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| | + | Please put any requests you have here. |
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| | + | == Standard notation for water depth == |
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| − | \paperid{}
| + | Should we have a standard notation for water depth? |
| − | \ccc{}
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| − | \lefthead{MEYLAN}
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| − | \righthead{Wave Response of an Ice Floe}
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| − | \received{November 2000}
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| − | \revised{May 2001}
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| − | \accepted{??}
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| − | \authoraddr{M. H. Meylan, Institute of Information and Mathematical Sciences, Massey
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| − | University, Auckland, New Zealand (email:~m.h.meylan@massey.ac.nz)}
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| − | \slugcomment{Submitted to ''Journal of Geophysical Research'' , November 2000.}
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| − | = The Wave Response of an Ice Floe of Arbitrary Geometry. =
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| − | Michael H. Meylan
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| − | \affil{Institute of Information and Mathematical Sciences, Massey University,
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| − | Auckland, New Zealand}
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| | | | |
| | + | I have started the following page |
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| − | A fully three dimensional model for the motion and bending of a solitary ice
| + | [[Standard Notation]] |
| − | floe due to wave forcing is presented. This allows the scattering and wave
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| − | induced force for a realistic ice floe to be calculated. These are required
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| − | to model wave scattering and wave induced ice drift in the marginal ice
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| − | zone. The ice floe is modelled as a thin plate and its motion is expanded in
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| − | the thin plate modes of vibration. The modes are substituted into the
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| − | integral equation for the water. This gives a linear system of equations for
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| − | the coefficients used to expand the ice floe motion. Solutions are presented
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| − | for the ice floe displacement, the scattered energy, and the time averaged
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| − | force for a range of ice floe geometries and wave periods. It is found that
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| − | ice floe stiffness is the most important factor in determining ice floe
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| − | motion, scattering, and force. However, above a critical value of stiffness
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| − | the floe geometry also influences the scattering and force.
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| | | | |
| − | | + | [[Category:Administration]] |
| − | TCIMACRO{\TeXButton{Begin Article}{}}
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| − | BeginExpansion
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| − | | |
| − | EndExpansion
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| − | | |
| − | ==Introduction==
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| − | | |
| − | It is widely recognized that understanding the relationship between ocean
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| − | waves and sea ice requires a fully three-dimensional model for the motion
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| − | and bending of a solitary ice floe [[Squire_Review]]. Such an ice floe
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| − | model is derived in this paper. The model includes flexure and is for an ice
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| − | floe of arbitrary shape. It allows the wave scattering and wave induced
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| − | force for a realistic ice floe to be calculated for the first time.
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| − | | |
| − | The Marginal Ice Zone (MIZ) is an interfacial region, composed of an
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| − | aggregate of ice and water, which forms at the boundary of open and ice
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| − | covered seas. The major interaction between open and ice covered seas is
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| − | through waves. These waves are generated in the open water and are
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| − | responsible for breaking up the continuous pack ice. For waves to reach the
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| − | continuous pack ice they must pass through the MIZ.
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| − | | |
| − | Experimental measurements have shown that as ocean waves pass through the
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| − | MIZ their character is radically altered. [[attenuation88]] and
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| − | [[attenuationNature80]] found that there was strong and significant
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| − | exponential attenuation of wave energy. This attenuation decreased with
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| − | increasing wave period. The angular spreading of incoming ocean waves was
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| − | measured by [[directional86]] who found that from a narrow directional
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| − | spectrum in the open water the directional wave spectrum evolves to become
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| − | isotropic within the MIZ. Experimental measurements of the motion of
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| − | individual ice floes have shown that ice floe bending is significant.
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| − | [[Squireicefloe]] and [[SquireMartin]] measured the motion of ice
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| − | floes using strain gauges. They showed that ice floe flexure is significant
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| − | and must be included in ice floe models.
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| − | | |
| − | [[Masson_Le]] and [[jgrrealism]] have derived models for the
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| − | propagation of waves through the MIZ, based on the linear transport
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| − | equation. These models predict the exponential decay and directional
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| − | spreading of the incoming wave spectrum qualitatively. However, they were
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| − | quantitatively inaccurate because neither of them used a realistic model to
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| − | calculate the individual ice floe scattering. This was because no realistic
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| − | model existed. [[Masson_Le]] and [[Massondrift]] assumed that the
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| − | ice floe was a rigid body. [[Wadhams1986]] included ice floe flexure but
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| − | the model was two dimensional and the solution was only calculated
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| − | approximately. While \citet {jgrfloe1d} solved the Wadhams problem exactly,
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| − | two dimensional models give no information about the directional scattering
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| − | so cannot be used in wave scattering models. \citet {jgrfloecirc} extended
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| − | the two dimensional model to three dimensions but only for the unrealistic
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| − | case of a circular ice floe. This was the ice floe model used in
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| − | [[jgrrealism]].
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| − | | |
| − | In this paper a fully three-dimensional model for the motion and bending of
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| − | a solitary ice floe is developed as follows. The ice floe motion is expanded
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| − | in the free modes of vibration (the modes of vibration of the ice floe in
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| − | the absence of the water). These modes must be determined numerically and
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| − | this is done by using the finite element method. The equations of motion for
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| − | the water are transformed into an integral equation over the wetted surface
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| − | of the ice floe. This transformation was developed by [[john1,john2]]
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| − | and is standard in offshore engineering [[Sarp_Isa]]. The free modes of
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| − | vibration are then substituted into the integral equation for the water.
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| − | This gives a linear system of equations for the coefficients used to expand
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| − | the floe motion.
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| − | | |
| − | Solutions are presented for the ice floe displacements, the scattering of
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| − | wave energy, and the wave induced force for a range of ice floe geometries
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| − | and wave periods. The scattering is considered because of its importance in
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| − | MIZ scattering models [[Masson_Le,jgrrealism]]. The wave induced force
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| − | is included because of the importance of this term in models of ice floe
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| − | drift.
