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@ -103,9 +103,85 @@ We truncate the Fourier modes and assume that $\hat u_k=0$ if $|k_1|>K_1$ or $|k
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\mathcal K:=\{(k_1,k_2),\ |k_1|\leqslant K_1,\ |k_2|\leqslant K_2\}
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.
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\end{equation}
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\bigskip
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\subsubsection{Runge-Kutta methods}.
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To solve the equation numerically, we will use Runge-Kutta methods, which compute an approximate value $\hat u_k^{(n)}$ for $\hat u_k(t_n)$.
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{\bf Remark}:
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Since $U$ is real, $\hat U_{-k}=\hat U_k^*$, and so
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\begin{equation}
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\hat u_{-k}=\hat u_k^*
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.
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\label{realu}
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\end{equation}
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Similarly,
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\begin{equation}
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\hat g_{-k}=\hat g_k^*
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.
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\label{realg}
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\end{equation}
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Thus,
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\begin{equation}
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T(\hat u,-k)
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=
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T(\hat u,k)^*
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.
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\label{realT}
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\end{equation}
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In order to keep the computation as quick as possible, we only compute and store the values for $k_1\geqslant 0$.
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(In fact, if we do not enforce the reality conditions, the computation has been found to be unstable.)
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\subsection{Reversible equation}
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\indent The reversible equation is similar to\-~(\ref{ins}) but instead of fixing the viscosity, we fix the enstrophy\-~\cite{Ga22}.
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It is defined directly in Fourier space:
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\begin{equation}
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\partial_t\hat U_k=
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-\frac{4\pi^2}{L^2}\alpha(\hat U) k^2\hat U_k+\hat G_k
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-i\frac{2\pi}L\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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(q\cdot\hat U_p)\hat U_q
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,\quad
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k\cdot\hat U_k=0
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\end{equation}
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where $\alpha$ is chosen such that the enstrophy is constant.
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In terms of $\hat u$\-~(\ref{udef}), (\ref{gdef}), (\ref{T}):
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\begin{equation}
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\partial_t\hat u_k=
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-\frac{4\pi^2}{L^2}\alpha(\hat u) k^2\hat u_k
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+\hat g_k
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+\frac{4\pi^2}{L^2|k|}T(\hat u,k)
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.
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\label{rns_k}
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\end{equation}
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To compute $\alpha$, we use the constancy of the enstrophy:
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\begin{equation}
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\sum_{k\in\mathbb Z^2}k^2\hat U_k\cdot\partial_t\hat U_k
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=0
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\end{equation}
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which, in terms of $\hat u$ is
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\begin{equation}
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\sum_{k\in\mathbb Z^2}k^2\hat u_k^*\partial_t\hat u_k
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=0
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\end{equation}
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that is
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\begin{equation}
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\frac{4\pi^2}{L^2}\alpha(\hat u)\sum_{k\in\mathbb Z^2}k^4|\hat u_k|^2
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=
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\sum_{k\in\mathbb Z^2}k^2\hat u_k^*\hat g_k
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+\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}|k|\hat u_k^*T(\hat u,k)
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\end{equation}
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and so
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\begin{equation}
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\alpha(\hat u)
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=\frac{\frac{L^2}{4\pi^2}\sum_k k^2\hat u_k^*\hat g_k+\sum_k|k|\hat u_k^*T(\hat u,k)}{\sum_kk^4|\hat u_k|^2}
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.
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\label{alpha}
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\end{equation}
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Note that, by\-~(\ref{realu})-(\ref{realT}),
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\begin{equation}
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\alpha(\hat u)\in\mathbb R
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.
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\end{equation}
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\subsection{Runge-Kutta methods}.
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To solve these equations numerically, we will use Runge-Kutta methods, which compute an approximate value $\hat u_k^{(n)}$ for $\hat u_k(t_n)$.
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{\tt nstrophy} supports the 4th order Runge-Kutta ({\tt RK4}) and 2nd order Runge-Kutta ({\tt RK2}) algorithms.
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In addition, several variable step methods are implemented:
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\begin{itemize}
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@ -190,30 +266,8 @@ It can be made by specifying the parameter {\tt adaptive\_cost}.
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These cost functions are selected by choosing {\tt adaptive\_cost=k3} and {\tt adaptive\_cost=k32} respectively.
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\end{itemize}
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\subsubsection{Reality}.
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Since $U$ is real, $\hat U_{-k}=\hat U_k^*$, and so
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\begin{equation}
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\hat u_{-k}=\hat u_k^*
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.
