Abstract
A theory of homogeneous, isotropic turbulence in an incompressible fluid is formulated. This theory provides a basis for the dynamics of fully developed turbulence. A hierarchy of equations to determine the time evolution of single-time moments of the velocity field is derived. Various properties of this hierarchy are exhibited, such as positivity of the dynamically determined energy density. The theory is applied to the study of the inertial range, where the Kolmogoroff theory is justified. A generalized inertial-range theory is formulated. This “semilocal” theory is characterized by local energy transfer and nonlocal relaxation of triple moments. The generalized theory is applied to the study of hydromagnetic turbulence, which is shown to possess a semilocal inertial range.
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