Nearly fifty years ago, Roberts (1978) postulated that the Earth’s magnetic field, which is generated by turbulent motions of liquid metal in its outer core, likely results from a subcritical dynamo instability characterised by a dominant balance between Coriolis, pressure and Lorentz forces (requiring a finite-amplitude magnetic field). Here, we numerically explore subcritical convective dynamo action in a spherical shell, using techniques from optimal control and dynamical systems theory to uncover the nonlinear dynamics of magnetic field generation. Through nonlinear optimisation, via direct-adjoint looping, we identify the minimal seed – the smallest magnetic field that attracts to a nonlinear dynamo solution. Additionally, using the Newton-hookstep algorithm, we converge stable and unstable travelling wave solutions to the governing equations. By combining these two techniques, complex nonlinear pathways between attracting states are revealed, providing insight into a potential subcritical origin of the geodynamo. This paper showcases these methods on the widely studied benchmark of Christensen et al. (2001, Phys. Earth Planet. Inter., vol. 128, pp. 25–34), laying the foundations for future studies in more extreme and realistic parameter regimes. We show that the minimal seed reaches a nonlinear dynamo solution by first approaching an unstable travelling wave solution, which acts as an edge state separating a hydrodynamic solution from a magnetohydrodynamic one. Furthermore, by carefully examining the choice of cost functional, we establish a robust optimisation procedure that can systematically locate dynamo solutions on short time horizons with no prior knowledge of its structure.

We present a framework for parametric proper orthogonal decomposition (POD)-Galerkin reduced-order modelling (ROM) of fluid flows that accommodates variations in flow parameters and control inputs. As an initial step, to explore how the locally optimal POD modes vary with parameter changes, we demonstrate a sensitivity analysis of POD modes and their spanned subspace, respectively rooted in Stiefel and Grassmann manifolds. The sensitivity analysis, by defining distance between POD modes for different parameters, is applied to the flow around a rotating cylinder with varying Reynolds numbers and rotation rates. The sensitivity of the subspace spanned by POD modes to parameter changes is represented by a tangent vector on the Grassmann manifold. For the cylinder case, the inverse of the subspace sensitivity on the Grassmann manifold is proportional to the Roshko number, highlighting the connection between geometric properties and flow physics. Furthermore, the Reynolds number at which the subspace sensitivity approaches infinity corresponds to the lower bound at which the characteristic frequency of the Kármán vortex street exists (Noack & Eckelmann, J. Fluid Mech., 1994, vol. 270, pp. 297–330). From the Stiefel manifold perspective, sensitivity modes are derived to represent the flow field sensitivity, comprising the sensitivities of the POD modes and expansion coefficients. The temporal evolution of the flow field sensitivity is represented by superposing the sensitivity modes. Lastly, we devise a parametric POD-Galerkin ROM based on subspace interpolation on the Grassmann manifold. The reconstruction error of the ROM is intimately linked to the subspace-estimation error, which is in turn closely related to subspace sensitivity.

Experimental studies of natural convection in yield stress fluids have revealed transient behaviours that contradict predictions from viscoplastic models. For example, at a sufficiently large yield stress, these models predict complete motionlessness; below a critical value, yielding and motion onset can be delayed in viscoplastic models. In both cases, however, experiments observe immediate motion onset. We present numerical simulations of the transient natural convection of elastoviscoplastic (EVP) fluids in a square cavity with differentially heated side walls, exploring the role of elasticity in reconciling theoretical predictions with experimental observations. We consider motion onset in EVP fluids under two initial temperature distributions: (i) a linear distribution characteristic of steady pure conduction, and (ii) a uniform distribution representative of experimental conditions. The Saramito EVP model exhibits an asymptotic behaviour similar to the Kelvin-Voigt model as $t\to 0^+$, where material behaviour is primarily governed by elasticity and solvent viscosity. The distinction between motion onset and yielding, a hallmark of EVP models, is the key feature that bridges theoretical predictions with experimental observations. While motion onset is consistently immediate (as seen in experiments), yielding occurs with a delay (as predicted by viscoplastic models). Scaling analysis suggests that this delay varies logarithmically with the yield stress and is inversely proportional to the elastic modulus. The intensity of the initial pre-yield motion increases with higher yield stress and lower elastic modulus. The observed dynamics resemble those of under- and partially over-damped systems, with a power-law fit providing an excellent match for the variation of oscillation frequency with the elastic modulus.

