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== Standard notation for water depth ==


\paperid{}
Should we have a standard notation for water depth?
\ccc{}
\lefthead{MEYLAN}
\righthead{Wave Response of an Ice Floe}
\received{November 2000}
\revised{May 2001}
\accepted{??}
\authoraddr{M. H. Meylan, Institute of Information and Mathematical Sciences, Massey
University, Auckland, New Zealand (email:~m.h.meylan@massey.ac.nz)}
\slugcomment{Submitted to ''Journal of Geophysical Research'' , November 2000.}
= The Wave Response of an Ice Floe of Arbitrary Geometry. =
Michael H. Meylan
\affil{Institute of Information and Mathematical Sciences, Massey University,
Auckland, New Zealand}


I have started the following page


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
induced force for a realistic ice floe to be calculated. These are required
to model wave scattering and wave induced ice drift in the marginal ice
zone. The ice floe is modelled as a thin plate and its motion is expanded in
the thin plate modes of vibration. The modes are substituted into the
integral equation for the water. This gives a linear system of equations for
the coefficients used to expand the ice floe motion. Solutions are presented
for the ice floe displacement, the scattered energy, and the time averaged
force for a range of ice floe geometries and wave periods. It is found that
ice floe stiffness is the most important factor in determining ice floe
motion, scattering, and force. However, above a critical value of stiffness
the floe geometry also influences the scattering and force.


 
[[Category:Administration]]
TCIMACRO{\TeXButton{Begin Article}{}}
BeginExpansion
 
EndExpansion
 
==Introduction==
 
It is widely recognized that understanding the relationship between ocean
waves and sea ice requires a fully three-dimensional model for the motion
and bending of a solitary ice floe [[Squire_Review]]. Such an ice floe
model is derived in this paper. The model includes flexure and is for an ice
floe of arbitrary shape. It allows the wave scattering and wave induced
force for a realistic ice floe to be calculated for the first time.
 
The Marginal Ice Zone (MIZ) is an interfacial region, composed of an
aggregate of ice and water, which forms at the boundary of open and ice
covered seas. The major interaction between open and ice covered seas is
through waves. These waves are generated in the open water and are
responsible for breaking up the continuous pack ice. For waves to reach the
continuous pack ice they must pass through the MIZ.
 
Experimental measurements have shown that as ocean waves pass through the
MIZ their character is radically altered. [[attenuation88]] and
[[attenuationNature80]] found that there was strong and significant
exponential attenuation of wave energy. This attenuation decreased with
increasing wave period. The angular spreading of incoming ocean waves was
measured by [[directional86]] who found that from a narrow directional
spectrum in the open water the directional wave spectrum evolves to become
isotropic within the MIZ. Experimental measurements of the motion of
individual ice floes have shown that ice floe bending is significant.
[[Squireicefloe]] and [[SquireMartin]] measured the motion of ice
floes using strain gauges. They showed that ice floe flexure is significant
and must be included in ice floe models.
 
[[Masson_Le]] and [[jgrrealism]] have derived models for the
propagation of waves through the MIZ, based on the linear transport
equation. These models predict the exponential decay and directional
spreading of the incoming wave spectrum qualitatively. However, they were
quantitatively inaccurate because neither of them used a realistic model to
calculate the individual ice floe scattering. This was because no realistic
model existed. [[Masson_Le]] and [[Massondrift]] assumed that the
ice floe was a rigid body. [[Wadhams1986]] included ice floe flexure but
the model was two dimensional and the solution was only calculated
approximately. While \citet {jgrfloe1d} solved the Wadhams problem exactly,
two dimensional models give no information about the directional scattering
so cannot be used in wave scattering models. \citet {jgrfloecirc} extended
the two dimensional model to three dimensions but only for the unrealistic
case of a circular ice floe. This was the ice floe model used in
[[jgrrealism]].
 
In this paper a fully three-dimensional model for the motion and bending of
a solitary ice floe is developed as follows. The ice floe motion is expanded
in the free modes of vibration (the modes of vibration of the ice floe in
the absence of the water). These modes must be determined numerically and
this is done by using the finite element method. The equations of motion for
the water are transformed into an integral equation over the wetted surface
of the ice floe. This transformation was developed by [[john1,john2]]
and is standard in offshore engineering [[Sarp_Isa]]. The free modes of
vibration are then substituted into the integral equation for the water.
This gives a linear system of equations for the coefficients used to expand
the floe motion.
 
Solutions are presented for the ice floe displacements, the scattering of
wave energy, and the wave induced force for a range of ice floe geometries
and wave periods. The scattering is considered because of its importance in
MIZ scattering models [[Masson_Le,jgrrealism]]. The wave induced force
is included because of the importance of this term in models of ice floe
drift.
 
