  We consider a one dimensional transport equation with varying vector field
and a small viscosity coefficient, controlled by one endpoint of the interval.
We give upper and lower bounds on the minimal time needed to control to zero,
uniformly in the vanishing viscosity limit. We assume that the vector field
varies on the whole interval except at one point. The upper/lower estimates we
obtain depend on geometric quantities such as an Agmon distance and the
spectral gap of an associated semiclassical Schr{\"o}dinger operator. They
improve, in this particular situation, the results obtained in the companion
paper [LL21]. The proofs rely on a reformulation of the problem as a uniform
observability question for the semiclassical heat equation together with a fine
analysis of localization of eigenfunctions both in the semiclassically allowed
and forbidden regions [LL22], together with estimates on the spectral gap
[HS84, All98]. Along the proofs, we provide with a construction of biorthogonal
families with fine explicit bounds, which we believe is of independent
interest.
