We report for the first time the pattern dynamics in the vicinity of an inverse homoclinic bifurcation in an extended dissipative system. We observe, in direct numerical simulations of three dimensional Rayleigh-Bénard convection, a spontaneous breaking of a competition of two mutually perpendicular sets of oscillating cross rolls to one of two possible sets of oscillating cross rolls as the Rayleigh number is raised above a critical value. The time period of the cross-roll patterns diverges, and shows scaling behavior near the bifurcation point. This is an example of a transition from nonlocal to local pattern dynamics near an inverse homoclinic bifurcation. We also present a simple four-mode model that captures the pattern dynamics quite well. [9]. The selection of a pattern is a consequence of at least one broken symmetry of the system. Unbroken symmetries often introduce multiple patterns, which may lead to a transition from local to global pattern dynamics. The gluing [10] of two limit cycles on two sides of a saddle point in the phase space of a given system is an example of a local to nonlocal bifurcation. It occurs when two limit cycles simultaneously become homoclinic orbits of the same saddle point. This phenomenon has been recently observed in a variety of systems including liquid crystals [11], fluid dynamical systems [12], biological systems [13], optical systems [14], and electrical circuits [15], and is a topic of current research. The pattern dynamics in the vicinity of a homoclinic bifurcation has, however, not been investigated in a fluid dynamical system.A Rayleigh-Bénard system [16,17], where a thin layer of a fluid is heated uniformly from below and cooled uniformly from above, is a classical example of an extended dissipative system which shows a plethora of patternforming instabilities [2], chaos [18], and turbulence [19]. Low-Prandtl-number [20] and very low-Prandtl-number convection [21,22] show three-dimensional oscillatory behavior close to the instability onset. In addition, the Rayleigh-Bénard system possesses symmetries under translation and rotation in the horizontal plane that can introduce multiple sets of patterns. The possibility of a homoclinic bifurcation and the pattern dynamics in its vicinity are unexplored in three dimensional (3D) Rayleigh-Bénard convection.We report, in this article, for the first time the possibility of an inverse homoclinic bifurcation in direct numerical simulations (DNS) of three dimensional (3D) Rayleigh-Bénard convection (RBC) in low-Prandtlnumber fluids, and the results of our investigations of fluid patterns close to the bifurcation. We observe spontaneous breaking of a periodic competition of two mutually perpendicular sets of cross rolls to one set of oscillating cross rolls, as the Rayleigh number Ra is raised above a critical value Ra h . The time period of the oscillating patterns diverges, and shows scaling behavior in the close vicinity of the transition point. The exponents of scaling are asymmetric on the two sides of the transition po...

We present the effects of small Coriolis force on homoclinic gluing and ungluing bifurcations in low-Prandtl-number(0.025 ≤ P r) rotating Rayleigh-Bénard system with stress-free top and bottom boundaries. We have performed direct numerical simulations for a a wide range of Taylor number (5 ≤ T a ≤ 50) and reduce Rayleigh number r (≤ 1.25). We observe homoclinic ungluing bifurcation, marked by the spontaneous breaking of a larger limit cycle into two possible set of limit cycles in the phase space, for lower values of T a. Two unglued limit cycles merge together for higher values of T a, as Ra is raised sufficiently. The range of T a for which both gluing and ungluing bifurcation can be seen depends on the Prandtl number P r. The variation of the bifurcation points with T a is also investigated. We also present a low-dimensional model which qualitatively captures the dynamics of the system near the homoclinic bifurcation points. The model is used to study the variation of the homoclinic bifurcation points with Prandtl number for different values of T a.

