arXiv subjects

Edgar Knobloch

Publications and source records attributed to Edgar Knobloch.

At least 19 recordsLinked to original sources

arXiv2026

Nonuniform relaxation oscillations near SNIPER bifurcations

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.

nlin.PS↗

arXiv2026

Reduced wave number dynamics in the real and complex Ginzburg-Landau equations

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.

nlin.PS↗

arXiv2026

Localized spatiotemporal reaction-diffusion patterns on a line and a disk arising from a subcritical finite wavenumber Hopf instability

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.

nlin.PS↗

arXiv2026

Quasi-geostrophic Rayleigh-B\'enard convection on the tilted $f$-plane

Rapidly rotating Rayleigh-B\'enard convection on a $f$-plane at colatitude $\vartheta_f$ 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 $f$-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 $\vartheta_f$ 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.

physics.flu-dyn↗

arXiv2026

Accurate simulation of pulled and pushed fronts in the nonautonomous Fisher-KPP equation

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.

physics.flu-dyn↗

arXiv2025

Spontaneous generation of helical flows by salt fingers

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 spontaneously generated by strictly non-helical fluctuations.

physics.flu-dyn↗

arXiv2025

Two-dimensional turbulent condensates without bottom drag

The extent to which statistical equilibrium theory is applicable to driven dissipative dynamics remains an important open question in many systems. We use extensive direct numerical simulations 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 $Re_c\approx 4.19$. Close to this onset, we find a power-law scaling of the energy with $Re-Re_c$, 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 wave numbers 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 $Re\gg1$.

physics.flu-dyn↗

arXiv2024

Traveling spatially localized convective structures in an inclined porous medium

Multiple stationary, localized structures were recently found for inclined porous medium convection with constant-temperature boundaries. We analyze traveling asymmetric, localized convective structures, consisting of 1 to 5 pulses, in a 2D inclined porous layer with fixed temperature at the bottom and an imperfectly conducting boundary at the top, such that midplane reflection symmetry is broken. Direct numerical simulations (DNS) are performed with different Biot numbers at the top boundary. The drift velocity $c$ of pulses is measured for different values of the symmetry parameter $\kappa\geq0$ based on the Biot number, with perfect midplane reflection symmetry and $c=0$ at $\kappa=0$. In small domains, the drift velocity $c>0$ (upslope), increases monotonically with $\kappa$, while in large domains $c$ changes sign depending on parameters. We show that pulse tails, controlling interactions, depend on the dominant spatial eigenvalues, whose real part is closest to zero, with a transition at $\kappa_c>0$: below $\kappa_c$, both dominant eigenvalues are complex, and the tails are oscillatory. Above $\kappa_c$, the dominant spatial eigenvalue with a positive real part becomes real, and the downslope tail transitions from oscillatory to monotonic. Below $\kappa_c$, bound states with different numbers of pulses exist whose collisions are studied. Well above $\kappa_c$, adjacent pulses repel each other, spreading out and becoming equispaced in the domain. A reduced model is proposed based on the interaction via tails of adjacent pulses, reproducing the repulsion and collisions from DNS. The model shows that the transition from bound states to equidistant spreading occurs when the monotonic/oscillatory tails have the same slope. This study elucidates the motion of localized patterns in moderate-Rayleigh number convection in an inclined porous layer with an imperfectly conducting boundary.

physics.flu-dyn↗

arXiv2024

Rescaled Equations for Well-Conditioned Direct Numerical Simulations of Rapidly Rotating Convection

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\'enard 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 $Ek\gtrsim10^{-8}$), 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 $Ek\to 0$, 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 $Ek\to0$, 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 $Ek$.

physics.flu-dyn↗

arXiv2024

Bridging the Rossby number gap in rapidly rotating thermal convection

Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh-B\'enard 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 astro-/geophysical 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 incompressible Navier-Stokes equations (RiNSE) -- 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 $\widetilde{Ra}= Ra Ek^{4/3}$, 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 RiNSE 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 RiNSE.

physics.flu-dyn↗

arXiv2024

Time-dependent localized patterns in a predator-prey model

Numerical continuation is used to compute solution branches in a two-component reaction-diffusion model of Leslie--Gower type. %in the vicinity of a Turing-Hopf interaction. 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.

math.DS↗

arXiv2024

Emergence of rogue-like waves in a reaction-diffusion system: Stochastic output from deterministic dissipative dynamics

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-like wave 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.

nlin.PS↗

arXiv2023

Fixed-flux Rayleigh-B\'enard convection in doubly periodic domains: generation of large-scale shear

This work studies two-dimensional fixed-flux Rayleigh-B\'enard 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.

physics.flu-dyn↗

arXiv2023

Bifurcation delay and front propagation in the real Ginzburg-Landau equation on a time-dependent domain

This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equation (RGLE) with periodic boundary conditions on monotonically 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; 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 fronts are influenced by a time-dependent domain. 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.

