N. R. T. Biggs

Polynya Dynamics: a Review of Observations and Modeling

[1]xa0Polynyas are nonlinear-shaped openings within the ice cover, ranging in size from 10 to 105 km2. Polynyas play an important climatic role. First, winter polynyas tend to warm the atmosphere, thus affecting atmospheric mesoscale motions. Second, ocean surface cooling and brine rejection during sea ice growth in polynyas lead to vertical mixing and convection, contributing to the transformation of intermediate and deep waters in the global ocean and the maintenance of the oceanic overturning circulation. Since 1990, there has been an upsurge in polynya observations and theoretical models for polynya formation and their impact on the biogeochemistry of the polar seas. This article reviews polynya research carried out in the last 2 decades, focusing on presenting a state-of-the-art picture of the physical interactions between polynyas and the atmosphere-sea ice-ocean system. Observational and modeling studies, the surface heat budget, and water mass transformation within these features are addressed.

A multi-mode approximation to wave scattering by ice sheets of varying thickness

The problem of linear wave scattering by an ice sheet of variable thickness floating on water of variable quiescent depth is considered by applying the Rayleigh–Ritz method in conjunction with a variational principle. By using a multi-mode expansion to approximate the velocity potential that represents the fluid motion, Porter & Porter (J. Fluid Mech. vol. 509, 2004, p. 145) is extended and the solution of the problem may be obtained to any desired accuracy. Explicit solution methods are formulated for waves that are obliquely incident on two-dimensional geometry, comparisons are made with existing work and a range of new examples that includes both total and partial ice-cover is considered.

Polynya flux model solutions incorporating a parameterization for the collection thickness of consolidated new ice

Previous polynya flux models have specified a constant value for the collection thickness of frazil ice, H , at the polynya edge. In certain circumstances, this approach can cause the frazil ice depth, h , within the polynya, to exceed H , a result which violates assumptions made in the formulation of the ice flux balance equations at the polynya edge. To overcome this problem, a parameterization for H is derived in terms of the depth of frazil ice arriving at the polynya edge and the component, normal to the polynya edge, of the frazil ice velocity relative to the velocity of the consolidated ice pack. Thus, H is coupled to the unknown polynya edge. Using the new parameterization for H , an analysis of the unsteady one-dimensional opening of a coastal polynya is presented. Analytical solutions are also derived, using the new parameterization for H , for steady-state two-dimensional polynyas adjacent to a semi-infinite and finite-length coastal barrier, the latter case representing a prototype island. In all cases, the solutions show close qualitative and quantitative agreement with those derived using a constant value for H . However, the steady-state two-dimensional polynya edge can, in certain circumstances, exhibit a corner at the point where the offshore equilibrium width is reached. Precise conditions for the existence of a corner are derived in terms of the orientation of the frazil ice velocity ( u ) and the consolidated ice velocity ( U ). Upper and lower bounds are also obtained for the area of the steady-state island polynya, and it is shown that over a large range of orientations of u and U , the area exceeds that associated with the island polynya with constant H . Finally, two simulations of the St. Lawrence Island Polynya are presented using the new parameterization for H , and the results are compared with the H -constant theory.

A steady-state coupled ocean-polynya flux model of the North Water, Baffin Bay

Abstract A steady-state coupled wind-driven reduced-gravity polynya flux model is developed to study the maintenance of the North Water Polynya. The following features are incorporated into the coupled model: (i) arbitrary wind stress; (ii) entrainment of fluid into the upper active layer from the deep motionless layer; (iii) frazil ice trajectories determined from the free-drift ice momentum balance; (iv) the collection depth of frazil ice at the polynya edge is calculated using a parameterization in terms of the depth of frazil ice arriving at the edge and the normal component to the edge of the frazil ice velocity relative to the consolidated new ice velocity. Solutions are calculated for a range of values of the meridional mass flux across Nares Strait (the northern boundary of the domain), wind stress orientation, entrainment rates and the ice-water drag coefficient. The polynya edge is found to be insensitive to entrainment, in agreement with previous modeling studies. On the other hand, the polynya is extremely sensitive to the magnitude of the ice-water drag coefficient and the meridional mass transport across the northern boundary. Also included are two simulations of the North Water Polynya which incorporate forecasted wind stress data.

