  We extend the It\^o-Wentzell formula for the evolution of a time-dependent
stochastic field along a semimartingale to $k$-form-valued stochastic
processes. The result is the Kunita-It\^o-Wentzell (KIW) formula for $k$-forms.
We also establish a correspondence between the KIW formula for $k$-forms
derived here and a certain class of stochastic fluid dynamics models which
preserve the geometric structure of deterministic ideal fluid dynamics. This
geometric structure includes Eulerian and Lagrangian variational principles,
Lie--Poisson Hamiltonian formulations and natural analogues of the Kelvin
circulation theorem, all derived in the stochastic setting.
