Background

In this chapter, we give a background on the topics covered in the thesis.

Sea-Ice and the Marginal Ice Zone (MIZ)

Sea-ice surrounds the poles of our planet and occupies about 7% of the total area of the world's oceans (Wadhams 2000). These ice-covered seas represent the cold end of the enormous heat engine which enables our planet to sustain temperatures suitable for human life. Sea-ice creates a barrier between the ocean and the atmosphere and prevents the transfer of heat and gases across the interface. Water has a high heat capacity, and without a barrier can transfer a large amount of heat to the polar atmosphere. When sea-ice is present, the atmosphere can cool by up to [math]\displaystyle{ 30^\circ }[/math]C (Ruddiman 2001). Sea-ice also has a strong impact on the albedo and reflects up to 80% solar radiation while water only reflects 10% (Wadhams 2000). Increasing the sea-ice extent, increases the albedo which decreases the absorbed heat and increases cooling. Satellite data from 1978 to 2000, has shown that the sea-ice extent in the Arctic is retreating by 10% - 11% per decade (Cosimo et. al. 2008). It is unknown whether Arctic summer sea-ice will even exist in years to come. Sea-ice also has an influence on the ocean circulation system. The formation of sea-ice creates salty dense water which sinks and carries oxygen and dissolved carbon dioxide to the ocean depths. These convection regions drive a three-dimensional global pattern of deep and shallow currents which spread nutrient and oxygen rich waters throughout the globe (Wadhams 2000). Sea-ice also plays a major role in supporting polar marine ecosystems. Several species of copepods, amphipods and two species of krill have adapted to life in close association with sea-ice, where they feed on ice algae and seek shelter from predators. It is these organisms and their predators that are being affected most by warming (Smetacek and Nicol 2005).

The interfacial region between the open ocean and the ice-covered seas is known as the Marginal Ice Zone (MIZ). This region consists of neither open ocean, nor frozen ocean, but consists of a patchwork of ice floes and open water. The MIZ is a dynamic and at times turbulent region, which is formed by wave-induced breaking of continuous ice. This process is balanced by wave scatter within the MIZ which partially shields the continuous ice from the destructive wave energy. There are two aspects which need to be understood to model this process: the first is the wave-induced breaking of the continuous ice, and the second is the wave scattering in the MIZ (squire_etal95). The greatest amount of breakup, and thus the smallest floes, are found closest to the ice edge. If the wind is blowing towards the ice edge from the open sea, it compresses the MIZ and produces a compact ice field. If the wind is blowing away from the ice edge, most of the MIZ becomes diffuse. The outermost edge can sometimes organise itself into a series of compact ice edge bands which are separated by completely open water and lie with their long axes roughly perpendicular to the wind. In winter, new ice can form in these open water bands (wadhams00). The four major MIZ regions of the world are located in the Greenland Sea, Labrador Sea, Bering Sea and circumpolar Antarctic ice edge.

Wave--Ice Modelling

It is a well known fact that sea-ice has the effect of reducing incoming swell. The distance at which waves are found to penetrate into ice varies considerably. Modelling this process requires understanding of the physical processes involved in wave-attenuation. In this section, we summarise some of the wave--ice modelling techniques to date. We begin by briefly introducing some of the early concepts and research techniques. Following this we discuss in some detail wave scatter modelling, which is applicable to diffuse ice fields and is the focus of this thesis. Various solution methods and problems are discussed. Finally, we briefly discuss viscous models which are applicable to compact ice fields. A review of the interaction between waves and ice is summarised in squire_etal95 and squire07.

