  In homogeneous drift-wave (DW) turbulence, zonal flows (ZFs) can be generated
via a modulational instability (MI) that either saturates monotonically or
leads to oscillations of the ZF energy at the nonlinear stage. This dynamics is
often attributed as the predator--prey oscillations induced by ZF collisional
damping; however, similar dynamics is also observed in collisionless ZFs, in
which case a different mechanism must be involved. Here, we propose a
semi-analytic theory that explains the transition between the oscillations and
saturation of collisionless ZFs within the quasilinear Hasegawa--Mima model. By
analyzing phase-space trajectories of DW quanta (driftons) within the
geometrical-optics (GO) approximation, we argue that the parameter that
controls this transition is $N\sim\gamma_{{\rm MI}}/\omega_{{\rm DW}}$, where
$\gamma_{{\rm MI}}$ is the MI growth rate and $\omega_{{\rm DW}}$ is the linear
DW frequency. We argue that at $N\ll1$, ZFs oscillate due to the presence of
so-called passing drifton trajectories, and we derive an approximate formula
for the ZF amplitude as a function of time in this regime. We also show that at
$N\gtrsim1$, the passing trajectories vanish and ZFs saturate monotonically,
which can be attributed to phase mixing of higher-order sidebands. A
modification of $N$ that accounts for effects beyond the GO limit is also
proposed. These analytic results are tested against both quasilinear and
fully-nonlinear simulations. They also explain the earlier numerical results by
Connaughton $\textit{et al}$. [J. Fluid Mech. $\textbf{654}$, 207 (2010)] and
Gallagher $\textit{et al}$. [Phys. Plasmas $\textbf{19}$, 122115 (2012)] and
offer a revised perspective on what the control parameter is that determines
the transition from the oscillations to saturation of collisionless ZFs.

Plasma Physics;Fluid Dynamics;Geophysics