \underbrace{ \vphantom{\mathcal{R}\left\{ \overline{\ujphat \widehat{u^\prime_i \frac{\partial \overline{u}_j}{\partial x_i}} } \right\}}
  \frac{2}{Re} \overline{ \widehat{s_{ij}^\prime} \widehat{s_{ij}^\prime}^*}}_{\Dissips(y,k_z)}\nonumber\\
    &+ \underbrace{\vphantom{- \mathcal{R}\left\{  \overline{\ujphatc \widehat{  u_i^\prime \frac{\partial  u_j^\prime}{\partial x_i}  }} \right\}}
    \frac{2}{Re} \mathcal{R} \left\{  \overline{\frac{\partial}{\partial y} (\ujphatc \widehat{s_{yj}^\prime}) } \right\}}_{\Transvs(y,k_z)}
~  \underbrace{\vphantom{\mathcal{R}\left\{  \overline{\ujphatc \widehat{  u_i^\prime \frac{\partial  u_j^\prime}{\partial x_i}  }} \right\}}
 ~ - \mathcal{R}\left\{ \overline{\frac{\partial}{\partial y}  (\widehat{u_y^{\prime}}^*\widehat{p^{\prime}})} \right\} }_{\Transps(y,k_z)}
~
    \underbrace{- \mathcal{R}\left\{  \overline{\ujphatc \widehat{  u_i^\prime \frac{\partial  u_j^\prime}{\partial x_i}  }} \right\} }_{\Transnls (y,k_z)}
     \label{eq:tke_bal_spec}
\end{align}
where we revert from the general partial derivative $\partial/\partial x_i$ or subscript $i$ to the wall-normal coordinate $y$ when this is the only non-zero term.
\begin{itemize}[leftmargin=*,labelindent=8mm,labelsep=3mm, itemindent=0mm]
    \item[-] $\Advs$ is an interaction between mean velocity and fluctuations, corresponding to the spectral version of the advection term $\ubar \cdot \nabla K$;
    \item[-] $\Prods$ is the spectral production term, which is an interaction between the mean gradient and fluctuations at scale $k_z$;
    \item[-] $\Dissips$ is the viscous dissipation at mode $k_z$;
    \item[-] $\Transvs$, $\hat{T}_p$ are transfer terms to mode $k_z$ due to strain-velocity and pressure-velocity correlations;
    \item[-] $\Transnls$ is an inter-scale transfer to mode $k_z$ and position $y$ due to triad interactions. When summed over $k_z$ and integrated over $y$, this term is zero. \\

\end{itemize}

The forms of the pressure, viscous diffusion, dissipation and triadic terms are the same as they would be if the flow were uniform in $z$.
Only advection and production terms, which contain the inhomogeneous mean flow, do not simplify as in the uniform case, and instead require a convolution over wavenumbers.
In the usual analysis of uniform turbulence in a non-tilted box \citep{bolotnov2010spectral, cho2018scale, lee2019spectral},
 $\ubar$ reduces to $(U(y),0,0)$ and $\widehat{\ubar}= (U(y), 0, 0)$ for $k_z=0$ and is otherwise 0,
which simplifies the spectral balance. In particular, the advection term $\Advs$ vanishes, because in such cases:
\begin{align}
    \Advs (y, k_z) =  -\mathcal{R}\left\{ \overline{\ujphatc U(y) \widehat{ \frac{\partial u_j^\prime}{\partial x} }} \right\} = -\frac{1}{2} \mathcal{R}\left\{ U(y) \overline{ \frac{\partial}{\partial x} \ujphatc \ujphat} \right\} = 0
    \end{align}
(due to $x$ averaging). This is also true in the case of tilted uniform turbulence $\ubar=(U(y), 0 , W(y))$:
\begin{align}
    \Advs (y, k_z) = -\mathcal{R}\left\{W(y) i k_z  \overline{\ujphatc \ujphat} \right\} = 0
    \label{eq:adv_unif}
\end{align}
However, this is not true for a patterned mean flow $\ubar=(U(y,z), V(y,z), W(y,z))$ like the one shown in figure \ref{fig:mf_u_P}.

We furthermore introduce the balance equation for the spectral energy of the mean flow $\Esmean \equiv \frac{1}{2} \widehat{\ubar}^* \cdot \widehat{\ubar}$ at wavenumber $k_z$:

\begin{align}
  \underbrace{\vphantom{\mathcal{R}\left\{ \frac{\partial \ujmhatc}{\partial x_i} \widehat{\overline{u_i^\prime u_j^\prime}} \right\} } \frac{\partial \Esmean(y, k_z)}{\partial t} }_0
&+ \underbrace{\mathcal{R}\left\{ \ujmhatc  \widehat{ \overline{u}_i \frac{\partial \overline{u}_j}{\partial x_i} } \right\}  }_{-\Advms(y,k_z)}
       =\underbrace{ \mathcal{R}\left\{ \frac{\partial \ujmhatc}{\partial x_i} \widehat{\overline{u_i^\prime u_j^\prime}} \right\} }_{-\Prodms(y, k_z)}
   - \underbrace{\vphantom{\mathcal{R}\left\{ \frac{\partial \ujmhatc}{\partial x_i} \widehat{\overline{u_i^\prime u_j^\prime}} \right\} } ~
   \frac{2}{Re} \widehat{\overline{s}}_{ij} \widehat{\overline{s}}_{ij}^*}_{\Dissipms(y,k_z)}
   \nonumber\\
    &+  \underbrace{\frac{2}{Re} \mathcal{R}\left\{\frac{\partial}{\partial y} (\ujmhatc \widehat{\overline{s}}_{yj} ) \right\}}_{\Transvms(y,k_z)}
~  \underbrace{ - \mathcal{R}\left\{ \frac{\partial}{\partial y} \widehat{\overline{u}}_y^* \widehat{\overline{p}}  \right\} }_{\Transpms(y,k_z)}
~     \underbrace{- \mathcal{R}\left\{ \frac{\partial}{\partial y}(\ujmhatc \widehat{\overline{u_y^\prime u_j^\prime}} ) \right\}  }_{\Transnlms (y,k_z)}
     \label{eq:mean_bal_spec}
\end{align}

where:
\begin{itemize}[leftmargin=*,labelindent=8mm,labelsep=3mm, itemindent=0mm]
    \item[-] $\Advms$ is a non-linear transfer term for the mean flow. This is a spectral version of the advection term $\ubar \cdot \nabla \Emean$ appearing in the mean-flow balance equation \eqref{eq:mean_bal_phys}.
    \item[-] $\Prodms$ is the interaction between Reynolds stress at scale $k_z$ and the mean gradient at scale $k_z$, and hence is a form of production term.
    \item[-] $\Dissipms$ is a dissipation term for the mean flow energy;
    \item[-] $\Transvms$, $\Transpms$ are transfer terms due to correlations between mean strain and velocity, and mean pressure and velocity;
    \item[-] $\Transnlms$ is a flux term due to the interactions between the Reynolds stress and the mean flow. \\
\end{itemize}

In the rest of this section, we will focus on $y-$integrated TKE and mean-flow balance to characterise the
spectral distribution of total energy.
As the mean flow $\ubar$ is dominated by $\ubar_0$ and $\ubar_1$,
we write (\ref{eq:mean_bal_spec}) in $y$-integrated form for $k_z=0$ and $k_z=2\pi/L_z$ and obtain:
\begin{align}
    I + \Advms_0  -\Prodms_0 - \Dissipms_0 = 0  ~~~ \text{and} ~~~   \Advms_1 - \Prodms_1 - \Dissipms_1 = 0
    \label{eq:mf_bal}
\end{align}
where we have introduced
\begin{align}
    \widehat{\overline{\Pi}}_0 \equiv \int_{-1}^{1}\Prodms(y, 0) ~\text{d} y  ~~~~ \text{and} ~~~~ \widehat{\overline{\Pi}}_1 \equiv \int_{-1}^{1}\Prodms\left(y, \frac{2\pi}{L_z}\right) ~\text{d} y
    \label{eq:pi0_pi1}
\end{align}
with similar definitions for $\Advms_0$, $\Dissipms_0$, $\Advms_1$ and $\Dissipms_1$.
We have also introduced the total energy injection due to the action of the walls:
\begin{align}
    I &= \sum_{k_z} \int_{-1}^1 \Transvms(y, k_z) ~\text{d} y  =  \frac{2}{Re} \sum_{k_z}  \ujmhatc (k_z) ~\widehat{\overline{s}}_{yj} (k_z) \bigg \rvert_{-1}^1
    \label{eq:injection}
\end{align}
This term is non-zero only for mode $k_z=0$ because the applied wall velocity is uniform, so that
\begin{align}
    I &= \frac{2}{Re} \ujmhatc (k_z=0) ~\widehat{\overline{s}}_{yj} (k_z=0) \bigg \rvert_{-1}^1
    =  2 \frac{u_\tau^2}{U_{\rm wall}^2}
\end{align}
Note that $\Transpms$ and $\Transnlms$ integrate to zero, since both $\overline{u}_y$ and the Reynolds stress vanish at the walls.

