  We prove that for any finite Thurston-type ordering $<_{T}$ on the braid
group\ $B_{n}$, the restriction to the positive braid monoid
$(B_{n}^{+},<_{T})$ is a\ well-ordered set of order type
$\omega^{\omega^{n-2}}$. The proof uses a combi\ natorial description of the
ordering $<_{T}$. Our combinatorial description is \ based on a new normal form
for positive braids which we call the $\C$-normal fo\ rm. It can be seen as a
generalization of Burckel's normal form and Dehornoy's \ $\Phi$-normal form
(alternating normal form).
