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The role of phase dynamics in a stochastic model of a passively advected scalar

Collective synchronous motion of the phases is introduced in a model for the stochastic passive advection-diffusion of a scalar with external forcing. The model for the phase coupling dynamics follows the well known Kuramoto model paradigm of limit-cycle oscillators. The natural frequencies in the Kuramoto model are assumed to obey a given scale dependence through a dispersion relation of the drift-wave form $-β\frac{k}{1+k^2}$, where $β$ is a constant representing the typical strength of the gradient. The present aim is to study the importance of collective phase dynamics on the characteristic time evolution of the fluctuation energy and the formation of coherent structures. Our results show that the assumption of a fully stochastic phase state of turbulence is more relevant for high values of $β$, where we find that the energy spectrum follows a $k^{-7/2}$ scaling. Whereas for lower $β$ there is a significant difference between a-synchronised and synchronised phase states, and one could expect the formation of coherent modulations in the latter case.