nlin.PS↗

arXiv2023

Fluctuation-Induced Transitions in Anisotropic Two-Dimensional Turbulence

Two-dimensional (2D) turbulence features an inverse energy cascade that produces large-scale flow structures such as large-scale vortices (LSVs) and unidirectional jets. We investigate the dynamics of such structures using extensive direct numerical simulations (DNS) of randomly forced, viscously damped 2D turbulence within a periodic rectangular (Cartesian) domain $[0,L_x]\times[0,L_y]$. LSVs form and dominate the system when the domain aspect ratio $\delta = L_x/L_y \approx 1$, while unidirectional jets predominate at $\delta \gtrsim 1.1$. At intermediate $\delta$, both structures are metastable, with noise-induced transitions between LSVs and jets. We derive and verify predictions for the dependence of kinetic energy E and flow polarity on the nondimensional control parameters. We further collect detailed statistics on the lifetimes of LSVs and jets from DNS runs that are up to 10738 viscous diffusive times long. The distribution of the lifetimes is consistent with that of a memoryless process. Our DNS show an exponential dependence of the mean lifetime on $\delta$. Mean lifetimes depend sensitively on the Reynolds number Re: as Re increases, the energy gap between LSV (lower E) and jet states (higher E) arising from anisotropic dissipation increases, leading to an approximately exponential increase in lifetimes with Re for both LSVs and jets. Similarly, as the forcing scale decreases, transitions become less frequent. We study the transitions in detail, revealing that they occur in two stages: an initial, rapid redistribution of kinetic energy by nonlinear triadic interactions deforms LSVs into jets or vice versa. In the second stage, the energy of the newly formed structure slowly adjusts to its associated equilibrium value on a longer, viscous timescale, producing hysteresis. Our findings shed new light on the dynamics of coherent large-scale structures in anisotropic turbulence.

physics.flu-dyn↗

arXiv2023

Suppression of wall modes in rapidly rotating Rayleigh-B\'enard convection by narrow horizontal fins

The heat transport by rapidly-rotating Rayleigh-B\'enard convection is of fundamental importance to many geophysical flows. Laboratory measurements are impeded by robust wall modes which develop along vertical walls, significantly perturbing the heat flux. We show that narrow horizontal fins along the vertical walls efficiently suppress wall modes ensuring that their contribution to the global heat flux is negligible compared with bulk convection in the geostrophic regime, thereby paving the way for new experimental studies of geophysically relevant regimes of rotating convection.

physics.flu-dyn↗

arXiv2023

Front propagation and global bifurcations in a multivariable reaction-diffusion model

We study the existence and stability of propagating fronts in Meinhardt's multivariable reaction-diffusion model of branching in one spatial dimension. We identify a saddle-node-infinite-period (SNIPER) bifurcation of fronts that leads to episodic front propagation in the parameter region below propagation failure and show that this state is stable. Stable constant speed fronts exist only above this parameter value. We use numerical continuation to show that propagation failure is a consequence of the presence of a T-point corresponding to the formation of a heteroclinic cycle in a spatial dynamics description. Additional T-points are identified that are responsible for a large multiplicity of different unstable traveling front-peak states. The results indicate that multivariable models may support new types of behavior that are absent from typical two-variable models but may nevertheless be important in developmental processes such as branching and somitogenesis.

nlin.PS↗

arXiv2023

Collisions of localized patterns in a nonvariational Swift-Hohenberg equation

The cubic-quintic Swift-Hohenberg equation (SH35) has been proposed as an order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use numerical continuation, together with extensive direct numerical simulations, to study SH35 with an additional nonvariational quadratic term to model the effects of breaking the midplane reflection symmetry. The nonvariational structure of the model leads to the propagation of asymmetric spatially localized structures (LSs). An asymptotic prediction for the drift velocity of such structures is validated numerically. Next, we present an extensive study of possible collision scenarios between identical and nonidentical traveling structures, varying a temperature-like control parameter. The final state may be a simple bound state of the initial LSs or longer or shorter than the sum of the two initial states as a result of nonlinear interactions. The Maxwell point of the variational system is shown to have no bearing on which of these scenarios is realized. Instead, we argue that the stability properties of bound states are key. While individual LSs lie on a modified snakes-and-ladders structure in the nonvariational SH35, the multi-pulse bound states resulting from collisions lie on isolas in parameter space. In the gradient SH35, such isolas are always of figure-eight shape, but in the present non-gradient case they are generically more complex, some of which terminate in T-point bifurcations. A reduced model consisting of two coupled ordinary differential equations is proposed to describe the linear interactions between the tails of the LSs in which the model parameters are deduced using gradient descent optimization. For collisions leading to the formation of simple bound states, the reduced model reproduces the trajectories of LSs with high quantitative accuracy.

nlin.PS↗