4.1. Radiation damping due to time-harmonic point forcing on ice

We apply Green’s second identity to the Green function and its complex conjugate, denoted with an asterisk:

(4.3) \begin{equation} 2\mathrm{i} \, \mathrm{Im}\biggl \{\int _{S^I/S^\varepsilon }G G_z^*{\rm d}S + \int _{S^\infty }G G_r^*{\rm d}S\biggr \} = 0 . \end{equation}

To evaluate the first term of (4.3), we use $G = ( {1}/{k^O})(\ell _D^4 {\nabla} _{\boldsymbol{x}}^4 + 1 - \gamma k^O)G_z$ , valid for $\boldsymbol{x}\in S^I$ except at $\boldsymbol{x}=\boldsymbol{x}'$ , to find

(4.4) \begin{equation} 2\mathrm{i} \, \mathrm{Im}\biggl \{\int _{S^I/S^\varepsilon }G G_z^*{\rm d}S\biggr \} = 2\mathrm{i} \, \mathrm{Im}\biggl \{\int _{S^I}\frac {\ell _D^4}{k^O} ({\nabla} _{\boldsymbol{x}}^4G) G_z^*{\rm d}S - \int _{S^\varepsilon }\frac {\ell _D^4}{k^O} ({\nabla} _{\boldsymbol{x}}^4G) G_z^*{\rm d}S \biggr \} . \end{equation}

It can be shown using Green’s first identity and the free–free edge conditions that the first term on the right-hand side of (4.4) evaluates to zero because the edges are not a source or sink of power. We evaluate the second term by multiplying (4.1) by $G_z^*$ , integrating over $S^\varepsilon$ and taking the limit as $\varepsilon \to 0^+$ (where only the biharmonic term and delta function contribute), resulting in a final expression as

(4.5) \begin{equation} 2\mathrm{i} \, \mathrm{Im}\biggl \{\int _{S^I/S^\varepsilon }G G_z^*{\rm d}S\biggr \} = 2\mathrm{i} \frac {\omega }{\rho g} \mathrm{Re} \big\{ F G_z^*(\boldsymbol{x}',0;\boldsymbol{x}') \big\} . \end{equation}

It is straightforward to evaluate the second term of (4.3) using the far-field approximation (4.2). The complete expression resulting from (4.3) is

(4.6) \begin{equation} -\mathrm{Re}\{F G_z^*(\boldsymbol{x}',0;\boldsymbol{x}')\} = \rho g C_g^O \frac {2}{\pi k^O}\int _0^{2\pi }|a^\infty (\theta ;\boldsymbol{x}')|^2{\rm d}\theta . \end{equation}

Equation (4.6) resembles expressions describing the radiation damping of a floating rigid body in simple harmonic motion (see (8.6.13), Mei et al. (Reference Mei, Stiassnie and Yue2005)); however, the Green function evaluated at its source point takes the place of radiation potentials for discrete modes.

4.2. Reciprocity relation for a single incident direction

We now derive a reciprocity relation between the Green function and a velocity potential $\phi (\boldsymbol{r},z;\theta _0)$ satisfying (2.1) and the edge/corner conditions which represents a plane progressive incident wave propagating in direction $\theta _0$ interacting with the ice sheet. This potential has a far-field approximation for $k^Or\gg 1$ ,

(4.7) \begin{align} \phi \approx -\mathrm{i}\frac {\omega }{k^O}A^O\exp (k^Oz)\left ( \exp (\mathrm{i} k^Or \cos (\theta -\theta _0)) + \sqrt {\frac {2}{\pi k^O r}}\mathcal{A}(\theta ;\theta _0)\exp (\mathrm{i} k^O r - \mathrm{i} \pi /4)\right )\!, \end{align}

where $\mathcal{A}(\theta ;\theta _0)$ denotes the scattered wave amplitude in direction $\theta$ due to an incident wave from direction $\theta _0$ , normalised by $A^O$ , the amplitude of the incident wave in open water.

We once again apply Green’s second identity:

(4.8) \begin{equation} \int _{S^I/S^\varepsilon }(\phi G_z - G \phi _z){\rm d}S + \int _{S^\infty }(\phi G_r - G \phi _r){\rm d}S = 0. \end{equation}

We follow a similar procedure to § 4.1 to evaluate the first term of (4.8). For the second, far-field term involving (4.2) and (4.7), we recognize that the scattered wave field does not contribute and employ the method of stationary phase to evaluate in a straightforward manner. The reciprocity relation between the Green function and a unidirectional incident wave is

(4.9) \begin{equation} -\mathrm{i} \omega F \eta (\boldsymbol{x}';\theta _0) = \frac {4}{k^O}\rho g C_g^OA^Oa^\infty (\theta _0+ \pi ; \boldsymbol{x}'), \end{equation}

where we have used the kinematic boundary condition to relate $\phi$ and $\eta$ . Equation (4.9) resembles the Haskind–Hanoaka theorem (see (8.6.41), Mei et al. (Reference Mei, Stiassnie and Yue2005)), but instead relates the free surface elevation at $\boldsymbol{x} = \boldsymbol{x}'$ in the ice-covered region due to an incident plane wave to the far-field amplitude of the radiated wave generated by a time-harmonic point forcing applied at $\boldsymbol{x} = \boldsymbol{x}'$ .

