On the Abel Equation of the Second Kind with Sinusoidal Forcing

We consider the Abel equation of the second kind with sinusoidal forcing, which is a model equation for western boundary outflow in the Stommel model of the large scale ocean circulation. Series solutions of this equation indicate the presence of resonances at certain discrete values of a parameter which measures the nonlinearity of the flow, but numerical solutions using a standard scheme show no evidence of these resonances. We discuss and resolve this apparent contradiction

Parametric resonant triad interactions in a free shear layer

We investigate the weakly nonlinear evolution of a triad of nearly-neutral modes superimposed on a mixing layer with velocity profile u bar equals Um + tanh y. The perturbation consists of a plane wave and a pair of oblique waves each inclined at approximately 60 degrees to the mean flow direction. Because the evolution occurs on a relatively fast time scale, the critical layer dynamics dominate the process and the amplitude evolution of the oblique waves is governed by an integro-differential equation. The long-time solution of this equation predicts very rapid (exponential of an exponential) amplification and we discuss the pertinence of this result to vortex pairing phenomena in mixing layers

Poincare Indices of Rheoscopic Visualisations

Suspensions of small anisotropic particles, termed 'rheoscopic fluids', are used for flow visualisation. By illuminating the fluid with light of three different colours, it is possible to determine Poincare indices for vector fields formed by the longest axis of the particles. Because this vector field is non-oriented, half-integer Poincare indices are possible, and are observed experimentally. An exact solution for the direction vector appears to preclude the existence of topological singularities. However, we show that upon averaging over the random initial orientations of particles, singularities with half-integer Poincare index appear. We describe their normal forms.Comment: 4 pages, 4 figure

Laplace transforms and the American straddle

We address the pricing of American straddle options. We use partial Laplace transform techniques due to Evans et al. (1950) to derive a pair of integral equations giving the locations of the optimal exercise boundaries for an American straddle option with a constant dividend yield

A Green′s function for a convertible bond using the Vasicek model

We consider a convertible security where the underlying stock price obeys a lognormal random walk and the risk-free rate is given by the Vasicek model. Using a Laplace transform in time and a Mellin transform in the stock price, we derive a Green′s function solution for the value of the convertible bond

Chern - Simons Gauge Field Theory of Two - Dimensional Ferromagnets

A Chern-Simons gauged Nonlinear Schr\"odinger Equation is derived from the continuous Heisenberg model in 2+1 dimensions. The corresponding planar magnets can be analyzed whithin the anyon theory. Thus, we show that static magnetic vortices correspond to the self-dual Chern - Simons solitons and are described by the Liouville equation. The related magnetic topological charge is associated with the electric charge of anyons. Furthermore, vortex - antivortex configurations are described by the sinh-Gordon equation and its conformally invariant extension. Physical consequences of these results are discussed.Comment: 15 pages, Plain TeX, Lecce, June 199

Enhanced sedimentation of elongated plankton in simple flows

Negatively buoyant phytoplankton play an important role in the sequestration of CO_2 from the atmo-sphere and are fundamental to the health of the world’s fisheries. However, there is still much to discoveron transport mechanisms from the upper photosynthetic regions to the deep ocean. In contrast to intuitive expectations that mixing increases plankton residence time in light-rich regions, recent experimental and computational evidence suggests that turbulence can actually enhance sedimentation of negatively buoyant diatoms. Motivated by these studies we dissect the enhanced sedimentation mechanisms using the simplest possible two-dimensional flows, avoiding expensive computations and obfuscation. In particular, we find that in vertical shear, preferential flow alignment and aggregation in down-welling regions both increase sedimentation, whereas horizontal shear reduces the sedimentation due only to alignment. However the magnitude of the shear does not affect the sedimentation rate. In simple vertical Kolmogorov flow elongated particles also have an enhanced sedimentation speed as they spend more time in down-welling regions of the flow with vertically aligned orientation, an effect that increases with the magnitude of shear. An additional feature is identified in horizontal Kolomogorov flow, whereby the impact of shear-dependent sedimentation speed is to cause aggregation in regions of high-shear where the sedimentation speed is minimum. In cellular flow, there is an increase in mean sedimentation speed with aspect ratio and shear strength associated with aggregation in down-welling regions. Furthermore, spatially projected trajectories can intersect and give rise to chaotic dynamics, which is associated with a depletion of particles within so called retention zones