Iterative methods for Navier-Stokes inverse problems.

Even when the partial differential equation underlying a physical process can be evolved forward in time, the retrospective (backward in time) inverse problem often has its own challenges and applications. Direct adjoint looping (DAL) is the defacto approach for solving retrospective inverse problems, but it has not been applied to deterministic retrospective Navier-Stokes inverse problems in 2D or 3D. In this paper, we demonstrate that DAL is ill-suited for solving retrospective 2D Navier-Stokes inverse problems. Alongside DAL, we study two other iterative methods: simple backward integration (SBI) and the quasireversible method (QRM). As far as we know, our iterative SBI approach is novel, while iterative QRM has previously been used. Using these three iterative methods, we solve two retrospective inverse problems: 1D Korteweg-de Vries-Burgers (decaying nonlinear wave) and 2D Navier-Stokes (unstratified Kelvin-Helmholtz vortex). In both cases, SBI and QRM reproduce the target final states more accurately and in fewer iterations than DAL. We attribute this performance gap to additional terms present in SBI and QRM's respective backward integrations which are absent in DAL.

The solar dynamo begins near the surface.

The magnetic dynamo cycle of the Sun features a distinct pattern: a propagating region of sunspot emergence appears around 30° latitude and vanishes near the equator every 11 years (ref. 1). Moreover, longitudinal flows called torsional oscillations closely shadow sunspot migration, undoubtedly sharing a common cause2. Contrary to theories suggesting deep origins of these phenomena, helioseismology pinpoints low-latitude torsional oscillations to the outer 5-10% of the Sun, the near-surface shear layer3,4. Within this zone, inwardly increasing differential rotation coupled with a poloidal magnetic field strongly implicates the magneto-rotational instability5,6, prominent in accretion-disk theory and observed in laboratory experiments7. Together, these two facts prompt the general question: whether the solar dynamo is possibly a near-surface instability. Here we report strong affirmative evidence in stark contrast to traditional models8 focusing on the deeper tachocline. Simple analytic estimates show that the near-surface magneto-rotational instability better explains the spatiotemporal scales of the torsional oscillations and inferred subsurface magnetic field amplitudes9. State-of-the-art numerical simulations corroborate these estimates and reproduce hemispherical magnetic current helicity laws10. The dynamo resulting from a well-understood near-surface phenomenon improves prospects for accurate predictions of full magnetic cycles and space weather, affecting the electromagnetic infrastructure of Earth.

The photometric variability of massive stars due to gravity waves excited by core convection.

Massive stars die in catastrophic explosions that seed the interstellar medium with heavy elements and produce neutron stars and black holes. Predictions of the explosion's character and the remnant mass depend on models of the star's evolutionary history. Models of massive star interiors can be empirically constrained by asteroseismic observations of gravity wave oscillations. Recent photometric observations reveal a ubiquitous red noise signal on massive main sequence stars; a hypothesized source of this noise is gravity waves driven by core convection. We present three-dimensional simulations of massive star convection extending from the star's centre to near its surface, with realistic stellar luminosities. Using these simulations, we predict the photometric variability due to convectively driven gravity waves at the surfaces of massive stars, and find that gravity waves produce photometric variability of a lower amplitude and lower characteristic frequency than the observed red noise. We infer that the photometric signal of gravity waves excited by core convection is below the noise limit of current observations, and thus the red noise must be generated by an alternative process.

Scaling behaviour and control of nuclear wrinkling.

The cell nucleus is enveloped by a complex membrane, whose wrinkling has been implicated in disease and cellular aging. The biophysical dynamics and spectral evolution of nuclear wrinkling during multicellular development remain poorly understood due to a lack of direct quantitative measurements. Here, we characterize the onset and dynamics of nuclear wrinkling during egg development in the fruit fly when nurse cell nuclei increase in size and display stereotypical wrinkling behavior. A spectral analysis of three-dimensional high-resolution live imaging data from several hundred nuclei reveals a robust asymptotic power-law scaling of angular fluctuations consistent with renormalization and scaling predictions from a nonlinear elastic shell model. We further demonstrate that nuclear wrinkling can be reversed through osmotic shock and suppressed by microtubule disruption, providing tuneable physical and biological control parameters for probing mechanical properties of the nuclear envelope. Our findings advance the biophysical understanding of nuclear membrane fluctuations during early multicellular development.

Improved phase-field models of melting and dissolution in multi-component flows.