One-Dimensional Models for the Closure of a Coastal Latent Heat Polynya

Abstract One-dimensional models for the closure of a coastal latent heat polynya due to the onshore drift of frazil ice and consolidated ice are presented in this paper. The models predict the width of a polynya during an opening and closing cycle and include the cases in which the collection thickness of consolidated new ice at the polynya edge (during opening) or at the coast (during closing) is (a) a prescribed constant or (b) parameterized in terms of the depth of frazil ice arriving at the polynya edge (during opening) or the coast (during closing) and the relative velocity of the frazil ice with respect to the “convergence boundary” (i.e., the consolidated ice during opening or the stationary coastal wall during closing). Four dimensionless parameters specify the opening and closing cycle of a polynya, and throughout most of the four-dimensional parameter space the closing timescale is shorter than the opening timescale. In this case, it is found that the heat released from the polynya to the atmosp...

Acoustic trapped modes in a three-dimensional waveguide of slowly varying cross section

In this paper, we develop an asymptotic scheme to approximate the trapped mode solutions to the time harmonic wave equation in a three-dimensional waveguide with a smooth but otherwise arbitrarily shaped cross section and a single, slowly varying ‘bulge’, symmetric in the longitudinal direction. Extending previous research carried out in the two-dimensional case, we first use a WKBJ-type ansatz to identify the possible quasi-mode solutions that propagate only in the thicker region, and hence find a finite cut-on region of oscillatory behaviour and asymptotic decay elsewhere. The WKBJ expansions are used to identify a turning point between the cut-on and cut-off regions. We note that the expansions are non-uniform in an interior layer centred on this point, and we use the method of matched asymptotic expansions to connect the cut-on and cut-off regions within this layer. The behaviour of the expansions within the interior layer then motivates the construction of a uniformly valid asymptotic expansion. Finally, we use this expansion and the symmetry of the waveguide around the longitudinal centre, x=0, to extract trapped mode wavenumbers, which are compared with those found using a numerical scheme and seen to be extremely accurate, even to relatively large values of the small parameter.

EditorialWaves 2007 conference

This special volume contains a selection of papers, based on talks presented at the Eighth International Conference on Mathematical and Numerical Aspects of Waves (Waves 2007), which took place at the University of Reading, UK, organised jointly with INRIA, from 23rd to 27th July 2007. Wave phenomena are hugely important in many areas of science and engineering and at a whole variety of length scales, from the atomic to terrestrial and beyond. Applications include acoustic and electromagnetic scattering, seismicwave motions, the coupling of wind and ocean waves, refraction and imaging with photonic crystals, probing atomic structure with attosecond pulses, the design of low-loss optical wave guides, and travelling waves in the occurrence of epidemics. This conference is one of the main venues where significant advances in the analysis and computational modeling of wave phenomena and exciting new applications are presented, and this meeting was the eighth in a sequence which started in Strasbourg in 1991. Conference themes included forward and inverse scattering, nonlinearwave phenomena, fast computational techniques, numerical analysis, absorbing layers and approximate boundary conditions, analytic and semi-analytic techniques for wave problems, domain decomposition, guided waves and random media. The invited speakers were Mark Ablowitz (Colorado, USA), Annalisa Buffa (Pavia, Italy), Weng Cho Chew (Illinois, USA), Tom Hagstrom (NewMexico, USA), Andreas Kirsch (Karlsruhe, Germany), Ross McPhedran (Sydney, Australia) and John Toland (Bath, UK). In addition, there were approximately 180 varied and interesting contributed talks, with speakers travelling from all over the world to attend the meeting. The conference featured an embeddedWorkshop on High Frequency Propagation and Scattering, supported by the Isaac Newton Institute, Cambridge, UK, as a satellite meeting of the INI Programme on Highly Oscillatory Problems: Computation, Theory and Application. Waves 2007 was also a satellite conference of ICIAM 07, the 6th International Congress on Industrial andAppliedMathematics. In addition to theworkshop, therewere sevenminisymposia: brainwaves and cognitive neurodynamics; inverse problems; nonlinear waves; periodic and random media; photonic crystals and metamaterials; resonances and trappedmodes; time domainmethods. These areas, and others, are represented amongst the 51 papers that appear in this volume (from62 originally submitted), each ofwhichwas subjected to an extensive reviewprocess.Wewould like to thank the many anonymous referees who gave up their time to contribute to this effort. It is a pleasure also to thank the following for their direct or indirect financial support for theWaves 2007meeting and its embedded workshop: the Isaac Newton Institute, Cambridge; the Institute of Mathematics and its Applications; the Society for Industrial and Applied Mathematics (SIAM), UK section; INRIA, and the University of Reading Computational Sciences Theme. We sincerely hope you find this an interesting and illuminating volume of papers related to wave phenomena, and we look forward to seeing you at the Ninth International Conference on Mathematical and Numerical Aspects of Waves, which will take place in Pau, France, in June 2009.