Early Findings

As early as 1887, scientists developed an interest in understanding waves and how they interact with floating ice; greenhill1887 found an expression for the velocity of waves through ice where the ice sheet is modelled as a thin elastic beam. During the 1930's and 50's, Greenhill's ideas were developed through a series of field experiments and analyses Roethlisberger72. Waves were artificially generated by detonation blasts, and the elastic and flexural ice-coupled wave propagation in ice of various types was studied, where the ice was treated as a plate rather than a beam. The artificial waves were also used to measure the Young's modulus of ice. During the 50's, the theoretical aspect of wave propagation through ice was also studied. keller_weitz53 focused on the boundary value problem at the ice edge and the calculation of the transmission and reflection coefficients. Here the ice is treated as a floating material of uniform surface density with no elastic properties nor viscosity. This problem is solved completely and the solution method may be particularly useful in modelling wave propagation through frazil or pancake ice. During a voyage, robin63 confirmed visually that ice floes bend. It has consequently become standard to model ice floes as thin elastic plates.

==Scattering Models==(5) For ice conditions consisting of discrete solitary floes, it has been assumed that the major physical factor influencing wave-attenuation is the redistribution of wave energy due to scattering by the floes squire_etal95. Within each floe, energy propagates with an altered dispersion relation. The ice floe therefore scatters incoming ocean wave energy due to a mismatch between the mode of propagation beneath the raft as compared to that under an open water surface squire_etal95. This scattering energy generates an energy reflection and transmission wave.

Solitary plates and two semi-infinite plates

Initially the problem of wave scatter through floating elastic plates, was solved for two semi-infinite plates with identical properties. kouzov63 solved this problem explicitly using the Riemann--Hilbert technique. More recently this problem has been reconsidered by squire_dixon00 and williams_squire02, who solve it using a Green Function method. Their method is applicable to infinitely deep water and they extend the problem to oblique wave incidence. \\\\ A more challenging problem involves solving for two semi-infinite plates with arbitrary properties. The first significant work on this problem was done by evans_davies68, who presented a solution method for evaluating the transmission and reflection of waves, propagating from a semi-infinite region of open water into a semi-infinite region of a floating elastic plate. The method of solution was based on the Wiener--Hopf technique. evans_davies68, however, only solved explicitly for the case of shallow water. They presented only the formulation for the finite-depth case, as they were unable to compute the transmission and reflection coefficients. wadhams86 presents an alternative method (the eigenfunction expansion method), which he developed in the early 1970's. It was the first based on a solution for a single finite-floe surrounded by water. Unfortunately, no numerical method to solve the problem for a single floe had been developed at the time, and the reflection and transmission coefficients were only derived approximately, as only a subset of the evanescent waves were included. fox_squire94 returned to the problem of a wave propagating from a semi-infinite region of open water into a semi-infinite region covered by a floating elastic plate. Their method is derived from the work of wadhams86 and they extend by matching along the entire water column at the plate edge and solve for oblique incidence. Soon after, barrett_squire96 extended the solution of fox_squire94 to two plates of arbitrary properties. jgrfloe1d solve for a single ice floe using dry mode eigenfunctions to construct a Green function for the plate and used a second Green function for the water. A Fredholm integral equation is assembled for the velocity potential at the surface which is solved using the Nystrom method. Recently, the computational difficulties associated with the Wiener--Hopf solution of evans_davies68 have been overcome and solved explicitly by several authors, including balmforth_craster99,chakrabarti00,tkacheva01,chung_fox02. chung_linton05 have also solved the problem of open water and a semi-infinite plate using the residue calculus technique, a method which is closely related to the Wiener--Hopf method. \\\\ Attention has also been focused towards constructing physically more realistic ice sheets such as solving for pressure ridges, open and refrozen leads and sudden or gradual changes in thickness porter_porter04,williams_squire04,chung_linton05,gayen_etal07. Authors, including bennetts_etal07,porter_porter04 have incorporated sea-ice draft into their models. marchenko96,marchenko_voliak97, vaughan_etal07 attempt to estimate floe thickness and other parameters from the wave propagation through real sea-ice terrain using in situ wave data. The three-dimensional floating circular ice floe is solved in jgrfloecirc via an extension of jgrfloe1d and IJOPE04 solve for the finite depth case using matched eigenfunctions. jgr02 solves for arbitrary floe shapes where the dry modes are substituted into the integral equation for water to give a linear system of equations for the coefficients used to expand the ice floe motion. The modes are determined generally using a finite element approach.