==The Equation of Motion for the Ice Floe==
 
Ice floes range in size from much smaller to much larger than the dominant
wavelength of the ocean waves. However there are two reasons why solutions
for ice floes of intermediate size (a size similar to the wavelength) are
the most important. The first is that at these intermediate sizes ice floes
scatter significant wave energy. The second is that, since it is wave
induced flexure which determines the size of ice floes in the MIZ, ice floes
tend to form most often at this intermediate length.
 
The theory for an ice floe of intermediate size which is developed in this
paper obviously also applies to small or large floes. However, if the
solution for a small or large floe is required then the appropriate simpler
theory should be used. Small ice floes (ice floes much small than the
wavelength) should be modelled as rigid [[Masson_Le,Massondrift]]. Large
ice floes (ice floes much larger than the wavelength) should be modelled as
infinite and flexible [[FoxandSquire]]. In the intermediate region,
where the size of the wavelength is similar to the size of the ice floe, the
ice floe must be modelled as finite and flexible.
 
We model the ice floe as a thin plate of constant thickness and shallow
draft following [[Wadhams1986]] and [[Squire_Review]]. The thin
plate equation [[Hildebrand65]] gives the following equation of motion
for the ice floe
<center><math>
D\nabla ^{4}W+\rho _{i}h\frac{\partial ^{2}W}{\partial t^{2}}=p,
(plate)
</math></center>
where <math>W</math> is the floe displacement, <math>\rho _{i}</math> is the floe density, <math>h</math> is
the floe thickness, <math>p</math> is the pressure, and <math>D</math> is the modulus of rigidity
of the floe (<math>D=Eh^{3}/12(1-\nu ^{2})</math> where <math>E</math> is the Young's modulus and <math>
\nu <math> is Poisson's ratio). Visco-elastic effects can be included by making </math>
D </math> have some imaginary (damping)\ component but this will not be done here
to keep the presented results simpler. We assume that the plate is in
contact with the water at all times so that the water surface displacement
is also <math>W.</math> Equation ((plate)) is subject to the free edge boundary
conditions for a thin plate
<center><math>
\frac{\partial ^{2}W}{\partial n^{2}}+\nu \frac{\partial ^{2}W}{\partial
s^{2}}=0,\;\;\;=\textrm{and= }\mathrm{\;\;\;}\frac{\partial ^{3}W}{
\partial n^{3}}+\left( 2-\nu \right) \frac{\partial ^{3}W}{\partial
n\partial s^{2}}=0,  (boundaryplate)
</math></center>
[[Hildebrand65]] where <math>n</math> and <math>s</math> denote the normal and tangential
directions respectively.
 
The pressure, <math>p</math>, is given by the linearized Bernoulli's equation at the
water surface,
<center><math>
p=-\rho \frac{\partial \Phi }{\partial t}-\rho gW  (pressure)
</math></center>
where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density
of the water, and <math>g</math> is the acceleration due to gravity.
 
We now introduce non-dimensional variables. We non-dimensionalise the length
variables with respect to <math>a</math> where the surface area of the floe is <math>4a^{2}.</math>
We non-dimensionalise the time variables with respect to <math>\sqrt{g/a}</math> and
the mass variables with respect to <math>\rho a^{3}</math>. The non-dimensional
variables, denoted by an overbar, are
<center><math>
\bar{x}=\frac{x}{a},\;\;\bar{y}=\frac{y}{a},\;\;\bar{z}=\frac{z}{a},\;\;\bar{
W}=\frac{W}{a},\;\;\bar{t}=t\sqrt{\frac{g}{a}},\;\;=and= \;\;\bar{\Phi}=
\frac{\Phi }{a\sqrt{ag}}.
</math></center>
In the non-dimensional variables equations ((plate)) and ((pressure)
) become
<center><math>
\beta \nabla ^{4}\bar{W}+\gamma \frac{\partial ^{2}\bar{W}}{\partial \bar{t}
^{2}}=\frac{\partial \bar{\Phi}}{\partial \bar{t}}-\bar{W},  (n-d_ice)
</math></center>
where
<center><math>
\beta =\frac{D}{g\rho a^{4}}\;\;=and\ \ = \gamma =\frac{\rho _{i}h}{\rho
a}.
</math></center>
We shall refer to <math>\beta </math> and <math>\gamma </math> as the stiffness and mass
respectively.
 
