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

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of physics. There are many approaches to this problem, the two | of physics. There are many approaches to this problem, the two | ||

most popular being | most popular being | ||

− | the linear Boltzmann (or transport) equation, and | + | the linear Boltzmann (or transport) equation, and multiple scattering. |

− | + | However, the MIZ presents | |

a slightly more complex case than is usually encountered in scattering | a slightly more complex case than is usually encountered in scattering | ||

theories because of the random nature of the scatterers (i.e. the random | theories because of the random nature of the scatterers (i.e. the random | ||

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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 | + | derivation given in [[Meylan_Squire_Fox_1997a | 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 | ||

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we assume that wave energy is incoherent. | we assume that wave energy is incoherent. | ||

We consider the surface of an | We consider the surface of an | ||

− | infinitely deep ocean represented in Cartesian co-ordinates by | + | infinitely deep ocean represented in Cartesian co-ordinates by </math>{\bf r} =(x,y).</math> |

Wave energy is propagating across this surface in all directions so that, | 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 | at any point, we must consider the energy travelling in each direction. We | ||

− | introduce an intensity function | + | introduce an intensity function </math>I({\bf r},t,\theta)</math> which is the rate of flow of |

energy travelling in a given direction, per unit surface, per unit | energy travelling in a given direction, per unit surface, per unit | ||

angle. In the absence of scatterers, we assume that the waves continue to | 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 | propagate in the same direction and that the energy intensity satisfies the | ||

following equation, | following equation, | ||

− | + | <math>\label{no_scattering} | |

\frac{1}{c_{g}}\frac{\partial}{\partial t}I({\bf r},t,\theta)+\hat{\theta}.\nabla | \frac{1}{c_{g}}\frac{\partial}{\partial t}I({\bf r},t,\theta)+\hat{\theta}.\nabla | ||

I({\bf r},t,\theta)=0, | I({\bf r},t,\theta)=0, | ||

− | + | </math> | |

− | (\cite{phillips77}) where | + | (\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 |

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. | ||

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general equation for the propagation of wave energy through a | general equation for the propagation of wave energy through a | ||

scattering medium, | scattering medium, | ||

− | + | <math> | |

\frac{1}{c_{g}}\frac{\partial}{\partial t}I({\bf r},t,\theta)+\hat{\theta}.\nabla | \frac{1}{c_{g}}\frac{\partial}{\partial t}I({\bf r},t,\theta)+\hat{\theta}.\nabla | ||

I({\bf r},t,\theta)=-\beta({\bf r},\theta)I({\bf r},t,\theta)+\int_{0}^{2\pi}S({\bf r},\theta,\theta | I({\bf r},t,\theta)=-\beta({\bf r},\theta)I({\bf r},t,\theta)+\int_{0}^{2\pi}S({\bf r},\theta,\theta | ||

^{\prime})I({\bf r},t,\theta^{\prime})d\theta^{\prime}, \label{Howells}% | ^{\prime})I({\bf r},t,\theta^{\prime})d\theta^{\prime}, \label{Howells}% | ||

− | + | </math> | |

− | (\cite{howells60}) where | + | (\cite{howells60}) where </math>\beta</math> is the absorption coefficient and |

− | + | </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~(\ref{Howells}) 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 | ||

− | incoherently. The absorption coefficient, | + | incoherently. The absorption coefficient, </math>\beta({\bf r},\theta)</math>, is the |

fraction of energy lost by scattering and dissipative processes (assumed | fraction of energy lost by scattering and dissipative processes (assumed | ||

− | linear) from a pencil of radiation in direction | + | linear) from a pencil of radiation in direction </math>\theta</math>, per unit path length |

− | travelled in the medium. The scattering function | + | travelled in the medium. The scattering function </math>S({\bf r},\theta,\theta^{\prime |

− | }) | + | })</math> specifies the angular distribution of scattered energy in such a way |

that, | that, | ||

− | + | <math>\label{equation_S} | |

S({\bf r},\theta,\theta^{\prime})I({\bf r},t,\theta^{\prime})d\Omega dSd\Omega | S({\bf r},\theta,\theta^{\prime})I({\bf r},t,\theta^{\prime})d\Omega dSd\Omega | ||

^{\prime}, | ^{\prime}, | ||

− | + | </math> | |

is the rate at which energy is scattered from a pencil of radiation of | is the rate at which energy is scattered from a pencil of radiation of | ||

− | intensity | + | intensity </math>I({\bf r},t,\theta^{\prime})</math> at an angle </math>d\Omega^{\prime} </math> |

− | in direction | + | in direction </math>\theta^{\prime}</math>, by a surface </math>dS</math> at position </math>{\bf r}</math>, into an |

− | angle | + | 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 (\ref{Howells}) to wave scattering in the MIZ we must | ||

first estimate the scattering | first estimate the scattering | ||

− | function | + | function </math>S({\bf r},\theta,\theta^{\prime})</math> and |

− | the absorption coefficient | + | the absorption coefficient </math>\beta({\bf r},\theta)</math>. |

The scattering function is determined by | The scattering function is determined by | ||

calculating the scattering from a single ice floe. | calculating the scattering from a single ice floe. | ||

Each ice floe scatters energy, and the energy radiated per unit angle | Each ice floe scatters energy, and the energy radiated per unit angle | ||

− | per unit time in the | + | per unit time in the </math>\theta</math> direction for a wave incident in |

− | the | + | the </math>\theta^{\prime}</math> direction, </math>E</math>, is given by, |

− | + | <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},\label{energyrad}% | g}{4k}|D(\theta-\theta^{\prime})|^{2},\label{energyrad}% | ||

− | + | </math> | |

− | where | + | where </math>H</math> is the wave height, </math>\rho</math> is the water density, |

− | + | </math>\omega</math> and </math>k</math> are the radian frequency and wavenumber of the wave, | |

− | respectively, and | + | respectively, and </math>g</math> is the acceleration due to gravity. |

− | + | </math>D(\theta - \theta^{\prime})</math> is the scattered amplitude for which, | |