Two important comments can be made at this stage.
The first one starts from a
word of caution: all terms in \eqref{eq:mean_bal_spec} are not the Fourier transforms of those in \eqref{eq:mean_bal_phys}.
(This is a generalisation of the fact that $\Etot (k_z)$ is defined to be $\uhat (k_z)\cdot \uhat (k_z) /2$ and not $\widehat{\utot \cdot \utot}(k_z)/2$.)
This means in particular that although energy is injected only in the balance of $\ubar_0$ via $I$, the energy is not injected uniformly within the flow, as $\Transvm$ is not uniform in $z$ (see figure~\ref{fig:mean_bal_phys}). The connection with the physical injection of energy is indeed only through $z$ averaging:
\begin{equation}
    I= \frac{1}{L_z}\int_0^{L_z} \int_{-1}^1 \Transvm (y, z) ~\text{d}y ~ \text{d} z  \end{equation}

The second comment is about the way in which this injected energy is communicated to the TKE spectral balance. Contrary to the physical-space version of the energy balance, where the same production $\Prodp$ appears in the TKE (\ref{eq:tke_bal_phys}) and the mean flow (\ref{eq:mean_bal_phys})  equations, the spectral production terms appearing in  (\ref{eq:tke_bal_spec}) and (\ref{eq:mean_bal_spec}), $\Prods$ and $\Prodms$, are different. However, when summed over $k_z$ and integrated over $y$,
these two terms agree, so we can write the total production $\Pi$ as:
\begin{align}
    \Pi \equiv \sum_{k_z} \int_{-1}^1 \Prods(y, k_z) ~\text{d} y =  \sum_{k_z} \int_{-1}^1 \Prodms(y, k_z) ~\text{d} y
    \label{eq:total_prod}
\end{align}
Furthermore, in the physical-space representation,
\begin{align}
\Pi = \frac{1}{L_z}\int_{0}^{L_z} \int_{-1}^{1} \Prodp(y,z) ~\text{d}z ~\text{d}y = \frac{1}{L_z} \int_{0}^{L_z} \int_{-1}^{1} \Dissipp (y,z) ~\text{d}z ~\text{d}y
\end{align}
where the last equality follows since all transfer terms integrate to zero.
The equivalence (\ref{eq:total_prod}) is key to understanding how TKE and mean-flow energy are connected. This will be further developed in section \ref{sec:spec_results}.

\subsection{Results for the spectral energy balance}
\label{sec:spec_results}

\subsubsection{TKE balance}

\begin{figure}
    \centering
\subfloat[Pattern ($Re=400$), TKE]{ \includegraphics[width=0.5\columnwidth]{spec_balance_L40_R400_arrow.pdf}
\label{fig:tke_bal_spec_pattern}} ~
\subfloat[Pattern ($Re=400$), mean flow]{ \includegraphics[width=0.5\columnwidth]{mean_bal_spec_L40_R400.pdf} \label{fig:mean_bal_spec_pattern}} \\
\subfloat[Uniform ($Re=500$), TKE]{ \includegraphics[width=0.5\columnwidth]{spec_balance_L40_R500.pdf}
\label{fig:tke_bal_spec_unif}} ~
\subfloat[Uniform ($Re=500$), mean flow]{ \includegraphics[width=0.5\columnwidth]{mean_bal_spec_L40_R500.pdf} \label{fig:mean_bal_spec_unif}
}
\caption{(a) Spectral energy budget (\ref{eq:tke_bal_spec}) for a pattern at $Re=400$, integrated over $y\in[-1, 1]$. Viscous and pressure transfers are not shown as they integrate to 0. The grey circles indicate $\kLS$ and $\kSS$, which delimit the spectral region where transfer $\Transnls$ is negative.
(b) Spectral energy budget of the mean flow \eqref{eq:mean_bal_spec} integrated over $y\in[-1, 1]$, shown for $k_z=0$, $k_z=2\pi/L_z$ and summed over $k_z>2\pi/L_z$.
(c, d) show the same as (a, b) for a uniform state at $Re=500$.
}
\label{fig:tke_mean_bal}
\end{figure}

We examine the spectral balance of the TKE \eqref{eq:tke_bal_spec}, integrated over the cross-channel direction. This balance is presented for the patterned state in figure \ref{fig:tke_bal_spec_pattern} ($Re=400$) and for the uniform state on figure \ref{fig:tke_bal_spec_unif} ($Re=500$).
The transfer terms $\Transvs$ and $\Transps$ are not shown as they integrate to zero.
(The $y$ dependence of energy transfer will be discussed in section \ref{sec:y_spec}.)

We first focus on the similarities between patterned and uniform states. We observe a peak in the production and dissipation terms near the energy-containing scale $\ksmax$, as we saw for the total energy in figures \ref{fig:spec_Lz800_Re}
and \ref{fig:spec_MBU}.
At this scale, the non-linear transfer $\Transnls$ is negative and of large amplitude: scale $\ksmax$ produces much more than it dissipates, and the remainder is transferred to other $k_z$.
The non-linear transfer becomes positive above a small-scale wavenumber that we denote $\kSS$. ($\kSS \simeq3.6$ in the patterned state at $Re=400$ and the uniform state at $Re=500$.)
This positive transfer at small scales is indicative of a direct energy cascade to small dissipative scales.

The TKE balance for $k_z<\ksmax\simeq 1.41$ contrasts with that at large $k_z$. First, production decreases with decreasing $k_z$ and becomes negative for $k_z<0.5$. This {\em negative production} at large scales appears in both patterned and uniform states. It corresponds to energy transfer from the fluctuations to the mean flow.
The zone of negative production spans from $k_z=0$ to $k_z \simeq 0.47$ in both the patterned and uniform cases presented here. We note that this unusual sign of part of the production term has been also reported by \citet[]{symon2021energy} in spanwise-constant modes of channel flow in a minimal domain.
Second, energy in the range $k_z < 0.94$ is fuelled by a positive non-linear transfer $\Transnls$, which signifies a transfer from small to large scales. This is present in both patterned and uniform states. We denote the (large) scale at which this transfer becomes positive by $\kLS$ as seen in figures \ref{fig:tke_bal_spec_pattern},
\ref{fig:mean_bal_spec_unif} and \ref{fig:scales}.
In the part of the spectrum $k_z<\kLS$, the influx of energy from smaller scales is mostly balanced by dissipation, while only a relatively small amount of energy is yielded to the mean flow via negative production.

Now considering the differences between the patterned (figure \ref{fig:tke_bal_spec_pattern}) and uniform states (figure \ref{fig:tke_bal_spec_unif}), the advection term $\Advs$ plays a more significant role in redistributing energy between scales in the patterned state: it is positive for $k_z<1.1 < \ksmax$, negative near $\ksmax$, and negligible for $k_z>3$. This role is very similar to that of non-linear transfers $\Transnls$, but with weaker amplitude.
$\Advs$ is nearly zero in the uniform state.
This term should vanish when the mean flow is strictly uniform in $z$, see \eqref{eq:adv_unif}.
This is not exactly the case here, especially at $k_z \simeq \ksmax$, and this is probably due to the inexact uniformity of the mean flow.
Other differences are visible between the uniform and patterned states, especially regarding the shape and intensity of each individual curve: for instance, the behaviour of $\Transnls$ and $\Dissips$ is changed near $\ksmax$:
these are almost equal in the uniform case while $\Dissips$ exceeds $\Transnls$ in the patterned case.
The comparison between uniform and patterned states will be discussed further in section \ref{sec:bal_Re}.

