  The present work investigates the evolution of linear perturbations of
time-dependent ideal fluid flows with advected quantities, expressed in terms
of the second order variations of the action corresponding to a Lagrangian
defined on a semidirect product space. This approach is related to Jacobi
fields along geodesics and several examples are given explicitly to elucidate
our approach. Numerical simulations of the perturbation dynamics are also
presented.
