Difference between revisions of "Linear Boltzmann Model for Wave Scattering in the MIZ"

From WikiWaves
Jump to navigationJump to search
 
(21 intermediate revisions by 2 users not shown)
Line 1: Line 1:
=Introduction=
+
{{incomplete pages}}
 +
 
 +
==Introduction==
  
 
We present a linear Boltzmann equation to model
 
We present a linear Boltzmann equation to model
[[Wave Scattering in the Marginal Ice Zone]]. This model
+
[[:Category:Wave Scattering in the Marginal Ice Zone|Wave Scattering in the Marginal Ice Zone]]. This model
 
is three dimensional and requires the  
 
is three dimensional and requires the  
 
three-dimensionao solution for a [[Floating Elastic Plate]].
 
three-dimensionao solution for a [[Floating Elastic Plate]].
This model was given in [[Meylan_Masson_2006a | Meylan and Masson 2006]]
+
This model was given in [[Meylan and Masson 2006]]
 
(which also showed the equivalence with [[The Multiple Scattering Theory of Masson and LeBlond]]).
 
(which also showed the equivalence with [[The Multiple Scattering Theory of Masson and LeBlond]]).
The derivation closely follows that of [[Meylan_Squire_Fox_1997a|Meylan, Squire and Fox 1997]]
+
The derivation closely follows that of [[Meylan, Squire, and Fox 1997]]
 
with a correction.
 
with a correction.
  
Line 34: Line 36:
 
are not significant for the MIZ.
 
are not significant for the MIZ.
  
=The Linear Boltzmann equation for Wave Scattering in the MIZ.=
+
==The Linear Boltzmann equation for Wave Scattering in the MIZ==
  
 
In this section we present a derivation of the linear Boltzmann equation  
 
In this section we present a derivation of the linear Boltzmann equation  
 
for wave scattering in the MIZ that follows closely the
 
for wave scattering in the MIZ that follows closely the
derivation given in [[Meylan_Squire_Fox_1997a | Meylan, Squire and Fox 1997]] but corrects an error in this
+
derivation given in [[Meylan, Squire, and Fox 1997]] but corrects an error in this
 
earlier derivation. The linear Boltzmann equation is applicable  
 
earlier derivation. The linear Boltzmann equation is applicable  
 
to the propagation of wave energy through the MIZ over  
 
to the propagation of wave energy through the MIZ over  
Line 56: Line 58:
 
<math>
 
<math>
 
\frac{1}{c_{g}}\frac{\partial}{\partial t}I(r,t,\theta)+\hat{\theta}.\nabla
 
\frac{1}{c_{g}}\frac{\partial}{\partial t}I(r,t,\theta)+\hat{\theta}.\nabla
I(r,t,\theta)=0,
+
I(r,t,\theta)=0,\,\,\,(1)
 
</math>
 
</math>
  
(\cite{phillips77}) where <math>\hat{\theta}</math> is a unit vector in the <math>\theta</math> direction and  <math>c_{g}</math> is the speed of wave propagation (the deep water group speed). The
+
where <math>\hat{\theta}</math> is a unit vector in the <math>\theta</math> direction and  <math>c_{g}</math> is the speed of wave propagation (the deep water group speed). The
 
presence of the floes will modify this expression by scattering energy, i.e.
 
presence of the floes will modify this expression by scattering energy, i.e.
 
by changing the direction in which the energy is travelling.  
 
by changing the direction in which the energy is travelling.  
  
We modify equation (\ref{no_scattering}) to take into account the scattering
+
We modify equation (1) to take into account the scattering
 
effects of the ice floes using the
 
effects of the ice floes using the
 
general  equation for the propagation of wave energy through a  
 
general  equation for the propagation of wave energy through a  
Line 71: Line 73:
 
\frac{1}{c_{g}}\frac{\partial}{\partial t}I(r,t,\theta)+\hat{\theta}.\nabla
 
\frac{1}{c_{g}}\frac{\partial}{\partial t}I(r,t,\theta)+\hat{\theta}.\nabla
 
I(r,t,\theta)=-\beta(r,\theta)I(r,t,\theta)+\int_{0}^{2\pi}S(r,\theta,\theta
 
I(r,t,\theta)=-\beta(r,\theta)I(r,t,\theta)+\int_{0}^{2\pi}S(r,\theta,\theta
^{\prime})I(r,t,\theta^{\prime})d\theta^{\prime},  
+
^{\prime})I(r,t,\theta^{\prime})d\theta^{\prime},\,\,\,(2)
 
</math>
 
</math>
  
(\cite{howells60}) where </math>\beta</math> is the absorption coefficient and  
+
([[Howells 1960]]) where <math>\beta</math> is the absorption coefficient and  
</math>S</math> is the scattering
+
<math>S</math> is the scattering
 
function (assumed to be independent of time).
 
function (assumed to be independent of time).
Equation~(\ref{Howells}) depends on the assumption that
+
Equation (2) depends on the assumption that
 
each floe scatters independently
 
each floe scatters independently
 
and that the energy from different scatterers may be added
 
and that the energy from different scatterers may be added
Line 88: Line 90:
  
 
<math>
 
<math>
S(r,\theta,\theta^{\prime})I(r,t,\theta^{\prime})d\Omega dSd\Omega
+
S(r,\theta,\theta^{\prime})I(r,t,\theta^{\prime})d\Omega dSd\Omega^{\prime},,\,\,\,(3)
^{\prime},
 
