  This paper presents a geometric microcanonical ensemble perspective on
two-dimensional Truncated Euler flows, which contain a finite number of
(Fourier) modes and conserve energy and enstrophy. We explicitly perform phase
space volume integrals over shells of constant energy and enstrophy. Two
applications are considered. In a first part, we determine the average energy
spectrum for highly condensed flow configurations and show that the result is
consistent with Kraichnan's canonical ensemble description, despite the fact
that no thermodynamic limit is invoked. In a second part, we compute the
probability density for the largest-scale mode of a free-slip flow in a square,
which displays reversals. We test the results against numerical simulations of
a minimal model and find excellent agreement with the microcanonical theory,
unlike the canonical theory, which fails to describe the bimodal statistics.
This article is part of the theme issue "Mathematical problems in physical
fluid dynamics".