We present the results of our investigations of the primary instability and the flow patterns near onset in zero-Prandtl-number Rayleigh-B\'enard convection with uniform rotation about a vertical axis. The investigations are carried out using direct numerical simulations of the hydrodynamic equations with stress-free horizontal boundaries in rectangular boxes of size $(2\pi/k_x) \times (2\pi/k_y) \times 1$ for different values of the ratio $\eta = k_x/k_y$. The primary instability is found to depend on $\eta$ and $Ta$. Wavy rolls are observed at the primary instability for smaller values of $\eta$ ($1/\sqrt{3} \le \eta \le 2$ except at $\eta = 1$) and for smaller values of $Ta$. We observed K\"{u}ppers-Lortz (KL) type patterns at the primary instability for $\eta = 1/\sqrt{3}$ and $ Ta \ge 40$. The fluid patterns are found to exhibit the phenomenon of bursting, as observed in experiments [Bajaj et al. Phys. Rev. E {\bf 65}, 056309 (2002)]. Periodic wavy rolls are observed at onset for smaller values of $Ta$, while KL-type patterns are observed for $ Ta \ge 100$ for $\eta =\sqrt{3}$. In case of $\eta = 2$, wavy rolls are observed for smaller values of $Ta$ and KL-type patterns are observed for $25 \le Ta \le 575$. Quasi-periodically varying patterns are observed in the oscillatory regime ($Ta > 575$). The behavior is quite different at $\eta = 1$. A time dependent competition between two sets of mutually perpendicular rolls is observed at onset for all values of $Ta$ in this case. Fluid patterns are found to burst periodically as well as chaotically in time. It involved a homoclinic bifurcation. We have also made a couple of low-dimensional models to investigate bifurcations for $\eta = 1$, which is used to investigate the sequence of bifurcations.Comment: 50 pages, 22 figures and 3 table

We investigate the Lagrangian statistics of three-dimensional rotating turbulent flows through direct numerical simulations. We find that the emergence of coherent vortical structures because of the Coriolis force leads to a suppression of the "flight-crash" events reported by Xu, et al. [Proc. Natl. Acad. Sci. (U.S.A) 111, 7558 (2014)]. We perform systematic studies to trace the origins of this suppression in the emergent geometry of the flow and show why such a Lagrangian measure of irreversibility may fail in the presence of rotation.The irreversibility of fully developed, homogeneous and isotropic turbulence, as well as the non-trivial spatio-temporal structure of its (Eulerian) velocity field shows up in an interesting way in the statistics of the kinetic energy along Lagrangian trajectories. Xu, et al.[1], measured the kinetic energy of a tracer along its trajectory, as a function of time, to show that the gain in kinetic energy (over time) is gradual whereas the loss is rapid. (The average energy, statistically, is of course constant over time.) This behaviour of the energy fluctuations is quantified most conveniently by the statistics of energy increments (gain or loss) at small, but fixed, time intervals. In the limiting case, the rate of change of the kinetic energy, or power, serves as a useful probe to understand how Eulerian irreversibility manifests itself in the Lagrangian framework. Bhatnagar, et al. [2], extended this idea to the case of heavy, inertial particles, preferentially sampling the flow, to disentangle the effects of irreversibility and flow geometry.This feature of Lagrangian trajectories, dubbed as "flightcrash events" [1], is a consequence of the dissipative nature of turbulent flows as well as the spatial structure of the Eulerian field with its intense, though sparse, regions of vorticity and more abundant, though milder, regions of strain. However, so far, measurements have been confined only to flows which are statistically homogeneous and isotropic. Therefore it is natural to ask if flight-crash events are just as ubiquitous in turbulent settings with anisotropy and structures different from those seen in statistically homogeneous, isotropic turbulence. An obvious candidate for this is fully developed turbulent flows under rotation [3][4][5] which are seen in a variety of processes spanning scales ranging from the astrophysical [6][7][8], geophysical [9] to the industrial [10]. In all these phenomena, although the Coriolis force does no work, it leads to the formation of large-scale columnar vortices leading to dynamics quite different from non-rotating, three-dimensional flow. In particular, rotation gives rise to an enhanced accumulation of energy in modes perpendicular to the plane of rotation [11][12][13], an inverse energy cascade in 3D turbulence [14][15][16], generation of inertial waves [5,17,18], and an increase in length scales parallel to the axis of rotation [19]. Consequently, rotating turbulence has been the subject of much experimental [16][17][18][19][20][21][22...