 
</math>
 
</math>
  
Line 96: Line 97:
 
in direction <math>\theta^{\prime}</math>, by a surface <math>dS</math> at position <math>r</math>, into an
 
in direction <math>\theta^{\prime}</math>, by a surface <math>dS</math> at position <math>r</math>, into an
 
angle <math>d\Omega</math> in direction <math>\theta</math>.
 
angle <math>d\Omega</math> in direction <math>\theta</math>.
To apply equation (\ref{Howells}) to wave scattering in the MIZ we must
+
To apply equation (2) to wave scattering in the MIZ we must
 
first estimate the scattering
 
first estimate the scattering
 
function <math>S(r,\theta,\theta^{\prime})</math> and
 
function <math>S(r,\theta,\theta^{\prime})</math> and
 
the absorption coefficient <math>\beta(r,\theta)\,</math>.
 
the absorption coefficient <math>\beta(r,\theta)\,</math>.
  
 
+
=== Finding the scattering and absorbtion functions ===
== Finding the scattering and absorbtion functions ==
 
  
 
The scattering function is determined by
 
The scattering function is determined by
Line 112: Line 112:
 
<math>
 
<math>
 
E(\theta - \theta^{\prime})=\left(  \frac{H}{2}\right)  ^{2}\frac{\rho\omega
 
E(\theta - \theta^{\prime})=\left(  \frac{H}{2}\right)  ^{2}\frac{\rho\omega
g}{4k}|D(\theta-\theta^{\prime})|^{2},
+
g}{4k}|D(\theta-\theta^{\prime})|^{2},\,\,\,(4)
 
</math>
 
</math>
  
Line 125: Line 125:
  
 
<math>
 
<math>
\frac{H}{2}\frac{D\left(  \theta - \theta^{\prime}\right)  }{\sqrt{r}}.
+
\frac{H}{2}\frac{D\left(  \theta - \theta^{\prime}\right)  }{\sqrt{r}},\,\,\,(5)
 
</math>
 
</math>
  
Note that, in equations~(\ref{energyrad}) and (\ref{energyrad_D}), we have assumed
+
Note that, in equations (4) and (5), we have assumed
 
that the scattering is isotropic (depends only on the difference of angle).
 
that the scattering is isotropic (depends only on the difference of angle).
 
This will not necessarily be true for a given ice floe, but we expect this  
 
This will not necessarily be true for a given ice floe, but we expect this  
Line 134: Line 134:
 
the ice floes are oriented, and the floes are of random shape.
 
the ice floes are oriented, and the floes are of random shape.
  
We must now express the scattering kernel in equation (\ref{Howells}),  
+
We must now express the scattering kernel in equation (2),  
 
<math>S(r,\theta,\theta^{\prime})</math>,  in terms of <math>E</math>.
 
<math>S(r,\theta,\theta^{\prime})</math>,  in terms of <math>E</math>.
 
Given the definition of <math>S</math> (equation (\ref{equation_S})),  
 
Given the definition of <math>S</math> (equation (\ref{equation_S})),  
Line 152: Line 152:
 
S(\theta,\theta^{\prime})  =\left(  \frac{H}{2}\right)  ^{2}\frac{\rho\omega
 
S(\theta,\theta^{\prime})  =\left(  \frac{H}{2}\right)  ^{2}\frac{\rho\omega
 
g}{4k}|D(\theta-\theta^{\prime})|^{2}\frac{4f_{i}k}{\rho gH^{2}A_{f}\omega}
 
g}{4k}|D(\theta-\theta^{\prime})|^{2}\frac{4f_{i}k}{\rho gH^{2}A_{f}\omega}
</math>
+
=\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime})|^{2},\,\,\,(6)
 
 
<math>
 
=\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime})|^{2}.
 
 
</math>
 
</math>
  
 
This expression is not exactly the same as the equivalent
 
This expression is not exactly the same as the equivalent
expression in \cite{jgrrealism} because,
+
expression in [[Meylan, Squire, and Fox 1997]] because,
in the derivation for </math>S</math> in \cite{jgrrealism}, the
+
in the derivation for <math>S</math> in [[Meylan, Squire, and Fox 1997]], the
 
wave phase speed rather than the group speed was erroneously used.
 
wave phase speed rather than the group speed was erroneously used.
  
 
We can determine <math>\beta</math> from the absorption cross section,  
 
We can determine <math>\beta</math> from the absorption cross section,  
 
<math>\sigma_a</math>, and the ice cover fraction <math>f_i</math>
 
<math>\sigma_a</math>, and the ice cover fraction <math>f_i</math>
(M\&Le p. 58). The absorption cross section,  
+
he absorption cross section,  
 
<math>\sigma_a</math>, may be estimated from the total scattering or
 
<math>\sigma_a</math>, may be estimated from the total scattering or
 
from experimental measurements.  
 
from experimental measurements.  
Line 172: Line 169:
 
<math>
 
<math>
 
\beta = \int_{0}^{2\pi}\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime})|^{2}d\theta^{\prime}+\sigma_{a}
 
\beta = \int_{0}^{2\pi}\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime})|^{2}d\theta^{\prime}+\sigma_{a}
\frac{f_{i}}{A_{f}}.
+
\frac{f_{i}}{A_{f}},\,\,\,(7)
 
