  Wind forcing injects energy into the mesoscale eddies and near-inertial waves
(NIWs) in the ocean, and the NIW is believed to solve the puzzle of mesoscale
energy budget by absorbing energy from mesoscale eddies followed by a forward
cascade of NIW energy which finally dissipates at the ocean interior. This work
studies the turbulent energy transfer in the NIW--quasigeostrophic mean
mesoscale eddy coupled system based on a previously derived two-dimensional
model which has a Hamiltonian structure and inherits conserved quantities in
the Boussinesq equations (Xie \& Vanneste, \textit{J. Fluid Mech.}, vol. 774,
2015, pp. 147--169). Based on the conservation of energy, potential enstrophy
and wave action, we propose a heuristic argument predicting the existence of
phase transition with changing the relative strength between NIW and mean flow.
By running forced-dissipative numerical simulations with varying parameter $R$,
the ratio of the magnitude of NIW and mean-flow forcing, we justify the
existence of phase transition, which is found to be second-order, around
critical value $R_c$. When $0<R<R_c$, energy transfers bidirectionally, wave
action transfers downscale, and vorticity form strong cyclones. While when
$R>R_c$, energy transfers downscale, wave action transfers bidirectionally, and
vortex filaments are dominant. We find the catalytic wave induction (CWI)
mechanism where the NIW induces a downscale energy flux of the mean flow. The
CWI mechanism differs from the stimulated loss of balance by the absence of
energy conversion from the mesoscale eddy to NIW, and it is found to be
effective in the toy-model study, making it potentially important for ocean
energetics.
