  We consider a drift-diffusion process with a time-independent and
divergence-free random drift that is of white-noise character. We are
interested in the critical case of two space dimensions, where one has to
impose a small-scale cut-off for well-posedness, and is interested in the
marginally super-diffusive behavior on large scales.
  In the presence of an (artificial) large-scale cut-off at scale L, as a
consequence of standard stochastic homogenization theory, there exist harmonic
coordinates with a stationary gradient $F_L$; the merit of these coordinates
being that under their lens, the drift-diffusion process turns into a
martingale.
  It has recently been established that the second moments diverge as
$\mathbb{E}|F_L|^2\sim\sqrt{\ln L}$ for $L\uparrow\infty$. We quantitatively
show that in this limit, and in the regime of small P\'eclet number,
$|F_L|^2/\mathbb{E}|F_L|^2$ is not equi-integrable, and that $\mathbb{E}|{\rm
det}F_L|/\mathbb{E}|F_L|^2 $ is small. Hence the Jacobian matrix of the
harmonic coordinates is very peaked and non-conformal.
  We establish this asymptotic behavior by characterizing a proxy $\tilde F_L$
introduced in previous work as the solution of an It\^{o} SDE w. r. t. the
variable $\ln L$, and which implements the concept of a scale-by-scale
homogenization based on a variance decomposition and admits an efficient
calculus. For this proxy, we establish $\mathbb{E}|\tilde
F_L|^4\gg(\mathbb{E}|\tilde F_L|^2)^2$ and $\mathbb{E}({\rm det}\tilde
F_L-1)^2\ll 1$. In view of the former property, we assimilate this phenomenon
to intermittency. In fact, $\tilde F_L$ behaves like a tensorial stochastic
exponential, and as a field can be assimilated to multiplicative Gaussian
chaos.
