  We discover an instability mechanism in suspensions of self-propelled
particles that does not involve active stress. Instead, it is driven by a
subtle interplay of inertia, swimmer motility, and concentration fluctuations,
through a crucial time lag between the velocity and the concentration field.
The resulting time-persistent state seen in our high-resolution numerical
simulations consists of self-sustained waves of concentration and orientation,
transiting from regular oscillations to wave turbulence. We analyze the
statistical features of this active turbulence, including an intriguing
connection to the Batchelor spectrum of passive scalars.
