Pattern avoiding meandric permutations

We study and characterize meandric permutations avoiding one or more patterns of length three, and find explicit formulae for the cardinality of each of these sets. We determine the distribution of the descent statistic for the set of meandric permutations avoiding the pattern 231. The sets of meandric permutations avoiding any other pattern of length three can be either trivially determined, or deduced from the 231 case via the symmetries of the square. In the 231 case we provide a bijection with a set of Motzkin paths that maps the statistic "number of descents of a permutation" to the statistic "number of non-horizontal steps of a path".