Introduction
This thesis is concerned primarily with the determination of the scattering of ice-coupled waves by imperfections in large expanses of sea ice, with a possible future application being the development of a technique that is able to sense ice thickness remotely. Such ice sheets are found mainly in the Arctic Ocean but they can also be found close to the Antarctic continent, although the sheets there are of a more transient nature as they are regularly broken up by incoming waves from the surrounding Southern Ocean. They may also be dispersed by strong katabatic winds from the continent.
The sea ice of both the Arctic and the Antarctic plays a very important role in the earth's climate. Section 1 below presents a general overview of this. As well as describing the specific regions we intend to model, that section also describes how work involving wave interactions with sea ice can be generalized to provide results for certain situations that occur in marine engineering.
The following sections describe and justify our choice of model for polar sea ice (Section 2.1), review and discuss mathematical techniques used previously and currently in use in the wave scattering literature (Section 2.2), and provide the general outline that this thesis will follow (Section 3).
Arctic and Antarctic Sea Ice
This section begins by giving an overview of the important role that polar sea ice plays in the earth's environment (Section 1.1). This includes the reflection of sunlight and the production of bottom water which helps to drive global currents. Section 1.2 then proceeds to describe the specific regions we intend to model, while Section 1.3 discusses how work involving wave interactions with sea ice can be generalized to provide results for certain situations that occur in marine engineering.
Given the importance of sea ice in the environment, a convenient, remote ice-thickness-sensing technique for monitoring the condition of the polar ice sheets would be a valuable asset in monitoring the overall health of the planet. Techniques currently in use are submarine sonar, most recently coupled with LIDAR---the former is able to detect the under-sea profile of the ice, while the latter uses a laser operated from an aeroplane to detect the ice's upper profile---and EMS, electromagnetic sensing from a specially-fitted helicopter. If ocean wave scattering could be used to deduce average ice thickness, it might be possible to measure it using suitably placed strain gauges or tilt meters, which would eliminate the need for aeroplanes or submarines to be in operation for extended periods of time.
Recall also that by breaking up sea ice, waves affect the way in which the ice interacts with the rest of the climate system; this process should be included in climate models. Consequently, any understanding that wave-ice research can contribute to such models is extremely valuable.
Environmental Role of Sea Ice
The ice of both the Arctic and Antarctic regions is crucial in the maintenance of the earth's climate. Arctic ice particularly also plays a role in the lives of many human and animal populations---for example, the Sami, the indigenous people of Lapland, use the sea ice for reindeer herding, while polar bears use it as a hunting ground. Animals such as seals and penguins also use Antarctic sea ice as a base from which to search for fish.
Sea ice, the type of ice that this thesis is most concerned with, covers about 7% of the area of the world's ocean, providing a solid barrier between the ocean and atmosphere that hinders the free exchange of heat and moisture between the two. It is also involved in helping to drive currents, most notably the massive thermohaline conveyor which links the Atlantic and Indian with the Pacific and Southern Oceans, and, aided also by the freshwater terrestrial ice shelves of Antarctica and the Greenland ice cap, reflects a large amount of incoming solar radiation.
By reflecting so much solar energy, the ice provides some resistance to increasing global temperatures, and by melting it can absorb still more. However, the melting of the ice reduces the amount of energy that can be reflected, and so the rate of warming increases. In particular, this will allow the amount of heat entering the ocean to increase, contributing to sea level rise due to thermal expansion, a major source of sea level rise. The melting of terrestrial ice also increases sea levels, and decreases the salinity of polar waters.
Sea ice also has a role in driving the world's currents. Its ice formation in the Antarctic causes the production of dense salty water, which then sinks to become Antarctic bottom water (AABW). Similarly North Atlantic deep water (NADW) is partly produced by sea ice formation in the Greenland Sea. Deep water produced then spreads out across the ocean floor (i.e. away from the poles), rising to the surface because of mixing due to roughness in the sea bed and wind, and returning to the poles in wind-driven currents such as the Gulf Stream. Along with the thermal forcing of warm water from the tropics travelling to cooler waters, cooling and sinking, this is the origin of the massive thermohaline conveyor current, which travels between both poles.
Before the existence of plumes was discovered, it was thought that the above process was due entirely to convection, which is impossible in fresh water. (For convection to occur, one needs density to increase as temperature decreases---this is not the case with fresh water, which has a density maximum at [math]\displaystyle{ 4^\circ }[/math]C.) Therefore it was also thought that if the salinity of the polar oceans decreased enough, convection would shut off, with a disastrous effect on currents like the Gulf Stream which bring warm water to North Atlantic countries such as the United Kingdom and the United States of America.
However, deep water can also be formed by smaller scale events such as the refreezing of a newly formed lead. This process produces a plume---a "packet" of dense salty water that sinks to the sea floor. With the extra bottom water produced by these entities, global currents would not be as affected as was first thought by a decrease in polar salinity. Nonetheless, a significant decrease in the formation of sea ice, and thus bottom water, would inevitably affect global currents to some extent.
