Difference between revisions of "The Multiple Scattering Theory of Masson and LeBlond"

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=The Multiple Scattering Theory of Masson and LeBlond=
+
{{complete pages}}
 +
 
 +
== Introduction==
  
 
The scattering theory of [[Masson_LeBlond_1989a| Masson and LeBlond 1989]] was the first  
 
The scattering theory of [[Masson_LeBlond_1989a| Masson and LeBlond 1989]] was the first  
Line 11: Line 13:
 
wind generation, nonlinear coupling in frequency and wave breaking. However,
 
wind generation, nonlinear coupling in frequency and wave breaking. However,
 
what was original in their work was their equation
 
what was original in their work was their equation
for the scattering of wave energy by ice floes. [[Masson_LeBlond_1989a| Masson and LeBlond 1989]] began  
+
for the scattering of wave energy by ice floes.  
 +
 
 +
[[Meylan_Masson_2006a | Meylan and Masson 2006]] showed the equivalence
 +
of the multiple scattering theory of Masson and LeBlond with the
 +
[[Linear Boltzmann Model for Wave Scattering in the MIZ]]
 +
and this is included the final section.
 +
 
 +
== Equation for Wave Scattering==
 +
 
 +
[[Masson and LeBlond 1989]] began  
 
with the following equation for the evolution of wave scattering,
 
with the following equation for the evolution of wave scattering,
 
+
<center>
 
<math>
 
<math>
 
\frac{\partial I}{\partial t}+c_{g}\hat{\theta}\nabla  
 
\frac{\partial I}{\partial t}+c_{g}\hat{\theta}\nabla  
Line 21: Line 32:
 
}+S_{{\mathrm{ice}}},  
 
}+S_{{\mathrm{ice}}},  
 
</math>
 
</math>
 
+
</center>
 
where <math>S_{\mathrm{{in}}}</math> is the input of wave energy due
 
where <math>S_{\mathrm{{in}}}</math> is the input of wave energy due
 
to wind forcing, <math>S_{\mathrm{{ds}}}</math> is the dissipation of wave
 
to wind forcing, <math>S_{\mathrm{{ds}}}</math> is the dissipation of wave
 
energy due to wave breaking, <math>S_{\mathrm{{nl}}}</math> is the non-linear transfer
 
energy due to wave breaking, <math>S_{\mathrm{{nl}}}</math> is the non-linear transfer
 
of wave energy and <math>S_{\mathrm{{ice}}}</math> is the wave scattering.
 
of wave energy and <math>S_{\mathrm{{ice}}}</math> is the wave scattering.
Similarly, the terms <math>S_{\mathrm{{in}}},</math>
+
[[Masson and LeBlond 1989]]
<math>S_{\mathrm{{ds}}},</math> and <math>S_{\mathrm{{nl}}}</math> could be added to equation
+
solved this in the isotropic (no spatial
(\ref{Howells}). However, the purpose of this
 
paper is to derive a consistent equation for <math>S_{\mathrm{{ice }}}.</math> [[Masson_LeBlond_1989a| Masson and LeBlond 1989]]
 
solved equation (\ref{BoltzMasson}) in the isotropic (no spatial
 
 
dependence) case. Furthermore, they did not actually determine
 
dependence) case. Furthermore, they did not actually determine
 
<math>S_{\mathrm{{ice}}}</math> but derived a time stepping procedure to solve the
 
<math>S_{\mathrm{{ice}}}</math> but derived a time stepping procedure to solve the
 
isotropic solution using multiple scattering. We will derive <math>S_{\mathrm{{ice}}}</math>
 
isotropic solution using multiple scattering. We will derive <math>S_{\mathrm{{ice}}}</math>
from their time stepping equation.
+
from the time stepping equation.
 