The breakup and coalescence of particle aggregates confined at the interface of turbulent liquid layers are investigated experimentally and theoretically. In particular, we consider conductive fluid layers driven by Lorentz forces and laden with millimetre-scale floating particles. These form aggregates held together by capillary attraction and disrupted by the turbulent motion. The process is fully characterised by imaging at high spatio-temporal resolution. The breakup frequency $\varOmega$ is proportional to the mean strain rate and follows a power-law scaling $\varOmega \sim D^{3\text{/}2}$, where $D$ is the size of the aggregate, attributed to the juxtaposition of particle-scale strain cells. The daughter aggregate size distribution exhibits a robust U-shape, which implies erosion of small fragments as opposed to even splitting. The coalescence kernel $\varGamma$ between pairs of aggregates of size $D_{1}$ and $D_{2}$ scales as $\varGamma \sim ( D_{1} + D_{2} )^{2}$, which is consistent with gas-kinetic dynamics. These relations, which apply to regimes dominated both by capillary-driven aggregation and by drag-driven breakup, are implemented into the population balance equation for the evolution of the aggregate number density. Comparison with the experiments shows that the framework captures the observed distribution for aggregates smaller than the forcing length scale.

This study investigates the stability characteristics of rotating-disk boundary layers in rotor–stator cavities under the frameworks of local linear, global linear and global nonlinear analyses. The local linear stability analysis uses the Chebyshev polynomial method, the global linear stability analysis relies on the linearised incompressible Navier–Stokes (N–S) equations and the global nonlinear analysis involves directly solving the complete incompressible N–S equations. In the local linear framework, the velocity profile derived from the laminar self-similar solution on the rotating-disk side of an infinite rotor–stator cavity is mapped to the Bödewadt–Ekman–von Kármán theoretical model to establish a unified analytical framework. For the global stability study, we extend the methodological framework proposed by Appelquist et al. (J. Fluid Mech.,vol 765, 2015, pp. 612–631) for the von Kármán boundary layer, implementing pulsed disturbances and constructing a radial sponge layer to effectively capture the spatiotemporal evolution of perturbation dynamics while mitigating boundary reflection effects. The analysis reveals that the rotating-disk boundary layer exhibits two distinct instability regimes: convective instability emerges at ${\textit{Re}}=r^*/\sqrt {\nu ^*/\varOmega ^*}=204$ (where $r^*$ is the radius, $\nu ^*$ is the kinematic viscosity and $\varOmega ^*$ is the rotation rate of the system) with azimuthal wavenumber $\beta =27$, while absolute instability emerges at ${\textit{Re}}=409.6$ with azimuthal wavenumber $\beta =85$. Under pulsed disturbance excitation, an initial convective instability behaviour dominates in regions exceeding the absolute instability threshold. As perturbations propagate into the sponge layer’s influence domain, upstream mode excitation triggers the emergence of a global unstable mode, characterised by a minimum critical Reynolds number ${\textit{Re}}_{\textit{end}}=484.4$. Further analysis confirms that this global mode is an inherent property of the rotating-disk boundary layer and is independent of the characteristics of the sponge layer. Frequency-domain analysis establishes that the global mode frequency is governed by local stability characteristics at ${\textit{Re}}_{\textit{end}}$, while its growth rate evolution aligns with absolute instability trends. By further incorporating nonlinear effects, it was observed that the global properties of the global nonlinear mode remain governed by ${\textit{Re}}_{\textit{end}}$. The global temporal frequency corresponds to ${\textit{Re}}_{\textit{end}}=471.8$. When ${\textit{Re}}$ approaches 517.2, the spiral waves spontaneously generate ring-like vortices, which subsequently trigger localised turbulence. This investigation provides novel insights into the fundamental mechanisms governing stability transitions in the rotating-disk boundary layer of the rotor–stator cavity.