Regions to be Modelled
We are interested in modelling the scattering of waves by imperfections in large sea ice sheets, particularly keeping in mind the possibility of remotely estimating ice thickness. In the northern hemisphere, which we are most interested in, such sheets exist mainly in the central Arctic Ocean, due to its sheltered location, although there are also some shore fast ice sheets skirting the Arctic coastline.
The sea ice which forms in places like the Bering Sea or the Greenland Sea does not normally grow into very large sheets due to the presence of high amplitude waves originating in the Pacific and Atlantic Oceans respectively. The ice in both seas is broken up into pack ice, although much of the ice in the Greenland sea is older, thicker ice that has come from the Arctic Ocean. This ice serves to attenuate waves from the Atlantic, and filter out those with shorter periods. As a result, only waves with periods greater than about 6s penetrate into the Arctic Ocean from the Atlantic Hunkins 1962. The size of the Atlantic Ocean also limits the possible lengths of waves originating inside it, with a result that only periods less than about 20s are possible.
The ice in the Bering Sea also plays a similar role. However, the main barriers to waves from the Pacific Ocean reaching the Arctic Ocean are the Aleutian Islands and the Bering Strait. These only allow very long waves (those with periods greater than about 22s) to reach the Arctic from the Pacific (the greater fetches of the Pacific Ocean permits much longer waves to develop within it).
As well as this relative lack of short wave energy, another factor which aids in the growth of large ice sheets in the Arctic is the Beaufort Gyre current. This current circulates around the North Pole, trapping the ice in the centre of the ocean. The result is that the same ice is sometimes able to survive for several years, enabling it to grow to thicknesses of up to 10m, although most of the undeformed sheets are around 1m-3m Wadhams 1995.
The Arctic ice sheets are criss-crossed with sometimes quite large pressure ridges, cracks and open and refrozen leads. Cracks and leads form as adjacent ice sheets respond differently to forces such as those exerted by winds and currents, causing them to drift apart. As a lead opens up, the water exposed to the air freezes; this new, thinner ice then may contribute to the sail and keel of a pressure ridge as winds and currents change and the two ice sheets are pushed back together again. Pressure ridges may also be produced by rafting as ice sheets of similar thicknesses collide and buckle, or in shear as adjacent floes drift past one another.
Antarctic sea ice is not afforded as much protection from incoming waves and is thus generally younger and thinner than Arctic sea ice. (Except in very sheltered bays and inlets, and to the east of the Antarctic peninsula, Antarctic sea ice usually ranges from 0.5m to 1m in thickness.) In winter it is usually formed from roughly circular plates of pancake ice that freeze together to give larger and larger floes the further they are from the ice edge. Like the ice described above that occurs in the Bering Sea, the region near the margins of the ice cover (called the Marginal Ice Zone) is made up of pack ice which reflects and attenuates incoming waves, with the result that the sea ice closer to the continent usually only experiences low amplitude waves of periods in the range 10--20s. The floes are thus able to reach extremely large sizes, even to the point of being able to be described as a continuous ice sheet (although vestiges of their pancake ice origins are left in the form of abundant slightly raised circular edges, giving the ice a "stony field" appearance (Lange et al. 1989). Cracks and leads can be found between neighbouring ice sheets.
The Arctic process of cracks/leads forming then contributing to ridges also occur in the Antarctic. However, much fewer ridges are observed in the Antarctic because the ice is both thinner and less constrained. Substantial ridges are only observed in the ice of the Weddell Sea, where they are formed by ice buckling against the Antarctic Peninsula.
Engineering Applications of Ice Research
Although this thesis is primarily concerned with providing solutions for situations involving sea ice, some of its results, as well as similar results obtained by other researchers, can also be applied in marine engineering.
A simple application could be to scattering due to a series of floating breakwaters---this could be solved by the method given by Meylan 1993, which describes the effect of a series of ice floes modelled by floating elastic plates.
Another more topical example is provided by the 1000m[math]\displaystyle{ \times }[/math]121m[math]\displaystyle{ \times }[/math]3m floating "Mega-Float" platform that was constructed in 2000 in Tokyo Bay, Japan, with the intention of undertaking experiments to determine the suitability of such a structure as an airport (Kashiwagi 2000; Watanabe et al. 2004]]). (Such a structure is also referred to as a Very Large Floating Structure, or VLFS.) With property in Tokyo at such a premium, such an airport would avoid using valuable space on land.
Verification experiments performed included actual take-offs and landings, with pilots reporting experiencing no mechanical differences from landing at land airports (Kanzawa et al. 2002). Landing distances and other factors in safe landing were also found to be satisfactory, as were other considerations such as under-sea aircraft noise, which was found to be at a level that would not be detrimental to the Mega-Float's marine environment.