− | at a large distance, | + | at a large distance, </math>r</math>, from the scatterer, |

the asymptotic amplitude of the | the asymptotic amplitude of the | ||

− | outgoing wave in the | + | outgoing wave in the </math>\theta</math> direction, for an incident wave |

− | travelling in the | + | travelling in the </math>\theta^{\prime}</math> direction, is given by |

− | + | <math>\label{energyrad_D} | |

\frac{H}{2}\frac{D\left( \theta - \theta^{\prime}\right) }{\sqrt{r}}. | \frac{H}{2}\frac{D\left( \theta - \theta^{\prime}\right) }{\sqrt{r}}. | ||

− | + | </math> | |

Note that, in equations~(\ref{energyrad}) and (\ref{energyrad_D}), we have assumed | Note that, in equations~(\ref{energyrad}) and (\ref{energyrad_D}), 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). | ||

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We must now express the scattering kernel in equation (\ref{Howells}), | We must now express the scattering kernel in equation (\ref{Howells}), | ||

− | + | </math>S({\bf r},\theta,\theta^{\prime})</math>, in terms of </math>E</math>. | |

− | Given the definition of | + | Given the definition of </math>S</math> (equation (\ref{equation_S})), |

− | + | </math>S</math> can be found by | |

− | dividing | + | dividing </math>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 | of energy passing under the floe is given by the product of the wave energy | ||

− | density ( | + | density (</math>\frac{1}{8}\rho gH^{2},</math> since </math>H/2</math> is the wave amplitude and we |

are considering only the energy in the water), the | are considering only the energy in the water), the | ||

− | average area occupied by a floe ( | + | average area occupied by a floe (</math>A_{f}/f_{i},</math> where </math>A_{f}</math> is the |

average area of | average area of | ||

− | the floe and | + | the floe and </math>f_{i}</math> is the fraction |

of the surface area of the ice covered ocean which is covered in ice), | of the surface area of the ice covered ocean which is covered in ice), | ||

− | and the wave group speed ( | + | and the wave group speed (</math>\omega/2k)</math>. |

− | This gives the following expression for | + | This gives the following expression for </math>S</math>, |

\begin{align} | \begin{align} | ||

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 | ||

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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 \cite{jgrrealism} because, | ||

− | in the derivation for | + | in the derivation for </math>S</math> in \cite{jgrrealism}, 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 | + | We can determine </math>\beta</math> from the absorption cross section, |

− | + | </math>\sigma_a</math>, and the ice cover fraction </math>f_i</math> | |

(M\&Le p. 58). The absorption cross section, | (M\&Le p. 58). The absorption cross section, | ||

− | + | </math>\sigma_a</math>, may be estimated from the total scattering or | |

from experimental measurements. | from experimental measurements. | ||

− | The expression for | + | The expression for </math>\beta</math> is the following, |

− | + | <math>\label{beta_equation} | |

\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}}. | ||

− | + | </math> | |

Combining equations (\ref{Howells}), (\ref{scatteringfromeng2})\ | Combining equations (\ref{Howells}), (\ref{scatteringfromeng2})\ | ||

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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, | ||

− | + | <math>\label{boltzmann} | |

\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 | ||

= | = | ||

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\frac{f_{i}}{A_{f}} | \frac{f_{i}}{A_{f}} | ||

\right)I\left( \theta\right). | \right)I\left( \theta\right). | ||

− | + | </math> | |

\section{The Multiple Scattering Theory of M\&Le} | \section{The Multiple Scattering Theory of M\&Le} | ||

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for the scattering of wave energy by ice floes. M\&Le began | for the scattering of wave energy by ice floes. M\&Le began | ||

with the following equation for the evolution of wave scattering, | with the following equation for the evolution of wave scattering, | ||

− | + | <math> | |

\frac{\partial I}{\partial t}+c_{g}\hat{\theta}.\nabla I= \left( S_{\mathrm{{in}}% | \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{{ds}}}\right) \left( 1-f_{i}\right) +S_{\mathrm{{nl}}% | ||

}+S_{\mathrm{{ice}}}, \label{BoltzMasson}% | }+S_{\mathrm{{ice}}}, \label{BoltzMasson}% | ||

− | + | </math> | |

− | where | + | where </math>S_{\mathrm{{in}}}</math> is the input of wave energy due |

− | to wind forcing, | + | to wind forcing, </math>S_{\mathrm{{ds}}}</math> is the dissipation of wave |

− | energy due to wave breaking, | + | energy due to wave breaking, </math>S_{\mathrm{{nl}}}</math> is the non-linear transfer |

− | of wave energy and | + | of wave energy and </math>S_{\mathrm{{ice}}}</math> is the wave scattering. |

− | Similarly, the terms | + | Similarly, the terms </math>S_{\mathrm{{in}}},</math> |

− | + | </math>S_{\mathrm{{ds}}},</math> and </math>S_{\mathrm{{nl}}}</math> could be added to equation | |

(\ref{Howells}). However, the purpose of this | (\ref{Howells}). However, the purpose of this | ||

− | paper is to derive a consistent equation for | + | paper is to derive a consistent equation for </math>S_{\mathrm{{ice }}}.</math> M\&Le |

solved equation (\ref{BoltzMasson}) in the isotropic (no spatial | 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 | |

− | isotropic solution using multiple scattering. We will derive | + | isotropic solution using multiple scattering. We will derive </math>S_{\mathrm{{ice}}}</math> |

from their time stepping equation. | from their time stepping equation. | ||

M\&Le derived the following difference equation as | M\&Le derived the following difference equation as | ||

a discrete analogue of equation~(\ref{BoltzMasson}) | a discrete analogue of equation~(\ref{BoltzMasson}) | ||

− | + | <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 | ||

;t)+((S_{\mathrm{in}}+S_{\mathrm{ds}})(1-f_{i})+S_{\mathrm{nl}})\Delta | ;t)+((S_{\mathrm{in}}+S_{\mathrm{ds}})(1-f_{i})+S_{\mathrm{nl}})\Delta | ||

t],\label{timestep}% | t],\label{timestep}% | ||

− | + | </math> | |

− | where | + | where </math>f_{n}</math> is the wave frequency |

− | (M\&Le, equation~(51)). It is important to realise that | + | (M\&Le, equation~(51)). It is important to realise that </math>[\mathbf{T}]_{f_{n}}</math> is a |

function of | function of | ||

− | + | </math>\Delta t</math> in the above equation. We are interested only in the | |

wave scattering term so we will set the terms due to | wave scattering term so we will set the terms due to | ||

