Rapidly rotating Rayleigh-Bénard convection on a f-plane at colatitude ϑ_f is investigated numerically using an asymptotically reduced equation set valid in the limit of very rapid rotation. The equations provide a non-hydrostatic but quasi-geostrophic description in a non-orthogonal coordinate system. The tilt changes the structure of the large-scale barotropic condensate from large-scale vortices to zonal flows as the colatitude of the f-plane increases, with bistable states present for certain parameter ranges, extending prior work to a geophysically significant parameter regime. This behaviour is understood through the impact of broken rotation symmetry on the barotropic source terms resulting from baroclinic vortical stresses and baroclinic torque. As the tilt angle ϑ_f increases, global heat and momentum transport is reduced relative to upright-polar convection, a result that is explained through linear theory and nonlinear power maps both of which demonstrate increased attenuation of the domain of dynamically active spatial scales as the convective modes depart from a North-South alignment in the horizontal plane. A key finding is that the predominance of lateral thermal mixing allows for the maintenance of a persistent unstable mean temperature gradient that saturates at increasing forcing levels and remains insensitive to the colatitude.

The extent to which equilibrium statistical theory is applicable to driven dissipative dynamics remains an important open question in many systems. We use extensive direct numerical simulations (DNS) of the incompressible two-dimensional (2D) Navier-Stokes equation to examine the steady state of large-scale condensates in 2D turbulence at finite Reynolds number Re in the absence of bottom drag. Large-scale condensates appear above a critical Reynolds number Rec≈4.19. Close to this onset, we find a power-law scaling of the energy with Re-Rec, with the energy spectrum at large scales following the absolute equilibrium form proposed by Kraichnan. At larger Re, the energy spectrum deviates from this form, displaying a steep power-law range at low wavenumbers with exponent -5, with most of the energy dissipation occurring within the condensate at large scales. We show that this spectral exponent is consistent with the logarithmic radial vorticity profile of the condensate vortices predicted by quasi-linear theory for a viscously saturated condensate. Our findings shed new light on the classical problem of large-scale turbulent condensation in forced dissipative 2D flows in finite domains, showing that the large scales are close to equilibrium dynamics in weakly turbulent flows but not in the strong condensate regime with Re≫1. This article is part of the theme issue 'Frontiers of turbulence and statistical physics'.

Properties of spatially dependent relaxation oscillations near a SNIPER bifurcation are described. A SNIPER bifurcation creates a large-amplitude long-period periodic orbit via the annihilation of a pair of fixed points in a saddle-node bifurcation. We show that in spatially extended media, this orbit may undergo a long-wavelength instability, leading to spatially modulated oscillations that persist on both sides of the SNIPER. The oscillations take different forms depending on the system: a chimera state in a theta-reaction-diffusion model, and chaotic spiking in an activator-inhibitor-substrate model. The results are expected to have applications in a number of physical systems exhibiting SNIPER bifurcations, ranging from models of the nervous system through chemical reactions to nonlinear optics.

We study large-scale dynamics in the Ginzburg-Landau equation (GLE) using a reduced description derived from a WKB expansion. Rigorous mathematical results establishing that this reduced equation accurately approximates the full GLE are currently limited to the real GLE (RGLE) and exclude phase-slip dynamics. For the RGLE, we find that the reduced equation has conserved gradient form and show that, upon inclusion of a higher-order regularization, it admits exact stationary solutions. In the reduced dynamics, all nonuniform steady states are linearly unstable and among them, localized hole solutions identified through the reduced description differ from the classical hole solution of the RGLE due to Langer and Ambegaokar. In the Eckhaus-unstable regime, we derive a self-similar description of the approach to finite-time singularities in the reduced equation, with scaling exponents that agree with direct numerical simulations (DNS), and a similarity profile obtained from a nonlinear 4th-order boundary value problem. Extending the reduction to the complex GLE (CGLE) with nearly real coefficients introduces a Burgers nonlinearity that generates traveling shocks connecting two distinct plane-waves. We obtain exact expressions for the shock profile and perform extensive DNS to demonstrate convergence to the predicted profile in the appropriate large-scale, nearly real-coefficient limit of the CGLE. Away from this limit, the wave number profile loses monotonicity, which we explain in the framework of spatial dynamics. We further show that the exact shock solutions found here are qualitatively distinct from the Nozaki-Bekki solutions. Taken together, our results reveal how a single, scalar reduced equation elucidates unstable stationary states, self-similar collapse toward phase slips, and shock formation, providing an understanding large-scale phase dynamics in pattern-forming systems.