Given the apparent success of this first trial, the design of a 5\,km long Mega-Float has been completed, including trial landings with pilots using flight simulators. In order to do this, the hydroelastic response of the airport to waves must first be calculated. By approximating the Mega-Float by a thin elastic plate, work by \can{MS94}, which models a single ice floe, or alternatively by \can{Tkacheva02} could be used to estimate this response. Such models would approximate the Mega-Float as having the side perpendicular to the incident wave of infinite length. Such a solution would be most accurate closer to its centre. Three-dimensional solutions also exist (\citename{NO99}, \citeyear*{NO99}, who give a shallow water solution; \citename{WM03}, \citeyear*{WM03}; \citename{PM04}, \citeyear*{PM04}).
Work in this thesis could be used as well---there could easily be the possibility of a hinge joint or a similar type of joint in such a large structure, and strains near such a joint could be estimated by varying the working done in Chapter~\ref{chap-crack} \caf{SD00,EP03crack}{also solved in different situations by}.
Of course, when dealing with specific structures, one would need to refine our approaches somewhat. Inoue et al. \citeyear{Inoue02}
use a thin plate approximation to first obtain the water pressure and instantaneous accelerations at the surface, which are then incorporated into a finite element structural analysis. Clearly techniques that provide the hydrodynamic results quickly would be of great use in such a situation.
Mathematical Model and Existing Solution Techniques
In \Sr{sec-intro-mathmodel}, this section begins by presenting and justifying the mathematical model that is generally used for ice sheets, namely the thin elastic plate model. There is now considerable evidence that this model is an accurate one.
In \Sr{sec-math}, we then describe the various methods that have been and are being used in the solution of scattering problems involving thin plates floating on fluids. The geometries of the plates have become successively harder, often with several different approaches being attempted for each geometry before the most efficient method is settled upon.
Before proceeding to do this, however, it is worth defining some terms. The first is dispersion. Its name follows from the fact that waves of different periods travel at different speeds. The dispersion relation is the relationship that quantifies this, and forces the wavenumber, [math]\displaystyle{ \a_0 }[/math], to satisfy an equation that depends on the wave's radial frequency, [math]\displaystyle{ \om }[/math], and the properties of the ice (such as its Young's modulus, thickness and density). In general, [math]\displaystyle{ \a_0 }[/math] must be found numerically---an explicit formula in terms of the other quantities rarely exists. As a consequence of the dispersion relation, the phase velocity of the wave ([math]\displaystyle{ \om/\a_0 }[/math]) and also the wavelength ([math]\displaystyle{ 2\pi/\a_0 }[/math]) change with period ([math]\displaystyle{ \tau=2\pi/\om }[/math]) and ice properties. In particular, for constant ice properties the wavelength increases with period, and for constant period it increases with ice thickness.
The last term for the moment, which appears in the title of this thesis, is flexural gravity wave. Gravity waves are waves on a fluid surface that are driven by gravity and a restoring buoyancy force. Flexural gravity waves are waves where this interaction is modified by the inertia and the rigidity of an elastic material floating on the surface.
An Ice Sheet as a Thin Elastic Plate
An extensive amount of literature exists to support the modelling of an ice-sheet as a linear thin elastic plate, for which the governing equations are presented and discussed in Appendix~\ref{app-eqns} \caf{Mindlin51,Fung65}{also cf.}. \can{Greenhill87} was the first to propose modelling a floating ice sheet by a thin elastic beam on a fluid foundation, suggesting a dispersion relation based on the Euler-Bernoulli beam theory. He also wrote a further paper, humourously entitled ``Skating on thin ice" Greenhill16, in which he elaborated further on the dispersion of waves in ice, as well as touching on other related problems. (Another noteworthy comment that was made in that paper is that ice was the first material for which an experimental value of Young's modulus was obtained.)
This elastic model for ice has since been corroborated experimentally by numerous researchers, with slight improvements to Greenhill's theory being made with the passage of time. For example, during a series of seismic experiments carried out by Ewing et al. \citeyear{Ewing34}, the generation of flexural waves in thin lake/canal ice was observed in addition to the longitudinal and transverse waves being investigated at the time. Extending the theoretical component of Greenhill's work to include the compressibility of the water beneath the ice sheet and to model the ice as a thin plate rather than as a beam (i.e. as a three-dimensional structure rather than as a two-dimensional one), the measured group velocities of the flexural waves corresponded quite well to those predicted by their theory EC34.
A more extensive set of seismic experiments on lake ice to create ice-coupled waves artificially was done by Press et al. \citeyear{Press51}, who also observed air-coupled flexural waves travelling at the speed of sound \caf{PE51a}{discussed in further detail by}. Again, the measured flexural wave dispersion agreed quite well with the thin plate dispersion theory.
\can{PE51b} developed the elastic model of ice further by allowing for the horizontal and vertical displacements to vary arbitrarily in the vertical direction, although they neglect the effect of gravity. In the thin plate model, the horizontal displacements are neglected, while the vertical displacements are taken to be linear in the vertical coordinate. The latter approximations are valid when the thickness of the ice is small in comparison to the wavelength. Indeed, in the large wavelength limit, the dispersion relation of \can{PE51b} explicitly converges to the thin plate relation (allowing for the fact that the gravity term also becomes negligible for large wavelengths).
