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Inverse Problem in the Kuramoto Model with a Phase Lag: Application to the Sun

We solve the inverse problem in the Kuramoto model with three nonidentical oscillators and a phase lag. The model represents a three-cell-in-depth radial profile of the solar meridional circulation in each solar hemisphere and describes the synchronization between the two components (toroidal and poloidal) of the solar magnetic field. We reconstruct natural frequencies of the top and the bottom oscillators from the evolution of their phases when the oscillators are phase-locked. The phase-locking allows to solve the inverse problem when the phase of the middle oscillator is not available. We present the exact solution and investigate its stability. The model reveals a crucial role of the phase lag in the inverse problem solution. We apply the model to the reconstruction of the deep meridional circulation of the Sun and discuss peculiarities of its evolution in terms of solar dynamo.