 
[[Masson_LeBlond_1989a| Masson and LeBlond 1989]] derived the following difference equation as
 
a discrete analogue of equation~(\ref{BoltzMasson})
 
  
 +
[[Masson and LeBlond 1989]] derived the following difference equation
 +
<center>
 
<math>
 
<math>
 
I(f_{n},\theta;t+\Delta t)=[\mathbf{T}]_{f_{n}}[I(f_{n},\theta
 
I(f_{n},\theta;t+\Delta t)=[\mathbf{T}]_{f_{n}}[I(f_{n},\theta
Line 44: Line 51:
 
t]
 
t]
 
</math>
 
</math>
 
+
</center>
 
where <math>f_{n}</math> is the wave frequency  
 
where <math>f_{n}</math> is the wave frequency  
([[Masson_LeBlond_1989a| Masson and LeBlond 1989]], equation~(51)). It is important to realise that  
+
([[Masson and LeBlond 1989|Masson and LeBlond 1989 equation (51)]]. It is important to realise that  
 
<math>[\mathbf{T}]_{f_{n}}</math> is a  
 
<math>[\mathbf{T}]_{f_{n}}</math> is a  
 
function of
 
function of
Line 57: Line 64:
 
angle <math>\theta</math> into <math>n</math> evenly spaced angles <math>\theta_{i}</math> between <math>-\pi</math>
 
angle <math>\theta</math> into <math>n</math> evenly spaced angles <math>\theta_{i}</math> between <math>-\pi</math>
 
and <math>\pi</math>. <math>[\mathbf{T}]_{f_{n}}</math> is then given by
 
and <math>\pi</math>. <math>[\mathbf{T}]_{f_{n}}</math> is then given by
 
+
<center>
 
<math>
 
<math>
 
(T_{ij})_{f_{n}}=A^{2}\{\hat{\beta}|D(\theta_{ij})|^{2}\Delta
 
(T_{ij})_{f_{n}}=A^{2}\{\hat{\beta}|D(\theta_{ij})|^{2}\Delta
Line 64: Line 71:
 
\},
 
\},
 
</math>
 
</math>
 
+
</center>
 
where <math>\theta_{ij}=|(\theta_{i}-\theta_{j})|</math>  
 
where <math>\theta_{ij}=|(\theta_{i}-\theta_{j})|</math>  
([[Masson_LeBlond_1989a| Masson and LeBlond 1989]], equation~(42)). In equation~(\ref{wrong}), <math>\hat{\beta}</math>  
+
([[Masson_LeBlond_1989a| Masson and LeBlond 1989]], equation (42)). <math>\hat{\beta}</math>  
(this notation is chosen to follow from M\&Le
+
is a function of <math>\Delta t</math>
who used <math>\beta</math> and to avoid confusion with the expression for <math>\beta</math> in
 
equation (\ref{Howells}) and which
 
is used in \cite{howells60} and \cite{jgrrealism}) is a function of <math>\Delta t</math>
 
 
given by
 
given by
 
+
<center>
 
<math>
 
<math>
 
\hat{\beta}=\int_{0}^{c_{g}\Delta t}\rho_{e}(r)dr,
 
\hat{\beta}=\int_{0}^{c_{g}\Delta t}\rho_{e}(r)dr,
 
</math>
 
</math>
 
+
</center>
([[Masson_LeBlond_1989a| Masson and LeBlond 1989]] p. 68). The function <math>\rho_{e}(r)</math> gives the ''effective'' number of floes per unit  
+
([[Masson and LeBlond 1989]] p. 68). The function <math>\rho_{e}(r)</math> gives the ''effective'' number of floes per unit  
 
area effectively radiating waves under the single scattering approximation
 
area effectively radiating waves under the single scattering approximation
 
which is to assume that the amplitude of a wave scattered
 
which is to assume that the amplitude of a wave scattered
 
more than once is negligible. It is given by
 
more than once is negligible. It is given by
 
+
<center>
 
<math>
 
<math>
 
\rho_{e}(r)=\frac{2}{\sqrt{3}D_{\mathrm{{av}}}^{2}\left(  1-\frac{8a^{2}}{\sqrt
 
\rho_{e}(r)=\frac{2}{\sqrt{3}D_{\mathrm{{av}}}^{2}\left(  1-\frac{8a^{2}}{\sqrt
Line 87: Line 91:
 