</math>  
 
</math>  
  
Combining equations (\ref{Howells}), (\ref{scatteringfromeng2})\
+
Combining equations (2), (6)
and (\ref{beta_equation}),   
+
and (7),   
 
the following linear Boltzmann equation for  
 
the following linear Boltzmann equation for  
 
wave scattering in the MIZ is obtained,
 
wave scattering in the MIZ is obtained,
Line 189: Line 186:
 
\int_{0}^{2\pi}\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime})|^{2}d\theta^{\prime}+\sigma_{a}
 
\int_{0}^{2\pi}\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime})|^{2}d\theta^{\prime}+\sigma_{a}
 
\frac{f_{i}}{A_{f}}
 
\frac{f_{i}}{A_{f}}
\right)I\left(  \theta\right).
+
\right)I\left(  \theta\right),\,\,\,(8)
 
</math>
 
</math>
  
 +
==Determining the scattering amplitude==
  
 
+
We need to determine the scattering
=Determining the scattering amplitude=
 
 
 
The central difficulty in applying equations (\ref{boltzmann}) and
 
(\ref{M_Le_boltzmann2}) is the determination of the scattering
 
 
amplitude <math>D(\theta-\theta^{\prime})</math>.  
 
amplitude <math>D(\theta-\theta^{\prime})</math>.  
 
The scattering amplitude  is found by solving the
 
The scattering amplitude  is found by solving the
Line 206: Line 200:
 
the equations of motion depends on the model used to describe
 
the equations of motion depends on the model used to describe
 
an ice floe. The principal difference between the ice floe models
 
an ice floe. The principal difference between the ice floe models
used by \cite{jgrrealism} and M\&LE is the following. \cite{jgrrealism}
+
used by [[Meylan, Squire, and Fox 1997]] and
 +
[[Masson and LeBlond 1989]] is the following. [[Meylan, Squire, and Fox 1997]]
 
assumed that the
 
assumed that the
ice floes had negligible submergence but could flex, while M\&Le
+
ice floe was a [[Floating Elastic Plate]] while [[Masson and LeBlond 1989]]
 
assumed the floes were rigid but allowed for submergence. Both
 
assumed the floes were rigid but allowed for submergence. Both
 
models have different ranges of validity (although typical
 
models have different ranges of validity (although typical
 
ice floes tend to be relatively thin).  
 
ice floes tend to be relatively thin).  
 
Of course, either ice floe model could have been used in
 
Of course, either ice floe model could have been used in
the large scale scattering models derived by M\&Le and \cite{jgrrealism}.
+
the large scale scattering models derived by [[Masson and LeBlond 1989]] and [[Meylan, Squire, and Fox 1997]].
 
Here we simply present  
 
Here we simply present  
 
the equation for <math>D(\theta)</math> independent of the equation used to model
 
the equation for <math>D(\theta)</math> independent of the equation used to model
 
the ice floe.  
 
the ice floe.  
 
The water is assumed irrotational and inviscid and the wave
 
amplitude is assumed sufficiently small that we can linearise all the
 
equations. The water motion is represented by a velocity potential which is
 
denoted by <math>\phi</math>.  The coordinate system is the standard Cartesian coordinate system
 
with the <math>z</math> axis pointing vertically up. The water occupies the region
 
<math>-\infty<z<0.</math> We denote the free surface by <math>\Gamma_s</math> (located at <math>z=0</math>)
 
and the wetted
 
surface of the ice floe by <math>\Gamma_w</math>.
 
  
 
The linearised boundary value problem for the fluid velocity
 
The linearised boundary value problem for the fluid velocity
 
potential <math>\phi({r},z)</math> subject to an incoming
 
potential <math>\phi({r},z)</math> subject to an incoming
wave of frequency <math>\omega</math> is  
+
wave of frequency <math>\omega</math> is the [[Standard Linear Wave Scattering Problem]]
 +
in infinitely deep water which we briefly recap here.
  
 
<math>
 
<math>
\nabla^{2}\phi=0, \, -\infty<z<0,
+
\nabla^{2}\phi=0, \, -\infty<z<0,\,\,\,(9)
 
</math>
 
</math>
  
Line 240: Line 227:
  
 
<math>
 
<math>
  \frac{\partial\phi}{\partial z} = k\phi\,z\in\Gamma_s
+
  \frac{\partial\phi}{\partial z} = k\phi,\,z\in\Gamma_s,
 
</math>
 
</math>
  
 
<math>
 
<math>
  \frac{\partial\phi}{\partial z} = L\phi \, z\in\Gamma_w
+
  \frac{\partial\phi}{\partial z} = L\phi, \, z\in\Gamma_w,
 
</math>
 
</math>
(e.g. \cite{jgrfloecirc,IJOPE04} or M\&Le).
+
 
The Laplace's equation comes from the fact
+
(note that since the water is assumed infinitely deep <math>k_\infty = k</math>).
that the water is irrotational and inviscid. The vanishing of the normal
+
The free surface is <math>\Gamma_s</math> (located at <math>z=0</math>)
derivative at <math>z=-\infty</math> is the no-flow through the boundary
+
and the wetted
condition at the bottom of the infinitely deep ocean.
+
surface of the ice floe is <math>\Gamma_w</math>.  
The condition at the free surface is the standard linear free surface
+
At the wetted surface of the ice floe the exact
condition. At the wetted surface of the ice floe the exact
 
 
equation of motion depends on the way in which the floe is modelled (for
 
equation of motion depends on the way in which the floe is modelled (for
 
example whether it is flexible or rigid) and we represent this by the
 
example whether it is flexible or rigid) and we represent this by the
 
operator <math>L</math>.  
 
operator <math>L</math>.  
To actually solve equation~(\ref{bvp}) requires us to choose a  
+
To actually solve equation (9) requires us to choose a  
 
specific model for the ice floe and hence to determine <math>L</math>.
 
specific model for the ice floe and hence to determine <math>L</math>.
\cite{jgrrealism} assumed that the
 
ice floes had negligible submergence but could flex.
 
