Edgar Knobloch

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’.

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 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.

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.