4.3. Reciprocity theorem for isotropic incidence

We derive an expression connecting $G(\boldsymbol{x}',0;\boldsymbol{x}')$ to the free surface elevation at $\boldsymbol{x}'$ due to an incoherent, isotropic incident wave field by relating (4.6) and (4.9) in the far field. We multiply (4.9) by its complex conjugate and angle average with respect to $\theta _0$ by integrating from $0$ to $2\pi$ . The angle-averaged, far-field squared amplitude now appears in both expressions that we substitute to yield

(4.10) \begin{equation} |A^I(\boldsymbol{x}')|^2 = -\frac {2\rho g|A^O|^2}{(k^O)^2 \omega |F|^2}\mathrm{Re} \big\{F G_z^*(\boldsymbol{x}',0;\boldsymbol{x}') \big\}, \end{equation}

where $|A^I(\boldsymbol{x}')|^2 = ({1}/{2\pi })\int _0^{2\pi }|\eta (\boldsymbol{x}'; \theta _0)|^2 {\rm d}\theta _0$ . Note that (4.10) is applicable without frequency averaging and makes no assumption regarding $k^I\ell$ .

In general, the Green function cannot be expressed in closed form and is a function of the ice sheet plan form via the ice edge boundary conditions. However, the influence of the ice edge boundary conditions is limited to a region near the ice edge $S^I_e$ . Based on this intuition, we assume

(4.11) \begin{equation} \langle \mathrm{Re}\{F G_z^*(\boldsymbol{x}',0;\boldsymbol{x}')\}\rangle _\omega \approx \langle \mathrm{Re}\{F (G_z^\infty (\boldsymbol{x}',0;\boldsymbol{x}'))^*\} \rangle _\omega , \quad \boldsymbol{x}'\in S^I/S^I_e, \end{equation}

where $G^\infty$ is the Green function for an ice sheet of infinite spatial extent.

Fox & Chung (Reference Fox and Chung1999) derive a closed-form expression for $G^\infty$ in the deep-water limit which contains propagating and spatially decaying wave components. We are interested in only the real part per (4.10) to which only the propagating component contributes:

(4.12) \begin{equation} \mathrm{Re}\{F (G_z^\infty (\boldsymbol{x}',0;\boldsymbol{x}'))^*\} = -\frac {|F|^2\omega }{2\rho g}\frac {(k^I)^2}{5(\ell _Dk^I)^4 + 1-\gamma k^O}. \end{equation}

We frequency average the general reciprocity theorem for isotropic incidence (4.10) and (4.12) and substitute, which yields

(4.13) \begin{equation} \bigl \lt |A^I(\boldsymbol{x}')|^2 \bigr \gt _\omega = \biggl \lt \left (\frac {k^I}{k^O}\right )^2 \frac {|A^O|^2}{5 (\ell _Dk^I)^4 + 1 - \gamma k^O} \biggr \gt _\omega \equiv \bigl \lt |A^I_b|^2 \bigr \gt _\omega , \quad \boldsymbol{x}'\in S^I/S^I_e. \end{equation}

We can recast (4.13) using the definitions of intensity (3.3) to yield

(4.14) \begin{equation} \langle I^I(\boldsymbol{x}')\rangle _\omega = \biggl \lt \frac {k^I}{k^O}I^O \biggr \gt _\omega \equiv \langle I^I_b\rangle _\omega \;, \quad \boldsymbol{x}'\in S^I/S^I_e. \end{equation}

We use $|A^I_b|^2$ and $I^I_b$ to denote the theoretical predictions valid in the bulk of the ice sheet. We note that the Green function result here contains the SWA result of § 3 because (4.11) becomes valid everywhere in the ice sheet (except at the ice edge) in the limit as $k^I_c\ell \to \infty$ . As a consequence, (4.14) is equivalent to (3.10) in this limit.

