  Structure formation in turbulence is effectively an instability of "plasma"
formed by fluctuations serving as particles. These "particles" are quantumlike;
namely, their wavelengths are non-negligible compared to the sizes of
background coherent structures. The corresponding "kinetic equation" describes
the Wigner matrix of the turbulent field, and the coherent structures serve as
collective fields. This formalism is usually applied to manifestly quantumlike
or scalar waves. Here, we extend it to compressible Navier--Stokes turbulence,
where the fluctuation Hamiltonian is a five-dimensional matrix operator and
diverse modulational modes are present. As an example, we calculate these modes
for a sinusoidal shear flow and find two modulational instabilities. One of
them is specific to supersonic flows, and the other one is a
Kelvin--Helmholtz-type instability that is a generalization of the known
zonostrophic instability. This work serves as a stepping stone toward improving
the understanding of magnetohydrodynamic turbulence, which can be approached
similarly.