In the short wave limit, where gravity does have more of an effect, their dispersion relation reproduces the relation for an infinitely thick ice sheet with unattenuated Rayleigh waves travelling along the ice-air surface and with attenuated Rayleigh waves travelling along the ice-water interface. The disadvantage of the theory of \can{PE51b} is that they were only able to solve their dispersion relation in the above long and short wave limits.
Oliver et al. \citeyear{Oliver54} did a similar seismic investigation to Press et al. \citeyear{Press51} in the Beaufort Sea and the Arctic Ocean amongst Arctic sea ice, and applied the theory of \can{PE51b} in their data analysis. In particular, studies of flexural wave velocities in shore-fast ice off Barter Island showed that for smaller wavelengths dispersion results were in very good agreement with theoretical predictions. Experimental results for longer waves were still quite good, although not as good as for shorter waves. The deviation was attributed to sea-bottom effects due to the relative shallowness of the water (3\,m). Oliver et al. \citeyear{Oliver54} also compared measured ice thicknesses in different areas with those deduced from the group velocities of flexural waves or from the frequency at which the air-coupled waves occurred. Probably due to ice inhomogeneity, the flexural wave dispersion was not as reliable a guide to thickness as the frequency of the air-coupled wave. In general, however, the air-coupled frequency still significantly underpredicted the ice thickness (by around 11\
Clearly, in seismological studies, the effect of water compression will become more important, especially if the charge is exploded in the liquid. However, in less destructive studies of ice-ocean interaction, such as the one carried out in Notre Dame Bay, Newfoundland by \can{SA80} involving strain gauges, neglecting that effect is less consequential. Moreover, in the analysis of Press et al. \citeyear{Press51}, carried out before the work of \can{PE51b}, the success of the thin plate model in describing the observed dispersion seems to suggest that the thin plate assumptions about the horizontal and vertical motion are not too significant. In any event, \can{SA80}, were able to verify that the thin plate dispersion relation predicted wavelengths that agreed with measured ones to within experimental error. In addition, by considering the problem of a semi-infinite floating ice sheet, they were able to predict the variation of strain in the ice with distance from the ice edge, given the spectrum of the incident waves. Although no such data were available, they were able to estimate the incident wave spectrum by calculating the ratio of the measured strains to the theoretical strains given a uniform spectrum of unit amplitude. If the calculated incident wave spectrum was independent of the position of the strain-meter producing the experimental results, it would serve as a good check of the theory, although it would not confirm it absolutely. Results for three different meters were qualitatively quite similar for three different locations, and differences could possibly have been able to be attributed to the theoretical strains being based on an incomplete solution Wadhams73.
\can{Mindlin51} also develops a thick plate model which allows for rotational and shear effects inside the plate. For a semi-infinite ice sheet, theoretical predictions \caf{FS90,BC99}{e.g.} show that those effects make little difference, although they might be more important near a structure of smaller width such as a pressure ridge.
Existing Solution Techniques
In summarising the development of mathematical solution techniques, we will concentrate on two particular problems that have been the focus of a lot of previous sea ice research, before showing how these methods could be applied and extended to the treatment of irregularities in sea ice sheets.
Shore-fast Sea Ice
One particular problem that illustrates the progression of solution techniques for floating plate problems is that of the scattering of incoming ocean waves by shore-fast sea ice, modelled by a semi-infinite sheet of ice. The first attempt at a solution was by \can{WK50} who solved a simplified problem by using the Wiener-Hopf technique to obtain an extremely simple expression for the modulus of the reflection coefficient, [math]\displaystyle{ R }[/math]. This formula is given by
where [math]\displaystyle{ \a_0 }[/math] and [math]\displaystyle{ k_0 }[/math] are the wavenumbers of the propagating waves in the water and in the ice respectively (both are real), and was pointed out by both \can{SS53} and \can{KW53}. (Note that although \citename{Wadhams86}, \citeyear*{Wadhams86}, showed that there is an error in the paper by \citename{SS53}, their formula for [math]\displaystyle{ R }[/math] is still valid as that mistake only pertained to an energy conservation theorem and an expression for the modulus of the transmission coefficient.) Their solution was obtained by ignoring the elastic properties of the ice, and simply modelling the ice as a collection of independent point masses. This model is referred to as the ``mass loading model", which, although somewhat inaccurate for ice sheets (for example it predicts that wavelengths should shorten on moving from water into ice, Squire et al., \citeyear*{Squire95rev}),
has proved reasonably successful for wave propagation through frazil or pancake ice WH91. It effectively reduces the order of the thin plate boundary condition from five to one, eliminating the need to apply additional conditions at the ice-edge.
