  In this work, we extend existing well-posedness by noise results for the
stochastic transport and continuity equations by treating them as special cases
of the linear advection equation of $k$-forms, which arises naturally in
geometric fluid dynamics. In particular, we prove the existence and uniqueness
of weak $L^p$-solutions to the stochastic linear advection equation of
$k$-forms that is driven by a H\"older continuous, $W^{1,1}_{loc}$ drift and
smooth diffusion vector fields, such that the equation without noise admits
infinitely many solutions.
