Numerical modelling of solute dispersion in natural heterogeneous porous media is facing several challenges. Amongst these we highlight the challenge of accounting for high-frequency variability that is filtered out by homogenization at the subgrid scale and the uncertainty in the dispersive flux for transport under non-ergodic conditions. These two effects when combined lead to inaccurate representation of the dispersive fluxes. We propose to compensate for this deficiency by defining a block-scale dispersion tensor and modelling it as a random space function ℳij. The derived dispersion tensor is a function of several length scales and time. Grid blocks will be assigned dispersion coefficients generated from the ℳij distribution. We will show the dependence of ℳij on the spatial variability of the conductivity field, on the contaminant source size, on the travel time and on the grid-block scale. For an ergodic source, a statistically uniform conductivity field and very large grid blocks, ℳij is equal to the macrodispersion coefficients proposed by Dagan (J. Fluid Mech., vol. 145, 1984, p. 151) with zero variance. For an ergodic source and non-uniform conductivity field with a finite-size grid block, ℳij approaches the model proposed by Rubin et al. (J. Fluid Mech., vol. 395, 1999, p. 161). In both cases, ℳij is defined by its mean value with zero variance. ℳij is subject to uncertainty when the source is non-ergodic and when the grid block is defined by a finite scale. When the grid-block scale approaches zero, which means that the spatial variability is captured completely on the numerical grid, ℳij approaches zero with zero variance. In addition, we provide a complete statistical characterization of ℳij by invoking the concept of minimum relative entropy, thus providing upper bounds on the uncertainty associated with ℳij.

In this paper, we explore the strong rotation limit of the rotating and stratified Boussinesq equations with periodic boundary conditions when the stratification is order 1 ([Rossby number] Ro = ε, [Froude number] Fr = O(1), as ε → 0). Using the same framework of Embid & Majda (Geophys. Astrophys. Fluid Dyn., vol. 87, 1998, p. 1), we show that the slow dynamics decouples from the fast. Furthermore, we derive equations for the slow dynamics and their conservation laws. The horizontal momentum equations reduce to the two-dimensional Navier–Stokes equations. The equation for the vertically averaged vertical velocity includes a term due to the vertical average of the buoyancy. The buoyancy equation, the only variable to retain its three-dimensionality, is advected by all three two-dimensional slow velocity components. The conservation laws for the slow dynamics include those for the two-dimensional Navier–Stokes equations and a new conserved quantity that describes dynamics between the vertical kinetic energy and the buoyancy. The leading order potential enstrophy is slow while the leading order total energy retains both fast and slow dynamics. We also perform forced numerical simulations of the rotating Boussinesq equations to demonstrate support for three aspects of the theory in the limit Ro → 0: (i) we find the formation and persistence of large-scale columnar Taylor–Proudman flows in the presence of O(1) Froude number; after a spin-up time, (ii) the ratio of the slow total energy to the total energy approaches a constant and that at the smallest Rossby numbers that constant approaches 1 and (iii) the ratio of the slow potential enstrophy to the total potential enstrophy also approaches a constant and that at the lowest Rossby numbers that constant is 1. The results of the numerical simulations indicate that even in the presence of the low wavenumber white noise forcing the dynamics exhibit characteristics of the theory.

The Lighthill–Weis-Fogh ‘clap–fling–sweep’ mechanism for lift generation in insect flight is re-examined. The novelty of this mechanism lies in the change of topology (the ‘break’) that occurs at a critical instant tc when two wings separate at their ‘hinge’ point as ‘fling’ gives way to ‘sweep’, and the appearance of equal and opposite circulations around the wings at this critical instant. Our primary aim is to elucidate the behaviour near the hinge point as time t passes through tc. First, Lighthill's inviscid potential flow theory is reconsidered. It is argued that provided the linear and angular accelerations of the wings are continuous, the velocity field varies continuously through the break, although the pressure field jumps instantaneously at t = tc. Then, effects of viscosity are considered. Near the hinge, the local Reynolds number is very small and local similarity solutions imply a logarithmic (integrable) singularity of the pressure jump across the hinge just before separation, in contrast to the ‘negligible pressure jump’ of inviscid theory invoked by Lighthill. We also present numerical simulations of the flow using a volume penalization technique to represent the motion of the wings. For Reynolds number equal to unity (based on wing chord), the results are in good agreement with the analytical solution. At a realistic Reynolds number of about 20, the flow near the hinge is influenced by leading-edge vortices, but local effects still persist. The lift coefficient is found to be much greater than that in the corresponding inviscid flow.