  A meander is a topological configuration of a line and a simple closed curve
in the plane (or a pair of simple closed curves on the 2-sphere) intersecting
transversally. Meanders can be traced back to H. Poincar\'e and naturally
appear in various areas of mathematics, theoretical physics and computational
biology (in particular, they provide a model of polymer folding). Enumeration
of meanders is an important open problem. The number of meanders with 2N
crossings grows exponentially when N grows, but the longstanding problem on the
precise asymptotics is still out of reach. We show that the situation becomes
more tractable if one additionally fixes the topological type (or the total
number of minimal arcs) of a meander. Then we are able to derive simple
asymptotic formulas for the numbers of meanders as N tends to infinity. We also
compute the asymptotic probability of getting a simple closed curve on a sphere
by identifying the endpoints of two arc systems (one on each of the two
hemispheres) along the common equator. The new tools we bring to bear are based
on interpretation of meanders as square-tiled surfaces with one horizontal and
one vertical cylinders. The proofs combine recent results on Masur-Veech
volumes of the moduli spaces of meromorphic quadratic differentials in genus
zero with our new observation that horizontal and vertical separatrix diagrams
of integer quadratic differentials are asymptotically uncorrelated. The
additional combinatorial constraints we impose in this article yield explicit
polynomial asymptotics.
