Horizontal convection (HC), the flow in a box heated and cooled from the bottom (or top), is a canonical model of buoyancy-driven transport with direct relevance to subglacial lakes, ocean circulation and climate in general. While HC is commonly characterised by the Rayleigh number $Ra$ and Prandtl number $Pr$, the role of the domain aspect ratio $\varGamma$ (defined as horizontal extension over height) has received less attention. Here, we perform three-dimensional direct numerical simulations over a broad range of Rayleigh numbers and aspect ratios at fixed Prandtl number in order to quantify how geometry affects flow organisation and heat transport. We find that $\varGamma$ strongly influences both the development of the overturning circulation and the scaling of heat transfer (i.e. in dimensionless form, the Nusselt number $Nu$). As the aspect ratio increases, the emergence of an organised overturning flow is shifted to higher Rayleigh numbers. Moreover, with increasing $\varGamma$ the Nusselt number shows a stronger dependence on the Rayleigh number. The results clarify the gradual cross-over from a finite intermediate range compatible with the Rossby-type scaling $\textit{Nu}\sim \textit{Ra}^{1/5}$ (Rossby 1965 Deep-Sea Res. Oceanogr. Abstr. vol. 12, pp. 9–16) towards the $\textit{Nu}\sim \textit{Ra}^{1/4}$ behaviour suggested by Shishkina, Grossmann & Lohse (2016 Geophys. Res. Lett. vol. 43, pp. 1219–1225). It reflects the progressively decreasing vertical confinement of the kinetic boundary layer with increasing aspect ratio. These results show that the aspect ratio is an important control parameter in laminar HC and should be incorporated explicitly into scaling arguments for geophysical flows.

The translation of a body along the axis of a rotating fluid is a classical problem in rotating flows where discrepancies between experiments, theory and numerical simulations, particularly regarding Taylor column formation and forward influence, have remained unsolved. Theoretical and many numerical studies have relied on various assumptions, including that the flows are inviscid, linear or in unbounded domains, limiting their relevance to laboratory configurations. The flow of a sphere travelling at constant speed along the axis of a finite rotating cylinder filled with a viscous fluid is here investigated via numerical solutions of the Navier–Stokes equations with no-slip boundary conditions on both the co-rotating sphere and the cylinder walls. A wide range of Rossby and Taylor numbers, parameters giving the ratios of the three time scales associated with the background rotation rate, the viscous dissipation rate and the travel speed of the sphere, is explored using the experimental geometry of Pritchard (J. Fluid Mech., 39, 1969, pp. 443–464). By explicitly retaining axial and radial confinement effects, the results indicate the regions of parameter space where viscous, unsteady, ageostrophic and nonlinear effects are significant, addressing the limits of steady, linear and quasi-geostrophic assumptions.

The Atlantic Meridional Overturning Circulation (AMOC), partially driven by double-diffusive horizontal convection (DDHC), plays a key role in regulating the global climate. Indeed, it governs the transfer of heat, salinity and nutrients between the equator and polar regions. The present study investigates an idealised model system, ‘thermohaline circulation in a box’ or ‘AMOC in a box’, namely DDHC in a well-defined geometry, specifically the flow in a box with horizontal temperature and salinity gradients. By varying the temperature Rayleigh number $\textit{Ra}_T$ and the density ratio $\varLambda$, or equivalently the salinity Rayleigh number $\textit{Ra}_S$, four distinct regimes are found. These regimes are distinguished by the global response parameters of the system, namely the temperature Nusselt number $\textit{Nu}_T$, the salinity Nusselt number $\textit{Nu}_S$ and the friction Reynolds number $\textit{Re}_\tau$, as well as by the flow structures. The two limiting regimes of horizontal convection, at high and low $\varLambda$ values, follow the Shishkina-Grossmann-Lohse theory for horizontal convection. In the two regimes in between, in which strong competition between temperature and saline buoyancy occurs, a clear thermohaline layering and the presence of oscillating convected salt fingers are found.

We present a study on the melting dynamics of neighbouring ice bodies by means of idealised simulations, focusing on collective effects, with the goal of obtaining fundamental insight into how collective interactions influence the melting of ice. Two neighbouring (vertically or horizontally aligned), square-shaped and equally sized ice objects (size of the order of centimetres) are immersed in quiescent fresh water at a temperature of ${20}\,^\circ \textrm {C}$. By performing two-dimensional direct numerical simulations, and using the phase-field method to model the phase change, the collective melting of these objects is studied. When the objects are horizontally aligned, no significant influence of the neighbouring object on the melting time is observed. On the other hand, when vertically aligned, although the melting of the upper object is mostly unaffected, the melting time and the morphology of the lower ice body strongly depends on the initial inter-object distance. We report that the melting of the bottom object can be enhanced by more than 10 %, or delayed more than 20 %, displaying a non-monotonic dependence on the initial object size. We show that this behaviour results from a non-trivial competition between layering of cold fluid, which lowers the heat transfer, and convective flows, which favour mixing and heat transfer. For this melting in mixed convection, we were able to collapse our data onto a single curve.

The flow in a rapidly rotating cylinder forced by the harmonic oscillations of a small sphere along the rotation axis is explored numerically. For oscillation frequencies less than twice the cylinder rotation frequency, the forced response flows feature conical shear layers emitted from the critical latitudes of the sphere. These latitudes are where the characteristics of the hyperbolic system, arrived at by ignoring nonlinear, viscous and forcing terms in the governing equations, are tangential to the sphere. These conical shear layers vary continuously with the forcing frequency so long as it remains inertial. At certain values of the forcing frequency, linear inviscid inertial modes of the cylinder are resonated. Of all possible inertial modes, only those whose symmetries are compatible with the symmetry of the forced system are resonated. This all occurs even in the linear limit of vanishingly small forcing amplitude. As the forcing amplitude is increased, nonlinearity leads to non-harmonic oscillations and a non-zero mean flow which features a Taylor columnar structure extending from the sphere to the two endwalls in an axially invariant fashion.