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Convection is a ubiquitous process driving geophysical/astrophysical fluid flows, which are typically strongly constrained by planetary rotation on large scales. A celebrated model of such flows, rapidly rotating Rayleigh-Bénard convection, has been extensively studied in direct numerical simulations (DNS) and laboratory experiments, but the parameter values attainable by state-of-the-art methods are limited to moderately rapid rotation (Ekman numbers Ek≳10−8), while realistic geophysical/astrophysical Ek are significantly smaller. Asymptotically reduced equations of motion, the nonhydrostatic quasi-geostrophic equations (NHQGE), describing the flow evolution in the limit Ek→0, do not apply at finite rotation rates. The geophysical/astrophysical regime of small but finite Ek therefore remains currently inaccessible. Here, we introduce a new, numerically advantageous formulation of the Navier-Stokes-Boussinesq equations informed by the scalings valid for Ek→0, the \textit{Rescaled Rapidly Rotating incompressible Navier-Stokes Equations} (RRRiNSE). We solve the RRRiNSE using a spectral quasi-inverse method resulting in a sparse, fast algorithm to perform efficient DNS in this previously unattainable parameter regime. We validate our results against the literature across a range of Ek and demonstrate that the algorithmic approaches taken remain accurate and numerically stable at Ek as low as 10−15. Like the NHQGE, the RRRiNSE derive their efficiency from adequate conditioning, eliminating spurious growing modes that otherwise induce numerical instabilities at small Ek. We show that the time derivative of the mean temperature is inconsequential for accurately determining the Nusselt number in the stationary state, significantly reducing the required simulation time, and demonstrate that full DNS using RRRiNSE agree with the NHQGE at very small Ek. 

Abstract Localized coherent structures can form in externally driven dispersive optical cavities with a Kerr-type non-linearity. Such systems are described by the Lugiato–Lefever (LL) equation, which supports a large variety of dynamical states. Here, we review our current knowledge of the formation, stability and bifurcation structure of localized structures in the one-dimensional LL equation. We do so by focusing on two main regimes of operation: anomalous and normal second-order dispersion. In the anomalous regime, localized patterns are organized in a homoclinic snaking scenario, which is eventually destroyed, leading to a foliated snaking bifurcation structure. In the normal regime, localized structures undergo a different type of bifurcation structure, known as collapsed snaking. The effects of third-order dispersion and various dynamical regimes are also described.