We study the simplest problem of turbulence spreading corresponding to the spatio-temporal propagation of a patch of turbulence from a region where it is locally excited to a region of weaker excitation or even local damping. A single model equation for the local turbulence intensity, I(x, t), includes the effects of local linear growth and damping, spatially local nonlinear coupling to dissipation and spatial scattering of turbulence energy induced by nonlinear coupling. In the absence of dissipation, front propagation into the linearly stable zone occurs with the property of rapid progression at small t, followed by slower sub-diffusive progression at late times. The turbulence radial spreading into the linearly stable zone reduces the turbulent intensity in the linearly unstable zone and introduces an additional dependence on the ρ* ≡ ρi/a to the turbulent intensity and the transport scaling. These are in broad, semi-quantitative agreement with a number of global gyrokinetic simulation results with zonal flows and without zonal flows. Front propagation stops when the radial flux of fluctuation energy from the linearly unstable region is balanced by local dissipation in the linearly stable region.
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