N. J. Balmforth

Dynamics of roll waves

Shallow-water equations with bottom drag and viscosity are used to study the dynamics of roll waves. First, we explore the effect of bottom topography on linear stability of turbulent flow over uneven surfaces. Low-amplitude topography is found to destabilize turbulent roll waves and lower the critical Froude number required for instability. At higher amplitude, the trend reverses and topography stabilizes roll waves. At intermediate topographic amplitude, instability can be created at much lower Froude numbers due to the development of hydraulic jumps in the equilibrium flow. Second, the nonlinear dynamics of the roll waves is explored, with numerical solutions of the shallow-water equations complementing an asymptotic theory relevant near onset. We find that trains of roll waves undergo coarsening due to waves overtaking one another and merging, lengthening the scale of the pattern. Unlike previous investigations, we find that coarsening does not always continue to its ultimate conclusion (a single roll wave with the largest spatial scale). Instead, coarsening becomes interrupted at intermediate scales, creating patterns with preferred wavelengths. We quantify the coarsening dynamics in terms of linear stability of steady roll-wave trains.

Ocean waves and ice sheets

A complete analytical study is presented of the reflection and transmission of surface gravity waves incident on ice-covered ocean. The ice cover is idealized as a plate of elastic material for which flexural motions are described by the Timoshenko–Mindlin equation. A suitable non-dimensionalization extracts parameters useful for the characterization of ocean-wave and ice-sheet interactions, and for scaled laboratory studies. The scattering problem is simplified using Fourier transforms and the Wiener–Hopf technique; the solution is eventually written down in terms of some easily evaluated quadratures. An important feature of this solution is that the physical conditions at the edge of the ice sheet are explicitly built into the analysis, and power-flow theorems provide verification of the results. Asymptotic results for large and small values of the non-dimensional parameters are extracted and approximations are given for general parameter values.

Visco-plastic models of isothermal lava domes

The dynamics of expanding domes of isothermal lava are studied by treating the lava as a viscoplastic material with the Herschel{Bulkley constitutive law. Thin-layer theory is developed for radially symmetric extrusions onto horizontal plates. This provides an evolution equation for the thickness of the fluid that can be used to model expanding isothermal lava domes. Numerical and analytical solutions are derived that explore the eects of yield stress, shear thinning and basal sliding on the dome evolution. The results are briefly compared with an experimental study. It is found that it is dicult to unravel the combined eects of shear thinning and yield stress; this may prove important to studies that attempt to infer yield stress from morphology of flowing lava.

Building a better snail: Lubrication and adhesive locomotion

Many gastropods, such as slugs and snails, crawl via an unusual mechanism known as adhesive locomotion. We investigate this method of propulsion using two mathematical models: one for direct waves and one for retrograde waves. We then test the effectiveness of both proposed mechanisms by constructing two mechanical crawlers. Each crawler uses a different mechanical strategy to move on a thin layer of viscous fluid. The first uses a flexible flapping sheet to generate lubrication pressures in a Newtonian fluid, which in turn propel the mechanical snail. The second generates a wave of compression on a layer of Laponite, a non-Newtonian, finite-yield stress fluid with characteristics similar to those of snail mucus. This second design can climb smooth vertical walls and perform an inverted traverse.

Shallow viscoplastic flow on an inclined plane

Evolving viscoplastic flows upon slopes are an important idealization of many flows in a variety of geophysical situations where yield stress is thought to play a role. For such models, asymptotic expansions suitable for slowly moving shallow fluid layers (lubrication theory) reduce the governing equations to a simpler problem in terms of the fluid thickness. We consider the version of the theory for fluids described by the Herschel–Bulkley constitutive law, and provide a variety of solutions to the reduced equation, both numerical and analytical. For extruded inclined domes, we derive the characteristic temporal behaviour of measures of the domes dimensions.

Dynamics of interfaces and layers in a stratified turbulent fluid

This paper formulates a model of mixing in a stratied and turbulent fluid. The model uses the horizontally averaged vertical buoyancy gradient and the density of turbulent kinetic energy as variables. Heuristic ‘mixing-length’ arguments lead to a coupled set of parabolic dierential equations. A particular form of mechanical forcing is proposed; for certain parameter values the relationship between the buoyancy flux and the buoyancy gradient is non-monotonic and this leads to an instability of equilibria with linear stratication. The instability results in the formation of steps and interfaces in the buoyancy prole. In contrast to previous ones, the model is mathematically well posed and the interfaces have an equilibrium thickness that is much larger than that expected from molecular diusion. The turbulent mixing process can take one of three forms depending on the strength of the initial stratication. When the stratication is weak, instability is not present and mixing smoothly homogenizes the buoyancy. At intermediate strengths of stratication, layers and interfaces form rapidly over a substantial interior region bounded by edge layers associated with the fluxless condition of the boundaries. The interior pattern subsequently develops more slowly as interfaces drift together and merge; simultaneously, the edge layers advance inexorably into the interior. Eventually the edge layers meet in the middle and the interior pattern of layers is erased. Any remaining structure subsequently decays smoothly to the homogeneous state. Both the weak and intermediate stratied cases are in agreement with experimental phenomenology. The model predicts a third case, with strong stratication, not yet found experimentally, where the central region is linearly stable and no steps form there. However, the edge layers are unstable; mixing fronts form and then erode into the interior.

A shocking display of synchrony

Abstract This article explores the Kuramoto model describing the synchronization of a population of coupled oscillators. Two versions of this model are considered: a discrete version suitable for a population with a finite number of oscillators, and a continuum model found in the limit of an infinite population. When the strength of the coupling between the oscillators exceeds a threshold, the oscillators partially synchronize. We explore the transition in the continuum model, which takes the form of a bifurcation of a discrete mode from a continuous spectrum. We use numerical methods and perturbation theory to study the patterns of synchronization that form beyond transition, and compare with the synchronization predicted by the discrete model. There are similarities with instabilities in ideal plasmas and inviscid fluids, but these are superficial.

Roll waves in mud

The stability of a viscoplastic fluid film falling down an inclined plane is explored, with the aim of determining the critical Reynolds number for the onset of roll waves. The Herschel–Bulkley constitutive law is adopted and the fluid is assumed two-dimensional and incompressible. The linear stability problem is described for an equilibrium in the form of a uniform sheet flow, when perturbed by introducing an infinitesimal stress perturbation. This flow is stable for very high Reynolds numbers because the rigid plug riding atop the fluid layer cannot be deformed and the free surface remains flat. If the flow is perturbed by allowing arbitrarily small strain rates, on the other hand, the plug is immediately replaced by a weakly yielded ‘pseudo-plug’ that can deform and reshape the free surface. This situation is modelled by lubrication theory at zero Reynolds number, and it is shown how the fluid exhibits free-surface instabilities at order-one Reynolds numbers. Simpler models based on vertical averages of the fluid equations are evaluated, and one particular model is identified that correctly predicts the onset of instability. That model is used to describe nonlinear roll waves.

Bounds on horizontal convection

For a fluid layer heated and cooled differentially at its surface, we use a variational approach to place bounds on the viscous dissipation rate and a horizontal Nusselt measure based on the entropy production. With a general temperature distribution imposed at the top of the layer and a variety of thermal boundary conditions at the base of the layer, the horizontal Nusselt number is bounded by

Shear instability in shallow water

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