The solution to equation~(\ref{bvp}) for this case is described
 
in \cite{jgrfloecirc,JGR02,IJOPE04}.
 
M\&Le
 
assumed the floes were rigid but allowed for submergence and the solution
 
to equation~(\ref{bvp}) in this case is described in
 
\cite{Masson_Le,Sarp_Isa}.
 
  
Equation (\ref{bvp}) requires boundary conditions as <math>x</math> or <math>y</math> tend to
+
Equation (9) requires boundary conditions as <math>x</math> or <math>y</math> tend to
 
infinity which are found from the incident or driving wave, denoted
 
infinity which are found from the incident or driving wave, denoted
<math>\phi^{\mathrm{{In}}}</math>.
+
<math>\phi^{\mathrm{{In}}}</math> which we
We assume that <math>\phi^{\mathrm{{In}}}</math> 
+
assume is a plane wave travelling in the <math>x</math> direction,
is a plane wave travelling in the <math>x</math> direction,
 
  
 
<math>
 
<math>
\phi^{\mathrm{{In}}}({ r},z)=\frac{\omega H}{2k}e^{\rm{i}kx}e^{kz}.
+
\phi^{\mathrm{{In}}}({ r},z)=\frac{\omega}{k}e^{{\rm i}kx}e^{kz}.
 
</math>
 
</math>
  
The condition as <math>\left|{r}\right|\rightarrow
+
plus the [[Sommerfeld Radiation Condition]].
\infty</math>
 
is the standard Sommerfeld radiation condition (e.g. \cite{Weh_Lait})
 
<math>
 
\sqrt{|{r}|}\left(  \frac{\partial}{\partial|\mathbf{r}|}-\rm{i}k\right)
 
(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.}
 
</math>
 
  
 
Once we have found the solution to  
 
Once we have found the solution to  
equation~(\ref{bvp}), we obtain the absolute value
+
equation (9), we obtain the absolute value
 
of the scattering
 
of the scattering
 
amplitude <math>D</math> as
 
amplitude <math>D</math> as
 +
 
<math>
 
<math>
\left\vert D\left(  \theta\right)  \right\vert =\frac{1}{H/2}\left(  \frac
+
\left\vert D\left(  \theta\right)  \right\vert =\left(  \frac
 
{k}{2\pi}\right)  ^{1/2}\left(  \frac{\omega}{g}\right)  \left\vert \mathbf{H}\left(
 
{k}{2\pi}\right)  ^{1/2}\left(  \frac{\omega}{g}\right)  \left\vert \mathbf{H}\left(
 
\pi+\theta\right)\right\vert,
 
\pi+\theta\right)\right\vert,
 
</math>
 
</math>
where the Kochin function <math>\mathbf{H}(\tau)</math> is
+
 
 +
where the [[Kochin Function]] <math>\mathbf{H}(\tau)</math> is
  
 
<math>
 
<math>
Line 303: Line 276:
 
where <math>\delta/\delta n</math> is the inward normal derivative.
 
where <math>\delta/\delta n</math> is the inward normal derivative.
  
= Numerical Solution of the Transport Equation=
+
== Numerical Solution of the Transport Equation==
  
 
We present here a simple method to solve the  
 
We present here a simple method to solve the  
Line 317: Line 290:
 
MIZ occupies the region <math>x>b</math>, i.e. the ice edge is at <math>x=b.</math> This will allow
 
MIZ occupies the region <math>x>b</math>, i.e. the ice edge is at <math>x=b.</math> This will allow
 
us to consider a wave spectrum which enters the MIZ from the open ocean. Under
 
us to consider a wave spectrum which enters the MIZ from the open ocean. Under
these assumptions equations (\ref{boltzmann}) and (\ref{M_Le_boltzmann2})
+
these assumptions equation (8)  
become,
+
becomes,
 +
 
 
<math>
 
<math>
 
\frac{1}{c_{g}}\frac{\partial I}{\partial t}+\cos\theta\frac{\partial
 
\frac{1}{c_{g}}\frac{\partial I}{\partial t}+\cos\theta\frac{\partial
Line 327: Line 301:
 
0,\;\;x<b.
 
0,\;\;x<b.
 
\end{bmatrix}
 
\end{bmatrix}
\right.
+
\right.\,\,\,(10)
 
</math>
 
</math>
  
To solve equation (\ref{almostsolvable}) we convert the problem to a matrix
+
To solve equation (10) we convert the problem to a matrix
 
equation by introducing a discretisation in angle. We use a discrete ordinate
 
equation by introducing a discretisation in angle. We use a discrete ordinate
 
method (\cite{Case&Zweifel}) and represent the angular coordinate by a
 
method (\cite{Case&Zweifel}) and represent the angular coordinate by a
Line 336: Line 310:
 
(<math>\theta_{j}=\frac{2\pi j}{n},\;0\leq j\leq n-1,</math>).
 
(<math>\theta_{j}=\frac{2\pi j}{n},\;0\leq j\leq n-1,</math>).
 