Multiple plates

The next major development in wave scattering theory was in the consideration of wave propagation through multiple plates. squire_dixon01 extend the single crack problem to a multiple crack problem, in which the semi-infinite regions are separated by a region consisting of a finite number of plates of finite size with all plates having identical properties. evans_porter03,porter_evans05 consider the multiple crack problem for finitely-deep water and derive a simple solution. hermans04 also present a solution for multiple plates, based on an earlier solution for a single plate hermans03, and is for a set of finite elastic plates of arbitrary properties.

hermans04 solves the problem using Green's theorem to obtain an integral equation for the deflection.

ogasawara_sakai06 numerically solve for a set of arbitrary plates using a time-domain solution incorporating the boundary element method and the finite-element method proposed by liu_sakai02. bukatov_bukatov99 consider the influence of floating broken ice on the displacement of non-linear surface waves. Sophisticated three-dimensional models have also been developed. masson_leblond89,jgrrealism,meylan_mason06 couple the solutions for individual ice floes with a transport equation. Also, for a finite number of arbitrary plates, the three-dimensional problem is solved by greens_malte04. A number of works consider periodic infinite or semi-infinite arrays. chou98 solves for wave propagation through an infinite array of periodically arranged surface scatterers or plates by an extension of Floquet's Theorem. array07 solve for an infinite array using a periodic Green function, while JFM_malte06 and peter_meylan07 solve the infinite and semi-infinite array problem respectively using an interaction theory. \\\\ In this study, we consider a two-dimensional multiple floating elastic plate solution, which is solved exactly via an extension of fox_squire94's matched eigenfunction expansion method elastic_plates07. The only physical parameters which are considered are length, mass and elastic stiffness. All non-linear effects, floe collisions and ice-creep are neglected so that the problem is only applicable to discrete floes which are large relative to thickness and non-extreme wave conditions. We consider this simplified model for its flexibility, computational efficiency and as a practical tool to help understand the key physical processes in wave scattering.

Viscous Models(6)

In more compact ice fields, the interactions between floes increase and it may no longer be realistic to consider the ice field as being composed of individual floes wadhams00. Instead, floes collide or are held together by the stress of an on-ice wind or by freezing of brash, pancake or frazil ice. Here the ice field is approaching the condition of being a single entity, yet does not possess the bulk property of being a uniform elastic sheet wadhams00. It is tempting to ignore the detailed physics of the various energy consuming, ice-water and ice-ice interaction processes and model a material with its sea surface and wave-attenuation properties determined empirically. weber87 was the first to introduce the idea of considering such an ice-cover as a thin, highly viscous fluid. The viscosity is included in the free surface condition as a dampening term. liu_mollochristensen88 describe a physically more realistic model for wave decay which assumes that attenuation is due to the viscous boundary layer under ice. It is assumed that in a highly compact ice-cover, waves disperse as though propagating beneath a thin elastic sheet. An oscillating boundary layer develops under the ice, causing energy loss.

Wave Attenuation

Experiments have shown that waves attenuate exponentially with distance of propagation through ice, and the attenuation coefficient decreases with increasing wave period. There is evidence of a ``rollover", where a trend of decreasing attenuation occurs at period less than 6-8 s. This rollover may be explained by an increase in energy at short periods due to local generation of waves by wind squire_wadhams85.

The point of rollover depends on ice conditions, especially ice thickness liu_etal91.

\\\\ There are several factors which influence the attenuation of waves through ice. It is, however, not fully understood what these factors are and how much influence they have. The most influential factor in diffuse ice is thought to be due to wave scatter (Section 5). In compact ice, viscous losses are thought to have the most influence (Section 6). According to shen_squire98, other factors which may be relevant include: the absorption due to hysteresis as floes deform on the passing wave field, the absorption in the water column from processes such as wave breaking and the absorption due to collisions and other interactions between floes. shen_etal87 however find that collisional stresses are only small and mckenna_crocker90 conclude that floe collisions cannot account for the observed decrease in wave energy.