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| − | | |
| − | ==The Equation of Motion for the Ice Floe==
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| − | | |
| − | Ice floes range in size from much smaller to much larger than the dominant
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| − | wavelength of the ocean waves. However there are two reasons why solutions
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| − | for ice floes of intermediate size (a size similar to the wavelength) are
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| − | the most important. The first is that at these intermediate sizes ice floes
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| − | scatter significant wave energy. The second is that, since it is wave
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| − | induced flexure which determines the size of ice floes in the MIZ, ice floes
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| − | tend to form most often at this intermediate length.
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| − | | |
| − | The theory for an ice floe of intermediate size which is developed in this
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| − | paper obviously also applies to small or large floes. However, if the
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| − | solution for a small or large floe is required then the appropriate simpler
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| − | theory should be used. Small ice floes (ice floes much small than the
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| − | wavelength) should be modelled as rigid [[Masson_Le,Massondrift]]. Large
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| − | ice floes (ice floes much larger than the wavelength) should be modelled as
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| − | infinite and flexible [[FoxandSquire]]. In the intermediate region,
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| − | where the size of the wavelength is similar to the size of the ice floe, the
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| − | ice floe must be modelled as finite and flexible.
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| − | | |
| − | We model the ice floe as a thin plate of constant thickness and shallow
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| − | draft following [[Wadhams1986]] and [[Squire_Review]]. The thin
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| − | plate equation [[Hildebrand65]] gives the following equation of motion
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| − | for the ice floe
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| − | <center><math>
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| − | D\nabla ^{4}W+\rho _{i}h\frac{\partial ^{2}W}{\partial t^{2}}=p,
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| − | (plate)
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| − | </math></center>
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| − | where <math>W</math> is the floe displacement, <math>\rho _{i}</math> is the floe density, <math>h</math> is
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| − | the floe thickness, <math>p</math> is the pressure, and <math>D</math> is the modulus of rigidity
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| − | of the floe (<math>D=Eh^{3}/12(1-\nu ^{2})</math> where <math>E</math> is the Young's modulus and <math>
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| − | \nu <math> is Poisson's ratio). Visco-elastic effects can be included by making </math>
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| − | D </math> have some imaginary (damping)\ component but this will not be done here
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| − | to keep the presented results simpler. We assume that the plate is in
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| − | contact with the water at all times so that the water surface displacement
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| − | is also <math>W.</math> Equation ((plate)) is subject to the free edge boundary
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| − | conditions for a thin plate
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| − | <center><math>
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| − | \frac{\partial ^{2}W}{\partial n^{2}}+\nu \frac{\partial ^{2}W}{\partial
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| − | s^{2}}=0,\;\;\;=\textrm{and= }\mathrm{\;\;\;}\frac{\partial ^{3}W}{
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| − | \partial n^{3}}+\left( 2-\nu \right) \frac{\partial ^{3}W}{\partial
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| − | n\partial s^{2}}=0, (boundaryplate)
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| − | </math></center>
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| − | [[Hildebrand65]] where <math>n</math> and <math>s</math> denote the normal and tangential
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| − | directions respectively.
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| − | | |
| − | The pressure, <math>p</math>, is given by the linearized Bernoulli's equation at the
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| − | water surface,
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| − | <center><math>
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| − | p=-\rho \frac{\partial \Phi }{\partial t}-\rho gW (pressure)
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| − | </math></center>
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| − | where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density
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| − | of the water, and <math>g</math> is the acceleration due to gravity.
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| − | | |
| − | We now introduce non-dimensional variables. We non-dimensionalise the length
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| − | variables with respect to <math>a</math> where the surface area of the floe is <math>4a^{2}.</math>
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| − | We non-dimensionalise the time variables with respect to <math>\sqrt{g/a}</math> and
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| − | the mass variables with respect to <math>\rho a^{3}</math>. The non-dimensional
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| − | variables, denoted by an overbar, are
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| − | <center><math>
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| − | \bar{x}=\frac{x}{a},\;\;\bar{y}=\frac{y}{a},\;\;\bar{z}=\frac{z}{a},\;\;\bar{
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| − | W}=\frac{W}{a},\;\;\bar{t}=t\sqrt{\frac{g}{a}},\;\;=and= \;\;\bar{\Phi}=
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| − | \frac{\Phi }{a\sqrt{ag}}.
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| − | </math></center>
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| − | In the non-dimensional variables equations ((plate)) and ((pressure)
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| − | ) become
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| − | <center><math>
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| − | \beta \nabla ^{4}\bar{W}+\gamma \frac{\partial ^{2}\bar{W}}{\partial \bar{t}
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| − | ^{2}}=\frac{\partial \bar{\Phi}}{\partial \bar{t}}-\bar{W}, (n-d_ice)
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| − | </math></center>
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| − | where
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| − | <center><math>
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| − | \beta =\frac{D}{g\rho a^{4}}\;\;=and\ \ = \gamma =\frac{\rho _{i}h}{\rho
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| − | a}.
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| − | </math></center>
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| − | We shall refer to <math>\beta </math> and <math>\gamma </math> as the stiffness and mass
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| − | respectively.
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| − | | |
| − | We will determine the response of the ice floe to wave forcing of a single
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| − | frequency (the response for more complex wave forcing can be found by
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| − | superposition of the single frequency solutions). Since the equations of
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| − | motion are linear the displacement and potential must have the same single
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| − | frequency dependence. Therefore they can be expressed as the real part of a
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| − | complex quantity whose time dependence is <math>e^{-i\sqrt{\alpha }t}</math>
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| − | where
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| − | <math>\alpha </math> is the non-dimensional wavenumber and we write <math>\bar{W}(\bar{x},
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| − | \bar{y},\bar{t})={Re}\left[ w\left( \bar{x},\bar{y}\right) e^{-i\sqrt{
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| − | \alpha }\bar{t}}\right] \ <math>and</math>\;\Phi (\bar{x},\bar{y},\bar{z},\bar{t})=
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| − | {Re}\left[ \phi \left( \bar{x},\bar{y},\bar{z}\right) e^{-i\sqrt{\alpha
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| − | }\bar{t}}\right] .</math> In the complex variables the equation of motion of the
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| − | ice floe ((n-d_ice)) is
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| − | <center><math>
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| − | \beta \nabla ^{4}w+\alpha \gamma w=\sqrt{\alpha }\phi -w. (plate2)
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| − | </math></center>
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| − | From now on we will drop the overbar and assume all variables are
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| − | non-dimensional.