− | wind input ( | + | wind input (</math>S_{\mathrm{{in}}}</math>), wave breaking (</math>S_{\mathrm{{ds}}}</math>) and |

− | non-linear coupling ( | + | non-linear coupling (</math>S_{\mathrm{{nl}}}</math>) to zero. |

These terms can be readily included in any model if required. | These terms can be readily included in any model if required. | ||

M\&Le discretized the | M\&Le discretized the | ||

− | angle | + | angle </math>\theta</math> into </math>n</math> evenly spaced angles </math>\theta_{i}</math> between </math>-\pi</math> |

− | and | + | and </math>\pi</math>. </math>[\mathbf{T}]_{f_{n}}</math> is then given by |

− | + | <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 | ||

\theta+\delta(\theta_{ij})(1+|\alpha_{c}D(0)|^{2}% | \theta+\delta(\theta_{ij})(1+|\alpha_{c}D(0)|^{2}% | ||

)+\delta(\pi-\theta_{ij})|\alpha_{c}D(\pi)|^{2} | )+\delta(\pi-\theta_{ij})|\alpha_{c}D(\pi)|^{2} | ||

\},\label{wrong}% | \},\label{wrong}% | ||

− | + | </math> | |

− | where | + | where </math>\theta_{ij}=|(\theta_{i}-\theta_{j})|</math> |

− | (M\&Le, equation~(42)). In equation~(\ref{wrong}), | + | (M\&Le, equation~(42)). In equation~(\ref{wrong}), </math>\hat{\beta}</math> |

(this notation is chosen to follow from M\&Le | (this notation is chosen to follow from M\&Le | ||

− | who used | + | who used </math>\beta</math> and to avoid confusion with the expression for </math>\beta</math> in |

equation (\ref{Howells}) and which | equation (\ref{Howells}) and which | ||

− | is used in \cite{howells60} and \cite{jgrrealism}) is a function of | + | is used in \cite{howells60} and \cite{jgrrealism}) is a function of </math>\Delta t</math> |

given by | given by | ||

− | + | <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> | |

− | (M\&Le p. 68). The function | + | (M\&Le 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 | ||

− | + | <math>\label{rho_e} | |

\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 | ||

{3}D_{\mathrm{{av}}}^{2}}\right) ^{1/2}}\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}, | {3}D_{\mathrm{{av}}}^{2}}\right) ^{r/2a}, | ||

− | + | </math> | |

(M\&Le equation~(29), | (M\&Le 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 | + | where </math>D_{av}</math> is the average floe spacing and </math>a</math> is the floe radius |

− | (remembering that M\&Le considered circular floes). The energy factor | + | (remembering that M\&Le considered circular floes). The energy factor </math>A</math> is |

given by, | given by, | ||

− | + | <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}}\label{equation_A}, | }|D(\theta)|^{2}d\theta+f_{d})^{-\frac{1}{2}}\label{equation_A}, | ||

− | + | </math> | |

− | (M\&Le equation~(52)) where the term | + | (M\&Le equation~(52)) where the term </math>f_{d}</math> represents dissipation and is given by |

\[ | \[ | ||

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, | ||

\] | \] | ||

− | (M\&Le equation~(53)) and | + | (M\&Le equation~(53)) and </math>\alpha_{c}</math>, the ``coherent'' scattering coefficient, is given by |

− | + | <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 | ||

{\mathrm{{i}\pi}}{4}\right) \frac{2}{\sqrt{3}D_{\mathrm{{av}}}^{2}\left( | {\mathrm{{i}\pi}}{4}\right) \frac{2}{\sqrt{3}D_{\mathrm{{av}}}^{2}\left( | ||

Line 271: | Line 271: | ||

}\exp(\mathrm{{i}}kx_{s})\left( 1-\frac{8a^{2}}{\sqrt{3}D_{{av}}^{2}}\right) | }\exp(\mathrm{{i}}kx_{s})\left( 1-\frac{8a^{2}}{\sqrt{3}D_{{av}}^{2}}\right) | ||

^{x_{s}/2a}dx_{s}. | ^{x_{s}/2a}dx_{s}. | ||

− | + | </math> | |

− | It should be noted that the upper limit of integration for | + | It should be noted that the upper limit of integration for </math>\alpha_{c}</math> |

was given as infinity in M\&Le. This is appropriate in the steady case | was given as infinity in M\&Le. This is appropriate in the steady case | ||

− | only; it should have been changed to | + | 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 M\&Le scattering operator | + | We will transform the M\&Le scattering operator </math>\mathbf{T}</math> by |

− | taking the limit as the number of angles used to discretise | + | taking the limit as the number of angles used to discretise </math>\theta</math> tends to |

− | infinity. On taking this limit, the operator | + | infinity. On taking this limit, the operator </math>\mathbf{T}\left( \Delta t\right) </math> |

becomes | becomes | ||

\[ | \[ | ||

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The scattering theory of M\&Le depends on the | The scattering theory of M\&Le depends on the | ||

− | values of the time step | + | 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 | ||

find the equation in the limit of small time steps by taking the | find the equation in the limit of small time steps by taking the | ||

− | limit as | + | limit as </math>\Delta t</math> tends to zero. |

As we shall see, when this limit is taken, | As we shall see, when this limit is taken, | ||

there is a considerable simplification in the form of the equation. | there is a considerable simplification in the form of the equation. | ||

Line 299: | Line 299: | ||

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), | ||

\] | \] | ||

− | we obtain the following expression for the time derivative of | + | we obtain the following expression for the time derivative of </math>I</math>, |

\[ | \[ | ||

\frac{\partial I}{\partial t}=\lim_{\Delta t\rightarrow0}\left( | \frac{\partial I}{\partial t}=\lim_{\Delta t\rightarrow0}\left( | ||

