  In this paper, we prove the invariance of the spectrum of the basic Dirac
operator defined on a Riemannian foliation $(M,\mathcal{F})$ with respect to a
change of bundle-like metric. We then establish new estimates for its
eigenvalues on spin flows in terms of the O'Neill tensor and the first
eigenvalue of the Dirac operator on $M$. We discuss examples and also define a
new version of the basic Laplacian whose spectrum does not depend on the choice
of bundle-like metric.
