  In this article, we study the weak coupling limit of the following equation
in $\mathbb{R}^2$:
$$dX_t^\varepsilon=\frac{\hat{\lambda}}{\sqrt{\log\frac1\varepsilon}}\omega^\varepsilon(X_t^\varepsilon)dt+\nu
dB_t,\quad X_0^\varepsilon=0. $$ Here
$\omega^\varepsilon=\nabla^{\perp}\rho_\varepsilon*\xi$ with $\xi$ representing
the $2d$ Gaussian Free Field (GFF) and $\rho_\varepsilon$ denoting an
appropriate identity. $B_t$ denotes a two-dimensional standard Brownian motion,
and $\hat{\lambda},\nu>0$ are two given constants. We use the approach from
\cite{Cannizzaro.2023} to show that the second moment of $X_t^\varepsilon$
under the annealed law converges to $(c(\nu)^2+2\nu^2)t$ with a precisely
determined constant $c(\nu)>0$, which implies a non-trivial limit of the drift
terms as $\varepsilon$ vanishes. We also prove that in this weak coupling
regime, the sequence of solutions converges in distribution to
$\left(\sqrt{\frac{c(\nu)^2}{2}+\nu^2}\right)\widetilde{B}_t$ as $\varepsilon$
vanishes, where $\widetilde{B}_t$ is a two-dimensional standard Brownian
motion.