\subsubsection{Mean-flow balance}

The spectral energy balance of the mean flow \eqref{eq:mf_bal} is presented in figure \ref{fig:mean_bal_spec_pattern} and  \ref{fig:mean_bal_spec_unif} for both patterned and uniform states. In each case, the three panels correspond, from left to right, to modes $\ubar_0$, $\ubar_1$ and $\ubar_{>1}$. In both patterned and uniform cases, $\ubar_0$ is fueled by the mean strain via injection term $I$ (purple cross). This energy is dissipated (blue square) and also transferred to the fluctuations via the production $\Prodms_0$ (red circle). Note that $\Prodms > 0$ corresponds to usual positive production and hence a sink of energy with respect to the mean flow: production appears as $-\Prodms$ in the mean balance equation \eqref{eq:mean_bal_spec}.

For $\ubar_1$ in the patterned state (middle panel of figure \ref{fig:mean_bal_spec_pattern}), the main source of energy is the advective term $\Advms_1$, with some energy coming from the negative production $\Prodms_1 < 0$.
Thus, the $\ubar_1$ component of the mean flow is fuelled to some extent by a negative transfer from fluctuations back to mean flow, but the advective contribution dominates.
The two sources are balanced by dissipation.
For the uniform state (middle panel of figure \ref{fig:mean_bal_spec_unif}), the terms are more than an order of magnitude smaller than in the patterned state, and the production term has the opposite sign.
The remaining scales in the mean spectral balance $k_z>2\pi/L_z$ (right panels of figures \ref{fig:mean_bal_spec_pattern} and \ref{fig:mean_bal_spec_unif}) are very weak compared to the first two components.

Our results show that the advection term $\Advms$ plays a crucial role in the mean-flow balance in the patterned state.
Since this term represents a transfer due to non-linearities, its sum over $k_z$ and $y$ vanishes.
At $Re=400$, we find that
$\Advms_0\simeq -9.0\times10^{-5}$, $\Advms_1\simeq9.4\times10^{-5}$, and $\sum_{k_z>2\pi/L_z} \int_{-1}^1 \Advms(y, k_z) ~\text{d}y \simeq -4\times10^{-6}$
Hence we have the following approximate equality:
\begin{align}
    \Advms_0 \approx - \Advms_1.
    \label{eq:adv_mf}
\end{align}
Even though the advection is negligible compared with the dominant terms in the $\ubar_0$ balance, it
is the dominant source of energy
at the pattern scale.
In the uniform case, $ \Advms_0  \approx \Advms_1 \approx 0$.

\subsubsection{Connection between TKE and mean flow}

We now investigate the connection between the TKE and mean flow, focusing particularly on the spectral production terms $\Prodms$ and $\Prods$.
While these production terms take different forms in the TKE and mean-flow spectral balances (eq. \eqref{eq:tke_bal_spec} and \eqref{eq:mean_bal_spec}), upon integration over $y$ and summation over $k_z$ (equation \eqref{eq:total_prod}), they give the same total production $\Pi$.

We decompose the total production in two ways: first by writing the total TKE production $\Pi$ as a sum of its positive and negative parts, and second by considering the dominant contributions from $\ubar_0$ and $\ubar_1$ in the mean-flow production $\Prodms$:
\begin{align}
     \Pi = \widehat{\Pi}^{<0} + \widehat{\Pi}^{>0} \simeq \Prodms_0 + \Prodms_1
     \label{eq:total_prod_appro}
\end{align}
where:
\begin{align}
    \widehat{\Pi}^{>0} \equiv \sum_{k_z=0}^{\infty} \int_{-1}^1\Prods (y,k_z) \text{ d}y ~ \Theta \left(  \int_{-1}^1 \Prods (y, k_z)  \text{ d}y \right) \\
    ~~~\text{and}~~~
    \widehat{\Pi}^{<0} \equiv \sum_{k_z=0}^{\infty} \int_{-1}^1 \Prods (y, k_z) \text{ d}y~ \Theta \left(- \int_{-1}^1 \Prods (k_z) \text{ d}y \right),
\end{align}
where $\Theta$ is the Heaviside function.
We recall that figure \ref{fig:tke_bal_spec_pattern} shows that $\int_{-1}^1 \Prods(y, k_z) \text{ d}y <0$ occurs mostly at large scales.
\begin{comment}
\begin{align}
    \widehat{\Pi}^{>0} \equiv \sum_{k_z=0}^{\infty} \Prods_y (k_z)  ~ \Theta (  \Prods_y (k_z)) ~~~\text{and}~~~  \widehat{\Pi}^{<0} \equiv \sum_{k_z=0}^{\infty} \Prods_y (k_z)~ \Theta (-\Prods_y (k_z)),
\end{align}
with $\Prods_y = \int_{-1}^1\Prods (y,k_z) ~\text{d}y $ the $y$-integrated production and $\Theta$ the Heaviside function.
We recall that figure \ref{fig:tke_bal_spec_pattern} shows that $\Prods_y<0$ occurs mostly at large scales.
\end{comment}
Each term in (\ref{eq:total_prod_appro}) in the patterned and uniform states is displayed in table \ref{tab:total_prod} for various values of $Re$.

\begin{table}
  \begin{center}
\def~{\hphantom{0}}
    \begin{tabular}{ccccccc}
       State~ & $Re$~~ & $\Pi$ & $\Prodms_0$ &  $\Prodms_1$ & $\widehat{\Pi}^{>0}$ & $\widehat{\Pi}^{<0}$
       \\[3pt]
           Pattern~ & 400~~ & $3.71\times10^{-3}$ & $3.77\times10^{-3}$ & $-5.44 \times 10^{-5}$ & $3.76\times10^{-3}$ & $-5.34\times10^{-5}$ \\
           Pattern~ & 430~~ & $3.82\times10^{-3}$ & $3.87\times10^{-3}$ & $-4.10\times10^{-5}$ & $3.87\times10^{-3}$ & $-5.36\times10^{-5}$ \\
            Uniform~ & 430~~ & $4.14\times10^{-3}$ & $4.14\times10^{-3}$ &
            $O(10^{-6})$ &$4.20\times10^{-3}$ & $-6.30\times10^{-5}$ \\
            Uniform~ & 500~~ & $4.12\times10^{-3}$ & $4.11\times10^{-3}$ & $O(10^{-6})$ & $4.17\times10^{-3}$ & $-5.64\times10^{-5}$ \\

     \end{tabular}
   \caption{Production terms appearing in the mean flow (\ref{eq:mean_bal_spec}) and the TKE (\ref{eq:tke_bal_spec}) balance, as decomposed in (\ref{eq:total_prod_appro}).}
  \label{tab:total_prod}
  \end{center}
\end{table}

