arXiv subjects

Adrian van Kan

Publications and source records attributed to Adrian van Kan.

At least 19 recordsLinked to original sources

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

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

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-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 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↗

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

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-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 $Ek\gtrsim10^{-8}$), 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↗

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↗

arXiv2025

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 $κ\geq0$ based on the Biot number, with perfect midplane reflection symmetry and $c=0$ at $κ=0$. In small domains, the drift velocity $c>0$ (upslope), increases monotonically with $κ$, 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 $κ_c>0$: below $κ_c$, both dominant eigenvalues are complex, and the tails are oscillatory. Above $κ_c$, the dominant spatial eigenvalue with a positive real part becomes real, and the downslope tail transitions from oscillatory to monotonic. Below $κ_c$, bound states with different numbers of pulses exist whose collisions are studied. Well above $κ_c$, 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↗

arXiv2025

Phase Transitions in Anisotropic Turbulence

Turbulence is a widely observed state of fluid flows, characterized by complex, nonlinear interactions between motions across a broad spectrum of length and time scales. While turbulence is ubiquitous, from teacups to planetary atmospheres, oceans and stars, its manifestations can vary considerably between different physical systems. For instance, three-dimensional (3D) turbulent flows display a forward energy cascade from large to small scales, while in two-dimensional (2D) turbulence, energy cascades from small to large scales. In a given physical system, a transition between such disparate regimes of turbulence can occur when a control parameter reaches a critical value. The behavior of flows close to such transition points, which separate qualitatively distinct \textit{phases} of turbulence, has been found to be unexpectedly rich. Here, we survey recent findings on such transitions in highly anisotropic turbulent fluid flows, including turbulence in thin layers and under the influence of rapid rotation. We also review recent work on transitions induced by turbulent fluctuations, such as random reversals and transitions between large-scale vortices and jets, among others. The relevance of these results and their ramifications for future investigations are discussed.

physics.flu-dyn↗

arXiv2024

Large-scale self-organisation in dry turbulent atmospheres

How turbulent convective fluctuations organise to form large-scale structures in planetary atmospheres remains a question that eludes quantitative answers. The assumption that this process is the result of an inverse cascade was suggested half a century ago in two-dimensional fluids, but its applicability to atmospheric and oceanic flows remains heavily debated, hampering our understanding of the energy balance in planetary systems. We show with direct numerical simulations of spatial resolutions of 122882 $\times$ 384 points that rotating and stratified flows can support a bidirectional cascade of energy, in three dimensions, with a ratio of Rossby to Froude numbers comparable to that of the Earth's atmosphere. Our results establish that in dry atmospheres spontaneous order can arise via an inverse cascade to the largest spatial scales.

physics.flu-dyn↗

arXiv2024

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 $δ= L_x/L_y \approx 1$, while unidirectional jets predominate at $δ\gtrsim 1.1$. At intermediate $δ$, 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 $δ$. 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

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↗

arXiv2022

1/f noise and anomalous scaling in Lévy noise-driven on-off intermittency

On-off intermittency occurs in nonequilibrium physical systems close to bifurcation points and is characterised by an aperiodic switching between a large-amplitude "on" state and a small-amplitude "off" state. Lévy on-off intermittency is a recently introduced generalisation of on-off intermittency to multiplicative Lévy noise, which depends on a stability parameter $α$ and a skewness parameter $β$. Here, we derive two novel results on Lévy on-off intermittency by leveraging known exact results on the first-passage time statistics of Lévy flights. First, we compute anomalous critical exponents explicitly as a function of arbitrary Lévy noise parameters $(α,β)$ for the first time, by a heuristic method, complementing previous results. The predictions are verified using numerical solutions of the fractional Fokker-Planck equation. Second, we derive the power spectrum $S(f)$ of Lévy on-off intermittency and show that it displays a power law $S(f)\propto f^κ$ at low frequencies $f$, where $κ\in (-1,0)$ depends on the noise parameters $α,β$. An explicit expression for $κ$ is obtained in terms of $(α,β)$. The predictions are verified using long time series realisations of Lévy on-off intermittency. Our findings help shed light on instabilities subject to non-equilibrium, power-law-distributed fluctuations, emphasizing that their properties can differ starkly from the case of Gaussian fluctuations.

physics.flu-dyn↗

arXiv2022

Spontaneous suppression of inverse energy cascade in instability-driven 2D turbulence

Instabilities of fluid flows often generate turbulence. Using extensive direct numerical simulations, we study two-dimensional turbulence driven by a wavenumber-localised instability superposed on stochastic forcing, in contrast to previous studies of state-independent forcing. As the contribution of the instability forcing, measured by a parameter $γ$, increases, the system undergoes two transitions. For $γ$ below a first threshold, a regular large-scale vortex condensate forms. Above this threshold, shielded vortices (SVs) emerge within the condensate. At a second, larger value of $γ$, the condensate breaks down, and a gas of weakly interacting vortices with broken symmetry spontaneously emerges, characterised by preponderance of vortices of one sign only and suppressed inverse energy cascade. The latter transition is shown to depend on the damping mechanism. The number density of SVs in the broken symmetry state slowly increases via a random nucleation process. Bistability is observed between the condensate and mixed SV-condensate states. Our findings provide new evidence for a strong dependence of two-dimensional turbulence phenomenology on the forcing.