\can{ED68} subsequently used the Wiener-Hopf method Noble58,Roos69 to solve the full problem. At the time the infinite products that formed a large part of the solution proved to be too difficult to compute and so they were not able to present any results, except for certain limiting cases (such as the shallow water limit). Incomplete mode-matching approaches were then taken \citeaffixed{HW63,Wadhams73,Squire78,Squire84,Wadhams86}{e.g.}, before \can{FS90} computed the solution for the full set of eigenfunctions using a conjugate gradient technique. \can{FS94} used the same method to complete further studies on this problem, investigating the strain in the ice, and also the effect of shore fast ice on an incoming directional wave spectrum of specified structure.
\can{GM898 presented a later Wiener-Hopf solution for infinite depth, but again only certain limiting situations were discussed. However, in the late 1990's and early [math]\displaystyle{ 21^{\rm st8 }[/math] century improvements in computing power enabled other authors to compute results using the Wiener-Hopf solution of \can{ED688. \can{BC998 turned the required infinite products into integrals which were evaluated by quadrature, while \can{CF02wtr8 showed that the products themselves could be evaluated directly with relatively little effort. Ironically \can{Tkacheva01fd8 finally showed that if the inertia term in the thin plate equation was neglected for normally incident waves (as it can for most wavelengths), then [math]\displaystyle{ |8| }[/math] could be calculated by simply using the correct value for [math]\displaystyle{ k_0 }[/math], the wavenumber in the ice, in the formula (8|8|8) given by \can{KW538.
Other authors have returned to the mode-matching approach of \can{FS90}. Sahoo et al. \citeyear{Sahoo01} defined an inner product enabling the solution to be found by using [math]\displaystyle{ N }[/math] eigenfunctions and inverting an [math]\displaystyle{ N\times N }[/math] matrix, while \can{LC03} effectively showed that the equations Sahoo et al. \citeyear{Sahoo01} had set up could be solved analytically using residue calculus. In the process they also demonstrated its equivalence to the Wiener-Hopf solution, and confirmed Tkacheva's (2001) formula. Earlier \can{Chakrabarti00} had used an infinite depth mode-matching scheme to set up a singular integral equation, the equivalent problem to a residue calculus problem when the eigenvalue spectrum is continuous. This singular integral equation was solved by transforming it into a Hilbert problem Roos69, a generalization of the Wiener-Hopf problem (although in practice both are solved in exactly the same way).
An Ice Strip of Finite Width in Open Water
A second problem that has generated a lot of the mathematics used herein is that of the scattering of water waves by an ice strip of finite width surrounded by open water. By restricting the incoming waves to normal incidence, it was initially intended as a two-dimensional model for a single ice floe such as might occur in the MIZ Meylan93,MS94. Although modelling of ice floes has since moved away from this approach towards a more three-dimensional one (e.g. Meylan et al., \citeyear*{Meylan97}; \citename{PM04}, \citeyear*{PM04}), the results and techniques are still quite applicable to situations where the length of the floating object is large in comparison to other characteristic lengths of the problem. Examples include the Mega-Float discussed in Section~4 (Kanzawa et al., \citeyear*{Kanzawa02}).\\
The solution in the shallow water limit is relatively straightforward Stoker57 and simply involves the inversion of an [math]\displaystyle{ 8\times8 }[/math] matrix. In his Ph.D. thesis, \can{Meylan93} solved the problem for both finite and infinite depth by using Green's theorem and the two-dimensional open water Green's function to set up an integral equation over the floe \citeaffixed{MS94}{the infinite depth solution was also published in}. It is also possible to extend the mode-matching technique of Sahoo et al. \citeyear{Sahoo01} to treat this problem, and the author is aware of several researchers who have done this, although such a solution is still unpublished. This technique could be improved by using the residue calculus method in the same way that is it is used in Appendix~\ref{app-rescalc} of this thesis, which treats the problem of the scattering by three adjacent ice sheets of different thicknesses. With minor adjustments, either or both of the outer ice sheets may be replaced with open water (cf. \Sr{sec-floe}; published papers on the residue calculus method include those by \citename{LC03}, \citeyear*{LC03} and \citename{CL05}, \citeyear*{CL05}).\\
\can{Tkacheva02} solved a pair of coupled Wiener-Hopf equations to produce another approach to this problem, using a similar method to the one that is used in \C8 of this thesis, and \can{Meylan02} solved the time dependent problem by using a spectral approach.
