  This paper reports several new classes of weakly unstable recurrent solutions
of the 2+1-dimensional Euler equation on a square domain with periodic boundary
conditions. These solutions have a number of remarkable properties which
distinguish them from analogous solutions of the Navier-Stokes equation
describing transitional flows. First of all, they come in high-dimensional
continuous families. Second, solutions of different types are connected, e.g.,
an equilibrium can be smoothly continued to a traveling wave or a time-periodic
state. Third, and most important, many of these solutions are dynamically
relevant for turbulent flow at high Reynolds numbers. Specifically, we find
that turbulence in numerical simulations exhibits large-scale coherent
structures resembling some of our time-periodic solutions both frequently and
over long temporal intervals. Such solutions are analogous to exact coherent
structures originally introduced in the context of transitional flows.
