  In this paper we study the asymptotic behavior of the spectral flow of a
one-parameter family $\{D_s\}$ of Dirac operators acting on the spinor bunldle
$S$ twisted by a vector bundle $E$ of rank $k$, with the parameter $s\in [0,r]$
when $r$ gets sufficiently large. Our method uses the variation of eta
invariant and local index theory technique. The key is a uniform estimate of
the eta invariant $\bar{\eta}(D_r)$ which is established via local index theory
technique and heat kernel estimate.