We observe that in the patterned case the positive production is very close to $\Prodms_0$ and the negative production is very close to
$\Prodms_1$, {\em i.e.}
$\Prodms_0  \simeq \widehat{\Pi}^{>0}$ and
$\Prodms_1 \simeq \widehat{\Pi}^{<0}$.
In the uniform case, $\Prodms_1$ is very small and $\Prodms_0$ accounts for essentially all the production, so it is the sum of the positive and negative parts.
In other words:
\begin{align}
\Prodms_0  ~
\begin{dcases}
    \approx \widehat{\Pi}^{>0}  & \text{in patterned state}\\
    \approx \widehat{\Pi}^{>0} + \Pi^{<0} & \text{in uniform state  }
 \end{dcases}
 ~~~~~ \Prodms_1 ~
 \begin{dcases}
    \approx \Pi^{<0}   & \text{in patterned state} \\
    \ll \Pi^{<0}& \text{in uniform state  }
 \end{dcases}
\end{align}
This supports an essential connection between the TKE and the mean-flow production terms: in the patterned state, almost all negative TKE production goes to $\ubar_1$, and almost all positive TKE production comes from $\ubar_0$; in the uniform state, the negative TKE production is absorbed by $\ubar=\ubar_0$.
(In all cases, the negative production, $\Pi^{<0}$, represents less than $1.5\%$ of $\Pi$: $-\Pi^{<0}/\Pi \simeq 1.46\% $ at $Re=400$ and $1.37\%$ at $Re=500$.)

At this stage, we can draw the following conclusions, illustrated in figures \ref{fig:sketch_unif} and \ref{fig:sketch_pattern}:

\begin{enumerate}[leftmargin=*,labelindent=8mm,labelsep=3mm, itemindent=0mm, label=(\arabic*)]
    \item Most of the energy flows into the mean flow and then to TKE according to the usual picture from developed shear flows: energy is injected to $\ubar_0$ by viscous stress, and is transferred to fluctuations via positive production. Energy is mostly produced at the scale of the energy-containing eddies (here, streaks and rolls) and is dissipated to the smaller scales through a positive transfer term.

   \item An important modification to this usual picture is the presence of an inverse transfer of some TKE to large scales via triad interactions $\Transnls$.
   This energy is not entirely dissipated and instead feeds back to the mean flow via negative production $\Prods^{<0}$.

   \item Although weak compared to total production $\Pi$, this negative production $\Prods^{<0}$ fuels $\ubar_1$ in the patterned state.

   \item $\Advms_1$ is the main source of energy of $\ubar_1$: non-linearities of the mean flow play a stronger role than negative production.
\end{enumerate}

We have defined large scales as those for which the non-linear transfer is negative:
$k_z < \kLS$ in figures \ref{fig:tke_bal_spec_pattern} and \ref{fig:tke_bal_spec_unif}.
This separates the large and small scales in figure \ref{fig:sketch}. Note, however, the scales at which production becomes negative are even larger $k_z \lesssim 0.5 < \kLS$ in figures \ref{fig:tke_bal_spec_pattern} and \ref{fig:tke_bal_spec_unif}. We do not distinguish these distinct notions of large scales in figure \ref{fig:sketch}.

\begin{figure}
    \centering
\subfloat{ \includegraphics[width=0.5\columnwidth]{schema_unif.pdf}     \label{fig:sketch_unif}}~
\subfloat{ \includegraphics[width=0.5\columnwidth]{schema_pattern.pdf}    \label{fig:sketch_pattern}} \\
\caption{Illustration of the mean flow - TKE interaction for (a) the uniformly turbulent state
and (b) the patterned state.
In the uniform state, $\Prodms_0 \approx \Prods^{<0} + \Prods^{>0}$, while in the patterned state, $\Prodms_0 \approx  \Prods^{>0}$ and $ \Prodms_1 \approx \Prods^{<0} $.}
\label{fig:sketch}
\end{figure}

We extend these considerations of transfers across scales by considering the quantities
\begin{align}
    \widehat{\Pi}_{<k}(k_z) \equiv \sum_{k_z' < k_z }\int_{-1}^1 \Prods (y, k_z') ~ \text{d}y, ~~~
    \widehat{D}_{<k}(k_z) \equiv \sum_{k_z' < k_z }\int_{-1}^1 \Dissips (y, k_z') ~ \text{d}y \nonumber \\
    \widehat{\Phi}_{<k}(k_z) \equiv \sum_{k_z' < k_z }\int_{-1}^1 \Transnls (y, k_z') ~ \text{d}y, ~~~
    \widehat{A}_{<k}(k_z) \equiv \sum_{k_z' < k_z }\int_{-1}^1 \Advs (y, k_z') ~ \text{d}y
    \label{eq:k_spec_bal}
\end{align}
These scale-to-scale quantities
are shown in figure~\ref{fig:k_spec_bal}. $\widehat{\Phi}_{<k}$ is the non-linear energy flux across a wavenumber $k_z$.
This integrated picture conveys the presence of a zone of inverse flux of energy to large scales ($\widehat{\Phi}_{<k} >0$ for $k_z<1.88$). For $k_z < O(1)$, this inverse transfer is the dominant source and is mostly balanced by dissipation. Starting at $k_z > O(1)$, production comes into play and eventually is the only source.

\begin{figure}
    \centering
\includegraphics[width=0.5\columnwidth]{k_spec_bal_L40_R400.pdf}
 \caption
 Cumulative energy balance \eqref{eq:k_spec_bal} integrated over $y\in[-1, 1]$ in the patterned case at $Re=400$.}
\label{fig:k_spec_bal}
\end{figure}

We emphasise that this strong inverse transfer does not correspond to an \emph{inverse cascade} per se
because it does not lead to an accumulation of energy towards the largest available scale in the system. Indeed, simulations in Large Slender Boxes have emphasized the presence of a small range of energetic large scales, around $\lambda_z \simeq 40$ (figure~\ref{fig:spec_Lz800_Re}). It is expected that
there is a build-up making energy condense around this finite scale.

\section{Evolution with Reynolds number}
\label{sec:bal_Re}

We now address the dependence of the global energy balance on $Re$. Unlike previous studies \citep[]{tuckerman2011patterns, rolland2011ginzburg}, we do not focus on an order parameter for the transition between uniform turbulence and patterns, but rather compute the Reynolds decomposition for each of the two states through the transition.
Figure \ref{fig:Re_evol} presents the evolution of several quantities computed in a Minimal Band Unit of $L_z=40$ for the uniform states at higher $Re$ and for the patterned states at lower $Re$.  For fluctuating bistable cases at intermediate $Re$ (e.g $Re=430$, as shown in figure \ref{fig:probes_MBU_R430}), conditional averaging has been carried out over selected time windows during which the state is either patterned or uniform. To compare across different values of $Re$, we will normalise these quantities either by viscous wall units or by global quantities (total injection $I$ or total production $\Pi$).

We start with the evolution of energies with $Re$ and consider the following quantities: average energy $\left<E\right>^+$, relative energy of $\ubar_0$ given by $\left<{\Emean}_{0}\right>/\left<E\right>\equiv \left<\ubar_0\cdot\ubar_0/2\right>/\left<E\right>$,  average TKE $\left<K\right>^+$, and average dissipation $\left<\epsilon\right>^+$.
(Here, $\left<\cdot\right>$ denotes a $(y,z)$ average of the $(x,t)$-averaged quantities introduced in sections \ref{sec:phys_balance} and \ref{sec:spec_balance}.)
Quantities are defined in terms of wall units (i.e.\ normalised by $Re~u_\tau^4 / U_{\rm wall}^4$).
As seen in figure \ref{fig:Re_evol}, all of these quantities change discontinuously when the flow switches from the uniform to the patterned state as $Re$ is decreased.

Next we consider the terms appearing in the mean balance and show their evolution with $Re$. We normalise all terms by injection rate $I$ except for $I$ itself, which we normalise by wall units and hence plot $I^+$.
The injection, dissipation and advection are discontinuous, with $\Advms_1/I$ undergoing an especially dramatic increase, by a factor of nearly ten, when going from uniform turbulence to the patterned state.
In contrast, $\Pi/I$ is approximately continuous through the transition from uniform to patterned states, meaning that at a given $Re$ the transfer of energy between mean flow and TKE is independent of whether the flow is uniformly turbulent or patterned.
Interestingly, $\Pi/I$ decreases with decreasing $Re$, meaning that relatively less energy is transferred to turbulence when $Re$ is reduced, and hence more energy is retained by the mean flow at lower $Re$.
While $\Pi/I$ is continuous through the transition,
the relative portion of negative production
is not continuous:
the patterned state shows a smaller $-\widehat{\Pi}^{<0}/\Pi$ than the uniform state.
What this means is that, surprisingly, the relative influx of energy from turbulence to the mean flow is lower in the patterned state than in the uniform state.

\begin{figure}
    \centering
\includegraphics[width=\columnwidth]{Re_evol.pdf}
\caption{Evolution with $Re$ of various energetic quantities defined throughout the text (equations \eqref{eq:injection}, \eqref{eq:mf_bal}, and \eqref{eq:pi0_pi1}).}
    \label{fig:Re_evol}
\end{figure}

We now turn to the evolution of transfer terms with $Re$. For this purpose, we focus only on the non-linear transfers into large scales at $k_z < \kLS$, and into small dissipative scales at $k_z > \kSS$. (See figure \ref{fig:tke_mean_bal}.) We define the total nonlinear transfer to large scales $\widehat{T}_{LS}$ and to small scales $\widehat{T}_{SS}$ by
\begin{align}
\widehat{T}_{LS} =  \sum_{k_z \leq \kLS} \int_{-1}^1 \widehat{T}_{nl}~\text{d}y \qquad
\widehat{T}_{SS} =  \sum_{k_z> k_{SS}} \int_{-1}^1 \widehat{T}_{nl}~\text{d}y ,
\label{eq:Tss}
\end{align}