==Wave Attenuation Models==(7) wadhams86 was a significant paper, as it showed how to estimate the transmitted energy through a set of floes in terms of the transmitted energy through a single floe, so that an attenuation coefficient could be estimated. The possible paths of each wave vector through the ice field are considered, and the final forward vector summed through all possible multiple reflections. dixon_squire01 model wave scatter in the Marginal Ice Zone, using the coherent potential approximation to compute the energy transport velocity and derive an attenuation coefficient, which unfortunately does not compare well to experimental results. liu_mollochristensen88 describe a model for wave decay, which assumes attenuation is due to the viscous boundary layer under ice (Section 6). They parametrise the energy loss by a tuning parameter, the eddy viscosity, which is related to actual flow conditions. Their model agrees well with experimental data. perrie_hu96 develop a model which is based on masson_leblond89's model and use a rigid cylindrical floe model isaacson82. The scattering model is incorporated into an operational wave model. The model suffers from the requirement that the ice floes be small enough to be modelled as rigid cylinders so that it is only applicable to ice fields with small floes. For the most sophisticated wave--ice models, significant work is required to reach the point where predictions of the attenuation coefficients are possible and such predictions would require large computational resources so that no summary of the attenuation coefficient as a function of various parameters has been possible. It would be interesting to see how perrie_hu96's model would perform using a more sophisticated three-dimensional ice floe model, such as the model of jgr02. \\\\ The attenuation model we present in this thesis is also limited, as the only physics considered is the elastic bending of the floes. The numerical values for the wave-attenuation, however, can be determined relatively straightforwardly and without using a tuning parameter (Section 171) kohout_meylanjgr08.

==Wave Attenuation Field Experiments==(8) Theory and experiment need to work together to understand complicated geophysical phenomena such as wave--ice interaction. It is important to realise that recently, for wave--ice interaction, there has been much more progress with modelling than with experiments. This is highly unsatisfactory, and from a modeller's perspective we have great need for more experimental results.\\

===Methods===(9) It is not at all surprising that models have surpassed experiments in this field. Scientist have to work in remote locations and contend with some of the world's toughest environmental conditions including freezing air temperatures, icy seas and at times wild winds and turbulent waves. In such conditions, scientists must calculate the wave spectrum simultaneously (or near simultaneously) at constant intervals from the ice edge to deep into the ice zone. The measurements need to be along the direction of the swell and the scientist need to record a thorough analysis of the ice conditions at the time of the experiment. A number of different methods are discussed below. \\\\ {\it Early measurements}\\

The first measurements of wave decay in the zone of discrete ice floes near an unconfined ice margin were made by shipborne wave recorder robin63 and reported in wadhams79,wadhams86. Later measurements were made by upward-looking echo sounder from a submerged hovering submarine wadhams72,wadhams78 and by airborne laser profilometer wadhams75. \\ \\ {\it Wave buoys}\\ Wave buoys have been the standard instrument for measuring waves for many years. The most recent buoys use ultra-sensitive tiltmeters and novel re-zeroing techniques to autonomously gather wave data. These modern buoys use Iridium satellite communication systems to recover data continuously and to remotely control the instrument. For over two years now, there have been several of these buoys drifting independently and successfully recording wave data. Unfortunately, while these buoys can successfully record many useful properties, they can not record wave-attenuation data. Attaining wave-attenuation data requires deployment of a series of wave buoys along the direction of swell to measure the local wave spectrum. The ice conditions at the time of the experiment, such as floe thickness and size, must also be measured. It is difficult to obtain accurate wave-attenuation data using wave buoys, as it requires extensive logistical support to measure wave properties simultaneously (or near simultaneously), and results can be skewed by changes in the swell conditions during the experiment and difference in the swell direction for different periods or multi-directional swell. \\\\ {\it Remote Sensing}\\ Remote sensing can be used to obtain sea-ice information on a continuous basis. Due to their all weather capability, microwave sensors like synthetic aperture radar (SAR) or the radiometer play an important role in this context. Radiometric systems like the Special Sensor Microwave Imager (SSM/I) Bjorgo_etal97 with a resolution between 10 and 50 km, are mainly used to measure sea-ice coverage and sea-ice type. SAR imagery as acquired by the European Remote Sensing Satellite (ERS), has a resolution of about 20 m and thus allows the study of processes in the MIZ on a smaller scale schulz_lehner02. SAR can be used to observe the spatial properties of a wave field in sea-ice larouche_Cariou92 and is useful for collecting wave direction information. SAR has been used to examine waves propagating through pancake ice fields wadhams_etal02 and to estimate pancake ice thickness wadhams_etal04. Unfortunately, however, the use of SAR has not been found to be practical in measuring wave-attenuation through discrete floes.

