The Multiple Scattering Theory of Masson and LeBlond

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.

Transport Equation

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. Similarly, the terms [math]\displaystyle{ S_{\mathrm{{in}}}, }[/math] [math]\displaystyle{ S_{\mathrm{{ds}}}, }[/math] and [math]\displaystyle{ S_{\mathrm{{nl}}} }[/math] could be added to equation (\ref{Howells}). However, the purpose of this paper is to derive a consistent equation for [math]\displaystyle{ S_{\mathrm{{ice }}}. }[/math] Masson and LeBlond 1989 solved equation (\ref{BoltzMasson}) 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 their time stepping equation.

Masson and LeBlond 1989 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] }[/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)). In equation~(\ref{wrong}), [math]\displaystyle{ \hat{\beta} }[/math] (this notation is chosen to follow from M\&Le who used [math]\displaystyle{ \beta }[/math] and to avoid confusion with the expression for [math]\displaystyle{ \beta }[/math] in equation (\ref{Howells}) and which is used in \cite{howells60} and \cite{jgrrealism}) 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>\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 Masson and LeBlond 1989 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>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 equation~(\ref{M_Le_boltzmann1}) by using equation~(\ref{rho_e}). 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].

If we substitute our expressions for [math]\displaystyle{ \rho_e(0) }[/math] in equation (\ref{M_Le_boltzmann1}) and 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 equations (\ref{boltzmann}) and (\ref{M_Le_boltzmann2}) 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]. As shown in equation (\ref{rho}),

the effective density

is related to the number density as [math]\displaystyle{ \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]\displaystyle{ \Delta t }[/math] tends to zero in the scattering equation of 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).