Irregularities in a Continuous Ice Sheet
The first type of irregularity in a continuous ice sheet considered was an abrupt change in ice properties, i.e. the scattering by a joint/edge separating two semi-infinite ice sheets, each with different properties. \can{BS96} solved this problem numerically by extending the method of \can{FS90} used in dealing with the shore fast ice problem, while \can{CF02ice} solved it by extending their own Wiener-Hopf solution CF02wtr.\\
A special case of the above problem is that of the scattering by a crack, i.e. by a free edge separating two identical semi-infinite ice sheets. \can{SD00} extended the work of \can{Meylan93} to derive the infinite depth Green's function for an ice sheet; this was then used to obtain an analytical expression for the reflection and transmission coefficients for a normally incident wave upon a crack (in ice floating on water of infinite depth). \can{WS02} later used the same method to derive similar expressions for the same coefficients when the incident wave arrived at an oblique angle. The finite depth problem was solved by \can{EP03crack} who used two different methods to calculate the finite depth solution---the first involved using a mode-matching technique and the other used the finite depth Green's function for an ice sheet.\\
\can{CL05} extended this mode-matching approach to treat a crack of finite width (also known as an open lead), using the residue calculus technique to improve their solution. \can{WS-isope04} also presented results using a mode-matching technique for an open lead (although without detailing their method). \citename{WS-isope04} could also have used the variable thickness method outlined in the same paper, and first presented in their \citeyear*{WS-RS} paper, by simply setting the thickness of the variable region to be constantly zero.\\
\can{Marchenko97} also solved the problem of cracks in ice for infinite depth, by taking a Fourier transform directly (as opposed to finding the transform of the Green's function), allowing the ice to have arbitrarily many (parallel) cracks. The scattering by many cracks was also treated by \can{DS01}, who considered the effect of randomising the separation of cracks, while \can{EP03cracks} applied Floquet's theorem to investigate the scattering by an infinite series of identically spaced parallel cracks.\\
An additional problem treated by \can{Marchenko97} was the problem of scattering by a pressure ridge, in which he modelled the ridge as an elastic beam joined to the surrounding ice cover by an elastic hinge joint. Approaching the problem in this manner enables the solution to be found in much the same way as the crack solution was found. \can{WS-RS} treated the ridge as a thin elastic plate of finite width, but with a varying thickness, and used Green's theorem to reduce the problem to an integral equation over the ridge's width (also presented in Chapter~\ref{chap-ridge} of this thesis). \can{DS01} had previously solved this problem in the special case where the thickness of the ridge was constant by replacing the open water Green's function in the method of \can{Meylan93} \caf{MS94}{also reported in} with the thin plate Green's function. Both \can{WS-RS} and \can{DS01} neglected the submergence of the ridge/berg, although this effect could be allowed for in the latter paper in the same way that \can{Meylan93} solved the problem of a submerged floe.\\
A different approach to the ridge problem was taken by \can{PP04}, who used a variational approach and a mild slope approximation to reduce it to a sixth order ordinary differential equation. Their method allowed a pressure ridge of arbitrary thickness and keel depth to be considered by assuming that a negligible amount of evanescent wave action was produced as the incident wave was scattered. In addition, an arbitrary sea floor topography could be treated simultaneously. (Evanescent waves are waves with non-real wavenumbers whose amplitudes decay exponentially as the distance from their point of generation increases. Their wavenumbers are generally pure imaginary, although two complex wavenumbers are possible in ice; the waves corresponding to those wavenumbers are sometimes distinguished from the others by being called damped traveling waves, or simply complex modes. Traveling or propagating waves have purely real wavenumbers.)\\
A surprising implication of the work of \can{PP04} is that for a large enough water depth, removing the submergence would have negligible impact on the scattering by a ridge as long as its overall rigidity is preserved. This means that, assuming that the gradient of the ridge's rigidity is not too large, the integral equation method of \can{WS-RS} will produce accurate results for such depths, as long as the total ice thickness is increased to allow for the keel.\\
\can{WS-RS} presented results for two ridges as well, comparing the exact solutions to a wide-spacing approximation which becomes more accurate as the separation between the ridges is increased. This approximation assumes that the evanescent waves generated at each ridge have decayed to a negligible amplitude by the time they have reached the next. For this reason it is called the No Evanescent Waves (NEW) approximation, although it is actually equivalent to the wide spacing approximation used, for example, by \can{Newman65} or \can{Evans90}. Equivalent results were presented by \can{Meylan93} for two floes.\\
The NEW approximation was extended by \can{WS-RS} to allow for arbitrarily many ridges. In addition, when the effect of the ridge separations was averaged out, the result was extremely well approximated by the so-called serial approximation. The serial approximation is obtained by simply multiplying the transmission coefficients for each ridge to obtain an overall transmission coefficient. It was introduced to allow the general scattering properties of a given ice field to be studied without the complicated interference produced in the presence of many ridges.\\
\can{WS-isope04} extended the work of \can{WS-RS} to allow the ice field to contain leads and cracks, and the serial approximation also proved to describe the average scattering behaviour of fields containing those features accurately.\\
This result was taken by \can{Williams-iahr04} to suggest that the potential for using flexural waves in remote ice-thickness-sensing could initially be considered by simply using the serial approximation. In that paper, reflection curves for a given set of ridges were compared for different ice thicknesses and were found to be fairly different over a relatively wide range of periods (from about 2\,s to 11\,s). Thus there seemed to be some hope that scattering results for different thicknesses would be sufficiently different to use those results in determining the ice thickness.\\
Outline of Thesis
In Chapter~\ref{chap-model} we begin by presenting the governing equations that we wish our velocity potential to satisfy. These equations are derived and discussed in Appendix~\ref{app-eqns}. Then, using the Green's functions that are discussed in Chapter~\ref{chap-Green} for both finite and infinite depth, we proceed to set up an integral equation in Chapter~\ref{chap-inteq} in terms of the displacement at the ice-water interface (or at the air-water interface---such as in the problem of an open lead). This integral equation may either be over a finite or an infinite interval.