Line 313: | Line 313: | ||

\] | \] | ||

− | + | <math>\label{M_Le_boltzmann1} | |

=c_{g}\rho_{e}\left( 0\right) \int | =c_{g}\rho_{e}\left( 0\right) \int | ||

_{0}^{2\pi}|D(\theta-\theta^{\prime})|^{2}I\left( \theta^{\prime}\right) | _{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^{\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). | |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 equation~(\ref{M_Le_boltzmann1}) by using equation~(\ref{rho_e}). | We can simplify equation~(\ref{M_Le_boltzmann1}) by using equation~(\ref{rho_e}). | ||

− | The value of | + | The value of </math>\rho_{e}\left( 0\right)</math> is given by |

− | + | <math>\label{rho} | |

\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( | ||

1-\frac{8a^{2}}{\sqrt{3}D_{\mathrm{{av}}}^{2}}\right) ^{1/2}} | 1-\frac{8a^{2}}{\sqrt{3}D_{\mathrm{{av}}}^{2}}\right) ^{1/2}} | ||

=\frac{f_{i}}{A_{f}\sqrt{1-4f_{i}/\pi}}, | =\frac{f_{i}}{A_{f}\sqrt{1-4f_{i}/\pi}}, | ||

− | + | </math> | |

− | where we have used the fact that | + | 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>. | |

− | If we substitute our expressions for | + | If we substitute our expressions for </math>\rho_e(0)</math> |

in equation (\ref{M_Le_boltzmann1}) and | in equation (\ref{M_Le_boltzmann1}) and | ||

include the spatial term | include the spatial term | ||

(which was not in M\&Le since they assumed | (which was not in M\&Le since they assumed | ||

− | isotropy) and divide by | + | isotropy) and divide by </math>c_g</math>, we obtain the following |

linear Boltzmann equation | linear Boltzmann equation | ||

\begin{align}\label{M_Le_boltzmann2} | \begin{align}\label{M_Le_boltzmann2} | ||

Line 348: | Line 348: | ||

If we compare equations (\ref{boltzmann}) and (\ref{M_Le_boltzmann2}) | If we compare equations (\ref{boltzmann}) and (\ref{M_Le_boltzmann2}) | ||

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. | |

This difference comes from the fact that, in M\&Le, multiple | This difference comes from the fact that, in M\&Le, multiple | ||

− | scattering is neglected by using an effective density, | + | scattering is neglected by using an effective density, </math>\rho_{e}</math>, |

− | in lieu of the number density | + | in lieu of the number density </math>\rho_{o}</math>. As shown in equation (\ref{rho}), |

the effective density | the effective density | ||

− | is related to the number density as | + | is related to the number density as </math> \rho_{e}(0) = \rho_{o}/ |

− | \sqrt{1-4f_{i}/\pi} | + | \sqrt{1-4f_{i}/\pi}</math>. |

In summary, we have shown that, by taking the limit as the number of | In summary, we have shown that, by taking the limit as the number of | ||

− | angles tend to infinity and as the time step | + | angles tend to infinity and as the time step </math>\Delta t</math> tends to |

zero in the scattering equation of M\&Le, | zero in the scattering equation of M\&Le, | ||

we obtain a linear Boltzmann equation equivalent | we obtain a linear Boltzmann equation equivalent | ||

Line 366: | Line 366: | ||

The central difficulty in applying equations (\ref{boltzmann}) and | The central difficulty in applying equations (\ref{boltzmann}) and | ||

(\ref{M_Le_boltzmann2}) is the determination of the scattering | (\ref{M_Le_boltzmann2}) is the determination of the scattering | ||

− | amplitude | + | amplitude </math>D(\theta-\theta^{\prime})</math>. |

The scattering amplitude is found by solving the | The scattering amplitude is found by solving the | ||

boundary value problem which arises when an isolated floe | boundary value problem which arises when an isolated floe | ||

Line 383: | Line 383: | ||

the large scale scattering models derived by M\&Le and \cite{jgrrealism}. | the large scale scattering models derived by M\&Le and \cite{jgrrealism}. | ||

Here we simply present | Here we simply present | ||

− | the equation for | + | the equation for </math>D(\theta)</math> independent of the equation used to model |

the ice floe. | the ice floe. | ||

Line 389: | Line 389: | ||

amplitude is assumed sufficiently small that we can linearise all the | amplitude is assumed sufficiently small that we can linearise all the | ||

equations. The water motion is represented by a velocity potential which is | equations. The water motion is represented by a velocity potential which is | ||

− | denoted by | + | denoted by </math>\phi</math>. The coordinate system is the standard Cartesian coordinate system |

− | with the | + | 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 | and the wetted | ||

− | surface of the ice floe by | + | 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 | + | potential </math>\phi({\bf r},z)</math> subject to an incoming |

− | wave of frequency | + | wave of frequency </math>\omega</math> is |

− | + | <math>\label{bvp} | |

\left. | \left. | ||

\begin{array} | \begin{array} | ||

Line 408: | Line 408: | ||

\end{array} | \end{array} | ||

\right\} % | \right\} % | ||

− | + | </math> | |

(e.g. \cite{jgrfloecirc,IJOPE04} or M\&Le). | (e.g. \cite{jgrfloecirc,IJOPE04} or M\&Le). | ||

The Laplace's equation comes from the fact | The Laplace's equation comes from the fact | ||

that the water is irrotational and inviscid. The vanishing of the normal | that the water is irrotational and inviscid. The vanishing of the normal | ||

− | derivative at | + | derivative at </math>z=-\infty</math> is the no-flow through the boundary |

condition at the bottom of the infinitely deep ocean. | condition at the bottom of the infinitely deep ocean. | ||

The condition at the free surface is the standard linear free surface | The condition at the free surface is the standard linear free surface | ||

Line 418: | Line 418: | ||

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 | + | operator </math>L</math>. |

To actually solve equation~(\ref{bvp}) requires us to choose a | To actually solve equation~(\ref{bvp}) requires us to choose a | ||