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| − | | |
| − | ==Equations of Motion for the Water==
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| − | | |
| − | We require the equation of motion for the water to solve equation (\ref
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| − | {plate2}). We begin with the non-dimensional equations of potential theory
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| − | which describe linear surface gravity waves
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| − | <center><math> (bvp)
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| − | \left.
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| − | \begin{matrix}{rr}
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| − | \nabla ^{2}\phi =0, & -\infty <z<0, \\
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| − | {\frac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\
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| − | {\frac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\;
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| − | \mathbf{x}\in \Delta , \\
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| − | {\frac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{
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| − | x}\notin \Delta ,
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| − | \end{matrix}
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| − | \right\} (bvp_nond)
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| − | </math></center>
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| − | ([[Weh_Lait]]). As before, <math>w</math> is the displacement of the floe and <math>p</math>
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| − | is the pressure at the water surface. The vector <math>\mathbf{x=(}x,y)</math> is a
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| − | point on the water surface and <math>\Delta </math> is the region of the water surface
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| − | occupied by the floe. The water is assumed infinitely deep. A schematic
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| − | diagram of this problem is shown in Figure (vibration).
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| − | \begin{figure}[tbp]
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| − | \begin{center}
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| − | \epsfbox{vibration.eps}
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| − | \end{center}
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| − | \caption{{The schematic diagram of the boundary value problem and the
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| − | coordinate system used in the solution.}}
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| − | (vibration)
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| − | \end{figure}
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| − | | |
| − | The boundary value problem ((bvp)) is subject to an incident wave which
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| − | is imposed through a boundary condition as <math>\left| \mathbf{x}\right|
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| − | \rightarrow \infty </math>. This boundary condition, which is called the
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| − | Sommerfeld radiation condition, is essentially that at large distances the
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| − | potential consists of a radial outgoing wave (the wave generated by the ice
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| − | floe motion) and the incident wave. It is expressed mathematically as
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| − | <center><math>
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| − | \lim_{\left| \mathbf{x}\right| \rightarrow \infty }\sqrt{|\mathbf{x}|}\left(
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| − | \frac{\partial }{\partial |\mathbf{x}|}-i\alpha \right) (\phi -\phi ^{
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| − | \mathrm{In}})=0, (summerfield)
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| − | </math></center>
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| − | [[Weh_Lait]]. The incident potential (i.e. the incoming wave) <math>\phi ^{
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| − | \mathrm{In}}</math> is
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| − | <center><math>
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| − | \phi ^{\mathrm{In}}(x,y,z)=\frac{A}{\sqrt{\alpha }}e^{i\alpha (x\cos \theta
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| − | +y\sin \theta )}e^{\alpha z}, (input)
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| − | </math></center>
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| − | where <math>A</math> is the non-dimensional wave amplitude.
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| − | | |
| − | The standard solution method to the linear wave problem is to transform the
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| − | boundary value problem into an integral equation using a Green function
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| − | \citep{john1,
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| − | john2,Sarp_Isa,jgrfloecirc}. Performing such a transformation, the boundary
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| − | value problem ((bvp)) and ((summerfield)) becomes
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| − | <center><math>
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| − | \phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x}
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| − | ;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x}
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| − | )\right) dS_{\mathbf{y}}. (water)
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| − | </math></center>
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| − | The Green function <math>G_{\alpha }</math> is
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| − | <center><math>
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| − | G_{\alpha }(\mathbf{x};\mathbf{y)}=\frac{1}{4\pi }\left( \frac{2}{|\mathbf{x}
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| − | -\mathbf{y}|}-\pi \alpha \left( \mathbf{H_{0}}(\alpha |\mathbf{x}-\mathbf{y}
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| − | |)+Y_{0}(\alpha |\mathbf{x}-\mathbf{y}|)\right) +2\pi i\alpha J_{0}(\alpha |
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| − | \mathbf{x}-\mathbf{y}|)\right) ,
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| − | </math></center>
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| − | [[Weh_Lait,jgrfloecirc]], where <math>J_{0}</math> and <math>Y_{0}</math> are respectively
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| − | Bessel functions of the first and second kind of order zero, and <math>\mathbf{
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| − | H_{0}}</math> is the Struve function of order zero [[abr_ste]]. A solution for
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| − | water of finite depth could be found by simply using the depth dependent
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| − | Green function [[Weh_Lait]].
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| − | | |
| − | The integral equation ((water)) will be solved using numerical
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| − | integration. The only difficulty arises from the non-trivial nature of the
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| − | kernel of the integral equation (the Green function). However, the Green
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| − | function has no <math>z</math> dependence due to the shallow draft approximation and
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| − | depends only on <math>|\mathbf{x}-\mathbf{y}|.</math> This means that the Green
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| − | function is one dimensional and the values which are required for a given
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| − | calculation can be looked up in a previously computed table.
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| − | | |
| − | ==Solving for the Wave Induced Ice Floe Motion==
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| − | | |
| − | To determine the ice floe motion we must solve equations ((plate2)) and (
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| − | (water)) simultaneously. We do this by expanding the floe motion in the
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| − | free modes of vibration of a thin plate. The major difficulty with this
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| − | method is that the free modes of vibration can be determined analytically
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| − | only for very restrictive geometries, e.g. a circular thin plate. Even the
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| − | free modes of vibration of a square plate with free edges must be determined
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| − | numerically. This is the reason why the solution of [[jgrfloecirc]] was
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| − | only for a circular floe.