We use bifurcation analysis and numerical methods to study dynamic patterns in a complex Swift-Hohenberg equation focusing on structures present on a 1D interval with periodic boundary conditions and on a 2D disk with Neumann boundary conditions. In 1D, the equation features Hopf bifurcations from the trivial state u ≡ 0 at finite wave number k ≠ 0 (also known as wave bifurcations), leading to a bifurcation problem with O(2)×S1 symmetry, i.e., symmetry under translations and reflections in space, and phase rotations. The O(2) symmetry implies the simultaneous appearance of traveling waves and standing waves, and we consider the case where both bifurcations are subcritical and yield secondary bifurcations to spatially localized structures in the form of modulated traveling waves and localized standing waves, respectively. We use numerical continuation to show that some of the solutions thus obtained exhibit homoclinic snaking associated with their gradual growth in spatial extent. We exploit gauge symmetry to compute these dynamic patterns as relative equilibria, which allows us to locate efficiently tertiary bifurcations to two-frequency localized standing waves and to localized drifting waves. The intricate behavior identified in 1D is shown to provide a "road map" for the organization of wall-attached states on 2D disks in the form of rotating and standing waves at the wall, which may further localize in angle, yielding rotating spots or stationary breathing spots at the wall, and tertiary bifurcations from these. Some direct numerical simulations are used to identify further structures, including dynamics present in the disk bulk.

We introduce a novel numerical method for direct simulation of front propagation in the Fisher-KPP equation with a time-dependent parameter on an infinite domain. The method computes a time-dependent boundary condition that accurately captures the leading-edge dynamics by coupling the nonlinear simulation region to a linear approximation region in which the dynamics can be solved exactly via the Green's function of the linearized equation. This approach enables precise front velocity measurements on relatively small computational domains for a variety of nonautonomous regimes and initial conditions for which existing numerical methods break down. We apply the method to pulled and pushed fronts in the Fisher-KPP equation with quadratic and quadratic-cubic nonlinearities, finding that it improves the accuracy of the simulated front velocity even for constant parameters and a fixed domain size. For pulled fronts with a diffusion coefficient that increases algebraically in time, our results reveal a deviation from the natural asymptotic velocity predicted by linear theory, whose explanation requires nonlinear theory. For pushed fronts with constant parameters, the method reproduces the exponential convergence to the theoretical asymptotic front speed and profile with improved precision. For a slowly time-varying linear growth parameter, we find that the pushed front velocity follows the changing parameter adiabatically if the asymptotic pushed velocity remains faster than the natural asymptotic pulled velocity. As the growth parameter moves toward the pushed–pulled transition point, the competition between the pushed and pulled fronts can result in both delayed and even premature onset of the pushed–pulled transition, depending on the form of parameter growth. The numerical method presented here proves to be an effective tool for analyzing front propagation in nonautonomous systems.

