  We prove an A'Campo type formula for the tame monodromy zeta function of a
smooth and proper variety over a discretely valued field $K$. As a first
application, we relate the orders of the tame monodromy eigenvalues on the
$\ell$-adic cohomology of a $K$-curve to the geometry of a relatively minimal
$sncd$-model, and we show that the semi-stable reduction theorem and Saito's
criterion for cohomological tameness are immediate consequences of this result.
As a second application, we compute the error term in the trace formula for
smooth and proper $K$-varieties. We see that the validity of the trace formula
would imply a partial generalization of Saito's criterion to arbitrary
dimension.