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| − | | |
| − | Since the operator <math>\nabla ^{4},</math> subject to the free edge boundary
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| − | conditions, is self adjoint a thin plate must possess a set of modes <math>w_{i}</math>
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| − | which satisfy the free boundary conditions and the following eigenvalue
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| − | equation
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| − | <center><math>
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| − | \nabla ^{4}w_{i}=\lambda _{i}w_{i}.
| |
| − | </math></center>
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| − | The modes which correspond to different eigenvalues <math>\lambda _{i}</math> are
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| − | orthogonal and the eigenvalues are positive and real. While the plate will
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| − | always have repeated eigenvalues, orthogonal modes can still be found and
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| − | the modes can be normalized. We therefore assume that the modes are
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| − | orthonormal, i.e.
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| − | <center><math>
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| − | \iint_{\Delta }w_{i}\left( \mathbf{Q}\right) w_{j}\left( \mathbf{Q}\right)
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| − | dS_{\mathbf{Q}}=\delta _{ij}
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| − | </math></center>
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| − | where <math>\delta _{ij}</math> is the Kronecker delta. The eigenvalues <math>\lambda _{i}</math>
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| − | have the property that <math>\lambda _{i}\rightarrow \infty </math> as <math>i\rightarrow
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| − | \infty </math> and we order the modes by increasing eigenvalue. These modes can be
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| − | used to expand any function over the wetted surface of the ice floe <math>\Delta </math>
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| − | .
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| − | | |
| − | We expand the displacement of the floe in a finite number of modes <math>N,</math> i.e.
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| − | <center><math>
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| − | w\left( \mathbf{x}\right) =\sum_{i=1}^{N}c_{i}w_{i}\left( \mathbf{x}\right) .
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| − | (expansion)
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| − | </math></center>
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| − | From the linearity of ((water)) the potential can be written in the
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| − | following form
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| − | <center><math>
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| − | \phi =\phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i} (expansionphi)
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| − | </math></center>
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| − | where <math>\phi _{0}</math> and <math>\phi _{i}</math> satisfy the integral equations
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| − | <center><math>
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| − | \phi _{0}(\mathbf{x})=\phi ^{\mathrm{In}}(\mathbf{x})+\iint_{\Delta }\alpha
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| − | G_{\alpha }(\mathbf{x};\mathbf{y})\phi (\mathbf{y})dS_{\mathbf{y}}
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| − | (phi0)
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| − | </math></center>
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| − | and
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| − | <center><math>
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| − | \phi _{i}(\mathbf{x})=\iint_{\Delta }G_{\alpha }(\mathbf{x};\mathbf{y}
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| − | )\left( \alpha \phi _{i}(\mathbf{x})+i\sqrt{\alpha }w_{i}(\mathbf{y})\right)
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| − | dS_{\mathbf{y}}. (phii)
| |
| − | </math></center>
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| − | The potential <math>\phi _{0}</math> represents the potential due the incoming wave
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| − | assuming that the displacement of the ice floe is zero. The potentials <math>\phi
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| − | _{i}</math> represent the potential which is generated by the plate vibrating with
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| − | the <math>i</math>th mode in the absence of any input wave forcing.
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| − | | |
| − | We substitute equations ((expansion)) and ((expansionphi)) into
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| − | equation ((plate2)) to obtain
| |
| − | <center><math>
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| − | \beta \sum_{i=1}^{N}\lambda _{i}c_{i}w_{i}-\alpha \gamma
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| − | \sum_{i=1}^{N}c_{i}w_{i}=i\sqrt{\alpha }\left( \phi
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| − | _{0}+\sum_{i=1}^{N}c_{i}\phi _{i}\right) -\sum_{i=1}^{N}c_{i}w_{i}.
| |
| − | (expanded)
| |
| − | </math></center>
| |
| − | To solve equation ((expanded)) we multiply by <math>w_{j}</math> and integrate over
| |
| − | the plate (i.e. we take the inner product with respect to <math>w_{j})</math> taking
| |
| − | into account the orthogonality of the modes <math>w_{i}</math>, and obtain
| |
| − | <center><math>
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| − | \beta \lambda _{j}c_{j}+\left( 1-\alpha \gamma \right) c_{j}=\iint_{\Delta }i
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| − | \sqrt{\alpha }\left( \phi _{0}\left( \mathbf{Q}\right)
| |
| − | +\sum_{i=1}^{N}c_{i}\phi _{i}\left( \mathbf{Q}\right) \right) w_{j}\left(
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| − | \mathbf{Q}\right) dS_{\mathbf{Q}} (final)
| |
| − | </math></center>
| |
| − | which is a matrix equation in <math>c_{i}.</math>
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| − | | |
| − | We cannot solve equation ((final)) without determining the modes of
| |
| − | vibration of the thin plate <math>w_{i}</math> (along with the associated eigenvalues <math>
| |
| − | \lambda _{i})</math> and solving the integral equations ((phi0)) and (\ref
| |
| − | {phii}). We use the finite element method to determine the modes of
| |
| − | vibration [[Zienkiewicz]] and the integral equations ((phi0)) and (
| |
| − | (phii)) are solved by a constant panel method [[Sarp_Isa]]. The same
| |
| − | set of nodes is used for the finite element method and to define the panels
| |
| − | for the integral equation.
