  We formulate a statistical wave-mechanical approach to describe dissipation
and instabilities in two-dimensional turbulent flows of magnetized plasmas and
atmospheric fluids, such as drift and Rossby waves. This is made possible by
the existence of Hilbert space, associated with the electric potential of
plasma or stream function of atmospheric fluid. We therefore regard such
turbulent flows as macroscopic wave-mechanical phenomena, driven by the
non-Hermitian Hamiltonian operator we derive, whose anti-Hermitian component is
attributed to an effect of the environment. Introducing a wave-mechanical
density operator for the statistical ensembles of waves, we formulate master
equations and define observables: such as the enstrophy and energy of both the
waves and zonal flow as statistical averages. We establish that our open system
can generally follow two types of time evolution, depending on whether the
environment hinders or assists the system's stability and integrity. We also
consider a phase-space formulation of the theory, including the
geometrical-optic limit and beyond, and study the conservation laws of physical
observables. It is thus shown that the approach predicts various mechanisms of
energy and enstrophy exchange between drift waves and zonal flow, which were
hitherto overlooked in models based on wave kinetic equations.
