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1. Stratospheric Planetary Flows from the Perspective of the Euler Equation on a Rotating Sphere

   https://doi.org/10.1007/s00205-022-01791-3  

   Constantin, A. ; Germain, P. ( May 2022 , Archive for Rational Mechanics and Analysis)

   This article is devoted to stationary solutions of Euler’s equation on a rotating sphere, and to their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system and in polar regions of the Earth. For the Euler equation, under appropriate conditions, rigidity results are established, ensuring that the solutions are either zonal or rotated zonal solutions. A natural analogue of Arnold’s stability criterion is proved. In both cases, the lowest mode Rossby–Haurwitz stationary solutions (more precisely, those whose stream functions belong to the sum of the first two eigenspaces of the Laplace-Beltrami operator) appear as limiting cases. We study the stability properties of these critical stationary solutions. Results on the local and global bifurcation of non-zonal stationary solutions from classical Rossby–Haurwitz waves are also obtained. Finally, we show that stationary solutions of the Euler equation on a rotating sphere are building blocks for travelling-wave solutions of the 3D system that describes the leading order dynamics of stratospheric planetary flows, capturing the characteristic decrease of density and increase of temperature with height in this region of the atmosphere.

    

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2. Barotropic instability of shear flows

   https://doi.org/10.1111/sapm.12297  

   Lin, Zhiwu ; Yang, Jincheng ; Zhu, Hao ( February 2020 , Studies in Applied Mathematics)

   We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm‐Liouville theory and hypergeometric functions.

    

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3. Spatiotemporal relationships between the cell shape and the actomyosin cortex of periodically protruding cells

   https://doi.org/10.1002/cm.21229  

   Driscoll, Meghan K. ; Losert, Wolfgang ; Jacobson, Ken ; Kapustina, Maryna ( August 2015 , Cytoskeleton)

   We investigate the dynamics of cell shape and analyze the actin and myosin distributions of cells exhibiting cortical density traveling waves. These waves propagate by repeated cycles of cortical compression (folding) and dilation (unfolding) that lead to periodic protrusions (oscillations) of the cell boundary. The focus of our detailed analysis is the remarkable periodicity of this phenotype, in which both the overall shape transformation and distribution of actomyosin density are repeated from cycle to cycle even though the characteristics of the shape transformation vary significantly for different regions of the cell. We show, using correlation analysis, that during traveling wave propagation cortical actin and plasma membrane densities are tightly coupled at each point along the cell periphery. We also demonstrate that the major protrusion appears at the wave trailing edge just after the actin cortex density has reached a maximum. Making use of the extraordinary periodicity, we employ latrunculin to demonstrate that sequestering actin monomers can have two distinct effects: low latrunculin concentrations can trigger and enhance traveling waves but higher concentrations of this drug retard the waves. The fundamental mechanism underlying this periodically protruding phenotype, involving folding and unfolding of the cortex‐membrane couple, is likely to hold important clues for diverse phenomena including cell division and amoeboid‐type migration. © 2015 The Authors. Cytoskeleton Published by Wiley Periodicals, Inc.

    

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4. Quantum scattering states in a nonlinear coherent medium

   https://doi.org/10.1103/PhysRevA.108.023314  

   Brattley, Allison ; Huang, Hongyi ; Das, Kunal K. ( August 2023 , Physical Review A)

   We present a comprehensive study of stationary states in a coherent medium with a quadratic or Kerr nonlinearity in the presence of localized potentials in one dimension for both positive and negative signs of the nonlinear term as well as for barriers and wells. The description is in terms of the nonlinear Schrödinger equation and hence applicable to a variety of systems, including interacting ultracold atoms in the mean field regime and light propagation in optical fibers. We determine the full landscape of solutions in terms of a potential step and build solutions for rectangular barrier and well potentials. It is shown that all the solutions can be expressed in terms of a Jacobi elliptic function with the inclusion of a complex-valued phase shift. Our solution method relies on the roots of a cubic polynomial associated with a hydrodynamic picture, which provides a simple classification of all the solutions, both bounded and unbounded, while the boundary conditions are intuitively visualized as intersections of phase space curves. We compare solutions for open boundary conditions with those for a barrier potential on a ring, and also show that numerically computed solutions for smooth barriers agree qualitatively with analytical solutions for rectangular barriers. A stability analysis of solutions based on the Bogoliubov equations for fluctuations shows that persistent instabilities are localized at sharp boundaries and are predicated by the relation of the mean density change across the boundary to the value of the derivative of the density at the edge. We examine the scattering of a wave packet by a barrier potential and show that at any instant the scattered states are well described by the stationary solutions we obtain, indicating applications of our results and methods to nonlinear scattering problems. 

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5. Traveling spiral wave chimeras in coupled oscillator systems: emergence, dynamics, and transitions

   https://doi.org/10.1088/1367-2630/acfd4f  

   Bataille-Gonzalez, M. ; Clerc, M. G. ; Knobloch, E. ; Omel’chenko, O. E. ( October 2023 , New Journal of Physics)

   Systems of coupled nonlinear oscillators often exhibit states of partial synchrony in which some of the oscillators oscillate coherently while the rest remain incoherent. If such a state emerges spontaneously, in other words, if it cannot be associated with any heterogeneity in the system, it is generally referred to as a chimera state. In planar oscillator arrays, these chimera states can take the form of rotating spiral waves surrounding an incoherent core, resembling those observed in oscillatory or excitable media, and may display complex dynamical behavior. To understand this behavior we study stationary and moving chimera states in planar phase oscillator arrays using a combination of direct numerical simulations and numerical continuation of solutions of the corresponding continuum limit, focusing on the existence and properties of traveling spiral wave chimeras as a function of the system parameters. The oscillators are coupled nonlocally and their frequencies are drawn from a Lorentzian distribution. Two cases are discussed in detail, that of a top-hat coupling function and a two-parameter truncated Fourier approximation to this function in Cartesian coordinates. The latter allows semi-analytical progress, including determination of stability properties, leading to a classification of possible behaviors of both static and moving chimera states. The transition from stationary to moving chimeras is shown to be accompanied by the appearance of complex filamentary structures within the incoherent spiral wave core representing secondary coherence regions associated with temporal resonances. As the parameters are varied the number of such filaments may grow, a process reflected in a series of folds in the corresponding bifurcation diagram showing the drift speedsas a function of the phase-lag parameterα.

    

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