We plot the ratio $\widehat{T}_{LS}/ \widehat{T}_{SS}$ in figure \ref{fig:Re_evol}.
For the uniform state, relatively more energy is transferred to large scales as $Re$ decreases. The ratio undergoes a discontinuous drop at the transition to patterns, where relatively less transfer goes to large scales.
We find that $\widehat{T}_{LS}/ \widehat{T}_{SS} \sim 1$ in the patterned state.

Lastly, in the two bottom-right plots of figure \ref{fig:Re_evol} we focus on the streak/roll scale $\ksmax$. We consider three quantities
\begin{comment}
\begin{equation}
\widehat{\Pi}_{\text{rolls}} = \int_{-1}^1 \Prods(\ksmax, y) ~\text{d}y, \quad\widehat{D}_{\text{rolls}} = \int_{-1}^1 \Dissips (\ksmax, y) ~\text{d}y, \quad  \widehat{T}_{nl,{\text{rolls}}} = \int_{-1}^1 \Transnls(\ksmax, y) ~\text{d}y
\end{equation}
\end{comment}
\begin{equation}
\widehat{\Pi}_{\text{rolls}} = \int_{-1}^1 \Prods(y,\ksmax) ~\text{d}y, \quad\widehat{D}_{\text{rolls}} = \int_{-1}^1 \Dissips (y,\ksmax) ~\text{d}y, \quad  \widehat{T}_{nl,{\text{rolls}}} = \int_{-1}^1 \Transnls(y,\ksmax) ~\text{d}y
\end{equation}
and plot the ratios $\widehat{\Pi}_{\text{rolls}}/\Pi$ and $\widehat{T}_{nl,{\text{rolls}}}/\widehat{D}_{\text{rolls}}$. We recall that $\widehat{\Pi}_{\text{rolls}}$ is the maximum of $\Prods$ visible in figure~\ref{fig:tke_bal_spec_pattern} and \ref{fig:tke_bal_spec_unif}.
We observe that the plotted ratios are approximately constant in each of the uniform and patterned states  over the $Re$ range considered, but each ratio exhibits a strong discontinuity at the transition between states.
In the uniform case, $\widehat{T}_{nl,{\text{rolls}}}/\widehat{D}_{\text{rolls}} > 1$, meaning that at the scale $\ksmax$, energy is transferred to other scales more than it is directly dissipated. This relationship is reversed in the patterned state. We note that $\ksmax$ is nearly constant with $Re$ in the range under study.

We summarise our findings:
 First, most quantities are discontinuous through the transition.
 The notable exception is the relative production, for which the uniform and patterned states at the same $Re$ exhibit the same exchange of energy between mean flow and TKE:
\begin{align}\frac{\Pi}{I} \Big\rvert_{\text{pattern}}  \approx \frac{\Pi}{I} \Big\rvert_{\text{uniform}} \end{align}
Moreover, we find that both negative production  $\Prods^{<0}$ and inverse transfer to large scales $\widehat{T}_{LS}$ are lower in the patterned state than in the uniform states.

It seems counter-intuitive to associate a stronger mean flow $\ubar$ with a reduced fueling of the mean flow by the fluctuations (via negative production $\Prods^{<0}$). This is, however, what we observe in comparing the patterned state to the uniform state at $Re=430$ and 440.
Referring to figure \ref{fig:sketch},
the exchange between $\ubar$ and TKE is more directional in the patterned state: the negative production is directly transferred to mode $\ubar_1$, whereas it is diverted into $\ubar_0$ in the uniform state.
Furthermore, the patterned state is more balanced
\begin{equation}
    \Prods^{<0}\approx \Prodms_1, \quad \Prods^{>0}\approx \Prodms_0, \quad \widehat{T}_{LS} \approx \widehat{T}_{SS}
\end{equation}
than the uniform state.

The patterned state could be seen as more adapted to an increasingly dissipative environment when $Re$ decreases:
as the mean and fluctuations dissipate more their energy, they interact less with each other via the production terms and the fluctuations transfer less of their energy to other scales.
Relatively to the amount of injected energy, less energy goes to turbulent fluctuations and the mean flow is therefore stronger in intensity.
The patterned state diverts this mean-flow energy from the uniform mean flow $\ubar_0$ to the large-scale flow $\ubar_1$.
Whether these energy processes have a role in the transition from uniform to patterned state is speculative at this stage.

\section{Wall-normal dependence of spectral balance}
\label{sec:y_spec}

\subsection{Energy balance at various $y$ locations}

Up to this point, nothing has been said about the location of the energy transfers in the wall-normal direction and no distinction has been made between near-wall and bulk effects on the mean flow and turbulent energies.
In this section, we present results on the TKE balance and subsequently the mean-flow balance for the patterned state at $Re=400$.

\begin{figure}
    \centering
 \includegraphics[width=1\columnwidth]{spec_balance_y_L40_R400.pdf} \\
  \caption{TKE spectral balance (\ref{eq:tke_bal_spec}) at different $y$ locations (left: mid-plane, $y=0$; middle: $y=0.63$;  right: near-wall, $y=0.92$). Shown is a patterned case at $Re=400$.}
 \label{fig:tke_bal_spec_y}
\end{figure}
\begin{figure}
    \centering
 \includegraphics[width=1\columnwidth]{mean_bal_spec_L40_R400_y.pdf}
\caption{Evolution of the mean-flow energy balance (\ref{eq:mean_bal_spec}) with $y$, for $k_z=0$, $k_z=2\pi/L_z$, and summed over $k_z>2\pi/L_z$. (Values for $y<0$ are obtained by reflection in $y=0$.) Shown is a patterned case at $Re=400$.}
\label{fig:mean_bal_spec_y}
\end{figure}

Figure \ref{fig:tke_bal_spec_y} shows the spectral TKE balance at different $y$ locations: the mid-plane ($y=0$, $y^+= 29.6$, left panel), the layer of maximal spectral production $\Prods$ ($y=0.63$, $y^+\simeq 11$, middle panel) and the near-wall region ($y=0.92$, $y^+\simeq 2.4$, right panel).

The balance in the near-wall region is simple because it is dominated by viscous effects, with injection of energy via the rate-of-strain compensated by dissipation.
A small portion of the energy comes from a positive transfer $\Transnls$.
In the plane $y=0.63$, the production term $\Prods$ is maximal (as will be shown in \S \ref{subsec:y_kz_P_T}.). Production peaks at the roll scale $\ksmax$, while the dissipation, viscous diffusion and non-linear transfers are all negative with similar magnitudes near this scale. Production becomes negative and non-linear transfers positive at long length scales (small $k_z$), similar to what we showed for $y$-integrated quantities in \S \ref{sec:spec_results}.
The spectral balance at the mid-plane is qualitatively similar to that at the plane $y=0.63$, with the notable exception that the viscous diffusion $\Transvs$ vanishes due to reflection symmetry about the midplane.
$\Prods$ and $\Transnl$ are smaller in the mid-plane than in the plane $y=0.63$, while $\Dissips$ and $\Advs$ have nearly the same magnitude in both planes.

The $y$-dependence of the mean-flow energy balance (\ref{eq:mean_bal_spec}) is displayed in figure \ref{fig:mean_bal_spec_y}. In line with our previous observations on $y$-integrated quantities (\S \ref{sec:spec_results}), figure \ref{fig:mean_bal_spec_y} reveals different phenomenology depending on the wavenumber ($k_z=0$, $k_z=2\pi/L_z$ or $k_z>2\pi/L_z$). The gain in energy in $\ubar_0$ (left panel) due to the viscous transfer term $\Transvms$ is large near the wall where energy is injected into the flow, whereas the two terms involving the Reynolds stress, $\Prodms$ and $\Transnlms$, are dominant and in approximate balance at the mid-plane. Note that while $\Transnlms$ integrates to zero, it has a local influence: flow above $y=0.63$ transfers energy to flow below.

The balance of $\ubar_1$ (middle panel) presents a complex and interesting behaviour. We know from \S \ref{sec:phys_balance} that when integrated over $y$, the balance for mode $\ubar_1$ is such that $\Prodms_1<0$ and that this mode extracts energy from TKE. However, the $y$-dependence of this term shows a change in sign:
the production is only negative (i.e.\ $-\Prodms_1>0$)
for $-0.6 \lesssim y \lesssim 0.6$.
$\Transnlms$ undergoes a change in sign at approximately the same $y$ value, with similar behaviour, although their $y$ integrals differ (the integral of $\Transnlms$ vanishes whereas that of $\Prodms$ is negative). $\Transnlms$ dominates the energy source at $y\simeq0.8$. At the wall ($y=1$), the energy balance is between viscous diffusion and dissipation.
The advection term $\Advms$ is always positive.

The situation at $k_z>2\pi/L_z$ is perhaps of negligible importance because of the small amplitude of the energy at this scale. However, we note that the balance near the wall (i.e.\ $0.7 \lesssim y \lesssim 1$) is qualitatively similar to that of mode $\ubar_1$, dominated by viscous diffusion, dissipation, and triad interaction.
In the bulk, energy comes from $\Prodms$ and is diverted towards the other terms

\subsection{Production and non-linear transfers in the $(y, k_z)$ plane}
\label{subsec:y_kz_P_T}

Figure \ref{fig:prod_y_kz} and \ref{fig:trans_y_kz} show, respectively, $\Prods^+(y,k_z)$ and $\Transnls^+(y,k_z)$ (i.e.\ $\Prods$ and $\Transnls$ normalised by $Re~u_\tau^4/U_{\rm wall}^4$) for different states and $Re$ ranging from 380 to 1000.
We focus on these terms because of their unusual signs in the balance at large scales (small $k_z$).
The zone of negative production at large scales is encircled by the dashed contour. We note that negative production spans the range $y\in[0,  0.8]$ at low $Re$, whereas it is more concentrated between $y=0.6$ and 0.9 at $Re=1000$. In viscous units, it approximately spans from $y^+=5$ to $y^+ \approx30$ at all $Re$.