We will determine the response of the ice floe to wave forcing of a single
frequency (the response for more complex wave forcing can be found by
superposition of the single frequency solutions). Since the equations of
motion are linear the displacement and potential must have the same single
frequency dependence. Therefore they can be expressed as the real part of a
complex quantity whose time dependence is <math>e^{-i\sqrt{\alpha }t}</math>
where
<math>\alpha </math> is the non-dimensional wavenumber and we write <math>\bar{W}(\bar{x},
\bar{y},\bar{t})={Re}\left[ w\left( \bar{x},\bar{y}\right) e^{-i\sqrt{
\alpha }\bar{t}}\right] \ <math>and</math>\;\Phi (\bar{x},\bar{y},\bar{z},\bar{t})=
{Re}\left[ \phi \left( \bar{x},\bar{y},\bar{z}\right) e^{-i\sqrt{\alpha
}\bar{t}}\right] .</math> In the complex variables the equation of motion of the
ice floe ((n-d_ice)) is
<center><math>
\beta \nabla ^{4}w+\alpha \gamma w=\sqrt{\alpha }\phi -w.  (plate2)
</math></center>
From now on we will drop the overbar and assume all variables are
non-dimensional.
 
==Equations of Motion for the Water==
 
We require the equation of motion for the water to solve equation (\ref
{plate2}). We begin with the non-dimensional equations of potential theory
which describe linear surface gravity waves
<center><math>  (bvp)
\left.
\begin{matrix}{rr}
\nabla ^{2}\phi =0, & -\infty <z<0, \\
{\frac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\
{\frac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\;
\mathbf{x}\in \Delta , \\
{\frac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{
x}\notin \Delta ,
\end{matrix}
\right\}    (bvp_nond)
</math></center>
([[Weh_Lait]]). As before, <math>w</math> is the displacement of the floe and <math>p</math>
is the pressure at the water surface. The vector <math>\mathbf{x=(}x,y)</math> is a
point on the water surface and <math>\Delta </math> is the region of the water surface
occupied by the floe. The water is assumed infinitely deep. A schematic
diagram of this problem is shown in Figure (vibration).
\begin{figure}[tbp]
\begin{center}
\epsfbox{vibration.eps}
\end{center}
\caption{{The schematic diagram of the boundary value problem and the
coordinate system used in the solution.}}
(vibration)
\end{figure}
 
The boundary value problem ((bvp)) is subject to an incident wave which
is imposed through a boundary condition as <math>\left| \mathbf{x}\right|
\rightarrow \infty </math>. This boundary condition, which is called the
Sommerfeld radiation condition, is essentially that at large distances the
potential consists of a radial outgoing wave (the wave generated by the ice
floe motion) and the incident wave. It is expressed mathematically as
<center><math>
\lim_{\left| \mathbf{x}\right| \rightarrow \infty }\sqrt{|\mathbf{x}|}\left(
\frac{\partial }{\partial |\mathbf{x}|}-i\alpha \right) (\phi -\phi ^{
\mathrm{In}})=0,  (summerfield)
</math></center>
[[Weh_Lait]]. The incident potential (i.e. the incoming wave) <math>\phi ^{
\mathrm{In}}</math> is
<center><math>
\phi ^{\mathrm{In}}(x,y,z)=\frac{A}{\sqrt{\alpha }}e^{i\alpha (x\cos \theta
+y\sin \theta )}e^{\alpha z},  (input)
</math></center>
where <math>A</math> is the non-dimensional wave amplitude.
 
The standard solution method to the linear wave problem is to transform the
boundary value problem into an integral equation using a Green function
\citep{john1,
john2,Sarp_Isa,jgrfloecirc}. Performing such a transformation, the boundary
value problem ((bvp)) and ((summerfield)) becomes
<center><math>
\phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x}
;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x}
)\right) dS_{\mathbf{y}}.  (water)
</math></center>
The Green function <math>G_{\alpha }</math> is
<center><math>
G_{\alpha }(\mathbf{x};\mathbf{y)}=\frac{1}{4\pi }\left( \frac{2}{|\mathbf{x}
-\mathbf{y}|}-\pi \alpha \left( \mathbf{H_{0}}(\alpha |\mathbf{x}-\mathbf{y}
|)+Y_{0}(\alpha |\mathbf{x}-\mathbf{y}|)\right) +2\pi i\alpha J_{0}(\alpha |
\mathbf{x}-\mathbf{y}|)\right) ,
</math></center>
[[Weh_Lait,jgrfloecirc]], where <math>J_{0}</math> and <math>Y_{0}</math> are respectively
Bessel functions of the first and second kind of order zero, and <math>\mathbf{
H_{0}}</math> is the Struve function of order zero [[abr_ste]]. A solution for
water of finite depth could be found by simply using the depth dependent
Green function [[Weh_Lait]].
 