− | specific model for the ice floe and hence to determine | + | specific model for the ice floe and hence to determine </math>L</math>. |

\cite{jgrrealism} assumed that the | \cite{jgrrealism} assumed that the | ||

ice floes had negligible submergence but could flex. | ice floes had negligible submergence but could flex. | ||

Line 430: | Line 430: | ||

\cite{Masson_Le,Sarp_Isa}. | \cite{Masson_Le,Sarp_Isa}. | ||

− | Equation (\ref{bvp}) requires boundary conditions as | + | Equation (\ref{bvp}) 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>. | |

− | We assume that | + | We assume that </math>\phi^{\mathrm{{In}}}</math> |

− | is a plane wave travelling in the | + | is a plane wave travelling in the </math>x</math> direction, |

− | + | <math> | |

\phi^{\mathrm{{In}}}({\bf r},z)=\frac{\omega H}{2k}e^{\rm{i}kx}e^{kz}. | \phi^{\mathrm{{In}}}({\bf r},z)=\frac{\omega H}{2k}e^{\rm{i}kx}e^{kz}. | ||

− | + | </math> | |

− | The condition as | + | The condition as </math>\left|\mathbf{r}\right|\rightarrow |

− | \infty | + | \infty</math> |

is the standard Sommerfeld radiation condition (e.g. \cite{Weh_Lait}) | is the standard Sommerfeld radiation condition (e.g. \cite{Weh_Lait}) | ||

− | + | <math> | |

\sqrt{|\mathbf{r}|}\left( \frac{\partial}{\partial|\mathbf{r}|}-\rm{i}k\right) | \sqrt{|\mathbf{r}|}\left( \frac{\partial}{\partial|\mathbf{r}|}-\rm{i}k\right) | ||

(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.} | (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.} | ||

\label{sommerfield}% | \label{sommerfield}% | ||

− | + | </math> | |

Once we have found the solution to | Once we have found the solution to | ||

equation~(\ref{bvp}), we obtain the absolute value | equation~(\ref{bvp}), we obtain the absolute value | ||

of the scattering | of the scattering | ||

− | amplitude | + | amplitude </math>D</math> as |

\[ | \[ | ||

\left\vert D\left( \theta\right) \right\vert =\frac{1}{H/2}\left( \frac | \left\vert D\left( \theta\right) \right\vert =\frac{1}{H/2}\left( \frac | ||

Line 458: | Line 458: | ||

\pi+\theta\right)\right\vert, | \pi+\theta\right)\right\vert, | ||

\] | \] | ||

− | where the Kochin function | + | where the Kochin function </math>\mathbf{H}(\tau)</math> is |

− | + | <math> | |

\mathbf{H}(\tau)=\iint_{\Gamma_s}\left( -\frac{\delta\phi}{\delta n} | \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 | + \phi\frac{\delta}{\delta n}\right) e^{kz}e^{ik(x\cos\tau | ||

+y\sin\tau)}dS, | +y\sin\tau)}dS, | ||

− | + | </math> | |

− | where | + | where </math>\delta/\delta n</math> is the inward normal derivative. |

\section{Numerical Solution of the Transport Equation} | \section{Numerical Solution of the Transport Equation} | ||

Line 474: | Line 474: | ||

equation in angle. We begin by | equation in angle. We begin by | ||

assuming that the solution is only a | assuming that the solution is only a | ||

− | function of the | + | function of the </math>x</math> spatial co-ordinate and time, i.e. there is no </math>y</math> |

dependence of the solution. We also consider a uniform MIZ so that the | dependence of the solution. We also consider a uniform MIZ so that the | ||

− | scattering function, | + | scattering function, </math>S</math>, is a function only of </math>\theta</math> and </math>\theta^{\prime}</math> |

− | and | + | and </math>\beta</math> is a constant. The only variation we allow spatially is that the |

− | MIZ occupies the region | + | 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 equations (\ref{boltzmann}) and (\ref{M_Le_boltzmann2}) | ||

become, | become, | ||

− | + | <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 | ||

I}{\partial x}=\left\{ | I}{\partial x}=\left\{ | ||

Line 493: | Line 493: | ||

\end{array} | \end{array} | ||

\right. \label{almostsolvable}% | \right. \label{almostsolvable}% | ||

− | + | </math> | |

To solve equation (\ref{almostsolvable}) we convert the problem to a matrix | To solve equation (\ref{almostsolvable}) 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 | ||

− | discrete set of | + | discrete set of </math>n</math> angles evenly spaced between </math>0</math> and </math>2\pi</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, | (\ref{almostsolvable}) to the following equation, | ||

− | + | <math> | |

\frac{1}{c_{g}}\frac{\partial\vec{I}}{\partial t}-\frac{\partial}{\partial | \frac{1}{c_{g}}\frac{\partial\vec{I}}{\partial t}-\frac{\partial}{\partial | ||

x}\mathbf{{D}}\vec{I}=\mathbf{\left\{ | x}\mathbf{{D}}\vec{I}=\mathbf{\left\{ | ||

Line 511: | Line 511: | ||

\end{array} | \end{array} | ||

\right. } \label{matrix}% | \right. } \label{matrix}% | ||

− | + | </math> | |

− | In equation (\ref{matrix}) the intensity | + | In equation (\ref{matrix}) the intensity </math>\vec{I}</math> is now a vector of functions |

− | of | + | of </math>x</math> and </math>t</math> |

− | for each angle | + | for each angle </math>\theta_{j}</math> and the elements of the matrices </math>\mathbf{{D}}</math> and |

− | + | </math>\mathbf{{S}}</math> are given by, | |

− | + | <math> | |

d_{ij}=\left\{ | d_{ij}=\left\{ | ||

\begin{array} | \begin{array} | ||

Line 524: | Line 524: | ||

\end{array} | \end{array} | ||

\right. | \right. | ||

− | + | </math> | |

and | and | ||

− | + | <math> | |

s_{ij}=\left\{ | s_{ij}=\left\{ | ||

\begin{array} | \begin{array} | ||

Line 534: | Line 534: | ||

\end{array} | \end{array} | ||

\right. | \right. | ||

− | + | </math> | |

respectively. Equation (\ref{matrix}) can be easily solved in the stationary (no time | respectively. Equation (\ref{matrix}) 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. \cite{jgrrealism} | ||