Of particular interest in Chapter~\ref{chap-Green} is the presentation of calculation techniques for the infinite depth Green's function for oblique incidence. These techniques include Fast Fourier Transform (FFT), Taylor series expansions, and numerical integration. Each of these methods have particular advantages and disadvantages, depending on the number of points that the Green's function needs to be evaluated at and the distance of the points from the singularity. They are described in more detail in Appendix~\ref{app-Green}. Another appendix that is related to this chapter is Appendix~\ref{app-roots}, which discusses the behaviour of the roots of the shallow water, finite depth and infinite depth dispersion relations. The small period behaviour of the latter two sets of roots has never been presented before and, although it generally only occurs at physically impractical periods, is nevertheless extremely interesting mathematically.
In Chapter~\ref{chap-inteq} the thickness profile of the ice is allowed to vary arbitrarily over a finite strip with the ice sheets of different but constant thicknesses. The method used to include a variable thickness profile was first reported by \can{WS-RS}, but the generalization of this approach to include different thicknesses on each side of the strip is new and is yet to be published. This chapter also presents a proof that the potential may always be represented by an eigenfunction expansion.
Chapters~5, 6, 7 and 8 solve the integral equation derived in \Cr{chap-inteq} in a variety of different situations, and are arranged in increasing order of mathematical difficulty. Chapters~5 and 6 can both be considered as treating special cases of the problem presented in \C7, both in terms of the actual complexity of the physical problems that they solve, and in the methods of solution that they employ. In physical terms, the problem solved in \C8 is also a special case of the one in \C7, but mathematically the use of the Wiener-Hopf technique in particular is more involved in \C8 than in \C7, in which the Wiener-Hopf technique is first introduced. Consequently, \C8 provides a more analytical solution method than \C7, but to a smaller range of physical problems; although \C7 reports a more numerical method, it can as a result deal with a wider range of problems.
The simplest problem is that of a single crack in a single uniform ice sheet. As mentioned in the previous section, this problem was solved for infinitely deep water and normally incident waves by \can{SD00}, obliquely incident waves by \can{WS02}, and finitely deep water by \can{EP03crack}. However, the solution follows in only one or two steps from the final result in \C4, which, as discussed above, solves a considerably more general problem. Moreover, it was also thought that the simplicity of the results obtained provide a good opportunity to investigate their behaviour with some of the more fundamental wave/ice parameters such as ice thickness, wave period, angle of incidence and water depth. For example, it is shown that the effects of the former two can be approximately combined into a single parameter, the nondimensional period, while in the discussion of the effect of water depth, criteria with regard to the accuracy of the shallow and infinite depth approximations are established.
\C6 describes scattering by a pressure ridge. It is modelled as a thin plate of variable thickness with zero submergence WS-RS,WS-isope04,Williams-iahr04, and, as in those papers, the problem is solved by setting up an integral equation over the interval in which the ice thickness is varying. This equation is solved numerically.
The results presented in this chapter are similar to the single-ridge results presented in the aforementioned papers. However, those results are adjusted to allow for the increased thickness due to the ridge keel. In light of the recent work of \can{PP04}, changing the thickness in this way should improve the accuracy of the \can{WS-RS} results for large water depths, but without significant changes to that method. Appendix~\ref{app-keel} also treats the topic of scattering by a pressure ridge, proposing a method by which submergence could be included in full, although no results are able to be presented yet.
\C7 extends the variable thin plate formulation developed in \C6 by allowing the ice thickness to the right of the variable region (which in this chapter we will refer to as a ramp) to differ from the ice thickness to its left. To do this we must solve two coupled integral equations---the first is much the same as the one solved in \C6, but the second is over a semi-infinite range---over the ice-water interface to the right of the ramp. Proceeding in exactly the same way as in \C6, we proceed to solve the first integral equation numerically for the displacement in the ramp, expressing it in terms of the displacement of the ice in the semi-infinite region. The second integral equation is one of the Wiener-Hopf type, and so we are able to write an analytical expression for the displacement in the semi-infinite region in terms of the displacement within the ramp. We can now eliminate the displacement in the right hand region from the two sets of equations, solve for the displacement in the ramp, and then regenerate the displacement to the right.
The method in \C7 is rather novel, as it combines a numerical integral quadrature scheme with the analytical Wiener-Hopf technique. Previously, authors have tended to favour either one approach or the other, and so this result also represents a philosophical breakthrough in that the combined weight of both methods can be brought to bear on a single problem most effectively. The numerical approach is able to deal with the arbitrariness of the ice thickness within the ramp, while the Wiener-Hopf technique is ideal for the semi-infinite interval over which the second integral equation must be solved---such intervals are extremely hard to deal with numerically.
