  The paper is concerned with the principal eigenvalue of some linear elliptic
operators with drift in two dimensional space. We provide a refined description
of the asymptotic behavior for the principal eigenvalue as the drift rate
approaches infinity. Under some non-degeneracy assumptions, our results
illustrate that these asymptotic behaviors are completely determined by some
connected components in the omega-limit set of the system of ordinary
differential equations associated with the drift term, which includes stable
fixed points, stable limit cycles, hyperbolic saddles connecting homoclinic
orbits, and families of closed orbits. Some discussions on degenerate cases are
also included.