Line 540: | Line 540: | ||

(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 | + | equation (\ref{matrix}) reduces to, setting the ice edge to </math>b=0,</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. \label{spaceonly}% | \;\;x>0. \label{spaceonly}% | ||

− | + | </math> | |

For the isotropic case, equation (\ref{matrix}) reduces to, setting the ice edge | For the isotropic case, equation (\ref{matrix}) reduces to, setting the ice edge | ||

− | to | + | to </math>b=-\infty</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. \label{timeonly}% | \vec{I},\;\;t>0. \label{timeonly}% | ||

− | + | </math> | |

Equations (\ref{spaceonly}) and (\ref{timeonly}) can be solved by | Equations (\ref{spaceonly}) and (\ref{timeonly}) can be solved by | ||

straightforward matrix methods (e.g. \cite{ishimaru}). Equation (\ref{spaceonly}) | straightforward matrix methods (e.g. \cite{ishimaru}). Equation (\ref{spaceonly}) | ||

− | requires boundary conditions (the wave spectrum at the ice edge | + | requires boundary conditions (the wave spectrum at the ice edge </math>x=0</math> |

− | and a condition as | + | and a condition as </math>x\to\infty</math>) and |

equation (\ref{timeonly}) requires an initial condition (the wave spectrum | equation (\ref{timeonly}) requires an initial condition (the wave spectrum | ||

− | at | + | at </math>t=0</math>). |

## Revision as of 04:07, 17 May 2006

# 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 The equation is derived by two methods, the first based on Masson and LeBlond 1989 and the second based on Meylan, Squire and Fox 1997.

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>{\bf 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>I({\bf 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{ \label{no_scattering} \frac{1}{c_{g}}\frac{\partial}{\partial t}I({\bf r},t,\theta)+\hat{\theta}.\nabla I({\bf r},t,\theta)=0, }[/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 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 (\ref{no_scattering}) 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({\bf r},t,\theta)+\hat{\theta}.\nabla I({\bf r},t,\theta)=-\beta({\bf r},\theta)I({\bf r},t,\theta)+\int_{0}^{2\pi}S({\bf r},\theta,\theta ^{\prime})I({\bf r},t,\theta^{\prime})d\theta^{\prime}, \label{Howells}% }[/math] (\cite{howells60}) where </math>\beta</math> is the absorption coefficient and </math>S</math> is the scattering function (assumed to be independent of time). Equation~(\ref{Howells}) depends on the assumption that each floe scatters independently and that the energy from different scatterers may be added incoherently. The absorption coefficient, </math>\beta({\bf r},\theta)</math>, is the fraction of energy lost by scattering and dissipative processes (assumed linear) from a pencil of radiation in direction </math>\theta</math>, per unit path length travelled in the medium. The scattering function </math>S({\bf r},\theta,\theta^{\prime })</math> specifies the angular distribution of scattered energy in such a way that, [math]\displaystyle{ \label{equation_S} S({\bf r},\theta,\theta^{\prime})I({\bf r},t,\theta^{\prime})d\Omega dSd\Omega ^{\prime}, }[/math] is the rate at which energy is scattered from a pencil of radiation of intensity </math>I({\bf r},t,\theta^{\prime})</math> at an angle </math>d\Omega^{\prime} </math> in direction </math>\theta^{\prime}</math>, by a surface </math>dS</math> at position </math>{\bf r}</math>, into an angle </math>d\Omega</math> in direction </math>\theta</math>. To apply equation (\ref{Howells}) to wave scattering in the MIZ we must first estimate the scattering function </math>S({\bf r},\theta,\theta^{\prime})</math> and the absorption coefficient </math>\beta({\bf r},\theta)</math>.

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

the asymptotic amplitude of the

outgoing wave in the </math>\theta</math> direction, for an incident wave travelling in the </math>\theta^{\prime}</math> direction, is given by [math]\displaystyle{ \label{energyrad_D} \frac{H}{2}\frac{D\left( \theta - \theta^{\prime}\right) }{\sqrt{r}}. }[/math] Note that, in equations~(\ref{energyrad}) and (\ref{energyrad_D}), 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 (\ref{Howells}), </math>S({\bf r},\theta,\theta^{\prime})</math>, in terms of </math>E</math>. Given the definition of </math>S</math> (equation (\ref{equation_S})),

</math>S</math> can be found by

dividing </math>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>\frac{1}{8}\rho gH^{2},</math> since </math>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>A_{f}/f_{i},</math> where </math>A_{f}</math> is the average area of the floe and </math>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>\omega/2k)</math>. This gives the following expression for </math>S</math>, \begin{align} 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} \nonumber \label{scatteringfromeng2}\\ & =\frac{f_{i}}{A_{f}}|D(\theta-\theta^{\prime})|^{2}. \end{align} This expression is not exactly the same as the equivalent expression in \cite{jgrrealism} because, in the derivation for </math>S</math> in \cite{jgrrealism}, the wave phase speed rather than the group speed was erroneously used.