Rapidly rotating Rayleigh–Bénard convection on an f f $f$ -plane at colatitude theta Subscript f ϑ f $\vartheta _{\!f}$ is investigated numerically using an asymptotically reduced equation set valid in the limit of very rapid rotation. The equations provide a non-hydrostatic but quasi-geostrophic description in a non-orthogonal coordinate system. The tilt changes the structure of the large-scale barotropic condensate from large-scale vortices to zonal flows as the colatitude of the f f $f$ -plane increases, with bistable states present for certain parameter ranges, extending prior work to a geophysically significant parameter regime. This behaviour is understood through the impact of broken rotation symmetry on the barotropic source terms resulting from baroclinic vorticity stresses and baroclinic torque. As the tilt angle theta Subscript f ϑ f $\vartheta _{\!f}$ increases, global heat and momentum transport is reduced relative to upright-polar convection, a result that is explained through linear theory and nonlinear power maps both of which demonstrate increased attenuation of the domain of dynamically active spatial scales as the convective modes depart from a north–south alignment in the horizontal plane. A key finding is that the predominance of lateral thermal mixing allows for the maintenance of a persistent unstable mean temperature gradient that saturates at increasing forcing levels and remains insensitive to the colatitude.

Spatiotemporal localized and extended structures associated with a subcritical finite wavenumber Hopf bifurcation are studied in the Purwins model (a three-variable FitzHugh-Nagumo version). Steady and time-dependent numerical continuation procedures are used to investigate snaking behavior of localized standing and traveling waves on the real line, and the results are corroborated using weakly nonlinear theory. The results shed light on the origin of so-called jumping oscillons and the organization of a nontypical homoclinic snaking structure of traveling pulses. The computations are extended to moderate size disks and used to identify wall-attached spots that travel along the disk boundary as well as wall-attached spots that oscillate in place and wall-attached jumping oscillons. The one-dimensional results are shown to be useful in interpreting the two-dimensional results. Domain-filling and mixed structures are also studied, demonstrating the variety of extended and localized states that emerge in two-space dimensions, ranging from periodic to disordered. The latter are potentially important for observations of waves in far-from-equilibrium media, such as those often observed in cell biology.

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-B & eacute;nard convection, has been extensively studied in direct numerical simulations (DNS) and laboratory experiments, but the parameter values attainable by state-ofthe-art methods are limited to moderately rapid rotation (Ekman numbers Ek & Agrave; 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 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 in sufficiently large domains 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 leading to improved stability of our numerical formulation. We furthermore demonstrate that full DNS using RRRiNSE agree with the NHQGE at very small Ek.

Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh–Bénard convection (RRRBC) provides a paradigm for experiments and direct numerical simulations (DNS) of such flows, but the accessible parameter space remains restricted to moderately fast rotation rates (Ekman numbers ${ {Ek}} \gtrsim 10^{-8}$ ), while realistic ${Ek}$ for geo- and astrophysical applications are orders of magnitude smaller. On the other hand, previously derived reduced equations of motion describing the leading-order behaviour in the limit of very rapid rotation ( $ {Ek}\to 0$ ) cannot capture finite rotation effects, and the physically most relevant part of parameter space with small but finite ${Ek}$ has remained elusive. Here, we employ the rescaled rapidly rotating incompressible Navier–Stokes equations (RRRiNSE) – a reformulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for ${Ek}\to 0$ , recently introduced by Julien et al. (2024) – to provide full DNS of RRRBC at unprecedented rotation strengths down to $ {Ek}=10^{-15}$ and below, revealing the disappearance of cyclone–anticyclone asymmetry at previously unattainable Ekman numbers ( ${Ek}\approx 10^{-9}$ ). We also identify an overshoot in the heat transport as ${Ek}$ is varied at fixed $\widetilde { {Ra}} \equiv {Ra}{Ek}^{4/3}$ , where $Ra$ is the Rayleigh number, associated with dissipation due to ageostrophic motions in the boundary layers. The simulations validate theoretical predictions based on thermal boundary layer theory for RRRBC and show that the solutions of RRRiNSE agree with the reduced equations at very small ${Ek}$ . These results represent a first foray into the vast, largely unexplored parameter space of very rapidly rotating convection rendered accessible by RRRiNSE.