\begin{figure}
    \centering
\subfloat[Pattern, $Re=400$ ($Re_\tau = 29.7$)]{ \includegraphics[width=0.5\columnwidth]{Prod+_pattern_R400.pdf}     \label{fig:prod_y_kz_380p}}~
\subfloat[Pattern, $Re=430$ ($Re_\tau = 31.9$)]{ \includegraphics[width=0.5\columnwidth]{Prod+_pattern_R430.pdf}    \label{fig:prod_y_kz_430p}} \\
\subfloat[Uniform, $Re=430$ ($Re_\tau = 33.0$)]{ \includegraphics[width=0.5\columnwidth]{Prod+_unif_R430.pdf} \label{fig:prod_y_kz_430u}}
~
\subfloat[Uniform, $Re=1000$ ($Re_\tau = 66.4$)]{ \includegraphics[width=0.5\columnwidth]{Prod+_unif_R1000.pdf}    \label{fig:prod_y_kz_1000u} }
\caption{Visualisations of production $\Prods^+(y, k_z)$ for different $Re$ and states. The cross-channel range is from the mid-plane ($y=0$, $y^+=Re_\tau$, lower axis) to the wall ($y=1$, $y^+=0$, upper axis). Dashed line separates positive from negative value for small $k_z$.}
 \label{fig:prod_y_kz}
\end{figure}

\begin{figure}
    \centering
\subfloat[Pattern, $Re=400$ ($Re_\tau = 29.7$)]{ \includegraphics[width=0.5\columnwidth]{Tnl+_pattern_R400.pdf}     \label{fig:trans_y_kz_380p}}~
\subfloat[Pattern, $Re=430$ ($Re_\tau = 31.9$)]{ \includegraphics[width=0.5\columnwidth]{Tnl+_pattern_R430.pdf}   \label{fig:trans_y_kz_430p}} \\
\subfloat[Uniform, $Re=430$ ($Re_\tau = 33.0$)]{ \includegraphics[width=0.5\columnwidth]{Tnl+_unif_R430.pdf} \label{fig:trans_y_kz_430u}} ~
\subfloat[Uniform, $Re=1000$ ($Re_\tau = 66.4$)]{ \includegraphics[width=0.5\columnwidth]{Tnl+_unif_R1000.pdf} \label{fig:trans_y_kz_1000u} }
\caption{Visualisations of  of non-linear transfer $\Transnls(y, k_z)$ for different $Re$ and states. The cross-channel range  is from the mid-plane ($y=0$, $y^+=Re_\tau$, lower axis) to the wall ($y=1$, $y^+=0$, upper axis).
}
\label{fig:trans_y_kz}
\end{figure}
The triadic interaction term $\Transnls^+$ is shown on figure \ref{fig:trans_y_kz}.
Inverse transfers are present from $k_z^+=0$ up to $k_z^+ \approx 0.07$ in the patterned cases, and $k_z^+ \approx 0.05$ in the uniform case at $Re=1000$ ($Re_\tau = 66.4$), i.e.\ scales smaller than that of rolls and streaks ($\ksmax\simeq1.41$, $\ksmax^+\simeq 0.04$ for $Re\leq430$).
However, this small-scale part of the inverse transfer is localised only near the wall ($y^+<8$), while for $k_z^+< 0.02$, the inverse transfer concerns the whole $y$ domain.

We see two caveats that prevent further quantitative comparisons to other studies in non-tilted domains, for both transitional and non-transitional regimes.
First, the imposition of an angle ($\theta=24^\circ$)
is completely arbitrary for
uniform turbulence, and along with the short domain size $L_x$, the streak spacing is imposed in our numerical domain. In Appendix \ref{app:mfu}, we present results in a non-oblique flow unit $(L_{\text{strm}}, L_{\text{span}})=(30,20)$ to confirm our observations in the Minimal Band Unit in the non-transitional case $Re=1000$ ($Re_\tau=66.2$).
Second, the reduction to one dimension
can miss the two-dimensionality
of energy transfers: inter-scale transfers can actually be orientational, i.e.\ they may differ for
wavenumbers $(k_x, k_z)$ with the same modulus but different orientations. Therefore, inverse transfers in a one-dimensional spectrum can be misleading as they mix transfers between different orientations and transfers between different scales $|\boldsymbol{k}|$.

\section{Conclusion}

Wall-bounded turbulence at low Reynolds numbers is marked by a strong scale separation between the streak/roll scale of the self-sustaining process that comprises the turbulence, and the large-scale flow associated with oblique laminar-turbulent patterns. In this article, we have computed the spectral energy balances for both the mean flow and the turbulent fluctuations in a Minimal Band Unit, thus revealing the energy transfers connecting the different scales in transitional plane Couette flow.

As expected, TKE production is maximal at the scale of streaks and rolls, and a direct cascade sends energy to smaller dissipative scales.
However, part of the TKE is also transferred to large scales via non-linear interaction. At large scales, this energy is partly sent to the mean flow, via negative production.
Negative production has not received much attention although it has been reported for spanwise-constant modes at $Re_\tau=180$ by \citet[]{symon2021energy}.
We have found negative production at large scales for $Re_\tau\lesssim 66$, which, although of weak intensity, plays a role in feeding the inhomogeneous mean flow in transitional patterns.

The intense large-scale flow along laminar-turbulent bands appears in the single trigonometric component of the mean flow $\ubar_1$.  The main energy source for $\ubar_1$ is the non-linear interaction with the uniform component $\ubar_0$ (via the term called $\Advms_1$ in this article). This interaction is due to the mean advection, which plays a significant role in both spatial and spectral transfers of mean-flow energy. Interestingly, the $\ubar_1$ component of the mean flow is also fueled by negative production transferring energy from fluctuations to mean flow. However, this is only a secondary driver of $\ubar_1$, as negative production accounts for only approximately $20 \%$ of its energy sources (see figure \ref{fig:mean_bal_spec_pattern}).

We have found that the ratio of total TKE dissipation to total energy injection is continuous through the evolution from uniform turbulence to patterns as $Re$ decreases.
Our results indicate that
as the environment becomes more dissipative, the energy is reorganised  such as to balance large-scale and small-scale transfers and to direct negative production into large-scale flow $\ubar_1$.

The large scales characterising the transitional regime are probably of a different nature than large-scale motions observed in uniform shear flows at higher
$Re_\tau$ \citep[and references therein]{jimenez1998largest, smits2011high, lee2018extreme},
which are typically streamwise-elongated modes dictated by inertial effects far from the wall (in the outer zone).
To the extent of our knowledge, large-scale fluctuating motions in fully developed turbulence do not feed the mean flow via negative production.
The inverse transfers from small to large scales, which we report here in low-$Re$ wall-bounded turbulence, echo those observed at higher $Re$ \citep[]{ cimarelli2013paths, mizuno2016spectra,  aulery2017spectral, cho2018scale, lee2019spectral, kawata2021scale}. Understanding the role of these transfers in the autonomous mechanisms governing wall-bounded turbulence is an active research topic.
In fully developed turbulence, these inverse transfers are weaker than those reported here in transitional turbulence, and are essentially concentrated near the wall, while we observe inverse transfers over the whole shear layer that dominate the TKE budget at large scales.

Our analysis of energy budgets
does not directly invoke a dynamical mechanism, such as the self-sustaining process governing wall-bounded turbulence and related mechanisms describing large scales in developed turbulence \citep[]{hwang2010self, hwang2016self, de2017streak, cho2018scale}.
Further investigations are required to understand whether the strong inverse transfers and the negative production that we observed are connected to the non-linear regeneration of rolls in the self-sustaining process.
Note that the energetic imprint of the self-sustaining process in developed wall-bounded turbulence was recently analysed by \citet[]{cho2018scale} and \citet{kawata2021scale}, the latter emphasising the role of non-linear transfers.

Although the oblique simulation domain
is very useful for the study of inter-scale distribution of energy in patterned transitional turbulence, further confirmation via simulations
in large streamwise-spanwise domains is also required: our simulation domain restricts the flow in a number of ways, such as imposing an orientation as well as a mean streak spacing due to the restrained short size $L_x=10$. These features do not seem to alter the robust observations that we have made about mean-turbulent interaction and inverse transfers (see Appendix \ref{app:mfu}).
However, it would be beneficial to disentangle the streamwise and spanwise directions in the energy budget and to compute inter-component transfers, so as to better understand the role of the self-sustaining process in the generation of transitional large-scale structures.

In \citet[Part 2]{gome2}, the energy cycle described above will be essential to understand the selection of a finite wavelength of transitional patterns.

\section*{Acknowledgements}

The calculations for this work were performed using high performance computing resources
provided by the Grand Equipement National de Calcul Intensif at the Institut du D\'eveloppement
et des Ressources en Informatique Scientifique (IDRIS, CNRS) through Grant No. A0102A01119.
This work was supported by a grant from the Simons Foundation (Grant No. 662985).
The authors wish to thank Yohann Duguet, Santiago Benavides, Anna Frishman and Tobias Grafke for fruitful discussions.

\section*{Declaration of Interests}
The authors report no conflict of interest.