This approximation converts equation
 
This approximation converts equation
(\ref{almostsolvable}) to the following equation,
+
(10) to the following equation,
  
 
<math>
 
<math>
Line 345: Line 319:
 
0,\;\;x<b.
 
0,\;\;x<b.
 
\end{bmatrix}
 
\end{bmatrix}
\right. }
+
\right. }\,\,\,(11)
 
</math>
 
</math>
  
In equation (\ref{matrix}) the intensity <math>\vec{I}</math> is now a vector of functions
+
In equation (11) the intensity <math>\vec{I}</math> is now a vector of functions
 
of <math>x</math> and </math>t</math>  
 
of <math>x</math> and </math>t</math>  
 
for each angle <math>\theta_{j}</math> and the elements of the matrices <math>\mathbf{{D}}</math> and
 
for each angle <math>\theta_{j}</math> and the elements of the matrices <math>\mathbf{{D}}</math> and
Line 374: Line 348:
 
</math>
 
</math>
  
respectively. Equation (\ref{matrix}) can be easily solved in the stationary (no time
+
respectively. Equation (11) can be easily solved in the stationary (no time
dependence), or isotropic (no spatial dependence) case. \cite{jgrrealism}
+
dependence), or isotropic (no spatial dependence) case. [[Meylan, Squire, and Fox 1997]]
solved the stationary problem and M\&Le solved the isotopic problem
+
solved the stationary problem and [[Masson and LeBlond 1989]] solved the isotopic problem
 
(with wind forcing etc.).
 
(with wind forcing etc.).
 
In the stationary case
 
In the stationary case
equation (\ref{matrix}) reduces to, setting the ice edge to <math>b=0,</math>
+
equation (11) reduces to, setting the ice edge to <math>b=0,</math>
  
 
<math>
 
<math>
 
-\frac{\partial}{\partial x}\mathbf{D}\vec{I}=-\beta\vec{I}+\mathbf{S}\vec{I},
 
-\frac{\partial}{\partial x}\mathbf{D}\vec{I}=-\beta\vec{I}+\mathbf{S}\vec{I},
\;\;x>0.
+
\;\;x>0.\,\,\,(12)
 
</math>
 
</math>
  
Line 391: Line 365:
 
<math>
 
<math>
 
\frac{1}{c_{g}}\frac{\partial}{\partial t}\vec{I}=-\beta\vec{I}+\mathbf{S}
 
\frac{1}{c_{g}}\frac{\partial}{\partial t}\vec{I}=-\beta\vec{I}+\mathbf{S}
\vec{I},\;\;t>0.
+
\vec{I},\;\;t>0.\,\,\,(13)
 
</math>
 
</math>
  
Equations (\ref{spaceonly}) and (\ref{timeonly}) can be solved by
+
Equations (12) and (13) can be solved by
straightforward matrix methods (e.g. \cite{ishimaru}). Equation (\ref{spaceonly})
+
straightforward matrix methods. Equation (12)
 
requires boundary conditions (the wave spectrum at the ice edge <math>x=0</math>  
 
requires boundary conditions (the wave spectrum at the ice edge <math>x=0</math>  
 
and a condition as <math>x\to\infty</math>) and
 
and a condition as <math>x\to\infty</math>) and
equation (\ref{timeonly}) requires an initial condition (the wave spectrum
+
equation (13) requires an initial condition (the wave spectrum
 
at <math>t=0</math>).
 
at <math>t=0</math>).
 +
 +
 +
[[Category: Wave Scattering in the Marginal Ice Zone]]

Latest revision as of 09:30, 20 October 2009


Introduction

We present a linear Boltzmann equation to model Wave Scattering in the Marginal Ice Zone. This model is three dimensional and requires the three-dimensionao solution for a Floating Elastic Plate. This model was given in Meylan and Masson 2006 (which also showed the equivalence with The Multiple Scattering Theory of Masson and LeBlond). The derivation closely follows that of Meylan, Squire, and Fox 1997 with a correction.

The wave scattering which occurs in the MIZ is due to the scattering effects of the individual ice floes which comprise the MIZ. To understand the process of wave scattering we need to understand the scattering which any individual ice floe produces. However, the equation for the propagation of wave energy, while dependent on the scattering from individual floes, will take a form quite different to the equation of scattering from an individual floe.

Determining the scattering at the large scale from the scattering from an individual scatterer is important in many areas of physics. There are many approaches to this problem, the two most popular being the linear Boltzmann (or transport) equation, and multiple scattering. However, the MIZ presents a slightly more complex case than is usually encountered in scattering theories because of the random nature of the scatterers (i.e. the random geometry of ice floes) and the constant motion of the ice floes. This means that it is extremely unlikely that any kind of coherent scattering effects will be observed. However, all large scale scattering theories require as input the individual scatterers and often assume that all scatterers are identical. This can lead, in the scattering theory, to coherent scattering effects which we believe are not significant for the MIZ.

