Abstract We analyze the dynamics of evolving lava‐fed deltas through the use of shallow‐layer mathematical models and analog laboratory experiments. Numerical and asymptotic solutions are calculated for the cases of planar and three‐dimensional flows fed by a point source upstream of the shoreline. We consider several modes of delta formation: a reduction in the driving buoyancy force; an enhanced viscosity of the submerged material; and the production of a granular subaqueous platform, over which a subaerial current can propagate. These modes of delta formation result in different behaviors. Under a steady supply of fluid upstream, the buoyancy‐driven case develops a solution with a steady subaerial delta and a subaqueous current which propagates at a constant speed, while the granular platform model extends the delta indefinitely. We determine a late‐time power‐law relation for the shoreline extent with time in this case. When the viscosity contrast is large, the model with an enhanced subaqueous viscosity is shown to mimic the initial dynamics of the granular platform model, but ultimately reaches a steady shoreline extent at sufficiently late times, as for the buoyancy‐driven model. The distinct behaviors of these models are further illustrated through laboratory experiments, utilizing the gelling reaction of sodium alginate solution in the presence of calcium ions as a novel analog for the abrupt rheological changes that occur when lava makes contact with water. These experiments provide quantitative verification of the buoyancy‐driven model in the absence of the reaction, and demonstrate the effects of a subaqueous platform qualitatively in its presence.
This article continues an exploration of instabilities of jets in two-dimensional, inviscid fluid on the beta-plane. At onset, for particular choices of the physical parameters, the normal modes responsible for instability have critical levels that coalesce along the axis of the jet. Matched asymptotic expansion (critical layer theory) is used to derive a reduced model describing the dynamics of these modes. Because the velocity profile is locally parabolic in the vicinity of the critical levels the dynamics is richer than in standard critical layer problems. The model captures the inviscid saturation of unstable modes, the excitation of neutral Rossby waves, and the decay of disturbances when there are no discrete normal modes. Inviscid saturation occurs when the vorticity distribution twists up into vortical structures that take the form of either a pair of ‘cat's eye’ patterns straddling the jet axis, or a single row of vortices. The addition of weak viscosity destroys these cat's eye structures and causes the critical layer to spread diffusively. The results are compared with numerical simulations of the governing equations.
Matched asymptotic expansions are used to obtain a reduced description of the nonlinear and viscous evolution of small, localized vorticity defects embedded in a Couette flow. This vorticity defect approximation is similar to the Vlasov equation, and to other reduced descriptions used to study forced Rossby wave critical layers and their secondary instabilities. The linear stability theory of the vorticity defect approximation is developed in a concise and complete form. The dispersion relations for the normal modes of both inviscid and viscous defects are obtained explicitly. The Nyquist method is used to obtain necessary and sufficient conditions for instability, and to understand qualitatively how changes in the basic state alter the stability properties. The linear initial value problem is solved explicitly with Laplace transforms; the resulting solutions exhibit the transient growth and eventual decay of the streamfunction associated with the continuous spectrum. The expansion scheme can be generalized to handle vorticity defects in non-Couette, but monotonic, velocity profiles.
A variety of problems in engineering and geology involve spreading cooling non-Newtonian fluids. If the fluid is relatively shallow and spreads slowly, lubrication-style asymptotic approximations can be used to build reduced models for the spreading dynamics. The centrepiece of such models is a nonlinear diffusion equation for the local fluid thickness, and ideally this should become coupled to a correspondingly simple equation determining the local temperature field. However, when heat diffuses relatively slowly as the fluid flows, we cannot usefully reduce the temperature equation, and the asymptotic reduction couples the local thickness equation to an advection diffusion equation that crucially involves diffusion in the vertical. We present an efficient computational algorithm for numerically solving this more complicated type of lubrication model, and describe a suite of solutions that illustrate the dynamics captured by the model in the case of an expanding Bingham fluid with a temperature-dependent viscosity. Based on these solutions, we evaluate two simpler models that further approximate the temperature equation: a vertically isothermal theory, and a 'skin theory'. The latter is based on the integral-balance method of heat-transfer theory, and demands that the vertical structure of the temperature field has the form of an advancing boundary layer, or skin. The vertically isothermal model performs well when the thermal conductivity is relatively large. The skin theory reproduces the full dynamics qualitatively, if not quantitatively, for all thermal conductivities. The main errors in both models arise near the fluid edge, where the numerical solutions show that chilled fluid is overridden as the fluid expands, creating an underlying collar of cold material. Encouraged by the success of the skin model, we extend the theory by incorporating extensional stresses in the skin, which emerge when cooling induces an extreme rheological change in the material, such as an exponential rise in the viscosity. The model predicts that when skin stresses are sufficiently strong, the skin is brought to rest, whilst hotter fluid expands underneath.
We explore the level of saturation of instabilities in a two-species plasma using a combination of matched asymptotic expansion and numerical computation. The plasma is assumed to be spatially periodic, and the domain size is chosen to allow a single mode to become unstable when a bump is added to the tail of the distribution of the lighter species. We consider two versions of the problem, arising when the mass ratio of the two species is either very small, or of the order of unity. For small mass ratios, the initial saturation level of the mode amplitude, as measured by the electric field disturbance, follows the ‘trapping scaling’. For mass ratios of order unity, nonlinear effects become important at the level predicted by Crawford and Jayaraman, but the instability does not saturate there and continues to grow. In both cases, the initial onset of nonlinearity does not reflect the longer- time evolution of the system. In fact, the system passes through multiple stages of evolution in which the electric field amplitude is not simply predicted; none of the previously published scalings are adequate. Eventually, for both cases, the distribution of the lighter ions becomes significantly rearranged, and much (though not all) of the destabilizing bump is flattened. A better predictor of the strength of the instability is given by the extent of these rearrangements.
The 2016 GFD Program theme was fluid-Structure Interaction in the Living Environment with Professors Mike Shelley of New York University and Anette 'Peko' Hosol of the Massachusetts Institute of Technology serving as principal lecturers. Together they introduced the audience in the cottage and on the porch to a fascinating mixture of topics ranging from swimming and swarming to cycling and sprinting, with Professor Jun Zhang of New York University interjecting some more traditional GFD (and art) part way through. The first ten chapters of this volume document these lectures, each prepared by pairs of the summer's GFD fellow. Following the principal lecture notes are the written reports of the follows' own research projects.
Strato-rotational instability (SRI) is normally interpreted as the resonant interactions between normal modes of the internal or Kelvin variety in three-dimensional settings in which the stratification and rotation are orthogonal to both the background flow and its shear. Using a combination of asymptotic analysis and numerical solution of the linear eigenvalue problem for plane Couette flow, it is shown that such resonant interactions can be destroyed by certain singular critical levels. These levels are not classical critical levels, where the phase speed $c$ of a normal mode matches the mean flow speed $U$ , but are a different type of singularity where $(c-U)$ matches a characteristic gravity-wave speed $\pm N/k$ , based on the buoyancy frequency $N$ and streamwise horizontal wavenumber $k$ . Instead, it is shown that a variant of SRI can occur due to the coupling of a Kelvin or internal wave to such ‘baroclinic’ critical levels. Two characteristic situations are identified and explored, and the conservation law for pseudo-momentum is used to rationalize the physical mechanism of instability. The critical level coupling removes the requirement for resonance near specific wavenumbers $k$ , resulting in an extensive continuous band of unstable modes.
A combined theoretical and experimental study is presented for rapid flow-induced compaction of a fibrous porous medium. The model is used to understand the dynamics of a standard test for pulp suspensions: the Canadian Standard Freeness test, which depends sensitively on the solid bulk viscosity.
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