is the same for positive and negative $k$. Thus, in what follows we restrict ourselves to the case $k\geq 1$. An equivalent formulation of the above problem is sometimes called *scattering formulation*, and can be derived as follows. We denote by $S(t)$ the solution operator corresponding to the transport operator $U\de_x$. More explicitly, $$\begin{aligned}
g(x,y)\mapsto (S(t)g)(x,y)=g(x+tU(y),y).\end{aligned}$$ Let $U,B$ be given, and let $\omega,\psi$ be the solution to . Accordingly, we define the *scattered vorticity* and *scattered streamfunction* by $$\begin{aligned}
F(t)=S(-t)\omega(t), \qquad \Psi(t)=S(-t)\psi(t).\end{aligned}$$ It is not hard to check that $\omega,\psi$ solve if and only if $F,\Psi$ solve $$\begin{aligned}
\label{eq:Eulerkscat}
\begin{cases}
\de_t F =ikB(y)\Psi,   \quad &\mbox{in } I, \\
E_t \Psi:=(-k^2+(\de_y-iktU'(y))^2)\Psi  = F, &\mbox{in } I,\\
ik\Psi= 0, &\mbox{on } \de I,
\end{cases}\end{aligned}$$ with $$\begin{aligned}
F(0,y)=\omega^{in}(y),  \quad \mbox{in } I.\end{aligned}$$ The function $F$ is sometimes called “profile”, and it is often studied in dispersive equations. In this context, it is the object which measures the difference between the passive scalar and full linearized (or nonlinear) dynamics.

\[remark:boundary\_data\] We note that the equations decouple in $k$ and the evolution of $F$ is trivial for $k=0$. Hence, we consider $k \in \frac{2\pi}{L} \mathbb{Z} \setminus\{0\}$ as a given parameter. In particular, in this case 