redpajama_set_name: RedPajamaArXiv


This food and wine experience is best served with another pairing: California sunshine and a good conversation. Enjoy!
Chop watermelon into large cubes, removing as many seeds as possible. Puree in blender, leaving out a few cubes to use as garnish or just for snacking. Quarter the tomatoes and remove seeds. Peel and roughly chop cucumbers. Place tomatoes and cucumbers in blender and puree with watermelon. Put the mixture through a sieve to remove extra pulp. Transfer to a large bowl. Add the cilantro, basil, serrano pepper, sugar and lime. Salt to taste. Chill for at least 30 minutes. To serve, add extra watermelon cubes and mint leaves. Summerland's fruit-forward Sauvignon Blanc pulls up the fresh notes of summer fruit and mint in this dish, while balancing its spice with low acidity.
Do you have a favorite recipe that you've paired with one of our wines? Share it with us on one of our social networks and you may just earn a free bottle of Summerland Wine!


This time year, with the Kentucky Derby and the Preakness coming up, some networks are busy promoting horse racing. Of course, NBC is worst offender, as it televises this nonsense every year.
Thankfully, we are seeing actual news coverage of horse racing. We just action in response to these stories.
The article reports an extremely troubling statistics. An average of 24 horses die at tracks across the US – EACH WEEK!
PETA released the following report this week, with undercover video concerning the horrors of horse racing. The New York Times focused on the news as well.
When profit motives are mixed with animals – most notably here with animal racing – the outcome is routinely horrible for the animals.
Now is the time to ban horse racing and greyhound racing. For every day that passes, horses and dogs are dying at alarming rates, while countless more are suffering.


A filmmaker looks to chronicle the history of The Birdman in games.
"Pretending I'm a Superman" did not meet its funding target but that's not stopping the producers.
We deliver hands-on impressions and media on the Dreamcast incarnation of the popular skating game.
WGamer gets profound with Ideaworks3D's CEO and resident visionary.
[UPDATE] Nokia gives game deck a second chance at success; enlists wide array of retailers to offer rebates.
N-Gage QD for under $99?
Nokia's redesigned game deck is reportedly being offered for $60 and under in Europe. But will the savings make it to America?
Nokia announces the results of its European N-Gage Challenge, which attracted around 150,000 competitors from 16 countries.
Tony Hawk is coming to the N-Gage, and we have a demo version of the game. Can Tony go portable while still staying polygonal? Find out inside.
Crave confirms that Activision's latest PlayStation skater is coming to the Dreamcast.
No forum topics for Tony Hawk's Pro Skater yet. Want to start us off? Create a new topic.


text:
Are You Prepared to Go Back to School Next Month?
Published on: July 16, 2020 | by Don Turner Legal Team
We are well into July, which means most parents in the state of Georgia are well into taking care of their back to school shopping needs. Even if the novel coronavirus or COVID-19 is still a concern, it is equally important that kids get back into the classroom. At the time of this writing, Georgia school districts are making plans to reopen safely, while addressing the safety concerns raised by the novel coronavirus. Like or follow the Don Turner Legal Team on Facebook for further updates on Georgia school districts reopening as they occur.
How and When Will My Child's School Reopen?
The Georgia Department of Education and the Georgia Department of Public Health has cooperated to complete a set of guidelines to support school districts, depending on the number of reported coronavirus cases in the individual district. These guidelines are meant to allow Georgia schools to reopen, while ensuring students can enjoy a safe 2020-2021 school year. They are not state-mandated, and school districts will be allowed to use their discretion on how and when to reopen.
At the time of this writing, a number of school districts in North Georgia have announced when they plan to reopen their doors, although not every district has announced how they are planning to reopen. The following school districts have announced some level of plans for reopening:
Atlanta Public Schools: School begins August 10. No concrete plans for how to handle reopening have been announced at this time.
Buford City Schools: School begins August 5. Students will have the option of in-person or virtual instruction, and parents will need to make their choice by July 15. Students will be required to wear a mask while riding the bus.
Cherokee County Schools: School begins on August 3. Students will not be required to wear masks, but teachers will.
Clayton County Schools:School begins August 3. Students will have a mixture of in-person and virtual instruction.
Cobb County Schools: School begins August 17, with the option of in-person or virtual instruction. Parents will need to choose by July 22. Face masks will be required for school staff are recommended for students.
Coweta County Schools: School begins August 6, with online and in-person options. The deadline for signing up for online classes is July 17.
Fayette County Schools: School begins August 10, with option of virtual instruction or in-classroom instruction for parents.
Forsyth County Schools: School begins August 6. Virtual classes will be offered for kindergarten through fifth grade students, while grades 6-12 have the option of attending Forsyth Virtual Academy. The deadline for parents to make a choice was on July 10. Students attending in-classroom lessons will not be required to wear face masks.
Fulton County Schools: School begins August 17. Parents will have the option to register for virtual classes if they are not comfortable sending their kids back to school. The deadline to register for online instruction is July 17.
Griffin-Spalding Schools: School begins September 8. Lessons will be in-person, but virtual lessons will be available for the first nine weeks of the school year for those uncomfortable sending their children back to school right away.
Gwinnett County Schools: School begins August 12. Parents were given a July 12 deadline to choose between virtual instruction or in classroom learning. Masks will be mandatory for students returning to the classroom.
Hall County Schools: School begins August 7. Lessons will be in person, but the district is willing to work with families who are unwilling to send their children back to school on an individual basis.
Henry County Schools:School begins August 3. Parents will need to decide between virtual classrooms or in-person instruction by July 17.
Marietta City Schools: School begins August 3, and parents can choose between virtual classrooms or sending their children back to school. Masks will be required.
Rockdale County Schools: School will begin virtually on August 24. The district plans to shift to a hybrid of in-person teaching and virtual learning on the first official day of school, September 8.
Be aware that these dates are subject to change as school districts continue to figure out how to handle the upcoming 2020-2021 school year. Reopening dates have already been modified for a few districts, and other districts are still deciding whether to mandate masks and how to handle other social distancing concerns. Make sure to keep an eye on your school district's website so you can be certain of when your child's school will reopen and what safety measures they will be required to abide by.
Reopening Georgia schools in this time of COVID-19 may be a complex and ongoing process, but with parents and students eager to go back to class, be assured that school districts across the state will come to a solution. Make sure to keep an eye on our Facebook page for additional developments on school districts' reopening plans. And if you or someone you love is exposed to COVID-19, be certain to contact Georgia's coronavirus hotline at (844) 442-2681.
Updated: January 7, 2021 11:38 am

redpajama_set_name: RedPajamaCommonCrawl


text:
Blogs, Intellectual Property
The Rule of Reasonableness:...
The Rule of Reasonableness: Non-Compete Provisions in California Business Contracts
by James Kachmar
The IP Law Blog
The California Supreme Court in the 2008 case, Edwards v. Arthur Andersen LLP, ruled that a provision in an employment agreement that prevented an employee from competing with his former employer following the termination of his employment was an invalid restraint on trade in violation of section 16600 of the California Business and Professions Code. The Court held that subject to certain statutory exceptions, i.e., to protect the value of goodwill in connection with the sale of one's business interest, section 16600 invalidated all contractual provisions that constituted a restraint on an employee's ability to practice his or her trade or profession. What the Court has not addressed since that 2008 decision was whether provisions that acted as a restraint on trade in business contracts (i.e. exclusive distribution agreements, franchise agreements, etc.) would suffer a similar fate. On August 3, 2020, the California Supreme Court issued its decision in Ixchel Pharma, LLC v. Biogen, Inc., and ruled that non-compete provisions in business to business contracts were not per se invalid, but rather subject to a rule of reasonableness.