{3}D_{\mathrm{{av}}}^{2}}\right)  ^{r/2a},
 
{3}D_{\mathrm{{av}}}^{2}}\right)  ^{r/2a},
 
</math>
 
</math>
 
+
</center>
([[Masson_LeBlond_1989a| Masson and LeBlond 1989]] equation~(29),  
+
([[Masson and LeBlond 1989]] equation (29),  
 
although there is a typographical error in their equation which we have corrected)
 
although there is a typographical error in their equation which we have corrected)
 
where <math>D_{av}</math> is the average floe spacing and <math>a</math> is the floe radius
 
where <math>D_{av}</math> is the average floe spacing and <math>a</math> is the floe radius
(remembering that [[Masson_LeBlond_1989a| Masson and LeBlond 1989]] considered circular floes).  
+
(remembering that [[Masson and LeBlond 1989]] considered circular floes).  
 
The energy factor <math>A</math> is
 
The energy factor <math>A</math> is
 
given by,
 
given by,
 
+
<center>
 
<math>
 
<math>
 
A=(1+|\alpha_{c}D(0)|^{2}+|\alpha_{c}D(\pi)|^{2}+\hat{\beta}\int_{0}^{2\pi
 
A=(1+|\alpha_{c}D(0)|^{2}+|\alpha_{c}D(\pi)|^{2}+\hat{\beta}\int_{0}^{2\pi
 
}|D(\theta)|^{2}d\theta+f_{d})^{-\frac{1}{2}},
 
}|D(\theta)|^{2}d\theta+f_{d})^{-\frac{1}{2}},
 
</math>
 
</math>
 
+
</center>
([[Masson_LeBlond_1989a| Masson and LeBlond 1989]] equation (52)) where the term  
+
([[Masson and LeBlond 1989]] equation (52)) where the term  
 
<math>f_{d}</math> represents dissipation and is given by
 
<math>f_{d}</math> represents dissipation and is given by
 
+
<center>
 
<math>
 
<math>
 
f_{d}=e^{\frac{f_{i}}{A_{f}}\sigma_{a}c_{g}\Delta t}-1,
 
f_{d}=e^{\frac{f_{i}}{A_{f}}\sigma_{a}c_{g}\Delta t}-1,
 
</math>
 
</math>
 
+
</center>
([[Masson_LeBlond_1989a| Masson and LeBlond 1989]] equation (53)) and </math>\alpha_{c}</math>, the ``coherent'' scattering coefficient, is given by
+
([[Masson and LeBlond 1989]] equation (53)) and <math>\alpha_{c}</math>, the ''coherent'' scattering coefficient, is given by
 
+
<center>
 
<math>
 
<math>
 
\alpha_{c}=\left(  \frac{2\pi}{k}\right)  ^{1/2}\exp\left(  \frac
 
\alpha_{c}=\left(  \frac{2\pi}{k}\right)  ^{1/2}\exp\left(  \frac
Line 116: Line 120:
 
^{x_{s}/2a}dx_{s}.
 
^{x_{s}/2a}dx_{s}.
 
</math>
 
</math>
 
+
</center>
 
It should be noted that the upper limit of integration for <math>\alpha_{c}</math>  
 
It should be noted that the upper limit of integration for <math>\alpha_{c}</math>  
was given as infinity in [[Masson_LeBlond_1989a| Masson and LeBlond 1989]]. This is appropriate in the steady case
+
was given as infinity in [[Masson and LeBlond 1989]]. This is appropriate in the steady case
 
only; it should have been changed to <math>c_{g}\Delta t</math> in the time dependent case. However, this correction  
 
only; it should have been changed to <math>c_{g}\Delta t</math> in the time dependent case. However, this correction  
 
leads to only negligible quantitative changes to the results.
 
leads to only negligible quantitative changes to the results.
  