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\label{realu}
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\end{equation}
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Similarly,
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\begin{equation}
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\hat g_{-k}=\hat g_k^*
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.
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\label{realg}
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\end{equation}
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Thus,
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\begin{equation}
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T(\hat u,-k)
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=
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T(\hat u,k)^*
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.
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\label{realT}
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\end{equation}
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In order to keep the computation as quick as possible, we only compute and store the values for $k_1\geqslant 0$.
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\subsubsection{FFT}. We compute T using a fast Fourier transform, defined as
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\subsection{Computation of $T$: FFT}. We compute T using a fast Fourier transform, defined as
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\begin{equation}
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\mathcal F(f)(n):=\sum_{m\in\mathcal N}e^{-\frac{2i\pi}{N_1}m_1n_1-\frac{2i\pi}{N_2}m_2n_2}f(m_1,m_2)
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\end{equation}
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@ -257,6 +311,11 @@ Therefore,
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\right)(k)
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\end{equation}
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\subsection{Observables}
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\indent
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We define the following observables.
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\bigskip
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\subsubsection{Energy}.
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We define the energy as
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\begin{equation}
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@ -367,7 +426,7 @@ The enstrophy is defined as
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.
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\end{equation}
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\subsubsection{Lyapunov exponents}
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\subsection{Lyapunov exponents}
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\indent
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To compute the Lyapunov exponents, we must first compute the Jacobian of $\hat u^{(n)}\mapsto\hat u^{(n+1)}$.
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This map is always of Runge-Kutta type, that is,
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@ -453,62 +512,6 @@ The Lyapunov exponents at time $\alpha\mathfrak L_r$ are then
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.
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\end{equation}
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\subsubsection{Numerical instability}.
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In order to prevent the algorithm from blowing up, it is necessary to impose the reality of $u(x)$ by hand, otherwise, truncation errors build up, and lead to divergences.
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It is sufficient to ensure that the convolution term $T(\hat u,k)$ satisfies $T(\hat u,-k)=T(\hat u,k)^*$.
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After imposing this condition, the algorithm no longer blows up, but it is still unstable (for instance, increasing $K_1$ or $K_2$ leads to very different results).
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\subsection{Reversible equation}
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\indent The reversible equation is similar to\-~(\ref{ins}) but instead of fixing the viscosity, we fix the enstrophy\-~\cite{Ga22}.
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It is defined directly in Fourier space:
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\begin{equation}
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\partial_t\hat U_k=
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-\frac{4\pi^2}{L^2}\alpha(\hat U) k^2\hat U_k+\hat G_k
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-i\frac{2\pi}L\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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(q\cdot\hat U_p)\hat U_q
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,\quad
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k\cdot\hat U_k=0
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\end{equation}
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where $\alpha$ is chosen such that the enstrophy is constant.
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In terms of $\hat u$\-~(\ref{udef}), (\ref{gdef}), (\ref{T}):
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\begin{equation}
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\partial_t\hat u_k=
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-\frac{4\pi^2}{L^2}\alpha(\hat u) k^2\hat u_k
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+\hat g_k
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+\frac{4\pi^2}{L^2|k|}T(\hat u,k)
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.
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\label{rns_k}
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\end{equation}
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To compute $\alpha$, we use the constancy of the enstrophy:
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\begin{equation}
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\sum_{k\in\mathbb Z^2}k^2\hat U_k\cdot\partial_t\hat U_k
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=0
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\end{equation}
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which, in terms of $\hat u$ is
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\begin{equation}
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\sum_{k\in\mathbb Z^2}k^2\hat u_k^*\partial_t\hat u_k
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=0
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\end{equation}
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that is
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\begin{equation}
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\frac{4\pi^2}{L^2}\alpha(\hat u)\sum_{k\in\mathbb Z^2}k^4|\hat u_k|^2
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=
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\sum_{k\in\mathbb Z^2}k^2\hat u_k^*\hat g_k
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+\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}|k|\hat u_k^*T(\hat u,k)
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\end{equation}
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and so
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\begin{equation}
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\alpha(\hat u)
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=\frac{\frac{L^2}{4\pi^2}\sum_k k^2\hat u_k^*\hat g_k+\sum_k|k|\hat u_k^*T(\hat u,k)}{\sum_kk^4|\hat u_k|^2}
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.
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\label{alpha}
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\end{equation}
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Note that, by\-~(\ref{realu})-(\ref{realT}),
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\begin{equation}
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\alpha(\hat u)\in\mathbb R
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.
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\end{equation}
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\vfill
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