| |
| − | | |
| − | ==Displacements==
| |
| − | | |
| − | We have developed a method to solve for the motion of an ice floe of
| |
| − | arbitrary shape and properties due to waves of arbitrary frequency. While we
| |
| − | will present the solutions in the non-dimensional variables we can still
| |
| − | vary <math>\alpha ,</math> <math>\beta ,</math> <math>\gamma </math> and floe geometry. We are therefore able
| |
| − | to only present a subset of possible solutions. We consider only the four
| |
| − | ice floe geometries which are shown in figure (shapes2) (along with
| |
| − | their numbering) which represent a range of floe geometries. We concentrate
| |
| − | on the effect of the stiffness <math>\beta </math> rather than the mass <math>\gamma </math>. This
| |
| − | is because the ice floes we consider are thin so that the mass term <math>\gamma </math>
| |
| − | must necessarily be small (i.e. <math>1-\alpha \gamma \approx 1).</math>
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{shapes2.eps}
| |
| − | \end{center}
| |
| − | \caption{The four ice floe geometries for which solutions will be calculated
| |
| − | and their numbering.}
| |
| − | (shapes2)
| |
| − | \end{figure}
| |
| − | | |
| − | Figures (motion1_revised)
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{motion1_revised.eps}
| |
| − | \end{center}
| |
| − | \caption{{}The displacement of an ice floe of geometry 1 for the times and
| |
| − | stiffness shown. The mass was <math>\gamma =0.005</math>, the wavenumber was <math>
| |
| − | \alpha =\pi <math> and the wave was travelling in the positive </math>x</math>
| |
| − | -direction. }
| |
| − | (motion1_revised)
| |
| − | \end{figure}
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{motion2_revised.eps}
| |
| − | \end{center}
| |
| − | \caption{{}The displacement of an ice floe of geometry 2 for the times and
| |
| − | stiffness shown. The mass was <math>\gamma =0.005</math>, the wavenumber was <math>
| |
| − | \alpha =\pi <math> and the wave was travelling in the positive </math>x</math>
| |
| − | -direction. }
| |
| − | (motion2_revised)
| |
| − | \end{figure}
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{motion3_revised.eps}
| |
| − | \end{center}
| |
| − | \caption{{}The displacement of an ice floe of geometry 3 for the times and
| |
| − | stiffness shown. The mass was <math>\gamma =0.005</math>, the wavenumber was <math>
| |
| − | \alpha =\pi <math> and the wave was travelling in the positive </math>x</math>
| |
| − | -direction. }
| |
| − | (motion3_revised)
| |
| − | \end{figure}
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{motion4_revised.eps}
| |
| − | \end{center}
| |
| − | \caption{{}The displacement of an ice floe of geometry 4 for the times and
| |
| − | stiffness shown. The mass was <math>\gamma =0.005</math>, the wavenumber was <math>
| |
| − | \alpha =\pi <math> and the wave was travelling in the positive </math>x</math>
| |
| − | -direction. }
| |
| − | (motion4_revised)
| |
| − | \end{figure}
| |
| − | is the displacement of an ice floe of geometry 1 to waves travelling in the <math>
| |
| − | x-<math>direction (this corresponds to </math>\theta =0</math> in equation ((input)))
| |
| − | with wavenumber <math>\alpha =\pi </math> (this corresponds to a non-dimensional
| |
| − | wavelength of <math>2)</math>. The values of the stiffness are <math>\beta =0.002,</math> <math>0.01,</math>
| |
| − | and <math>0.05</math> and the mass is <math>\gamma =0.005</math>. The motion of the ice floe is a
| |
| − | dynamic process in time and the displacement at <math>t=0</math> and <math>t=T/4</math> is plotted
| |
| − | (these correspond to the real and imaginary parts of <math>w</math> respectively). The
| |
| − | solution was calculated with 40 free plate modes and 900 panels. These
| |
| − | values were chosen by increasing the number of modes and panels until the
| |
| − | solution converges. Figures (motion2_revised), (motion3_revised) and
| |
| − | (motion4_revised) are the same as figure (motion1_revised) except
| |
| − | the floe geometries are 2, 3 and 4 respectively. The ragged edges in these
| |
| − | figures arise because the geometries are approximated by square panels.
| |
| − | | |
| − | Figures (motion1_revised) to (motion4_revised) show just how
| |
| − | complicated the motion of an ice floe can be and how much this motion
| |
| − | depends on details of the geometry of the ice floe. For example, the motion
| |
| − | of the triangular floe (figure (motion4_revised)) is far more
| |
| − | complicated than the motion of the square floe (figure (motion1_revised)
| |
| − | ). Obviously this high dependence on individual geometry would not hold for
| |
| − | floes which were much smaller or larger than the wavelength. Furthermore,
| |
| − | since most ice floes have intermediate values of stiffness, the rigid body
| |
| − | model is not appropriate.
| |
| − | | |
| − | ==Scattered Energy==
| |
| − | | |
| − | The wave energy scattering from the aggregate of ice floes that make up the
| |
| − | MIZ is controlled by the scattering from individual floes. One of the
| |
| − | problems in modelling wave propagation in the MIZ has been making an
| |
| − | accurate determination of the scattering by an individual floe. The
| |
| − | scattered energy can be expressed in terms of the Kochin function given by
| |
| − | <center><math>
| |
| − | H(\tau )=\iint_{\Delta }\left( \alpha \phi +i\sqrt{\alpha }w\right)
| |
| − | e^{i\alpha (x\cos \tau +y\sin \tau )}dS
| |
| − | </math></center>
| |
| − | [[Weh_Lait]], where <math>\phi </math> and <math>w</math> are the complex potential and
| |
| − | displacement as before. The non-dimensional radiated energy for a wave of
| |
| − | unit amplitude, per unit angle, per unit time, in the <math>\tau </math> direction is
| |
| − | <center><math>
| |
| − | \frac{E\left( \tau \right) }{A^{2}\rho a^{7/2}g^{3/2}}=\frac{\alpha ^{3}}{
| |
| − | A^{2}8\pi }|H(\pi +\tau )|^{2}
| |
| − | </math></center>
| |
| − | [[Weh_Lait]]. This is the wave energy which is generated by the motion
| |
| − | of the floe.