We will transform the [[Masson_LeBlond_1989a| Masson and LeBlond 1989]] scattering operator <math>\mathbf{T}</math> by
+
We will transform the scattering operator <math>\mathbf{T}</math> by
 
taking the limit as the number of angles used to discretise <math>\theta</math> tends to
 
taking the limit as the number of angles used to discretise <math>\theta</math> tends to
 
infinity. On taking this limit, the operator <math>\mathbf{T}\left(  \Delta t\right) </math>  
 
infinity. On taking this limit, the operator <math>\mathbf{T}\left(  \Delta t\right) </math>  
 
becomes
 
becomes
 
+
<center>
 
<math>
 
<math>
 
\mathbf{T}\left(  \Delta t\right)  I\left(  \theta\right)  =A^{2}\{\hat{\beta
 
\mathbf{T}\left(  \Delta t\right)  I\left(  \theta\right)  =A^{2}\{\hat{\beta
Line 132: Line 136:
 
}\right)  d\theta^{\prime}+ I\left(  \theta\right)  \}.
 
}\right)  d\theta^{\prime}+ I\left(  \theta\right)  \}.
 
</math>
 
</math>
 
+
</center>
 
+
The scattering theory of [[Masson and LeBlond 1989]] depends on the
The scattering theory of [[Masson_LeBlond_1989a| Masson and LeBlond 1989]] depends on the
 
 
values of the time step <math>\Delta t</math> and the correct solution
 
values of the time step <math>\Delta t</math> and the correct solution
 
is found for small time steps. We will now
 
is found for small time steps. We will now
Line 142: Line 145:
 
there is a considerable simplification in the form of the equation.
 
there is a considerable simplification in the form of the equation.
 
Since
 
Since
 
+
<center>
 
<math>
 
<math>
 
I(t+\Delta t)=\mathbf{T}\left(  \Delta t\right)  I\mathbf{(}t),
 
I(t+\Delta t)=\mathbf{T}\left(  \Delta t\right)  I\mathbf{(}t),
 
</math>
 
</math>
 
+
</center>
we obtain the following expression for the time derivative of </math>I</math>,
+
we obtain the following expression for the time derivative of <math>I</math>,
 
+
<center>
 
<math>
 
<math>
 
\frac{\partial I}{\partial t}=\lim_{\Delta t\rightarrow0}\left(
 
\frac{\partial I}{\partial t}=\lim_{\Delta t\rightarrow0}\left(
 
\frac{\mathbf{T}\left(  \Delta t\right)I(t)  - I(t)}{\Delta t}\right).
 
\frac{\mathbf{T}\left(  \Delta t\right)I(t)  - I(t)}{\Delta t}\right).
 
</math>
 
</math>
 
+
</center>
 
We can calculate this limit as follows,
 
We can calculate this limit as follows,
 
+
<center>
 
<math>
 
<math>
 
\lim_{\Delta t\rightarrow0}\left(  \frac{\mathbf{T}\left(  \Delta t\right)I(t)
 
\lim_{\Delta t\rightarrow0}\left(  \frac{\mathbf{T}\left(  \Delta t\right)I(t)
Line 163: Line 166:
 
\theta\right)  \}-I\left(  \theta\right)  }{\Delta t}\right)
 
\theta\right)  \}-I\left(  \theta\right)  }{\Delta t}\right)
 
</math>
 
</math>
 
+
</center>
 +
<center>
 
<math>
 
<math>
 
=c_{g}\rho_{e}\left(  0\right)  \int
 
=c_{g}\rho_{e}\left(  0\right)  \int
Line 170: Line 174:
 
|D(\theta-\theta^{\prime})|^{2}d\theta^{\prime}+\frac{f_{i}}{A_{f}}\sigma_{a}c_{g}I\left(  \theta\right).
 
|D(\theta-\theta^{\prime})|^{2}d\theta^{\prime}+\frac{f_{i}}{A_{f}}\sigma_{a}c_{g}I\left(  \theta\right).
 
</math>
 
</math>
 
+
</center>
We can simplify equation~(\ref{M_Le_boltzmann1}) by using equation~(\ref{rho_e}).
+
We can simplify this equation.
 