Unfortunately, the method does have the same drawback that our ridge model had---that submergence must be ignored. However, the method of \can{PP04} also incorporated a change in ice thickness from one side of the variable region to the other, and so we could again use their result that for large water depths the effect of submergence is negligible as long as the correct total thickness is used (and as long as the mild slope assumption is valid).
The physical problem solved in \C7, i.e. a ramp connecting a thinner ice sheet to a thicker one, has practical applications. Such a thickness profile could be taken to describe a sea ice/ice shelf transition such as the one observed in the Ross Sea, where the thickness of the sea ice beyond the ice shelf increases steadily to meet the thicker ice of the ice shelf itself. Alternatively, by setting the right hand ice thickness equal to zero, and by letting the incident wave arrive from the right instead of from the left, the problem could represent a breakwater shielding a VLFS from waves arriving from the open ocean.
\C8 treats a special case of the problems solved in Chapters~6 and 7. When the ice thickness over the variable region is constant, a more analytical approach may be taken. The problem may be written as two coupled Wiener-Hopf integral equations, producing a system of [math]\displaystyle{ M }[/math] linear equations in as many variables, where [math]\displaystyle{ M }[/math] is the number of evanescent waves generated at each end of the middle region that have not decayed sufficiently by the time they reach the other end to be able to be ignored. Since the evanescent waves decay exponentially with distance, the order of coupling [math]\displaystyle{ M }[/math] between the two semi-infinite regions decreases as the width of the middle region increases. If that region is wide enough, [math]\displaystyle{ M }[/math] becomes 1, and only the propagating waves generated need to be considered. Thus only a scalar matrix needs to be inverted, and the problem reduces to the well-known wide spacing approximation Evans90, abbreviated by \can{WS-RS} to the NEW approximation (No Evanescent Waves).
In the above special case, the approach of \C8 complements the numerical approach of the two earlier chapters well. When the central region is relatively wide, the method of \C8 becomes extremely efficient, while if [math]\displaystyle{ a }[/math], the width of that central region, is decreased and less evanescent modes can be neglected, less quadrature points are needed to obtain the numerical solution and that method would be preferable. Given the wave period, angle of incidence and ice thicknesses, the coupled Wiener-Hopf method can generate results for many different values of [math]\displaystyle{ a }[/math] quite rapidly, while if [math]\displaystyle{ a }[/math] is fixed and results are required for when the thickness of the central region is allowed to take several different values, the quadrature schemes would be more efficient.
Practically speaking again, the situation treated in \C8 is a good model for a lead separating two sheets of ice. Such features are extremely common in both the Arctic and Antarctic regions. Different stages of refreezing may be represented, ranging from a freshly formed lead which is still only open water, to an older lead which has frozen over, and whose edges may have even refrozen to the two larger ice sheets.
\C9 deals with the scattering by multiple features in a uniform ice sheet. After presenting some preliminary theory, it follows the structure of \can{WS-RS}, and starts by presenting results for the scattering at normal incidence by two pressure ridges or leads, mainly investigating the effect of the feature separation on the scattering. The results are similar to results for two cracks presented in \C8, but can be taken a little further in that we can also look at the effect of giving the two features different properties from each other. (All cracks are identical, so this couldn't be done in \C8.)
These results for two features are also compared with the NEW approximation, which is able to produce results for different feature separations much more quickly than can be obtained if an exact solution is sought. (This is also done in \C8.) Being a wide spacing approximation, it becomes more accurate as the separation is increased, the exact and approximate curves generally coalescing when the separation is about three quarters of one wavelength.
\C9 then seeks to average out the effect of separation by using a serial approximation which can produce results even more quickly than the NEW approximation can. Ridge separations are sampled from an observed type of distribution (log-normal in that case) and the median value of [math]\displaystyle{ |R| }[/math] (calculated using the NEW method for two identical ridges) is calculated and compared with the serial result. The two curves are almost identical, although the variance of the results about the median is quite large.
This experiment is repeated for a larger number of identical ridges with similar success; then when the ridge sail heights are also sampled from a random distribution; and it works just as well when large numbers of cracks or leads are used, and separations and lead widths are randomly sampled.
Having established that the serial approximation summarizes the general scattering properties of a series of features for normal incidence, we also attempt to use it to predict the effect of randomising the feature orientations, and also combine the results for individual features in an attempt to model an ice field populated with cracks, pressure ridges and leads, in the manner of \can{WS-isope04}. The scattering seems to be dominated by the effect of the leads.
This chapter goes a little further than either \can{WS-RS} or \can{WS-isope04}, however, in that it also presents results for pressure ridges when the flexural effects of a keel are accounted for, or when semi-refrozen leads are also present.
Finally, it investigates the effect of changing the background ice thickness on scattering results when either one or 100 ridges or leads are present. (The serial approximation is used in the latter case.) A similar result for ridges was given by \can{Williams-iahr04}, but since the leads seem to dominate the scattering by an ice field, results for them are also added to complete the chapter. These last results are presented with the intention of showing that the scattering changes enough with the background ice thickness that there is potential for it to be used to determine that thickness.
