Spontaneous capillary imbibition is a classical problem in interfacial fluid dynamics with a broad range of applications, from microfluidics to agriculture. Here we study the duration of the cross-over between an initial linear growth of the imbibition front to the diffusive-like growth limit of Washburn's law. We show that local-resistance sources, such as the inertial resistance and the friction caused by the advancing meniscus, always limit the motion of an imbibing front. Both effects give rise to a cross-over of the growth exponent between the linear and the diffusive-like regimes. We show how this cross-over is much longer than previously thought – even longer than the time it takes the liquid to fill the porous medium. Such slowly slowing-down dynamics is likely to cause similar long cross-over phenomena in processes governed by wetting.

The late-time behaviour of freely evolving quasi-geostrophic flows with initial characteristic length scale $L_0$ larger than or equal to the deformation radius $L_D$, $L_0/L_D \geq 1$, is studied. At late time the flows are dominated by large multi-level vortices consisting of ascending terraces of well-mixed potential vorticity (PV), i.e. PV staircases. We examine how the number of mixed PV levels depends on the initial conditions, in particular $L_0/L_D$. For sufficiently large values of $L_0/L_D \approx 5$, a complete staircase with regular steps forms, but as $L_0/L_D$ decreases, the staircase becomes more irregular, with fewer mixed levels and the appearance of a large step centred on zero PV, corresponding to large regions of near-zero PV separating the multi-level vortices. This occurs because weak PV features in the initial field with scales smaller than $L_D$ undergo filamentation and are coarse-grained away or homogenised. For all values of $L_0/L_D$ considered, inverse cascades of potential energy commence at sufficiently late times. The onset of these cascades, even when the flow is initialised well within the ‘asymptotic model’ (AM) regime, suggests that the AM regime is not self-consistent: when potential vorticity fronts are well-resolved, frontal dynamics eventually drive ongoing flow evolution.

This paper presents the effects of two-dimensional bathymetry on the evolution of skewness and kurtosis of the surface elevation and velocity field in long-crested irregular waves propagating over a circular shoal and a submerged bar with a semicircular step. For a circular shoal, we show that the surface elevation has a local maximum of skewness and kurtosis on top of the shoal. We find that the skewness of the horizontal velocity field has similar behaviour to the surface elevation, but the kurtosis of the horizontal velocity field has two local maxima at different locations on the lee side of the shoal. For a semicircular step, which acts similar to a converging lens, we find that refraction contributes to the statistical properties of the surface elevation and horizontal velocity field to deviate from Gaussian statistics.