The value of <math>\rho_{e}\left(  0\right)</math> is given by
 
The value of <math>\rho_{e}\left(  0\right)</math> is given by
 
+
<center>
 
<math>
 
<math>
 
\rho_{e}\left(  0\right)  =\frac{2}{\sqrt{3}D_{\mathrm{{av}}}^{2}\left(
 
\rho_{e}\left(  0\right)  =\frac{2}{\sqrt{3}D_{\mathrm{{av}}}^{2}\left(
Line 179: Line 183:
 
=\frac{f_{i}}{A_{f}\sqrt{1-4f_{i}/\pi}},
 
=\frac{f_{i}}{A_{f}\sqrt{1-4f_{i}/\pi}},
 
</math>
 
</math>
 
+
</center>
 
where we have used the fact that <math>f_{i}=2\pi a^{2}/\sqrt{3}D_{av}^{2}</math> and
 
where we have used the fact that <math>f_{i}=2\pi a^{2}/\sqrt{3}D_{av}^{2}</math> and
 
<math>A_{f}=\pi a^{2}</math>.
 
<math>A_{f}=\pi a^{2}</math>.
  
 +
== Equivalence with Linear Boltzmann Model==
  
 +
We show here that the equation above is very similar to the equation
 +
derived in [[Linear Boltzmann Model for Wave Scattering in the MIZ]].
 
If we substitute our expressions for <math>\rho_e(0)</math>
 
If we substitute our expressions for <math>\rho_e(0)</math>
in equation (\ref{M_Le_boltzmann1}) and
+
iand
 
include the spatial term  
 
include the spatial term  
 
(which was not in [[Masson_LeBlond_1989a| Masson and LeBlond 1989]] since they assumed  
 
(which was not in [[Masson_LeBlond_1989a| Masson and LeBlond 1989]] since they assumed  
 
isotropy) and divide by <math>c_g</math>, we obtain the following
 
isotropy) and divide by <math>c_g</math>, we obtain the following
 
linear Boltzmann equation  
 
linear Boltzmann equation  
 
+
<center>
 
<math>
 
<math>
 
\frac{1}{c_g}\frac{\partial I}{\partial t}+\hat{\theta}.\nabla I
 
\frac{1}{c_g}\frac{\partial I}{\partial t}+\hat{\theta}.\nabla I
Line 197: Line 204:
 
})|^{2}I\left(  \theta^{\prime}\right)  d\theta^{\prime}
 
})|^{2}I\left(  \theta^{\prime}\right)  d\theta^{\prime}
 
</math>
 
</math>
 
+
</center>
 +
<center>
 
<math>
 
<math>
 
- \left( \frac{1}{\sqrt{1-4f_{i}/\pi}}
 
- \left( \frac{1}{\sqrt{1-4f_{i}/\pi}}
Line 204: Line 212:
 
\right)I\left(  \theta\right).
 
\right)I\left(  \theta\right).
 
</math>
 
</math>
 
+
</center>
If we compare equations (\ref{boltzmann}) and (\ref{M_Le_boltzmann2})
+
If we compare this equations with
 +
the equivalent equation in [[Linear Boltzmann Model for Wave Scattering in the MIZ]]
 
we see that they are identical except for the factor  
 
we see that they are identical except for the factor  
 
<math>1 / \sqrt{1-4f_{i}/\pi}</math>  in the two components resulting from the scattering.
 
<math>1 / \sqrt{1-4f_{i}/\pi}</math>  in the two components resulting from the scattering.
 
This difference comes from the fact that, in [[Masson_LeBlond_1989a| Masson and LeBlond 1989]], multiple  
 
This difference comes from the fact that, in [[Masson_LeBlond_1989a| Masson and LeBlond 1989]], multiple  
 
scattering is neglected by using an effective density, <math>\rho_{e}</math>,  
 
scattering is neglected by using an effective density, <math>\rho_{e}</math>,  
in lieu of the number density <math>\rho_{o}</math>. As shown in equation (\ref{rho}),
+
in lieu of the number density <math>\rho_{o}</math>.  
the effective density
+
The effective density is related to the number density as <math> \rho_{e}(0) = \rho_{o}/
is related to the number density as <math> \rho_{e}(0) = \rho_{o}/
+
\sqrt{1-4f_{i}/\pi}</math>.
\sqrt{1-4f_{i}/\pi}</math>.  
+
 
In summary, we have shown that, by taking the limit as the number of
+
[[Category: Wave Scattering in the Marginal Ice Zone]]
angles tend to infinity and as the time step <math>\Delta t</math> tends to
 
zero in the scattering equation of [[Masson_LeBlond_1989a| Masson and LeBlond 1989]],
 
we obtain a linear Boltzmann equation equivalent
 
to the equation given in \cite{jgrrealism}
 
(once the error in this earlier work has been corrected).
 