| |
| − | | |
| − | The scattered energy <math>E\left( \tau \right) </math> is also a function of the
| |
| − | incoming wave angle <math>\theta .</math> To avoid plotting <math>E\left( \tau \right) </math> for
| |
| − | different values of <math>\theta </math> we average the scattering over the difference
| |
| − | in angle <math>\tau -\theta .</math> We expect the wave scattering in the MIZ to be
| |
| − | determined by such an averaging because in general ice floes do not appear
| |
| − | to be aligned. Figure (jgrscataveragelam2_revised)
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{jgrscataveragelam2_revised.eps}
| |
| − | \end{center}
| |
| − | \caption{{}The scattering as a function of angle for ice floe geometies 1
| |
| − | (solid), 2 (dashed), 3 (chained), and 4 (dotted). The values of the
| |
| − | stiffness were <math>\beta =</math>0.0004 (a), 0.002 (b), 0.01 (c), and 0.05
| |
| − | (d). <math>\alpha =\pi </math> and <math>\gamma =0.005.</math>}
| |
| − | (jgrscataveragelam2_revised)
| |
| − | \end{figure}
| |
| − | shows the average scattered energy for the four ice floe geometries (the
| |
| − | solid line is geometry 1, the dashed line is geometry 2, the chained line is
| |
| − | geometry 3 and the dotted line is geometry 4). The values of the stiffness
| |
| − | are <math>\beta =0.0004</math> (a), 0.002 (b), 0.01 (c), and 0.05 (d). The mass is <math>
| |
| − | \gamma =0.005.</math>
| |
| − | | |
| − | From figure (jgrscataveragelam2_revised) we can see that the scattering
| |
| − | is predominantly in the direction of the incoming wave. There is only
| |
| − | significant scattering in the backwards direction for higher values of <math>
| |
| − | \beta .<math> While there is a significant increase in the scattering from </math>\beta
| |
| − | =0.0004<math> to </math>\beta =0.002</math> there is little further increase, and for some
| |
| − | geometries a decrease, as <math>\beta </math> is increased further. For small values of
| |
| − | stiffness the scattering is independent of floe geometry but for higher
| |
| − | values of stiffness there is significant variation in the scattering between
| |
| − | the different geometries.
| |
| − | | |
| − | To understand figure (jgrscataveragelam2_revised) we plot the
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{jgrscatvarybeta_revised.eps}
| |
| − | \end{center}
| |
| − | \caption{{}The total scattering as a function of <math>\beta </math> for ice
| |
| − | floe geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). <math>
| |
| − | \alpha =\pi <math> (a and b) and </math>\pi /2<math> (c and d) and </math>
| |
| − | \gamma =0<math> (a and c) and </math>0.005</math> (b and d).}
| |
| − | (jgrscatvarybeta_revised)
| |
| − | \end{figure}
| |
| − | total scattered energy (the integral over all angles<math>)</math> as a function of <math>
| |
| − | \beta <math> for the four floe geometries. The wavenumber was </math>\alpha =\pi </math> (a
| |
| − | and b) and <math>\alpha =\pi /2</math> (c and d) with <math>\gamma =0</math> (a and c) and <math>\gamma
| |
| − | =0.005</math> (b and d). This figure shows that there is strong dependence of the
| |
| − | scattered energy on the parameter <math>\beta .\ </math>There is also a clear value of <math>
| |
| − | \beta ,</math> below which floe geometry is unimportant, and above which floe
| |
| − | geometry has a significant effect<math>.</math> Furthermore, for geometries and
| |
| − | wavelengths which are of a similar size (a and b) there is a stiffness of
| |
| − | maximum scattering. In the cases of non-zero mass <math>(b</math> and <math>d)</math> the low <math>
| |
| − | \beta </math> limit corresponds to the mass loading model of [[Keller53]]
| |
| − | which assumes zero floe stiffness.
| |
| − | | |
| − | ==Time-Averaged Force==
| |
| − | | |
| − | The time-averaged wave force on the floe is second order but it can be
| |
| − | determined by the first order solution we have calculated. The average force
| |
| − | components, for a wave of unit amplitude, in the <math>x</math> and <math>y</math> directions are
| |
| − | given by,
| |
| − | <center><math>
| |
| − | \frac{X_{av}}{A^{2}\rho a^{3}g}=\frac{\alpha ^{2}}{8\pi A^{2}}
| |
| − | \int_{0}^{2\pi }|H(\tau )|^{2}\cos \tau d\tau +\cos \theta \frac{\sqrt{
| |
| − | \alpha }}{2A^{2}}{Im}\left[ H(\pi +\theta )\right] ,
| |
| − | </math></center>
| |
| − | <center><math>
| |
| − | \frac{Y_{av}}{A^{2}\rho a^{3}g}=\frac{\alpha ^{2}}{8\pi A^{2}}
| |
| − | \int_{0}^{2\pi }|H(\tau )|^{2}\sin \tau d\tau +\sin \theta \frac{\sqrt{
| |
| − | \alpha }}{2A^{2}}\;{Im}\left[ H(\pi +\theta )\right] , (forcey)
| |
| − | </math></center>
| |
| − | [[Newman67,Maruo]]. There is a large variation in these components of
| |
| − | force as a function of incoming wave direction and we therefore consider the
| |
| − | total force <math>F</math> given by
| |
| − | <center><math>
| |
| − | F=\sqrt{\left( X_{av}\right) ^{2}+\left( Y_{av}\right) ^{2}}.
| |
| − | </math></center>
| |
| − | Figures (forcetotallam2_revised)
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{forcetotallam2_revised.eps}
| |
| − | \end{center}
| |
| − | \caption{{}The total force as a function of incoming waveangle for ice floe
| |
| − | geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). <math>
| |
| − | \beta =0.0004<math> (a), 0.002 (b), 0.01 (c), and 0.05 (d). </math>\gamma =0.005
| |
| − | <math> and </math>\alpha =\pi .</math>}
| |
| − | (forcetotallam2_revised)
| |
| − | \end{figure}
| |
| − | shows the total force as a function of incident wave angle for the four
| |
| − | values of stiffness <math>\beta =0.0004</math> (d), 0.002 (b), 0.01 (c), and <math>0.05</math>
| |
| − | (d). The wavenumber is <math>\alpha =\pi </math> and the mass is <math>\gamma =0.005.</math> Like
| |
| − | the scattering, below some value of <math>\beta </math> the force is independent of
| |
| − | geometry and above it geometry is significant. Figure \ref
| |
| − | {forcetotallam4_revised}
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{forcetotallam4_revised.eps}
| |
| − | \end{center}
| |
| − | \caption{{}The total force as a function of incoming waveangle for ice floe
| |
| − | geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). <math>
| |
| − | \beta =0.004<math> (a), 0.02 (b), 0.1 (c), and 0.5 (d). </math>\gamma =0.005</math>
| |
| − | and <math>\alpha =\pi /2.</math>}
| |
| − | (forcetotallam4_revised)
| |
| − | \end{figure}
| |
| − | shows the total force as a function of incident wave angle for the four
| |
| − | values of stiffness <math>\beta =0.004</math> (d), 0.02 (b), 0.1 (c), and <math>0.5</math> (d).
