  Active fluids, such as suspensions of microswimmers, are known to
self-organize into complex spatio-temporal flow patterns. An intriguing example
is mesoscale turbulence, a state of dynamic vortex structures exhibiting a
characteristic length scale. Here, we employ a minimal model for the effective
microswimmer velocity field to explore how the turbulent state develops from
regular vortex patterns when the strength of activity resp. related parameters
such as nonlinear advection or polar alignment strength - is increased. First,
we demonstrate analytically that the system, without any spatial constraints,
develops a stationary square vortex lattice in the absence of nonlinear
advection. Subsequently, we perform an extended stability analysis of this
nonuniform "ground state" and uncover a linear instability, which follows from
the mutual excitement and simultaneous growth of multiple perturbative modes.
This extended analysis is based on linearization around an approximation of the
analytical vortex lattice solution and allows us to calculate critical activity
parameters. Above these critical values, the vortex lattice develops into
mesoscale turbulence in numerical simulations. Utilizing the numerical
approach, we uncover an extended region of hysteresis where both patterns are
possible depending on the initial condition. Here, we find that turbulence
persists below the instability of the vortex lattice. We further determine the
stability of square vortex patterns as a function of their wavenumber and
represent the results analogous to the well-known Busse balloons known from
classical pattern-forming systems. Here, the region of stable periodic patterns
shrinks and eventually disappears with increasing activity parameters. Our
results show that the strength of activity plays a similar role for active
turbulence as the Reynolds number does in driven flow exhibiting inertial
turbulence.