We can determine </math>\beta</math> from the absorption cross section, </math>\sigma_a</math>, and the ice cover fraction </math>f_i</math> (M\&Le p. 58). The absorption cross section, </math>\sigma_a</math>, may be estimated from the total scattering or from experimental measurements. The expression for </math>\beta</math> is the following, [math]\displaystyle{ \label{beta_equation} \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}}. }[/math]

Combining equations (\ref{Howells}), (\ref{scatteringfromeng2})\ and (\ref{beta_equation}),

the following linear Boltzmann equation for

wave scattering in the MIZ is obtained, [math]\displaystyle{ \label{boltzmann} \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). }[/math]

\section{The Multiple Scattering Theory of M\&Le}

The scattering theory of M\&Le 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. M\&Le 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}}}, \label{BoltzMasson}% }[/math] where </math>S_{\mathrmTemplate:In}</math> is the input of wave energy due to wind forcing, </math>S_{\mathrmTemplate:Ds}</math> is the dissipation of wave energy due to wave breaking, </math>S_{\mathrmTemplate:Nl}</math> is the non-linear transfer of wave energy and </math>S_{\mathrmTemplate:Ice}</math> is the wave scattering. Similarly, the terms </math>S_{\mathrmTemplate:In},</math> </math>S_{\mathrmTemplate:Ds},</math> and </math>S_{\mathrmTemplate:Nl}</math> could be added to equation (\ref{Howells}). However, the purpose of this paper is to derive a consistent equation for </math>S_{\mathrmTemplate:Ice}.</math> M\&Le

solved equation (\ref{BoltzMasson}) in the isotropic (no spatial

dependence) case. Furthermore, they did not actually determine </math>S_{\mathrmTemplate:Ice}</math> but derived a time stepping procedure to solve the isotropic solution using multiple scattering. We will derive </math>S_{\mathrmTemplate:Ice}</math> from their time stepping equation.

M\&Le derived the following difference equation as
a discrete analogue of equation~(\ref{BoltzMasson})
[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],\label{timestep}%
}[/math]
where </math>f_{n}</math> is the wave frequency
(M\&Le, equation~(51)). It is important to realise that </math>[\mathbf{T}]_{f_{n}}</math> is a
function of
</math>\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>S_{\mathrmTemplate:In}</math>), wave breaking (</math>S_{\mathrmTemplate:Ds}</math>) and
non-linear coupling (</math>S_{\mathrmTemplate:Nl}</math>) to zero.
These terms can be readily included in any model if required.
M\&Le discretized the
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
[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}
\},\label{wrong}%
}[/math]
where </math>\theta_{ij}=|(\theta_{i}-\theta_{j})|</math>
(M\&Le, equation~(42)). In equation~(\ref{wrong}), </math>\hat{\beta}</math>
(this notation is chosen to follow from M\&Le
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
[math]\displaystyle{
\hat{\beta}=\int_{0}^{c_{g}\Delta t}\rho_{e}(r)dr,
}[/math]
(M\&Le 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

which is to assume that the amplitude of a wave scattered

more than once is negligible. It is given by
[math]\displaystyle{ \label{rho_e}
\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]
(M\&Le equation~(29),
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
(remembering that M\&Le considered circular floes). The energy factor </math>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}}\label{equation_A},
}[/math]
(M\&Le equation~(52)) where the term </math>f_{d}</math> represents dissipation and is given by
\[
f_{d}=e^{\frac{f_{i}}{A_{f}}\sigma_{a}c_{g}\Delta t}-1,
\]
(M\&Le equation~(53)) and </math>\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>\alpha_{c}</math>
was given as infinity in M\&Le. 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
leads to only negligible quantitative changes to the results.

We will transform the M\&Le scattering operator </math>\mathbf{T}</math> by 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> becomes \[ \mathbf{T}\left( \Delta t\right) I\left( \theta\right) =A^{2}\{\hat{\beta }\int_{0}^{2\pi}|D(\theta-\theta^{\prime})|^{2}I\left( \theta^{\prime }\right) d\theta^{\prime}+ I\left( \theta\right) \}. \]

The scattering theory of M\&Le depends on the
values of the time step </math>\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>\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
\[
I(t+\Delta t)=\mathbf{T}\left( \Delta t\right) I\mathbf{(}t),
\]
we obtain the following expression for the time derivative of </math>I</math>,
\[
\frac{\partial I}{\partial t}=\lim_{\Delta t\rightarrow0}\left(
\frac{\mathbf{T}\left( \Delta t\right)I(t) - I(t)}{\Delta t}\right).
\]
We can calculate this limit as follows,
\[
\lim_{\Delta t\rightarrow0}\left( \frac{\mathbf{T}\left( \Delta t\right)I(t)
-I(t)}{\Delta t}\right) =\lim_{\Delta t\rightarrow0}\left( \frac
{A^{2}\{\hat{\beta}\int_{0}^{2\pi}|D(\theta-\theta^{\prime})|^{2}I\left(
\theta^{\prime}\right) d\theta^{\prime}+I\left(
\theta\right) \}-I\left( \theta\right) }{\Delta t}\right)
\]

[math]\displaystyle{ \label{M_Le_boltzmann1} =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 equation~(\ref{M_Le_boltzmann1}) by using equation~(\ref{rho_e}). The value of </math>\rho_{e}\left( 0\right)</math> is given by [math]\displaystyle{ \label{rho} \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>f_{i}=2\pi a^{2}/\sqrt{3}D_{av}^{2}</math> and </math>A_{f}=\pi a^{2}</math>.

If we substitute our expressions for </math>\rho_e(0)</math>
in equation (\ref{M_Le_boltzmann1}) and
include the spatial term
(which was not in M\&Le since they assumed

isotropy) and divide by </math>c_g</math>, we obtain the following

linear Boltzmann equation \begin{align}\label{M_Le_boltzmann2} \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}\\ &- \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)\nonumber. \end{align} If we compare equations (\ref{boltzmann}) and (\ref{M_Le_boltzmann2}) 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. This difference comes from the fact that, in M\&Le, multiple 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}),

the effective density

is related to the number density as </math> \rho_{e}(0) = \rho_{o}/ \sqrt{1-4f_{i}/\pi}</math>. In summary, we have shown that, by taking the limit as the number of angles tend to infinity and as the time step </math>\Delta t</math> tends to zero in the scattering equation of M\&Le, we obtain a linear Boltzmann equation equivalent to the equation given in \cite{jgrrealism} (once the error in this earlier work has been corrected).

