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

Abstract

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 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.

Role of phase synchronisation in turbulence

“…Due to the co-evolving and interacting amplitudes and phases, a forcing interacting purely on the phases increases the number of phase locking events, during which the phases are locked to ± π/2. In contrast to the stochastically driven oscillator models studied previously 20 the present phase force may be thought of as a strongly coherent force that ultimately change the dynamics of both phases and amplitudes.…”

“…3. In the absence of the phase force, at a phase locking event no synchronization between the phases of |k|∈ [1,20] modes is obvious, and the phases of the |k|∈ [21,121] show synchronization to ± π/2 with equally divided numbers locked to π/2 and π/2. Here, we find that the quality of the synchronization in the high-k range of |k|∈ [121,700] is not as good.…”

Self-organisation of random oscillators with Lévy stable distributions

“…Here ω j is the natural frequency of the oscillator which is assumed to be distributed according to a Gaussian distribution f (ω) = exp(−ω 2 /2)/ √ 2π. J ij is the strength of the interactions between oscillators ith and jth and are assumed to be random constants distributed according to an α-stable distribution S(α, β, σ, µ) with characteristic exponent 0 < α 2, skewness β, scale σ and location μ [12,[25][26][27][28][29][30][31]. Here we chose β = 0, µ = 0, and 2) where F is the control parameter as in [6].…”

“…J ij is the strength of the interactions between oscillators ith and jth and are assumed to be random constants distributed according to an α-stable distribution S(α, β, σ, µ) with characteristic exponent 0 < α ≤ 2, skewness β, scale σ and location µ [12,[25][26][27][28][29][30][31]. Here we chose β = 0, µ = 0, and…”