The Linear Boltzmann equation for Wave Scattering in the MIZ

In this section we present a derivation of the linear Boltzmann equation for wave scattering in the MIZ that follows closely the derivation given in Meylan, Squire, and Fox 1997 but corrects an error in this earlier derivation. The linear Boltzmann equation is applicable to the propagation of wave energy through the MIZ over length and time scales large relative to the incident wavelength and wave period, respectively. Over such large scales, we assume that wave energy is incoherent. We consider the surface of an infinitely deep ocean represented in Cartesian co-ordinates by [math]\displaystyle{ r =(x,y). }[/math] Wave energy is propagating across this surface in all directions so that, at any point, we must consider the energy travelling in each direction. We introduce an intensity function [math]\displaystyle{ I(r,t,\theta) }[/math] which is the rate of flow of energy travelling in a given direction, per unit surface, per unit angle. In the absence of scatterers, we assume that the waves continue to propagate in the same direction and that the energy intensity satisfies the following equation,

[math]\displaystyle{ \frac{1}{c_{g}}\frac{\partial}{\partial t}I(r,t,\theta)+\hat{\theta}.\nabla I(r,t,\theta)=0,\,\,\,(1) }[/math]

where [math]\displaystyle{ \hat{\theta} }[/math] is a unit vector in the [math]\displaystyle{ \theta }[/math] direction and [math]\displaystyle{ c_{g} }[/math] is the speed of wave propagation (the deep water group speed). The presence of the floes will modify this expression by scattering energy, i.e. by changing the direction in which the energy is travelling.

We modify equation (1) to take into account the scattering effects of the ice floes using the general equation for the propagation of wave energy through a scattering medium,

[math]\displaystyle{ \frac{1}{c_{g}}\frac{\partial}{\partial t}I(r,t,\theta)+\hat{\theta}.\nabla I(r,t,\theta)=-\beta(r,\theta)I(r,t,\theta)+\int_{0}^{2\pi}S(r,\theta,\theta ^{\prime})I(r,t,\theta^{\prime})d\theta^{\prime},\,\,\,(2) }[/math]

(Howells 1960) where [math]\displaystyle{ \beta }[/math] is the absorption coefficient and [math]\displaystyle{ S }[/math] is the scattering function (assumed to be independent of time). Equation (2) depends on the assumption that each floe scatters independently and that the energy from different scatterers may be added incoherently. The absorption coefficient, [math]\displaystyle{ \beta(r,\theta) }[/math], is the fraction of energy lost by scattering and dissipative processes (assumed linear) from a pencil of radiation in direction [math]\displaystyle{ \theta }[/math], per unit path length travelled in the medium. The scattering function [math]\displaystyle{ S(r,\theta,\theta^{\prime }) }[/math] specifies the angular distribution of scattered energy in such a way that,

[math]\displaystyle{ S(r,\theta,\theta^{\prime})I(r,t,\theta^{\prime})d\Omega dSd\Omega^{\prime},,\,\,\,(3) }[/math]

is the rate at which energy is scattered from a pencil of radiation of intensity [math]\displaystyle{ I(r,t,\theta^{\prime}) }[/math] at an angle [math]\displaystyle{ d\Omega^{\prime} }[/math] in direction [math]\displaystyle{ \theta^{\prime} }[/math], by a surface [math]\displaystyle{ dS }[/math] at position [math]\displaystyle{ r }[/math], into an angle [math]\displaystyle{ d\Omega }[/math] in direction [math]\displaystyle{ \theta }[/math]. To apply equation (2) to wave scattering in the MIZ we must first estimate the scattering function [math]\displaystyle{ S(r,\theta,\theta^{\prime}) }[/math] and the absorption coefficient [math]\displaystyle{ \beta(r,\theta)\, }[/math].

Finding the scattering and absorbtion functions

The scattering function is determined by calculating the scattering from a single ice floe. Each ice floe scatters energy, and the energy radiated per unit angle per unit time in the [math]\displaystyle{ \theta }[/math] direction for a wave incident in the [math]\displaystyle{ \theta^{\prime} }[/math] direction, [math]\displaystyle{ E }[/math], is given by,

[math]\displaystyle{ E(\theta - \theta^{\prime})=\left( \frac{H}{2}\right) ^{2}\frac{\rho\omega g}{4k}|D(\theta-\theta^{\prime})|^{2},\,\,\,(4) }[/math]

where [math]\displaystyle{ H }[/math] is the wave height, [math]\displaystyle{ \rho }[/math] is the water density, [math]\displaystyle{ \omega }[/math] and [math]\displaystyle{ k }[/math] are the radian frequency and wavenumber of the wave, respectively, and [math]\displaystyle{ g }[/math] is the acceleration due to gravity. [math]\displaystyle{ D(\theta - \theta^{\prime}) }[/math] is the scattered amplitude for which, at a large distance, [math]\displaystyle{ r }[/math], from the scatterer, the asymptotic amplitude of the outgoing wave in the [math]\displaystyle{ \theta }[/math] direction, for an incident wave travelling in the [math]\displaystyle{ \theta^{\prime} }[/math] direction, is given by

[math]\displaystyle{ \frac{H}{2}\frac{D\left( \theta - \theta^{\prime}\right) }{\sqrt{r}},\,\,\,(5) }[/math]

Note that, in equations (4) and (5), we have assumed that the scattering is isotropic (depends only on the difference of angle). This will not necessarily be true for a given ice floe, but we expect this to be true in the MIZ since there are no special directions in which the ice floes are oriented, and the floes are of random shape.