\section{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>. 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 \cite{jgrrealism} and M\&LE is the following. \cite{jgrrealism} assumed that the ice floes had negligible submergence but could flex, while M\&Le 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 M\&Le and \cite{jgrrealism}. Here we simply present the equation for </math>D(\theta)</math> independent of the equation used to model 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 potential </math>\phi({\bf r},z)</math> subject to an incoming wave of frequency </math>\omega</math> is [math]\displaystyle{ \label{bvp} \left. \begin{array} [c]{rr}% \nabla^{2}\phi=0, & -\infty\lt z\lt 0,\\ {\dfrac{\partial\phi}{\partial z}=0}, & z\rightarrow-\infty,\\ \dfrac{\partial\phi}{\partial z} = k\phi & z\in\Gamma_s \\ \dfrac{\partial\phi}{\partial z} = L\phi & z\in\Gamma_w \end{array} \right\} % }[/math] (e.g. \cite{jgrfloecirc,IJOPE04} or M\&Le). The Laplace's equation comes from the fact that the water is irrotational and inviscid. The vanishing of the normal derivative at </math>z=-\infty</math> is the no-flow through the boundary

condition at the bottom of the infinitely deep ocean. The condition at the free surface is the standard linear free surface

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 example whether it is flexible or rigid) and we represent this by the operator </math>L</math>. To actually solve equation~(\ref{bvp}) requires us to choose a 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 infinity which are found from the incident or driving wave, denoted </math>\phi^{\mathrmTemplate:In}</math>. We assume that </math>\phi^{\mathrmTemplate:In}</math> is a plane wave travelling in the </math>x</math> direction,

[math]\displaystyle{ \phi^{\mathrm{{In}}}({\bf r},z)=\frac{\omega H}{2k}e^{\rm{i}kx}e^{kz}. }[/math]

The condition as </math>\left|\mathbf{r}\right|\rightarrow \infty</math>

is the standard Sommerfeld radiation condition (e.g. \cite{Weh_Lait})

[math]\displaystyle{ \sqrt{|\mathbf{r}|}\left( \frac{\partial}{\partial|\mathbf{r}|}-\rm{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.} \label{sommerfield}% }[/math]

Once we have found the solution to equation~(\ref{bvp}), we obtain the absolute value of the scattering amplitude </math>D</math> as \[ \left\vert D\left( \theta\right) \right\vert =\frac{1}{H/2}\left( \frac {k}{2\pi}\right) ^{1/2}\left( \frac{\omega}{g}\right) \left\vert \mathbf{H}\left( \pi+\theta\right)\right\vert, \] where the Kochin function </math>\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>\delta/\delta n</math> is the inward normal derivative.

\section{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>x</math> spatial co-ordinate and time, i.e. there is no </math>y</math> dependence of the solution. We also consider a uniform MIZ so that the scattering function, </math>S</math>, is a function only of </math>\theta</math> and </math>\theta^{\prime}</math> and </math>\beta</math> is a constant. The only variation we allow spatially is that the 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 these assumptions equations (\ref{boltzmann}) and (\ref{M_Le_boltzmann2}) become, [math]\displaystyle{ \frac{1}{c_{g}}\frac{\partial I}{\partial t}+\cos\theta\frac{\partial I}{\partial x}=\left\{ \begin{array} [c]{c}% \displaystyle -\beta I+\int_{0}^{2\pi}S(\theta-\theta^{\prime})I(\theta^{\prime}% )d\theta^{\prime},\;\;x\gt b,\\ 0,\;\;x\lt b. \end{array} \right. \label{almostsolvable}% }[/math]

To solve equation (\ref{almostsolvable}) 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>n</math> angles evenly spaced between </math>0</math> and </math>2\pi</math> (</math>\theta_{j}=\frac{2\pi j}{n},\;0\leq j\leq n-1,</math>). This approximation converts equation (\ref{almostsolvable}) 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{array} [c]{c}% -\beta \vec{I}+\mathbf{S}\vec{I},\;\;x\gt b,\\ 0,\;\;x\lt b. \end{array} \right. } \label{matrix}% }[/math] In equation (\ref{matrix}) the intensity </math>\vec{I}</math> is now a vector of functions of </math>x</math> and </math>t</math> for each angle </math>\theta_{j}</math> and the elements of the matrices </math>\mathbfTemplate:D</math> and </math>\mathbfTemplate:S</math> are given by, [math]\displaystyle{ d_{ij}=\left\{ \begin{array} [c]{rr}% -\cos(\theta_{i}), & i=j,\\ {0,} & i\neq j, \end{array} \right. }[/math] and [math]\displaystyle{ s_{ij}=\left\{ \begin{array} [c]{rr}% {-}\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{array} \right. }[/math] respectively. Equation (\ref{matrix}) can be easily solved in the stationary (no time dependence), or isotropic (no spatial dependence) case. \cite{jgrrealism} solved the stationary problem and M\&Le solved the isotopic problem (with wind forcing etc.). In the stationary case equation (\ref{matrix}) reduces to, setting the ice edge to </math>b=0,</math> [math]\displaystyle{ -\frac{\partial}{\partial x}\mathbf{D}\vec{I}=-\beta\vec{I}+\mathbf{S}\vec{I}, \;\;x\gt 0. \label{spaceonly}% }[/math] For the isotropic case, equation (\ref{matrix}) reduces to, setting the ice edge to </math>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. \label{timeonly}% }[/math] Equations (\ref{spaceonly}) and (\ref{timeonly}) can be solved by straightforward matrix methods (e.g. \cite{ishimaru}). Equation (\ref{spaceonly}) requires boundary conditions (the wave spectrum at the ice edge </math>x=0</math> and a condition as </math>x\to\infty</math>) and equation (\ref{timeonly}) requires an initial condition (the wave spectrum at </math>t=0</math>).

\section{Summary}

We have shown that the scattering theory of M\&Le can be reduced to the linear Boltzmann equation if the discrete equation is converted to a differential equation by taking the appropriate limits. We also showed that this linear Boltzmann equation is equivalent to the linear Boltzmann equation presented in \cite{jgrrealism} with an error corrected. The difference between the two theories is a term which comes from the fact that M\&Le explicitly neglected multiple scattering.

Finally, we have shown how the scattering

term is calculated from the equation of motion for an individual ice floe and how the linear Boltzmann equation can be solved in certain situations.

%\begin{acknowledgments}
%This reseach was supported by the Marsden Grant UOO004.
%\end{acknowledgments}