| |
| − | The wavenumber is <math>\alpha =\pi /2</math> and the mass is <math>\gamma =0.005.</math> Again
| |
| − | the variation of force with floe geometry is significant only for higher
| |
| − | stiffness. However, the variation in force with waveangle is less for the
| |
| − | smaller wavenumber (i.e. less in figure (forcetotallam4_revised) than in
| |
| − | figure (forcetotallam2_revised)).
| |
| − | | |
| − | The final component of force which we consider is the average yaw moment on
| |
| − | the body which is given by,
| |
| − | <center><math>
| |
| − | \frac{M_{av}}{\rho a^{2}g}=-\frac{\alpha }{8\pi }{Im}\left[
| |
| − | \int_{0}^{2\pi }H^{\ast }(\tau )H^{\prime }(\tau )d\tau \right] -\frac{1}{2
| |
| − | \sqrt{\alpha }}{Re}\left[ H^{\prime }(\pi +\theta )\right]
| |
| − | (forcexy)
| |
| − | </math></center>
| |
| − | [[Newman67]]. This moment is generally neglected in calculations of ice
| |
| − | floe motions. However the twisting of ice floes may be a significant cause
| |
| − | of floe collisions and may influence the total scattering by acting to align
| |
| − | the floes. Figures (jgryawlam2_revised) and (jgryawlam4_revised)
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{jgryawlam2_revised.eps}
| |
| − | \end{center}
| |
| − | \caption{{}The yaw moment as a function of incoming waveangle for ice floe
| |
| − | geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). <math>
| |
| − | \beta =0.0004<math> (a), 0.002 (b), 0.01 (c), and 0.05 (d). </math>\gamma =0.005
| |
| − | <math> and </math>\alpha =\pi .</math>}
| |
| − | (jgryawlam2_revised)
| |
| − | \end{figure}
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{jgryawlam4_revised.eps}
| |
| − | \end{center}
| |
| − | \caption{{}The yaw moment as a function of incoming waveangle for ice floe
| |
| − | geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). <math>
| |
| − | \beta =0.004<math> (a), 0.02 (b), 0.1 (c), and 0.5 (d). </math>\gamma =0.005</math>
| |
| − | and <math>\alpha =\pi /2.</math>}
| |
| − | (jgryawlam4_revised)
| |
| − | \end{figure}
| |
| − | show the yaw moment for the four ice floe geometries for mass <math>\gamma =0.005</math>
| |
| − | and wavenumber <math>\alpha =\pi </math> (figure (jgryawlam2_revised)) and <math>\alpha
| |
| − | =\pi /2<math> (figure (jgryawlam4_revised)). The stiffness is </math>\beta =0.0004</math>
| |
| − | (d), 0.002 (b), 0.01 (c), and <math>0.05</math> (d) (figure (jgryawlam2_revised))
| |
| − | and <math>\beta =0.004</math> (d), 0.02 (b), 0.1 (c), and <math>0.5</math> (d) (figure \ref
| |
| − | {jgryawlam4_revised})<math>.</math> The variation of yaw is markedly different for
| |
| − | different geometries regardless of the wavenumber and stiffness. This is to
| |
| − | be expected since the yaw moment is strongly dependent on the geometry of
| |
| − | the floe. The yaw figures are hard to interpret but it is apparent that for
| |
| − | certain geometries the yaw is a significant force.
| |
| − | | |
| − | ==Geophysical Implications==
| |
| − | | |
| − | It is well known that significant wave scattering occurs in the marginal ice
| |
| − | zone and that scattering plays an important role in controlling the break up
| |
| − | of pack ice. This wave scattering is controlled by the scattering from
| |
| − | individual ice floes. We are now in a position to better understand wave
| |
| − | scattering in the MIZ because we have a realistic model for individual ice
| |
| − | floe scattering. This ice floe model is required for any MIZ scattering
| |
| − | model such as the ones developed by [[Masson_Le]] and [[jgrrealism]]
| |
| − | . The ice floe model also allows the wave induced forces on ice floes to be
| |
| − | calculated. This allows the drift and inter-floe stress to be determined.
| |
| − | | |
| − | From figures (motion1_revised) to (motion4_revised) it is apparent
| |
| − | that measurements of strain, displacement or acceleration on an ice floe
| |
| − | will be highly dependent on ice floe geometry and should be interpreted with
| |
| − | extreme caution. Furthermore, if measurements are to be made, displacement
| |
| − | or acceleration measurements will work better than strain because the
| |
| − | variation in displacement is less than the variation in the second
| |
| − | derivative of displacement.
| |
| − | | |
| − | Figure (jgrscataveragelam2_revised) to (forcetotallam4_revised)
| |
| − | reveal that, for a given floe size, it is the ice floe stiffness, which
| |
| − | depends predominantly on the floe thickness and to a lesser extent on the
| |
| − | Young's modulus, which is the principle determinant of scattered energy and
| |
| − | force. This means that these parameters, as well as average floe size, must
| |
| − | be determined to accurately characterise the MIZ
| |
| − | | |
| − | Furthermore, figures (jgrscataveragelam2_revised) to \ref
| |
| − | {forcetotallam4_revised} show that there is a critical stiffness, below
| |
| − | which the scattering and force do not depend on floe geometry, and above
| |
| − | which the scattering and force do depend on floe geometry. The wave
| |
| − | scattering or floe drift can therefore be calculated by considering a simple
| |
| − | floe geometry (such as a circle) only for stiffness below the critical
| |
| − | stiffness. It is interesting to consider some "typical" ice floes and to
| |
| − | examine whether they lie above or below this critical stiffness. We take the
| |
| − | following as the floe parameters, the density <math>\rho _{i}=922.5</math>kgm<math>^{-3},</math>
| |
| − | the Young's modulus <math>E=6</math>GPa, Poisson's ratio <math>\nu =0.3</math> and floe thickness <math>
| |
| − | h=1<math>m. If we consider a 40,000m</math>^{2}<math> floe then </math>\beta =<math>5.4752</math>\times
| |
| − | 10^{-4}</math> which puts it below the critical value for the wavelengths in
| |
| − | figures (jgrscatvarybeta_revised) (which are <math>\lambda =100</math>m (a and b)
| |
| − | and <math>200</math>m (c and d)). However for a 10,000m<math>^{2}</math> floe <math>\beta =</math>8.7604<math>
| |
| − | \times 10^{-2}<math> which is above the critical value of </math>\beta <math> for </math>\lambda
| |
| − | =100<math>m and below for </math>\lambda =200</math>m. Therefore typical floes lies in both
| |
| − | regions and care must be taken if only a simple floe geometry is to be
| |
| − | considered.