Numerical continuation is used to study the interaction between a finite wave number Turing instability and a zero wave number Hopf instability in a two-species reaction-diffusion model of a semiconductor device. The model admits two such codimension-two interactions, both with a subcritical Turing branch that is responsible for the presence of spatially localized Turing states. The Hopf branch may also be subcritical. We uncover a large variety of spatially extended and spatially localized states in the vicinity of these points and by varying a third parameter show how disconnected branches of time-periodic spatially localized states can be "zipped up"into snaking branches of time-periodic oscillations. These are of two types: a Turing state embedded in an oscillating background, and a breathing Turing state embedded in a non-oscillating background. Stable two- frequency states resembling a mixture of these two states are also identified. Our results are complemented by direct numerical simulations. The findings explain the origin of the large multiplicity of localized steady and oscillatory patterns arising from the Turing-Hopf interaction and shed light on the competition between them.

The coexistence of multiple stationary, spatially localized structures has recently been reported for convection in an inclined porous layer, but the influence of imperfectly conducting boundaries has not been studied. This paper analyzes the traveling behavior of asymmetric, spatially localized convective structures ("convectons"), consisting of one or more pulses, in a two-dimensional inclined layer of a porous medium subject to a fixed temperature at the bottom and an imperfectly conducting boundary at the top, such that midplane reflection symmetry is broken. Extensive direct numerical simulations (DNS) are conducted for a wide range of Biot numbers associated with the upper boundary, revealing nontrivial relationships between the drift velocity c of spatially localized structures and the symmetry-breaking parameter kappa >= 0 based on the Biot number, with kappa = 0 corresponding to perfect midplane reflection symmetry associated with c = 0. In small domains, the drift velocity c is positive (corresponding to upslope motion) and increases monotonically with kappa, while for localized structures in larger domains consisting of a small number of pulses c can be of either sign depending on parameters. For longer structures c reverts to positive values and again increases monotonically with kappa. We show that the along-slope tails of pulses and the associated long-range interactions are governed by the dominant spatial eigenvalues, whose real part is of the smallest magnitude, and we uncover a transition at a finite symmetry-breaking strength kappa = kappa(c) > 0: for kappa < kappa(c), both dominant eigenvalues are complex, implying that the upslope and downslope exponential tails are oscillatory. In contrast, for kappa > kappa(c), the dominant spatial eigenvalue with a positive real part becomes real, implying that the downslope exponential tail transitions from an oscillatory profile to a monotonic one. As a result, for kappa < kappa(c), bound states consisting of different numbers of pulses are present. These display rich dynamical phenomena including inelastic collisions leading to other bound states, while for kappa >> kappa(c), adjacent pulses are found to repel one another and so tend to spread out, eventually becoming equispaced in the finite computational domain. The strength of the repulsive interaction increases with kappa, i.e., with increased symmetry breaking. A reduced description of this behavior is proposed based on the interaction between tails of adjacent localized structures, which accurately reproduces the repulsion and inelastic collisions observed in DNS. The reduced model indicates that the transition from bound states to an equidistant configuration occurs when the monotonic tail and the oscillatory tail have the same slope, which occurs at a value of kappa greater than kappa(c). The results presented here represent a step towards understanding the dynamics of spatially localized patterns in moderate-Rayleigh number convection in an inclined porous medium subject to an imperfectly conducting boundary.

We study the dynamics of salt fingers in the regime of slow salinity diffusion (small inverse Lewis number) and strong stratification (large density ratio), focusing on regimes relevant to Earth's oceans. Using three-dimensional direct numerical simulations in periodic domains, we show that salt fingers exhibit rich, multiscale dynamics in this regime, with vertically elongated fingers that are twisted into helical shapes at large scales by mean flows and disrupted at small scales by isotropic eddies. We use a multiscale asymptotic analysis to motivate a reduced set of partial differential equations that filters internal gravity waves and removes inertia from all parts of the momentum equation except for the Reynolds stress that drives the helical mean flow. When simulated numerically, the reduced equations capture the same dynamics and fluxes as the full equations in the appropriate regime. The reduced equations enforce zero helicity in all fluctuations about the mean flow, implying that the symmetry-breaking helical flow is generated spontaneously by strictly non-helical fluctuations.

