The energy and helicity of knotted magnetic flux tubes

Magnetic relaxation of a magnetic field embedded in a perfectly conducting incompressible fluid to minimum energy magnetostatic equilibrium states is considered. It is supposed that the magnetic field is confined to a single flux tube which may be knotted. A local non-orthogonal coordinate system, zero-framed with respect to the knot, is introduced, and the field is decomposed into toroidal and poloidal ingredients with respect to this system. The helicity of the field is then determined; this vanishes for a field that is either purely toroidal or purely poloidal. The magnetic energy functional is calculated under the simplifying assumptions that the tube is axially uniform and of circular cross-section. The case of a tube with helical axis is first considered, and new results concerning kink mode instability and associated bifurcations are obtained. The case of flux tubes in the form of torus knots is then considered and the ‘ground-state’ energy function ͞m ( h ) (where h is an internal twist parameter) is obtained; as expected, ͞m ( h ), which is a topological invariant of the knot, increases with increasing knot complexity. The function ͞m ( h ) provides an upper bound on the corresponding function m ( h ) that applies when the above constraints on tube structure are removed. The technique is applicable to any knot admitting a parametric representation, on condition that points of vanishing curvature are excluded.

Quasi-periodic solutions of the coupled nonlinear Schrödinger equations

We consider travelling periodic and quasi-periodic wave solutions of a set of coupled nonlinear Schrödinger equations. In fibre optics these equations can be used to model single mode fibres under the action of cross-phase modulation, with weak birefringence. The problem is reduced to the ‘1:2:1’ integrable case of the two-particle quartic potential. A general approach for finding elliptic solutions is given. New solutions which are associated with two-gap Treibich-Verdier potentials are found. General quasi-periodic solutions are given in terms of two dimensional theta functions with explicit expressions for frequencies in terms of theta constants. The reduction of quasi-periodic solutions to elliptic functions is discussed.

The influence of circulation on the stability of vortices to mode-one disturbances

The initial value problem for the two-dimensional inviscid vorticity equation, linearized about an azimuthal basic velocity field with monotonic angular velocity, is solved exactly for mode-one disturbances. The solution behaviour is investigated for large time using asymptotic methods. The circulation of the basic state is found to govern the ultimate fate of the disturbance: for basic state vorticity distributions with non-zero circulation, the perturbation tends to the steady solution first mentioned in Michalke & Timme (1967), while for zero circulation, the perturbation grows without bound. The latter case has potentially important implications for the stability of isolated eddies in geophysics.

The random packing of fibres in three dimensions

An investigation has been carried out of the limiting packing density of an array of long straight rigid fibres distributed randomly in space as a function of the length of the fibre. We derive an approximate relationship between the limiting volume fraction V f and the slenderness λ of the fibres defined as length divided by diameter. The formula agrees well with our experimental results and those found in the literature.

On sound energy scattered by a rigid body near a compliant surface

A theorem is derived which generalizes the classical extinction theorem (also known as the optical theorem) to cases where a rigid scatterer of arbitrary shape is lo­cated near a large compliant surface which has quite general mechanical properties, including dissipation and wave-bearing features, and where the acoustic media on both sides of the compliant surface may have different densities and sound speeds. The theorem relates the sound energy scattered from incident planar acoustic waves to the far field pressure in the specular reflection direction and that in the transmis­sion direction, determined by the Snell’s law. From this simple relation, the scattered energy can be found almost trivially from the far field pressures in these two partic­ular directions; the energy calculation then completely avoids integration of energy flux over control surfaces.

Travelling waves on a membrane: reflection and transmission at a corner of arbitrary angle. I

This article is the first part of a study into the reflection, transmission and scattering of waves on the semi-infinite faces of a compressible fluid wedge of arbitrary angle. This class of problems, in which the wedge surfaces are described by high order impedance conditions (that is, containing derivatives with respect to variables both normal and tangential to the boundary), is of great interest in structural acoustics and electromagnetism. Here, for mathematical convenience, the canonical problem of a fluid wedge with two plane membrane surfaces is examined. Forcing is taken as an unattenuated fluid coupled surface wave incident from infinity on one of the wedge faces. Explicit application of the edge-constraints allows the boundary value problem to be formulated as an inhomogeneous difference equation which is then solved in terms of Maliuzhinets special functions. An analytical solution is thus obtained for arbitrary wedge angle and the membrane wave reflection and transmission coefficients are deduced. The solution method is straightforward to apply and can easily be generalized to any boundary or edge conditions. Also in Part I, the solution obtained for the case of a wedge of angle 2π is compared with that determined by the Wiener-Hopf technique. The two methods are in complete agreement. In the second half of this work the reflection coefficients calculated here will be shown to confirm those given previously in the literature for certain specific wedge angles. A full numerical study, for a range of fluid-membrane parameter values, will also be presented in Part II.

Harmonic maps between three-spheres

It is shown that smooth maps f : S 3 → S 3 contain two countable families of harmonic representatives in the homotopy classes of degree zero and one.

Global existence and non-existence problem for a reaction—diffusion equation with a conserved first integral

In this paper, we prove the global existence and non-existence of solutions of the following problem: RDC{ u t = u xx + u 2 - ∫ u 2 ( x ) d x , x ϵ (0, 1), t > 0, u x (0, t ) = u x (1, t ) = t > 0, u ( x , 0) = u 0 ( x ), x ϵ (0, 1), ∫ 1 0 u ( x, t ) d x = 0, t > 0, Moreover, let u m ( x ) be a stationary solution of problem RDC with m zeros in the interval (0, 1) for m ϵ N , and if we take u 0 ( x ). Then we have that the solution exists globally if 0 < ϵ < 1, and blows up in finite time if ϵ > 1. This result verifies the numerical results of Budd et al . (1993, SIAM Jl appl. Math . 53, 718-742) that the non-zero stationary solutions are unstable.

Constraint-based failure assessment diagrams

Constraint-dependent toughness has been addressed by characterizing elastic-plastic crack tip fields and the associated toughness in terms of two parameters: J , and a parameter that indexes the level of constraint ( T/Q ). In the past, failure assessment diagrams have been developed on the basis of a single-parameter characterization of toughness. The present work modifies these diagrams to incorporate constraint effects and indicates the loadings where advantage can be taken of constraint enhanced toughness.

Erratum

Proc. R. Soc. Lond . A 451, 293-318 (1995) Stochastic backscatter of turbulence energy and scalar variance by random subgrid-scale fluxes By U. Schumann