| |
| − | | |
| − | ==Summary==
| |
| − | | |
| − | The wave induced motion of a flexible ice floe of arbitrary geometry has
| |
| − | been calculated. This solution was based on substituting the free modes of
| |
| − | vibration of the ice floe into the integral equation which describes the
| |
| − | water motion. Solutions were presented for four ice floe geometries and for
| |
| − | two wavenumbers. The complex nature of the motion of the ice floes was
| |
| − | apparent as was the significance of flexure.
| |
| − | | |
| − | The scattered energy was calculated and it was shown that the scattering was
| |
| − | most strongly dependent on ice floe stiffness. Further, it was shown that
| |
| − | there exists a critical value of stiffness, below which the scattered energy
| |
| − | is not a significant function of ice floe geometry, and above which the
| |
| − | average scattering is a significant function of ice floe geometry.
| |
| − | | |
| − | Finally the time averaged forces acting on the ice floe were calculated. The
| |
| − | total force showed a strong dependence on ice floe stiffness and also had a
| |
| − | critical stiffness value above which floe geometry become significant. The
| |
| − | results for the yaw moment were more difficult to interpret but showed that
| |
| − | for certain ice floe geometries this force is significant.
| |
| − | | |
| − | \bibliographystyle{agu}
| |
| − | \bibliography{mike,others}
| |
| − | \pagebreak
| |
| − | | |
| − | {\Large Figure Captions}
| |
| − | | |
| − | \textsc{Figure} 1. {The schematic diagram of the boundary value problem and
| |
| − | the coordinate system used in the solution.}
| |
| − | | |
| − | \textsc{Figure} 2. The four ice floe geometries for which solutions will be
| |
| − | calculated and their numbering.{\ }
| |
| − | | |
| − | \textsc{Figure} 3. The displacement of an ice floe of geometry 1 for the
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| − | times and stiffness shown. The mass was <math>\gamma =0.005</math>, the wavenumber was <math>
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| − | \alpha =\pi <math> and the wave was travelling in the positive </math>x</math>-direction.
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| − | | |
| − | \textsc{Figure} 4. The displacement of an ice floe of geometry 2 for the
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| − | times and stiffness shown. The mass was <math>\gamma =0.005</math>, the wavenumber was <math>
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| − | \alpha =\pi <math> and the wave was travelling in the positive </math>x</math>-direction.
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| − | | |
| − | \textsc{Figure} 5. The displacement of an ice floe of geometry 3 for the
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| − | times and stiffness shown. The mass was <math>\gamma =0.005</math>, the wavenumber was <math>
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| − | \alpha =\pi <math> and the wave was travelling in the positive </math>x</math>-direction.
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| − | | |
| − | \textsc{Figure} 6. The displacement of an ice floe of geometry 4 for the
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| − | times and stiffness shown. The mass was <math>\gamma =0.005</math>, the wavenumber was <math>
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| − | \alpha =\pi <math> and the wave was travelling in the positive </math>x</math>-direction.
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| − | | |
| − | \textsc{Figure} 7. The scattering as a function of angle for ice floe
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| − | geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). The values of
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| − | the stiffness were <math>\beta =</math>0.0004 (a), 0.002 (b), 0.01 (c), and 0.05 (d). <math>
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| − | \alpha =\pi <math> and </math>\gamma =0.005.</math>
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| − | | |
| − | \textsc{Figure} 8. The total scattering as a function of <math>\beta </math> for ice
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| − | floe geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). <math>\alpha
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| − | =\pi <math> (a and b) and </math>\pi /2<math> (c and d) and </math>\gamma =0<math> (a and c) and </math>0.005</math>
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| − | (b and d).
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| − | | |
| − | \textsc{Figure} 9. The total force as a function of incoming waveangle for
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| − | ice floe geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). <math>
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| − | \beta =0.0004<math> (a), 0.002 (b), 0.01 (c), and 0.05 (d). </math>\gamma =0.005<math> and </math>
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| − | \alpha =\pi .</math>
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| − | | |
| − | \textsc{Figure} 10. The total force as a function of incoming waveangle for
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| − | ice floe geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). <math>
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| − | \beta =0.004<math> (a), 0.02 (b), 0.1 (c), and 0.5 (d). </math>\gamma =0.005<math> and </math>
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| − | \alpha =\pi /2.</math>
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| − | | |
| − | \textsc{Figure} 11. The yaw moment as a function of incoming waveangle for
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| − | ice floe geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). <math>
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| − | \beta =0.0004<math> (a), 0.002 (b), 0.01 (c), and 0.05 (d). </math>\gamma =0.005<math> and </math>
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| − | \alpha =\pi .</math>
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| − | | |
| − | \textsc{Figure} 12. The yaw moment as a function of incoming waveangle for
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| − | ice floe geometries 1 (solid), 2 (dashed), 3 (chained), and 4 (dotted). <math>
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| − | \beta =0.004<math> (a), 0.02 (b), 0.1 (c), and 0.5 (d). </math>\gamma =0.005<math> and </math>
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| − | \alpha =\pi /2.</math>
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