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{\it Autonomous Underwater Vehicle (AUV)}\\ In these experiments, an upward looking Acoustic Doppler Current Profiler (ADCP) is mounted on an autonomous underwater vehicle (AUV). The AUV is a promising device which can sample at high resolution and can sample a large portion of the MIZ over short time scales hayes07.

Available Data Sets

{\it Early Data Sets}\\ The wave decay measurements from the 60's and early 70's are available and have been summarised and compared against wave-attenuation theory in wadhams86. \\\\ {\it The Scott Polar Research Institution (SPRI) Experiments}\\ The most substantial set of experiments to measure wave-attenuation in the MIZ were carried out by the SPRI in the late 1970's and early 1980's squire_moore80,wadhams_etal86,wadhams_etal88. During these experiments, a helicopter was used to visit floes at intervals within the ice fields along the major axis of the incoming wave spectrum. At each site a wave buoy was inserted between floes to measure the local wave spectrum. The flexural, heave and surge responses of the experimental floe were measured with accelerometers and strain-meters. A mean thickness of the floes was determined by coring at each of the experimental floes. Floe size distributions along the flight path into the ice were derived from overlapping vertical photographs taken from helicopter. The main sources of experimental error were changes in the swell conditions during the experiment, difficulties in determining the ice floe size distribution and thickness, and difference in the swell direction for different periods or multi-directional swell. \\\\ The attenuation coefficients from the experiments in the Greenland Sea in September 1978, September 1979 and July 1983 are given in wadhams_etal88 and are used in this study. We also use the attenuation coefficients from the Bering Sea in March 1979 and February 1983. Unfortunately, due to possible reflection or absorption of waves from the fjords, accurate attenuation coefficients could not be calculated from the 1978 Greenland Sea experiments. \\ \\ {\it The Labrador Ice Margin Experiment (LIMEX)}\\ Another set of experiments took place off the east coast of Newfoundland, Canada, in 1987 and 1989. These experiments provided synthetic aperture radar (SAR) imagery, wave buoy and ice property data. We attempted to obtain the results of this experiment, but unfortunately were not successful. \\ \\ {\it The Antarctic Peninsula Experiments}\\ Most recently an experiment took place in the MIZ of Antarctica. Autosub, an Autonomous Underwater Vehicle (AUV), was used to complete four missions west of the Antarctic Peninsula in the Bellinghausen Sea during 22 -- 25 March 2003. The Autosub is a promising new device. It is a battery-powered vehicle that follows a pre-programmed course and can travel up to a maximum range of 400 km. In water track mode, the Autosub's navigation frame of reference is the water rather than the seabed, hence for some missions, a mean current caused the sub to drift from the mission plan. Consequently, the line of travel of the sub was not necessarily along the direction of the swell nor at right angles to the ice edge. A thorough study of the ice conditions at the time of the experiments was not conducted; estimations from the ship were made for each mission. hayes07 notes that the attenuation coefficients for waves of period longer than 16 s may be compromised by possible surge response of the vehicle. A detailed description of these experiments can be found in hayes07. The results of these experiments were attained from D. Hayes and are used in this study.