Latest revision as of 01:42, 17 February 2010


Introduction

The scattering theory of Masson and LeBlond 1989 was the first model which properly accounted for the three dimensional scattering which occurs in the MIZ. The model was derived using multiple scattering and was presented in terms of a time step discretisation and only for ice floes with a circular geometry. Their scattering theory included the effects of wind generation, nonlinear coupling in frequency and wave breaking. However, what was original in their work was their equation for the scattering of wave energy by ice floes.

Meylan and Masson 2006 showed the equivalence of the multiple scattering theory of Masson and LeBlond with the Linear Boltzmann Model for Wave Scattering in the MIZ and this is included the final section.

Equation for Wave Scattering

Masson and LeBlond 1989 began with the following equation for the evolution of wave scattering,

[math]\displaystyle{ \frac{\partial I}{\partial t}+c_{g}\hat{\theta}\nabla I= \left(S_{\mathrm{in}}+S_{{\mathrm{ds}}}\right) \left(1-f_{i}\right) +S_{{\mathrm{nl}} }+S_{{\mathrm{ice}}}, }[/math]

where [math]\displaystyle{ S_{\mathrm{{in}}} }[/math] is the input of wave energy due to wind forcing, [math]\displaystyle{ S_{\mathrm{{ds}}} }[/math] is the dissipation of wave energy due to wave breaking, [math]\displaystyle{ S_{\mathrm{{nl}}} }[/math] is the non-linear transfer of wave energy and [math]\displaystyle{ S_{\mathrm{{ice}}} }[/math] is the wave scattering. Masson and LeBlond 1989 solved this in the isotropic (no spatial dependence) case. Furthermore, they did not actually determine [math]\displaystyle{ S_{\mathrm{{ice}}} }[/math] but derived a time stepping procedure to solve the isotropic solution using multiple scattering. We will derive [math]\displaystyle{ S_{\mathrm{{ice}}} }[/math] from the time stepping equation.

Masson and LeBlond 1989 derived the following difference equation

[math]\displaystyle{ I(f_{n},\theta;t+\Delta t)=[\mathbf{T}]_{f_{n}}[I(f_{n},\theta ;t)+((S_{\mathrm{in}}+S_{\mathrm{ds}})(1-f_{i})+S_{\mathrm{nl}})\Delta t] }[/math]

where [math]\displaystyle{ f_{n} }[/math] is the wave frequency (Masson and LeBlond 1989 equation (51). It is important to realise that [math]\displaystyle{ [\mathbf{T}]_{f_{n}} }[/math] is a function of [math]\displaystyle{ \Delta t }[/math] in the above equation. We are interested only in the wave scattering term so we will set the terms due to wind input ([math]\displaystyle{ S_{\mathrm{{in}}} }[/math]), wave breaking ([math]\displaystyle{ S_{\mathrm{{ds}}} }[/math]) and non-linear coupling ([math]\displaystyle{ S_{\mathrm{{nl}}} }[/math]) to zero. These terms can be readily included in any model if required. Masson and LeBlond 1989 discretized the angle [math]\displaystyle{ \theta }[/math] into [math]\displaystyle{ n }[/math] evenly spaced angles [math]\displaystyle{ \theta_{i} }[/math] between [math]\displaystyle{ -\pi }[/math] and [math]\displaystyle{ \pi }[/math]. [math]\displaystyle{ [\mathbf{T}]_{f_{n}} }[/math] is then given by

[math]\displaystyle{ (T_{ij})_{f_{n}}=A^{2}\{\hat{\beta}|D(\theta_{ij})|^{2}\Delta \theta+\delta(\theta_{ij})(1+|\alpha_{c}D(0)|^{2} )+\delta(\pi-\theta_{ij})|\alpha_{c}D(\pi)|^{2} \}, }[/math]

where [math]\displaystyle{ \theta_{ij}=|(\theta_{i}-\theta_{j})| }[/math] ( Masson and LeBlond 1989, equation (42)). [math]\displaystyle{ \hat{\beta} }[/math] is a function of [math]\displaystyle{ \Delta t }[/math] given by

