TY  - JOUR
A1  - Omel'chenko, Oleh
T1  - Mathematical framework for breathing chimera states
JF  - Journal of nonlinear science
N2  - About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence-incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott-Antonsen reduction technique.
KW  - Coupled oscillators
KW  - Breathing chimera states
KW  - Coherence-incoherence
KW  - patterns
KW  - Ott-Antonsen equation
KW  - Periodic solutions
KW  - Stability
Y1  - 2022
U6  - https://doi.org/10.1007/s00332-021-09779-1
SN  - 0938-8974
SN  - 1432-1467
VL  - 32
IS  - 2
PB  - Springer
CY  - New York
ER  - 
TY  - JOUR
A1  - Fürst, Johannes J.
A1  - Levermann, Anders
T1  - A minimal model for wind- and mixing-driven overturning  threshold behavior for both driving mechanisms
JF  - Climate dynamics : observational, theoretical and computational research on the climate system
N2  - We present a minimal conceptual model for the Atlantic meridional overturning circulation which incorporates the advection of salinity and the basic dynamics of the oceanic pycnocline. Four tracer transport processes following Gnanadesikan in Science 283(5410):2077-2079, (1999) allow for a dynamical adjustment of the oceanic pycnocline which defines the vertical extent of a mid-latitudinal box. At the same time the model captures the salt-advection feedback (Stommel in Tellus 13(2):224-230, (1961)). Due to its simplicity the model can be solved analytically in the purely wind- and purely mixing-driven cases. We find the possibility of abrupt transition in response to surface freshwater forcing in both cases even though the circulations are very different in physics and geometry. This analytical approach also provides expressions for the critical freshwater input marking the change in the dynamics of the system. Our analysis shows that including the pycnocline dynamics in a salt-advection model causes a decrease in the freshwater sensitivity of its northern sinking up to a threshold at which the circulation breaks down. Compared to previous studies the model is restricted to the essential ingredients. Still, it exhibits a rich behavior which reaches beyond the scope of this study and might be used as a paradigm for the qualitative behaviour of the Atlantic overturning in the discussion of driving mechanisms.
KW  - Meridional overturning circulation
KW  - Northern sinking
KW  - Critical freshwater threshold
KW  - Overturning sensitivity
KW  - Conceptual model
KW  - Stability
KW  - Atlantic meridional overturning circulation
KW  - Pycnocline depth
KW  - Driving mechanism
Y1  - 2012
U6  - https://doi.org/10.1007/s00382-011-1003-7
SN  - 0930-7575
VL  - 38
IS  - 1-2
SP  - 239
EP  - 260
PB  - Springer
CY  - New York
ER  -