Ixchel is a biotech company that develops drugs containing dimethyl fumarate ("DMF") to treat certain neurodegenerative disorders. Ixchel did not have the resources to develop these drug treatments independently so in 2016, it entered into a "Collaboration Agreement" with another biotech company, Forward. Under the agreement, Forward and Ixchel would work together to develop and test a potential new drug and they eventually began developing a plan for a trial study.
During that same time, Forward was involved in a patent dispute with another biotech company, Biogen, that related to the use of DMF in certain drugs. Biogen and Forward agreed to settle their dispute and enter into a settlement and license agreement. That agreement contained a provision (section 2.13) that required Forward to terminate its collaboration agreement with Ixchel and refrain from contracting with anyone to develop drugs with DMF. (Biogen was apparently worried that Forward's work with Ixchel (and/or others) would lead to the development of competing pharmaceutical drugs).
After entering into the settlement with Biogen, Forward notified Ixchel that it was terminating the Collaboration Agreement. Ixchel claimed that after Forward's termination, it was unable to find a new partner to help with the development of its pharmaceutical drug.
As a result, Ixchel sued Biogen alleging a variety of claims in connection with Forward's termination of the collaboration agreement. The Federal district court granted Biogen's motion to dismiss the complaint but granted Ixchel leave to amend its complaint. Ixchel then amended its complaint to allege, among other things, that Biogen violated section 16600 of the Business and Professions Code by entering into the settlement agreement with Forward that contained section 2.13. The district court again dismissed Ixchel's complaint, ruling that section 16600 did not apply outside employment matters. Ixchel appealed this decision to the Ninth Circuit which, after oral argument, asked the California Supreme Court to answer some legal questions about the state of California law in order to provide it guidance to rule on the matter. One of these questions that the Ninth Circuit certified to the California Supreme Court: "Does section 16600 of the California Business and Professions Code void a contract by which a business is restrained from engaging in a lawful trade or business with another business?"
Ixchel claimed that section 2.13 of the Forward/Biogen settlement agreement violated section 16600 which provides: "Except as provided in this chapter, every contract by which anyone is restrained from engaging in a lawful profession, trade or business of any kind is to that extent void." The California Supreme Court began by reframing the question that had been certified to it by the Ninth Circuit. It determined that the question as framed of whether section 16600 applied to contracts in the business context was not really in dispute. In fact, Biogen had acknowledged as much at oral argument before the Ninth Circuit. Rather, the issue that the California Supreme Court felt needed to be addressed was whether "contractual restraints on business operations or commercial dealings are subject to a reasonable standard under section 16600" or were per se invalid as in the employment context following the Edwards decision.
Biogen urged the California Supreme Court to apply the rule of reason similar to what is used to analyze antitrust violations in considering section 2.13 of the settlement agreement. The California Supreme Court recognized that this analysis focuses on "whether an agreement harms competition more than it helps" and requires courts to look at the following factors: (1) the facts peculiar to the business in which the restraint is to be applied; (2) the type of restraint at issue and its effects; (3) the history of the restraint; and (4) the reason for the restraint's adoption by the parties.
Ixchel, on the other hand, argued that the Court should not apply a reasonableness standard but rather to extend the Court's 2008 Edwards decision "and hold that any contract in restraint of trade is per se void." The California Supreme Court recognized that reading section 16600 "in isolation" might support such an argument. However, the Supreme Court cautioned that when considering statutes, the Court was required to analyze the statute by considering "the statutory framework as a whole in order to determine its scope and purpose." In doing so, the Supreme Court recognized that "section 16600 is best read not to render void per se all contractual restraints on business dealings, but rather to subject such restraints to a rule of reason."
The Court turned its attention to the history of section 16600 recognizing that its predecessor, Civil Code section 1673, had been enacted in 1872 "against the back drop of well-established common law prohibitions against restraints of trade." In fact, section 16600's predecessor should be considered in connection with another antitrust statute, California's Cartwright Act, which the Court had previously "construed to permit reasonable restraints of trade." Thus, the Court believed that it would be consistent to apply a reasonableness standard in considering business contracts under section 16600 similar to agreements that could raise antitrust issues under the Cartwright Act.
The Court also looked at the legislative history to section 16600 and its predecessor, Civil Code section 1673. In enacting section 1673 in 1872, the legislature "did not go so far as to say that Civil Code former section 1673 categorically replaced the common law standard of reasonableness with a per se [invalid] rule."
In addition, the Court recognized that during the next half century, it had never invoked a per se invalidity rule on contractual restraints on business operations and commercial dealings. Rather, throughout this time, the Court had "generally declared agreements in this context valid if the restraints they imposed were reasonable." In those cases where the Court invalidated a contractual provision that restricted trade, it did so after conducting a reasonableness inquiry and finding the contracts "to be invalid when their purpose was to restrain trade by creating a monopoly, restricting supply, or fixing prices."
The Court noted that although it had gradually evolved to evaluate contractual restraints in business contracts under a reasonableness standard, it had "often interpreted the statute more strictly when it came to agreements not to compete after the termination of employment or the sale of interest in a business." This line of cases culminated with the Court's 2008 decision in Edwards. In declining Ixchel's invitation to extend Edwards beyond the employment context to commercial contracts, the Court reasoned, "it is axiomatic that language in a judicial opinion is to be understood in accordance with the facts and issues before the Court" in a particular case. Furthermore, the Court's rationale in Edwards was focused largely on issues relating to employee mobility and competition in the marketplace. Thus, the California Supreme Court concluded that its holdings in Edwards and prior employment-specific cases should not be extended to similar restrictions in commercial dealings.
The California Supreme Court was also aware that if it adopted Ixchel's argument, this could have severe consequences and require it "to invalidate all contracts that limit the freedom to engage in commercial dealing." The Court noted that businesses often engage in legitimate partnerships and exclusive dealing arrangements or franchise arrangements "which limit the parties' freedom to engage in commerce with third parties." These types of arrangements allow businesses to maximize their complimentary capabilities; ensure stability and supply/demand; and protect R&D and marketing efforts from being exploited by their contractual partners. The Court recognized that such arrangements "can have pro-competitive effects" by allowing for "long term planning, giving protection against price fluctuations and offer predictable marketplaces." Faced with the potential consequences of adopting Ixchel's reasoning, the California Supreme Court concluded, "We decline to construe section 16600 to call such arrangements into question simply because they restrain trade in some way."
Thus, under California law, non-compete provisions in an employment contract/context will continue to be struck down unless they fall within one of the statutory exceptions to section 16600 such as the protection of the value of goodwill in connection with the sale of a business interest. However, restrictive covenants in commercial contracts, such as exclusive distributorship or franchise agreements, will be enforced provided they pass the rule of reasonableness.