  We perform a theoretical and numerical investigation of the time-average of
energy exchange among modes of Reduced Order Models (ROMs) of fluid flows. We
are interested in the statistical equilibrium problem, and especially in the
possible forward and backward average transfer of energy among ROM basis
functions (modes). We consider two types of ROM modes: eigenfunctions of the
Stokes operator and Proper Orthogonal Decomposition (POD) modes. We prove
analytical results for both types of ROM modes and we highlight the differences
between them. We also investigate numerically whether the time-average energy
exchange between POD modes is positive. To this end, we utilize the
one-dimensional Burgers equation as a simplified mathematical model, which is
commonly used in ROM tests. The main conclusion of our numerical study is that,
for long enough time intervals, the time-average energy exchange from low index
POD modes to high index POD modes is positive, as predicted by our theoretical
results.