We develop and analyse the first second-order phase-field model to combine melting and dissolution in multi-component flows. This provides a simple and accurate way to simulate challenging phase-change problems in existing codes. Phase-field models simplify computation by describing separate regions using a smoothed phase field. The phase field eliminates the need for complicated discretizations that track the moving phase boundary. However, standard phase-field models are only first-order accurate. They often incur an error proportional to the thickness of the diffuse interface. We eliminate this dominant error by developing a general framework for asymptotic analysis of diffuse-interface methods in arbitrary geometries. With this framework, we can consistently unify previous second-order phase-field models of melting and dissolution and the volume-penalty method for fluid-solid interaction. We finally validate second-order convergence of our model in two comprehensive benchmark problems using the open-source spectral code Dedalus.

The magnetorotational instability prefers three dimensions.

The magnetorotational instability (MRI) occurs when a weak magnetic field destabilizes a rotating, electrically conducting fluid with inwardly increasing angular velocity. The MRI is essential to astrophysical disc theory where the shear is typically Keplerian. Internal shear layers in stars may also be MRI-unstable, and they take a wide range of profiles, including near-critical. We show that the fastest growing modes of an ideal magnetofluid are three-dimensional provided the shear rate, S, is near the two-dimensional onset value, S c . For a Keplerian shear, three-dimensional modes are unstable above S ≈ 0.10S c , and dominate the two-dimensional modes until S ≈ 2.05S c . These three-dimensional modes dominate for shear profiles relevant to stars and at magnetic Prandtl numbers relevant to liquid-metal laboratory experiments. Significant numbers of rapidly growing three-dimensional modes remainy well past 2.05S c . These finding are significant in three ways. First, weakly nonlinear theory suggests that the MRI saturates by pushing the shear rate to its critical value. This can happen for systems, such as stars and laboratory experiments, that can rearrange their angular velocity profiles. Second, the non-normal character and large transient growth of MRI modes should be important whenever three-dimensionality exists. Finally, three-dimensional growth suggests direct dynamo action driven from the linear instability.

Anomalous Chained Turbulence in Actively Driven Flows on Spheres.

Recent experiments demonstrate the importance of substrate curvature for actively forced fluid dynamics. Yet, the covariant formulation and analysis of continuum models for nonequilibrium flows on curved surfaces still poses theoretical challenges. Here, we introduce and study a generalized covariant Navier-Stokes model for fluid flows driven by active stresses in nonplanar geometries. The analytical tractability of the theory is demonstrated through exact stationary solutions for the case of a spherical bubble geometry. Direct numerical simulations reveal a curvature-induced transition from a burst phase to an anomalous turbulent phase that differs distinctly from externally forced classical 2D Kolmogorov turbulence. This new type of active turbulence is characterized by the self-assembly of finite-size vortices into linked chains of antiferromagnetic order, which percolate through the entire fluid domain, forming an active dynamic network. The coherent motion of the vortex chain network provides an efficient mechanism for upward energy transfer from smaller to larger scales, presenting an alternative to the conventional energy cascade in classical 2D turbulence.

Rolling resistance of shallow granular deformation.

Experiments are conducted to measure the resistance experienced by light cylinders rolling over flat beds of granular media. Sand and glass spheres are used for the beds. The trajectories of the rolling cylinders are determined through optical tracking, and velocity and acceleration data are inferred through fits to these trajectories. The rolling resistance is dominated by a velocity-independent component, but a velocity-dependent drag exceeding the expected strength of air drag is also observed. The results are compared to a theoretical model based on a cohesionless Mohr-Coulomb rheology for a granular medium in the presence of gravity. The model idealizes the flow pattern underneath the rolling cylinder as a plastically deforming zone in front of a rigidly rotating plug attached to the cylinder, as proposed previously for cylinders rolling on perfectly cohesive plastic media. The leading-order, rate-independent rolling resistance observed experimentally is well reproduced by the model predictions.

Numerical simulations of internal wave generation by convection in water.

Water's density maximum at 4°C makes it well suited to study internal gravity wave excitation by convection: an increasing temperature profile is unstable to convection below 4°C, but stably stratified above 4°C. We present numerical simulations of a waterlike fluid near its density maximum in a two-dimensional domain. We successfully model the damping of waves in the simulations using linear theory, provided we do not take the weak damping limit typically used in the literature. To isolate the physical mechanism exciting internal waves, we use the spectral code dedalus to run several simplified model simulations of our more detailed simulation. We use data from the full simulation as source terms in two simplified models of internal-wave excitation by convection: bulk excitation by convective Reynolds stresses, and interface forcing via the mechanical oscillator effect. We find excellent agreement between the waves generated in the full simulation and the simplified simulation implementing the bulk excitation mechanism. The interface forcing simulations overexcite high-frequency waves because they assume the excitation is by the "impulsive" penetration of plumes, which spreads energy to high frequencies. However, we find that the real excitation is instead by the "sweeping" motion of plumes parallel to the interface. Our results imply that the bulk excitation mechanism is a very accurate heuristic for internal-wave generation by convection.