Editorial: Waves 2007 conference

This special volume contains a selection of papers, based on talks presented at the Eighth International Conference on Mathematical and Numerical Aspects of Waves (Waves 2007), which took place at the University of Reading, UK, organised jointly with INRIA, from 23rd to 27th July 2007. Wave phenomena are hugely important in many areas of science and engineering and at a whole variety of length scales, from the atomic to terrestrial and beyond. Applications include acoustic and electromagnetic scattering, seismicwave motions, the coupling of wind and ocean waves, refraction and imaging with photonic crystals, probing atomic structure with attosecond pulses, the design of low-loss optical wave guides, and travelling waves in the occurrence of epidemics. This conference is one of the main venues where significant advances in the analysis and computational modeling of wave phenomena and exciting new applications are presented, and this meeting was the eighth in a sequence which started in Strasbourg in 1991. Conference themes included forward and inverse scattering, nonlinearwave phenomena, fast computational techniques, numerical analysis, absorbing layers and approximate boundary conditions, analytic and semi-analytic techniques for wave problems, domain decomposition, guided waves and random media. The invited speakers were Mark Ablowitz (Colorado, USA), Annalisa Buffa (Pavia, Italy), Weng Cho Chew (Illinois, USA), Tom Hagstrom (NewMexico, USA), Andreas Kirsch (Karlsruhe, Germany), Ross McPhedran (Sydney, Australia) and John Toland (Bath, UK). In addition, there were approximately 180 varied and interesting contributed talks, with speakers travelling from all over the world to attend the meeting. The conference featured an embeddedWorkshop on High Frequency Propagation and Scattering, supported by the Isaac Newton Institute, Cambridge, UK, as a satellite meeting of the INI Programme on Highly Oscillatory Problems: Computation, Theory and Application. Waves 2007 was also a satellite conference of ICIAM 07, the 6th International Congress on Industrial andAppliedMathematics. In addition to theworkshop, therewere sevenminisymposia: brainwaves and cognitive neurodynamics; inverse problems; nonlinear waves; periodic and random media; photonic crystals and metamaterials; resonances and trappedmodes; time domainmethods. These areas, and others, are represented amongst the 51 papers that appear in this volume (from62 originally submitted), each ofwhichwas subjected to an extensive reviewprocess.Wewould like to thank the many anonymous referees who gave up their time to contribute to this effort. It is a pleasure also to thank the following for their direct or indirect financial support for theWaves 2007meeting and its embedded workshop: the Isaac Newton Institute, Cambridge; the Institute of Mathematics and its Applications; the Society for Industrial and Applied Mathematics (SIAM), UK section; INRIA, and the University of Reading Computational Sciences Theme. We sincerely hope you find this an interesting and illuminating volume of papers related to wave phenomena, and we look forward to seeing you at the Ninth International Conference on Mathematical and Numerical Aspects of Waves, which will take place in Pau, France, in June 2009.

Systems of integral equations with weighted difference kernels

Explicit expressions are derived for the inverses of operators of a particular class that includes the operator corresponding to a system of coupled integral equations having weighted difference kernels. The inverses are expressed in terms of a finite number of functions and a systematic way of generating different sets of these functions is devised. The theory generalizes those previously derived for a single integral equation and an integral-equation system with pure difference kernels. The connection is made between the finite generation of inverses and embedding. AMS 2000 Mathematics subject classification: Primary 45A05