A vector bundle flow ( Φ t , ϕ t ) ({\Phi ^t},{\phi ^t}) on the vector bundle E E over a compact metric space M M induces a one-parameter group { Φ t # } \{ \Phi _t^\# \} of bounded operators acting on the continuous sections of E E , with infinitesimal generator L L . An example is given by the tangent flow ( T ϕ t , ϕ t ) (T{\phi ^t},{\phi ^t}) , if ϕ t {\phi ^t} is a flow on a smooth manifold. In this article, the spectrum of the generator L L is used to study the exponential growth rates of bundle trajectories in the neighborhood of a fixed invariant subbundle, e.g. the tangent bundle of a submanifold of M M . Auxiliary normal and tangential spectra are introduced, and their relationship and fine structure are explored.