  Let $X$ be a compact connected CR manifold of dimension $2n-1, n\geq 2$. We
assume that there is a transversal CR locally free $S^1$ action on $X$. Let
$L^k$ be the $k$-th power of a rigid CR line bundle $L$ over $X$. Without any
assumption on the Levi-form of $X$, we obtain a scaling upper-bound for the
partial Szeg\H{o} kernel on $(0,q)$-forms with values in $L^k$. After
integration, this gives the weak Morse inequalities. By a refined spectral
analysis, we also obtain the strong Morse inequalities in CR setting. We apply
the strong Morse inequalities to show that the Grauert-Riemenschneider
criterion is also true in the CR setting.