[math]\displaystyle{ \hat{\beta}=\int_{0}^{c_{g}\Delta t}\rho_{e}(r)dr, }[/math]

(Masson and LeBlond 1989 p. 68). The function [math]\displaystyle{ \rho_{e}(r) }[/math] gives the effective number of floes per unit area effectively radiating waves under the single scattering approximation which is to assume that the amplitude of a wave scattered more than once is negligible. It is given by

[math]\displaystyle{ \rho_{e}(r)=\frac{2}{\sqrt{3}D_{\mathrm{{av}}}^{2}\left( 1-\frac{8a^{2}}{\sqrt {3}D_{\mathrm{{av}}}^{2}}\right) ^{1/2}}\left( 1-\frac{8a^{2}}{\sqrt {3}D_{\mathrm{{av}}}^{2}}\right) ^{r/2a}, }[/math]

(Masson and LeBlond 1989 equation (29), although there is a typographical error in their equation which we have corrected) where [math]\displaystyle{ D_{av} }[/math] is the average floe spacing and [math]\displaystyle{ a }[/math] is the floe radius (remembering that Masson and LeBlond 1989 considered circular floes). The energy factor [math]\displaystyle{ A }[/math] is given by,

[math]\displaystyle{ A=(1+|\alpha_{c}D(0)|^{2}+|\alpha_{c}D(\pi)|^{2}+\hat{\beta}\int_{0}^{2\pi }|D(\theta)|^{2}d\theta+f_{d})^{-\frac{1}{2}}, }[/math]

(Masson and LeBlond 1989 equation (52)) where the term [math]\displaystyle{ f_{d} }[/math] represents dissipation and is given by

[math]\displaystyle{ f_{d}=e^{\frac{f_{i}}{A_{f}}\sigma_{a}c_{g}\Delta t}-1, }[/math]

(Masson and LeBlond 1989 equation (53)) and [math]\displaystyle{ \alpha_{c} }[/math], the coherent scattering coefficient, is given by

[math]\displaystyle{ \alpha_{c}=\left( \frac{2\pi}{k}\right) ^{1/2}\exp\left( \frac {\mathrm{{i}\pi}}{4}\right) \frac{2}{\sqrt{3}D_{\mathrm{{av}}}^{2}\left( 1-\frac{8a^{2}}{\sqrt{3}D_{\mathrm{{av}}}^{2}}\right) ^{1/2}}\int_{0}^{c_g\Delta t }\exp(\mathrm{{i}}kx_{s})\left( 1-\frac{8a^{2}}{\sqrt{3}D_{{av}}^{2}}\right) ^{x_{s}/2a}dx_{s}. }[/math]

It should be noted that the upper limit of integration for [math]\displaystyle{ \alpha_{c} }[/math] was given as infinity in Masson and LeBlond 1989. This is appropriate in the steady case only; it should have been changed to [math]\displaystyle{ c_{g}\Delta t }[/math] in the time dependent case. However, this correction leads to only negligible quantitative changes to the results.

We will transform the scattering operator [math]\displaystyle{ \mathbf{T} }[/math] by taking the limit as the number of angles used to discretise [math]\displaystyle{ \theta }[/math] tends to infinity. On taking this limit, the operator [math]\displaystyle{ \mathbf{T}\left( \Delta t\right) }[/math] becomes

[math]\displaystyle{ \mathbf{T}\left( \Delta t\right) I\left( \theta\right) =A^{2}\{\hat{\beta }\int_{0}^{2\pi}|D(\theta-\theta^{\prime})|^{2}I\left( \theta^{\prime }\right) d\theta^{\prime}+ I\left( \theta\right) \}. }[/math]