physics.flu-dyn↗

arXiv2022

Bistability of the large-scale dynamics in quasi-two-dimensional turbulence

In many geophysical and astrophysical flows, suppression of fluctuations along one direction of the flow drives a quasi-2D upscale flux of kinetic energy, leading to the formation of strong vortex condensates at the largest scales. Recent studies have shown that the transition towards this condensate state is hysteretic, giving rise to a limited bistable range in which both the condensate state as well as the regular 3D state can exist at the same parameter values. In this work, we use direct numerical simulations of thin-layer flow to investigate whether this bistable range survives as the domain size and turbulence intensity are increased. By studying the time scales at which rare transitions occur from one state into the other, we find that the bistable range grows as the box size and/or Reynolds number Re are increased, showing that the bistability is neither a finite-size nor a finite-Re effect. We furthermore predict a crossover from a bimodal regime at low box size, low Re to a regime of pure hysteresis at high box size, high Re, in which any transition from one state to the other is prohibited at any finite time scale.

physics.flu-dyn↗

arXiv2021

Estimating nonlinear stability from time series data

Basin stability (BS) is a measure of nonlinear stability in multi-stable dynamical systems. BS has previously been estimated using Monte-Carlo simulations, which requires the explicit knowledge of a dynamical model. We discuss the requirements for estimating BS from time series data in the presence of strong perturbations, and illustrate our approach for two simple models of climate tipping elements: the Amazon rain forest and the thermohaline ocean circulation. We discuss the applicability of our method to observational data as constrained by the relevant time scales of total observation time, typical return time of perturbations and internal convergence time scale of the system of interest and other factors.

physics.data-an↗

arXiv2021

Energy cascades in rapidly rotating and stratified turbulence within elongated domains

We study forced, rapildy rotating and stably stratified turbulence in an elongated domain using an asymptotic expansion at simultaneously low Rossby number $Ro\ll1$ and large domain height compared to the energy injection scale, $h=H/\ell_{in}\gg1$. The resulting equations depend on the parameter $λ=(h Ro )^{-1}$ and the Froude number $Fr$. An extensive set of direct numerical simulations (DNS) is performed to explore the parameter space $(λ,Fr)$. We show that a forward energy cascade occurs in one region of this space, and a split energy cascade outside it. At weak stratification (large $Fr$), an inverse cascade is observed for sufficiently large $λ$. At strong stratification (small $Fr$) the flow becomes approximately hydrostatic and an inverse cascade is always observed. For both weak and strong stratification, we present theoretical arguments supporting the observed energy cascade phenomenology. Our results shed light on an asymptotic region in the phase diagram of rotating and stratified turbulence, which is difficult to attain by brute-force DNS.

physics.flu-dyn↗

arXiv2021

Geometric microcanonical theory of two-dimensional Truncated Euler flows

This paper presents a geometric microcanonical ensemble perspective on two-dimensional Truncated Euler flows, which contain a finite number of (Fourier) modes and conserve energy and enstrophy. We explicitly perform phase space volume integrals over shells of constant energy and enstrophy. Two applications are considered. In a first part, we determine the average energy spectrum for highly condensed flow configurations and show that the result is consistent with Kraichnan's canonical ensemble description, despite the fact that no thermodynamic limit is invoked. In a second part, we compute the probability density for the largest-scale mode of a free-slip flow in a square, which displays reversals. We test the results against numerical simulations of a minimal model and find excellent agreement with the microcanonical theory, unlike the canonical theory, which fails to describe the bimodal statistics. This article is part of the theme issue "Mathematical problems in physical fluid dynamics".

physics.flu-dyn↗

arXiv2021

Lévy on-off intermittency

We present a new form of intermittency, Lévy on-off intermittency, which arises from multiplicative $α$-stable white noise close to an instability threshold. We study this problem in the linear and nonlinear regimes, both theoretically and numerically, for the case of a pitchfork bifurcation with fluctuating growth rate. We compute the stationary distribution analytically and numerically from the associated fractional Fokker-Planck equation in the Stratonovich interpretation. We characterize the system in the parameter space $(α,β)$ of the noise, with stability parameter $α\in (0,2)$ and skewness parameter $β\in[-1,1]$. Five regimes are identified in this parameter space, in addition to the well-studied Gaussian case $α=2$. Three regimes are located at $1<α<2$, where the noise has finite mean but infinite variance. They are differentiated by $β$ and all display a critical transition at the deterministic instability threshold, with on-off intermittency close to onset. Critical exponents are computed from the stationary distribution. Each regime is characterized by a specific form of the density and specific critical exponents, which differ starkly from the Gaussian case. A finite or infinite number of integer-order moments may converge, depending on parameters. Two more regimes are found at $0<α\leq 1$. There, the mean of the noise diverges, and no critical transition occurs. In one case the origin is always unstable, independently of the distance $μ$ from the deterministic threshold. In the other case, the origin is conversely always stable, independently of $μ$. We thus demonstrate that an instability subject to non-equilibrium, power-law-distributed fluctuations can display substantially different properties than for Gaussian thermal fluctuations, in terms of statistics and critical behavior.

cond-mat.stat-mech↗