Numerical continuation is used to compute solution branches in a two-component reaction-diffusion model of Leslie-Gower type. Two regimes are studied in detail. In the first, the homogeneous state loses stability to supercritical spatially uniform oscillations, followed by a subcritical steady state bifurcation of Turing type. The latter leads to spatially localized states embedded in an oscillating background that bifurcate from snaking branches of localized steady states. Using two-parameter continuation, we uncover a novel mechanism whereby disconnected segments of oscillatory states zip up into a continuous snaking branch of time-periodic localized states, some of which are stable. In the second, the homogeneous state loses stability to supercritical Turing patterns, but steady spatially localized states embedded either in the homogeneous state or in a small amplitude Turing state are nevertheless present. We show that such behavior is possible when sideband Turing states are strongly subcritical and explain why this is so in the present model. In both cases, the observed behavior differs significantly from that expected on the basis of a supercritical primary bifurcation.

This work studies two-dimensional fixed-flux Rayleigh-Bénard convection with periodic boundary conditions in both horizontal and vertical directions and analyzes its dynamics using numerical continuation, secondary instability analysis and direct numerical simulation. The fixed-flux constraint leads to time-independent elevator modes with a well-defined amplitude. Secondary instability of these modes leads to tilted elevator modes accompanied by horizontal shear flow. For Pr=1, where Pr is the Prandtl number, a subsequent subcritical Hopf bifurcation leads to hysteresis behavior between this state and a time-dependent direction-reversing state, followed by a global bifurcation leading to modulated traveling waves without flow reversal. Single-mode equations reproduce this moderate Rayleigh number behavior well. At high Rayleigh numbers, chaotic behavior dominated by modulated traveling waves appears. These transitions are characteristic of high wavenumber elevator modes since the vertical wavenumber of the secondary instability is linearly proportional to the horizontal wavenumber of the elevator mode. At a low Pr, relaxation oscillations between the conduction state and the elevator mode appear, followed by quasiperiodic and chaotic behavior as the Rayleigh number increases. In the high Pr regime, the large-scale shear weakens, and the flow shows bursting behavior that can lead to significantly increased heat transport or even intermittent stable stratification.

Rogue waves are an intriguing nonlinear phenomenon arising across different scales, ranging from ocean waves through optics to Bose-Einstein condensates. We describe the emergence of rogue wave-like dynamics in a reaction-diffusion system that arise as a result of a subcritical Turing instability. This state is present in a regime where all time-independent states are unstable and consists of intermittent excitation of spatially localized spikes, followed by collapse to an unstable state and subsequent regrowth. We characterize the spatiotemporal organization of spikes and show that in sufficiently large domains the dynamics are consistent with a memoryless process.

Two-dimensional (2D) turbulence features an inverse energy cascade producing large-scale flow structures such as hurricane-like large-scale vortices (LSVs) and jets. We study the dynamics of such structures by direct numerical simulations (DNS) of stochastically forced, viscously damped 2D turbulence within a periodic rectangular domain $[0,L_x]\times[0,L_y]$. Stable LSVs form in the system when the aspect ratio $\delta = L_x/L_y \approx 1$, while jets predominate at $\delta\gtrsim 1.1$. At intermediate values of $\delta$, both structures are metastable, and noise-induced transitions occur between them. Based on large-scale energy balance, we derive and verify predictions for the parameter dependence of the total kinetic energy and the flow polarity. We collect detailed statistics on the lifetimes of LSVs and jets from long-time simulations, consistent with a memoryless process. Our DNS results support both an exponential and an algebraic dependence of the mean lifetime on $\delta$. We show that the mean lifetime depends strongly on the Reynolds number $Re$. As $Re$ increases, the energy gap between LSV (lower energy) and jet states (higher energy) grows, leading to increasing lifetimes following power laws in $Re$. Similarly, as the forcing scale decreases, transitions become less frequent. We study the transitions in terms of kinetic energy, flow polarity, and 2D phase-space diagrams, revealing that the transitions occur in two stages: an initial, efficient redistribution of kinetic energy by nonlinear triadic interactions facilitates a rapid transition from LSVs to jets and vice versa. In the second stage, the kinetic energy of the newly formed structure slowly adjusts to its associated equilibrium value on a longer, viscous timescale, leading to hysteresis. Our findings shed new light on the dynamics of coherent large-scale structures in anisotropic turbulence.

