Inhomogeneous Turbulence without Mean Flow

The diffusion of turbulence in a fluid at rest is studied. First, a simplified one-dimensional model is proposed and solved exactly. The change of the intensity and the length scale of turbulence with depth is calculated. Next, the equation of the energy spectrum function is derived and solved using two hypotheses: one expresses the non-linear interaction in terms of the eddy viscosity and the other describes the diffusion of turbulence using the notion of the eddy diffusivity. The energy spectrum function, the intensity of turbulence and the eddy diffusivity are calculated numerically.