The integral equation ((water)) will be solved using numerical
integration. The only difficulty arises from the non-trivial nature of the
kernel of the integral equation (the Green function). However, the Green
function has no <math>z</math> dependence due to the shallow draft approximation and
depends only on <math>|\mathbf{x}-\mathbf{y}|.</math> This means that the Green
function is one dimensional and the values which are required for a given
calculation can be looked up in a previously computed table.
 
==Solving for the Wave Induced Ice Floe Motion==
 
To determine the ice floe motion we must solve equations ((plate2)) and (
(water)) simultaneously. We do this by expanding the floe motion in the
free modes of vibration of a thin plate. The major difficulty with this
method is that the free modes of vibration can be determined analytically
only for very restrictive geometries, e.g. a circular thin plate. Even the
free modes of vibration of a square plate with free edges must be determined
numerically. This is the reason why the solution of [[jgrfloecirc]] was
only for a circular floe.
 
Since the operator <math>\nabla ^{4},</math> subject to the free edge boundary
conditions, is self adjoint a thin plate must possess a set of modes <math>w_{i}</math>
which satisfy the free boundary conditions and the following eigenvalue
equation
<center><math>
\nabla ^{4}w_{i}=\lambda _{i}w_{i}.
</math></center>
The modes which correspond to different eigenvalues <math>\lambda _{i}</math> are
orthogonal and the eigenvalues are positive and real. While the plate will
always have repeated eigenvalues, orthogonal modes can still be found and
the modes can be normalized. We therefore assume that the modes are
orthonormal, i.e.
<center><math>
\iint_{\Delta }w_{i}\left( \mathbf{Q}\right) w_{j}\left( \mathbf{Q}\right)
dS_{\mathbf{Q}}=\delta _{ij}
</math></center>
where <math>\delta _{ij}</math> is the Kronecker delta. The eigenvalues <math>\lambda _{i}</math>
have the property that <math>\lambda _{i}\rightarrow \infty </math> as <math>i\rightarrow
\infty </math> and we order the modes by increasing eigenvalue. These modes can be
used to expand any function over the wetted surface of the ice floe <math>\Delta </math>
.
 
We expand the displacement of the floe in a finite number of modes <math>N,</math> i.e.
<center><math>
w\left( \mathbf{x}\right) =\sum_{i=1}^{N}c_{i}w_{i}\left( \mathbf{x}\right) .
(expansion)
</math></center>
From the linearity of ((water)) the potential can be written in the
following form
<center><math>
\phi =\phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i}  (expansionphi)
</math></center>
where <math>\phi _{0}</math> and <math>\phi _{i}</math> satisfy the integral equations
<center><math>
\phi _{0}(\mathbf{x})=\phi ^{\mathrm{In}}(\mathbf{x})+\iint_{\Delta }\alpha
G_{\alpha }(\mathbf{x};\mathbf{y})\phi (\mathbf{y})dS_{\mathbf{y}}
(phi0)
</math></center>
and
<center><math>
\phi _{i}(\mathbf{x})=\iint_{\Delta }G_{\alpha }(\mathbf{x};\mathbf{y}
)\left( \alpha \phi _{i}(\mathbf{x})+i\sqrt{\alpha }w_{i}(\mathbf{y})\right)
dS_{\mathbf{y}}.  (phii)
</math></center>
The potential <math>\phi _{0}</math> represents the potential due the incoming wave
assuming that the displacement of the ice floe is zero. The potentials <math>\phi
_{i}</math> represent the potential which is generated by the plate vibrating with
the <math>i</math>th mode in the absence of any input wave forcing.
 
We substitute equations ((expansion)) and ((expansionphi)) into
equation ((plate2)) to obtain
<center><math>
\beta \sum_{i=1}^{N}\lambda _{i}c_{i}w_{i}-\alpha \gamma
\sum_{i=1}^{N}c_{i}w_{i}=i\sqrt{\alpha }\left( \phi
_{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>
\beta \lambda _{j}c_{j}+\left( 1-\alpha \gamma \right) c_{j}=\iint_{\Delta }i
\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(
\mathbf{Q}\right) dS_{\mathbf{Q}}  (final)
</math></center>
which is a matrix equation in <math>c_{i}.</math>
 
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
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.
 
\textsc{Figure} 4. 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.
 
\textsc{Figure} 5. 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.
 
\textsc{Figure} 6. 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.
 
\textsc{Figure} 7. The scattering as a function of angle for ice floe
geometries 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>
 
\textsc{Figure} 8. 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).
 
\textsc{Figure} 9. 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>
 
\textsc{Figure} 10. 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>
 
\textsc{Figure} 11. 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>
 
\textsc{Figure} 12. 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>
 
TCIMACRO{\TeXButton{End Article}{}}
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Latest revision as of 03:36, 22 August 2008

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