  Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension
$2n+1, n \ge 1$ with a transversal CR $S^1$-action on $X$. We introduce the
Fourier components of the Ray-Singer analytic torsion on $X$ with respect to
the $S^1$-action. We establish an asymptotic formula for the Fourier components
of the analytic torsion with respect to the $S^1$-action. This generalizes the
asymptotic formula of Bismut and Vasserot on the holomorphic Ray-Singer torsion
associated with high powers of a positive line bundle to strongly pseudoconvex
CR manifolds with a transversal CR $S^1$-action.