This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equation with periodic boundary conditions on isotropically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and show that the additional domain growth before the appearance of a pattern is independent of the growth time scale. We also quantify primary bifurcation delay on a shrinking domain; in contrast with a growing domain, the time scale of domain compression is reflected in the additional compression before the pattern decays. For secondary bifurcations such as the Eckhaus instability, we obtain a lower bound on the delay of phase slips due to a time-dependent domain. We also construct a heuristic model to classify regimes with arrested phase slips, i.e., phase slips that fail to develop. Then, we study how propagating fronts are influenced by a time-dependent domain. We identify three types of pulled fronts: homogeneous, pattern spreading, and Eckhaus fronts. By following the linear dynamics, we derive expressions for the velocity and profile of homogeneous fronts on a time-dependent domain. We also derive the natural "asymptotic" velocity and front profile and show that these deviate from predictions based on the marginal stability criterion familiar from fixed domain theory. This difference arises because the time dependence of the domain lifts the degeneracy of the spatial eigenvalues associated with speed selection and represents a fundamental distinction from the fixed domain theory that we verify using direct numerical simulations. The effect of a growing domain on pattern spreading and Eckhaus front velocities is inspected qualitatively and found to be similar to that of homogeneous fronts. These more complex fronts can also experience delayed onset. Lastly, we show that dilution-an effect present when the order parameter is conserved-increases bifurcation delay and amplifies changes in the homogeneous front velocity on time-dependent domains. The study provides general insight into the effects of domain growth on pattern onset, pattern transitions, and front propagation in systems across different scientific fields.

We study structure formation in two-dimensional turbulence driven by an external force, interpolating between linear instability forcing and random stirring, subject to nonlinear damping. Using extensive direct numerical simulations, we uncover a rich parameter space featuring four distinct branches of stationary solutions: large-scale vortices, hybrid states with embedded shielded vortices (SVs) of either sign, and two states composed of many similar SVs. Of the latter, the first is a dense vortex gas where all SVs have the same sign and diffuse across the domain. The second is a hexagonal vortex crystal forming from this gas when the instability is sufficiently weak. These solutions coexist stably over a wide parameter range. The late-time evolution of the system from small-amplitude initial conditions is nearly self-similar, involving three phases: initial inverse cascade, random nucleation of SVs from turbulence and, once a critical number of vortices is reached, a phase of explosive nucleation of SVs, leading to a statistically stationary state. The vortex gas is continued in the forcing parameter, revealing a sharp transition towards the crystal state as the forcing strength decreases. This transition is analysed in terms of the diffusion of individual vortices and tools from statistical physics. The crystal can also decay via an inverse cascade resulting from the breakdown of shielding or insufficient nonlinear damping acting on SVs. Our study highlights the importance of the forcing details in two-dimensional turbulence and reveals the presence of nontrivial SV states in this system, specifically the emergence and melting of a vortex crystal.

We present a detailed mathematical study of a truncated normal form relevant to the bifurcations observed in wake flow past axisymmetric bodies, with and without thermal stratification. We employ abstract normal form analysis to identify possible bifurcations and the corresponding bifurcation diagrams in parameter space. The bifurcations and the bifurcation diagrams are interpreted in terms of symmetry considerations. Particular emphasis is placed on the presence of attracting robust heteroclinic cycles in certain parameter regimes. The normal form coefficients are computed for several examples of wake flows behind buoyant disks and spheres, and the resulting predictions compared with the results of direct numerical flow simulations. In general, satisfactory agreement is obtained.