We must now express the scattering kernel in equation (2), [math]\displaystyle{ S(r,\theta,\theta^{\prime}) }[/math], in terms of [math]\displaystyle{ E }[/math]. Given the definition of [math]\displaystyle{ S }[/math] (equation (\ref{equation_S})), [math]\displaystyle{ S }[/math] can be found by dividing [math]\displaystyle{ E }[/math] by the rate of energy which is passing under the ice floe. The rate of energy passing under the floe is given by the product of the wave energy density ([math]\displaystyle{ \frac{1}{8}\rho gH^{2}, }[/math] since [math]\displaystyle{ H/2 }[/math] is the wave amplitude and we are considering only the energy in the water), the average area occupied by a floe ([math]\displaystyle{ A_{f}/f_{i}, }[/math] where [math]\displaystyle{ A_{f} }[/math] is the average area of the floe and [math]\displaystyle{ f_{i} }[/math] is the fraction of the surface area of the ice covered ocean which is covered in ice), and the wave group speed ([math]\displaystyle{ \omega/2k) }[/math]. This gives the following expression for [math]\displaystyle{ S }[/math],

[math]\displaystyle{ S(\theta,\theta^{\prime}) =\left( \frac{H}{2}\right) ^{2}\frac{\rho\omega g}{4k}|D(\theta-\theta^{\prime})|^{2}\frac{4f_{i}k}{\rho gH^{2}A_{f}\omega} =\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime})|^{2},\,\,\,(6) }[/math]

This expression is not exactly the same as the equivalent expression in Meylan, Squire, and Fox 1997 because, in the derivation for [math]\displaystyle{ S }[/math] in Meylan, Squire, and Fox 1997, the wave phase speed rather than the group speed was erroneously used.

We can determine [math]\displaystyle{ \beta }[/math] from the absorption cross section, [math]\displaystyle{ \sigma_a }[/math], and the ice cover fraction [math]\displaystyle{ f_i }[/math] he absorption cross section, [math]\displaystyle{ \sigma_a }[/math], may be estimated from the total scattering or from experimental measurements. The expression for [math]\displaystyle{ \beta }[/math] is the following,

[math]\displaystyle{ \beta = \int_{0}^{2\pi}\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime})|^{2}d\theta^{\prime}+\sigma_{a} \frac{f_{i}}{A_{f}},\,\,\,(7) }[/math]

Combining equations (2), (6) and (7), the following linear Boltzmann equation for wave scattering in the MIZ is obtained,

[math]\displaystyle{ \frac{1}{c_g}\frac{\partial I}{\partial t}+\hat{\theta}.\nabla I = \int_{0}^{2\pi}\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime })|^{2}I\left( \theta^{\prime}\right) d\theta^{\prime} - \left( \int_{0}^{2\pi}\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime})|^{2}d\theta^{\prime}+\sigma_{a} \frac{f_{i}}{A_{f}} \right)I\left( \theta\right),\,\,\,(8) }[/math]

Determining the scattering amplitude

We need to determine the scattering amplitude [math]\displaystyle{ D(\theta-\theta^{\prime}) }[/math]. The scattering amplitude is found by solving the boundary value problem which arises when an isolated floe is subject to linear wave forcing. The exact equations depend on the equations chosen to model the movement of the ice floe. The solution to the equations of motion depends on the model used to describe an ice floe. The principal difference between the ice floe models used by Meylan, Squire, and Fox 1997 and Masson and LeBlond 1989 is the following. Meylan, Squire, and Fox 1997 assumed that the ice floe was a Floating Elastic Plate while Masson and LeBlond 1989 assumed the floes were rigid but allowed for submergence. Both models have different ranges of validity (although typical ice floes tend to be relatively thin). Of course, either ice floe model could have been used in the large scale scattering models derived by Masson and LeBlond 1989 and Meylan, Squire, and Fox 1997. Here we simply present the equation for [math]\displaystyle{ D(\theta) }[/math] independent of the equation used to model the ice floe.

The linearised boundary value problem for the fluid velocity potential [math]\displaystyle{ \phi({r},z) }[/math] subject to an incoming wave of frequency [math]\displaystyle{ \omega }[/math] is the Standard Linear Wave Scattering Problem in infinitely deep water which we briefly recap here.

[math]\displaystyle{ \nabla^{2}\phi=0, \, -\infty\lt z\lt 0,\,\,\,(9) }[/math]

[math]\displaystyle{ \frac{\partial\phi}{\partial z}=0, \, z\to-\infty, }[/math]

[math]\displaystyle{ \frac{\partial\phi}{\partial z} = k\phi,\,z\in\Gamma_s, }[/math]

[math]\displaystyle{ \frac{\partial\phi}{\partial z} = L\phi, \, z\in\Gamma_w, }[/math]

(note that since the water is assumed infinitely deep [math]\displaystyle{ k_\infty = k }[/math]). The free surface is [math]\displaystyle{ \Gamma_s }[/math] (located at [math]\displaystyle{ z=0 }[/math]) and the wetted surface of the ice floe is [math]\displaystyle{ \Gamma_w }[/math]. At the wetted surface of the ice floe the exact equation of motion depends on the way in which the floe is modelled (for example whether it is flexible or rigid) and we represent this by the operator [math]\displaystyle{ L }[/math]. To actually solve equation (9) requires us to choose a specific model for the ice floe and hence to determine [math]\displaystyle{ L }[/math].

Equation (9) requires boundary conditions as [math]\displaystyle{ x }[/math] or [math]\displaystyle{ y }[/math] tend to infinity which are found from the incident or driving wave, denoted [math]\displaystyle{ \phi^{\mathrm{{In}}} }[/math] which we assume is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction,

[math]\displaystyle{ \phi^{\mathrm{{In}}}({ r},z)=\frac{\omega}{k}e^{{\rm i}kx}e^{kz}. }[/math]

plus the Sommerfeld Radiation Condition.