Strain and Floe Break-up

Incoming waves cause ice floes to bend. If the bending induces sufficient strain, fracture will occur squire93. The breaking of large continuous floes and land-fast ice, supplies the MIZ with ice floes and determines the floe size distribution of the MIZ langhorne_etal98. Observations suggest that cracks initially form within a few tens of metres on the ice edge and that sea-ice thickness is the principal determinant of crack position. Wavelength appears to play only a secondary role squire93. During ocean wave experiments, strain gauges were fixed to the upper surface of sea-ice floes in the Arctic which provided some direct measurements of sea-ice fracturing. Results from these experiments have shown an ice island to fracture at [math]\displaystyle{ 3\mathsf{x}10^{-5} }[/math] goodman80 and sea-ice to fracture from [math]\displaystyle{ 4.4\mathsf{x}10^{-5} }[/math] to [math]\displaystyle{ 8.5\mathsf{x}10^{-5} }[/math] squire_martin80. Based on a series of experiments carried out in the McMurdo Sound, Antarctica, langhorne_etal98 deduce that sea-ice fatigues when it is cyclically stressed. This fatigue can cause the ice to fracture and breakup at stresses well below its flexural strength. Using these experiments, langhorne_etal01 predict the lifetime of the sea-ice as a function of significant wave height and sea-ice brine fraction. We assume that, in general, if the strain is less than [math]\displaystyle{ 3\mathsf{x}10^{-5} }[/math], the ice will have an infinite resistance to failure (Squire, 1993; Personal correspondence with T. Haskell). \\\\ fox_squire91 were the first to completely and precisely model the strain in ice due to incoming ocean waves. The model is an extension of their fox_squire90 matched eigenfunction model, which is also summarised in fox_squire94, for the solution of the velocity potential. squire93 uses a variation on the fox_squire91 method and considers wave propagation into a viscoelastic ice sheet. This alters the travelling mode so that it attenuates exponentially by an amount which depends on the magnitude of the viscous damping term, which unfortunately is currently unknown. \\\\ In Section 175, we define the strain as defined in fox_squire91 and consider the effects it has on floe breakup for a given wave spectrum.

Very Large Floating Structures (VLFS)

The study of wave propagation through floating elastic plates can also be applied to the construction of Very Large Floating Structures (VLFS). (10) This has motivated much of the recent research in this field. Our world has a growing population and a corresponding expansion of urban development in land-scarce countries. Engineers have proposed the construction of VLFS for industrial space, airports, storage facilities and habitation watanabe_etal04. There are many examples of such structures already in place. Japan have constructed a Mega-Float (a VLFS test model for floating airport terminals and airstrips), a floating amusement facility, floating emergency rescue bases and floating oil storage systems. There have been floating bridges built in Japan, Canada, Norway and the United States. Canada also has a floating heliport and Vietnam a floating Hotel watanabe_etal04. \\\\ VLFS may be classified under two broad categories; the pontoon type and the semi-submersible type. The pontoon type is suited to calm sea conditions and is usually associated with naturally sheltered coastal formations. In open seas where wave heights are relatively large, the semi-submersible type of structure is required, to minimise the effects of waves watanabe_etal04. It is common for the semi-submersible type to be modelled as a column supported structure consisting of a thin upper deck and a great number of buoyancy elements. In Japan, most VLFS research has focused on the pontoon type structure kashiwagi00. Formulations for the pontoon type are closely related to ice plate formulations. For the pontoon type, the wavelengths are very small compared to the horizontal dimensions of the structure, hydroelastic responses are more important than the rigid-body motions due to the relatively small flexural rigidity of the structure, and they have a small draft kashiwagi00. Consequently, much of the floating ice and VLFS modelling methods are similar e.g. Matched Eigenfunction Expansion Methods, Mesh Methods, and Green function Methods. Recently, VLFS research has focused topics such as mooring systems, breakwaters, profiles of seabed and anti-motion devices. \\\\ In this thesis, we consider a VLFS which is constructed via connecting a series of plates, and solve using both the MEEM and GFM for various articulated edge conditions, including springed and hinged plates (Chapter 124).