The scattering theory of Masson and LeBlond 1989 depends on the values of the time step [math]\displaystyle{ \Delta t }[/math] and the correct solution is found for small time steps. We will now find the equation in the limit of small time steps by taking the limit as [math]\displaystyle{ \Delta t }[/math] tends to zero. As we shall see, when this limit is taken, there is a considerable simplification in the form of the equation. Since

[math]\displaystyle{ I(t+\Delta t)=\mathbf{T}\left( \Delta t\right) I\mathbf{(}t), }[/math]

we obtain the following expression for the time derivative of [math]\displaystyle{ I }[/math],

[math]\displaystyle{ \frac{\partial I}{\partial t}=\lim_{\Delta t\rightarrow0}\left( \frac{\mathbf{T}\left( \Delta t\right)I(t) - I(t)}{\Delta t}\right). }[/math]

We can calculate this limit as follows,

[math]\displaystyle{ \lim_{\Delta t\rightarrow0}\left( \frac{\mathbf{T}\left( \Delta t\right)I(t) -I(t)}{\Delta t}\right) =\lim_{\Delta t\rightarrow0}\left( \frac {A^{2}\{\hat{\beta}\int_{0}^{2\pi}|D(\theta-\theta^{\prime})|^{2}I\left( \theta^{\prime}\right) d\theta^{\prime}+I\left( \theta\right) \}-I\left( \theta\right) }{\Delta t}\right) }[/math]

[math]\displaystyle{ =c_{g}\rho_{e}\left( 0\right) \int _{0}^{2\pi}|D(\theta-\theta^{\prime})|^{2}I\left( \theta^{\prime}\right) d\theta^{\prime}-c_{g}\rho_{e}\left( 0\right) \int_{0}^{2\pi} |D(\theta-\theta^{\prime})|^{2}d\theta^{\prime}+\frac{f_{i}}{A_{f}}\sigma_{a}c_{g}I\left( \theta\right). }[/math]

We can simplify this equation. The value of [math]\displaystyle{ \rho_{e}\left( 0\right) }[/math] is given by

[math]\displaystyle{ \rho_{e}\left( 0\right) =\frac{2}{\sqrt{3}D_{\mathrm{{av}}}^{2}\left( 1-\frac{8a^{2}}{\sqrt{3}D_{\mathrm{{av}}}^{2}}\right) ^{1/2}} =\frac{f_{i}}{A_{f}\sqrt{1-4f_{i}/\pi}}, }[/math]

where we have used the fact that [math]\displaystyle{ f_{i}=2\pi a^{2}/\sqrt{3}D_{av}^{2} }[/math] and [math]\displaystyle{ A_{f}=\pi a^{2} }[/math].

Equivalence with Linear Boltzmann Model

We show here that the equation above is very similar to the equation derived in Linear Boltzmann Model for Wave Scattering in the MIZ. If we substitute our expressions for [math]\displaystyle{ \rho_e(0) }[/math] iand include the spatial term (which was not in Masson and LeBlond 1989 since they assumed isotropy) and divide by [math]\displaystyle{ c_g }[/math], we obtain the following linear Boltzmann equation

[math]\displaystyle{ \frac{1}{c_g}\frac{\partial I}{\partial t}+\hat{\theta}.\nabla I =\frac{1}{\sqrt{1-4f_{i}/\pi}} \int_{0}^{2\pi}\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime })|^{2}I\left( \theta^{\prime}\right) d\theta^{\prime} }[/math]

[math]\displaystyle{ - \left( \frac{1}{\sqrt{1-4f_{i}/\pi}} \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). }[/math]

If we compare this equations with the equivalent equation in Linear Boltzmann Model for Wave Scattering in the MIZ we see that they are identical except for the factor [math]\displaystyle{ 1 / \sqrt{1-4f_{i}/\pi} }[/math] in the two components resulting from the scattering. This difference comes from the fact that, in Masson and LeBlond 1989, multiple scattering is neglected by using an effective density, [math]\displaystyle{ \rho_{e} }[/math], in lieu of the number density [math]\displaystyle{ \rho_{o} }[/math]. The effective density is related to the number density as [math]\displaystyle{ \rho_{e}(0) = \rho_{o}/ \sqrt{1-4f_{i}/\pi} }[/math].