Once we have found the solution to equation (9), we obtain the absolute value of the scattering amplitude [math]\displaystyle{ D }[/math] as

[math]\displaystyle{ \left\vert D\left( \theta\right) \right\vert =\left( \frac {k}{2\pi}\right) ^{1/2}\left( \frac{\omega}{g}\right) \left\vert \mathbf{H}\left( \pi+\theta\right)\right\vert, }[/math]

where the Kochin Function [math]\displaystyle{ \mathbf{H}(\tau) }[/math] is

[math]\displaystyle{ \mathbf{H}(\tau)=\iint_{\Gamma_s}\left( -\frac{\delta\phi}{\delta n} + \phi\frac{\delta}{\delta n}\right) e^{kz}e^{ik(x\cos\tau +y\sin\tau)}dS, }[/math] where [math]\displaystyle{ \delta/\delta n }[/math] is the inward normal derivative.

Numerical Solution of the Transport Equation

We present here a simple method to solve the linear Boltzmann equation. It involves simplifying assumptions of the spatial or temporal independence of the solution as well as a discretisation of the equation in angle. We begin by assuming that the solution is only a function of the [math]\displaystyle{ x }[/math] spatial co-ordinate and time, i.e. there is no [math]\displaystyle{ y }[/math] dependence of the solution. We also consider a uniform MIZ so that the scattering function, [math]\displaystyle{ S }[/math], is a function only of [math]\displaystyle{ \theta }[/math] and [math]\displaystyle{ \theta^{\prime} }[/math] and [math]\displaystyle{ \beta }[/math] is a constant. The only variation we allow spatially is that the MIZ occupies the region [math]\displaystyle{ x\gt b }[/math], i.e. the ice edge is at [math]\displaystyle{ x=b. }[/math] This will allow us to consider a wave spectrum which enters the MIZ from the open ocean. Under these assumptions equation (8) becomes,

[math]\displaystyle{ \frac{1}{c_{g}}\frac{\partial I}{\partial t}+\cos\theta\frac{\partial I}{\partial x}=\left\{ \begin{bmatrix} -\beta I+\int_{0}^{2\pi}S(\theta-\theta^{\prime})I(\theta^{\prime} )d\theta^{\prime},\;\;x\gt b, \\ 0,\;\;x\lt b. \end{bmatrix} \right.\,\,\,(10) }[/math]

To solve equation (10) we convert the problem to a matrix equation by introducing a discretisation in angle. We use a discrete ordinate method (\cite{Case&Zweifel}) and represent the angular coordinate by a discrete set of [math]\displaystyle{ n }[/math] angles evenly spaced between [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ 2\pi }[/math] ([math]\displaystyle{ \theta_{j}=\frac{2\pi j}{n},\;0\leq j\leq n-1, }[/math]). This approximation converts equation (10) to the following equation,

[math]\displaystyle{ \frac{1}{c_{g}}\frac{\partial\vec{I}}{\partial t}-\frac{\partial}{\partial x}\mathbf{{D}}\vec{I}=\mathbf{\left\{ \begin{bmatrix} -\beta \vec{I}+\mathbf{S}\vec{I},\;\;x\gt b,\\ 0,\;\;x\lt b. \end{bmatrix} \right. }\,\,\,(11) }[/math]

In equation (11) the intensity [math]\displaystyle{ \vec{I} }[/math] is now a vector of functions of [math]\displaystyle{ x }[/math] and </math>t</math> for each angle [math]\displaystyle{ \theta_{j} }[/math] and the elements of the matrices [math]\displaystyle{ \mathbf{{D}} }[/math] and [math]\displaystyle{ \mathbf{{S}} }[/math] are given by,

[math]\displaystyle{ d_{ij}=\left\{ \begin{bmatrix} -\cos(\theta_{i}), & i=j,\\ {0,} & i\neq j, \end{bmatrix} \right. }[/math]

and

[math]\displaystyle{ s_{ij}=\left\{ \begin{bmatrix} {-}\beta{+}S(\theta_{i}-\theta_{i})\frac{2\pi}{n}{,} & i=j,\\ S(\theta_{i}-\theta_{j})\frac{2\pi}{n}{,} & i\neq j, \end{bmatrix} \right. }[/math]

respectively. Equation (11) can be easily solved in the stationary (no time dependence), or isotropic (no spatial dependence) case. Meylan, Squire, and Fox 1997 solved the stationary problem and Masson and LeBlond 1989 solved the isotopic problem (with wind forcing etc.). In the stationary case equation (11) reduces to, setting the ice edge to [math]\displaystyle{ b=0, }[/math]

[math]\displaystyle{ -\frac{\partial}{\partial x}\mathbf{D}\vec{I}=-\beta\vec{I}+\mathbf{S}\vec{I}, \;\;x\gt 0.\,\,\,(12) }[/math]

For the isotropic case, equation (\ref{matrix}) reduces to, setting the ice edge to [math]\displaystyle{ b=-\infty }[/math],

[math]\displaystyle{ \frac{1}{c_{g}}\frac{\partial}{\partial t}\vec{I}=-\beta\vec{I}+\mathbf{S} \vec{I},\;\;t\gt 0.\,\,\,(13) }[/math]

Equations (12) and (13) can be solved by straightforward matrix methods. Equation (12) requires boundary conditions (the wave spectrum at the ice edge [math]\displaystyle{ x=0 }[/math] and a condition as [math]\displaystyle{ x\to\infty }[/math]) and equation (13) requires an initial condition (the wave spectrum at [math]\displaystyle{ t=0 }[/math]).