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.

中文翻译：

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等离子体和大气中湍流的密度算子方法

我们制定了一种统计波机械方法来描述磁化等离子体和大气流体（例如漂移波和Rossby波）的二维湍流中的耗散和不稳定性。希尔伯特空间的存在使之成为可能，而希尔伯特空间与等离子体的电位或大气流体的流函数相关。因此，我们将这种湍流看作是宏观波浪力学现象，由我们推导出的非埃尔米特哈密顿算符驱动，其反埃尔米特分量归因于环境的影响。为波浪的统计集合引入了波浪机械密度算子，我们制定了主方程并定义了可观测值：例如波浪的涡旋和能量以及地带流均作为统计平均值。我们确定，我们的开放系统通常可以遵循两种类型的时间演化，具体取决于环境是阻碍还是辅助了系统的稳定性和完整性。我们还考虑了该理论的相空间公式化，包括几何光学极限及其他，并研究了物理观测值的守恒律。因此表明，该方法预测了漂移波和纬向流之间能量和内旋交换的各种机理，这些机理迄今在基于波动力学方程的模型中被忽略。