We ascertain the range of validity of the approximation (4.11) in § 5.1 by comparing the closed-form expression for $\langle \mathrm{Re}\{F (G_z^\infty (\boldsymbol{x}',0;\boldsymbol{x}'))^*\} \rangle _\omega$ with the numerical solution for $\langle \mathrm{Re}\{F G_z^*(\boldsymbol{x}',0;\boldsymbol{x}')\}\rangle _\omega$ for a circular ice sheet. We find that the approximation is valid in the bulk of the ice sheet, except near the ice edge which is influenced by the free–free edge conditions. We explain in § 5.4 that this edge-affected region is small compared with the bulk of the ice sheet not only for $k^I_c\ell \gg 1$ but also when $\ell /\ell _D\gg 1$ , regardless of $k^I_c\ell$ . Here $\ell /\ell _D\gg 1$ signifies that the size of the ice sheet is large compared with the flexural-gravity length scale $\ell _D$ which can be much less than $\lambda ^I_c$ for low ice stiffness. The Green function result, therefore, has a broader range of applicability than the SWA result.

5.1. Spatial distribution of intensity in the short wavelength regime

We consider cases for $k^I_cR\gg 1$ where both the SWA and Green function results are applicable to verify the theoretical results and establish its range of validity with respect to $k^I_cR$ . We plot the radial cross-sections of both intensity (figure 3 a) and squared amplitude (figure 3 b) for a moderately large ( $k^{I}R = 24$ ) ice sheet in an isotropic incident wave field. The radial cross-section is not a function of azimuthal angle $\theta$ since the geometry and incident wave forcing are axisymmetric.

Figure 3.

Radial distribution of (a) intensity and (b) squared amplitude for a circular ice sheet with $k^I_c\ell _D = 0.86$ , $k^O_c \gamma = 0.074$ and $k^I_c R = 24$ . We show direct computational results (red, ), $I^I_b$ ( $|A^I_b|^2$ ) (black, ) and $I^I_e$ ( $|A^I_e|^2$ ) (black, ) in (a) (and (b)). Inset depicts cross-section.

In figure 3, the intensity and squared amplitude are uniform in the bulk of the ice sheet except within $\lambda ^I_c$ of the ice edge. In figure 3(a), the intensity in the bulk is $\tilde {I}^I/I^O\approx k^I/k^O$ in agreement with the asymptotic result (3.10). With $k^I/k^O\lt 1$ in this case, the intensity reduction is the consequence of total reflection for an angular range of incident waves propagating from open water into the ice sheet as explained in § 3. In figure 3(b), the value of the squared amplitude in the bulk agrees with the theoretical result (4.13). The reduction in squared amplitude includes the effect of total reflection in addition to the change in group speed between open water and ice-covered regions and the difference in wave energy due to the flexural response and inertial effects of the ice.

Intensity and squared amplitude are related through the definitions of intensity (3.3). In general, $I^I = \sum _j \rho \omega ^2 (1 + \gamma k^I_j)C_{g,j}^I |A^I_j|^2/(2k_j^I)$ where $j$ refers to all the roots of the dispersion relation at frequency $\omega$ . However, our definition applies for $\boldsymbol{x}\in S^I/S^I_e$ and, hence, contains only the propagating component. We use $\tilde {I}^I$ for $\boldsymbol{x}\in S^I$ in figure 3(a) to indicate that our definition contains only the propagating component and is not strictly valid in $S^I_e$ . To avoid this ambiguity of definition, we present results in terms of squared amplitude exclusively for the remainder of this work. Note that, while it may appear that figure 3(a) is identical to figure 3(b) with the y-axis rescaled, the ratio $(\tilde {I}^I/I^O)/(|A^I|^2/|A^O|^2)$ varies spatially, particularly near the ice edge. This results from the frequency dependence of (3.3) upon frequency averaging.

We now consider the relation (4.10) between the ice response to an isotropic incident wave field and the Green function as well as the validity of the approximation (4.11) where we assume that, locally, $G$ behaves like $G^\infty$ . We verify (4.10) by comparing the direct computation of $|A^I|^2$ for an incident isotropic wave field via (5.1) (shown in figure 3 b) to the direct computation of $\mathrm{Re}\{F G_z^*(\boldsymbol{x}',0;\boldsymbol{x}')\}$ which yields $|A^I_{\textit{GF}}|^2$ via (4.10). We find that $|A^I|^2 = |A^I_{\textit{GF}}|^2$ everywhere in the ice sheet, even near the edges, as expected. For instance, in figure 3(b), $|A^I_{\textit{GF}}|^2$ (not shown) is indistinguishable from the direct computational results. Note that, while here we have compared the two results for a case with a narrow-banded spectrum and $k^{I}R\gg 1$ , we have also verified that (4.10) applies for general $k^{I}R$ and does not require frequency averaging.

While, in general, $G$ must be computed numerically, we are interested in the applicability of (4.11) which assumes the local response of $G$ can be approximated by $G^\infty$ . This is validated in this case by showing that $|A^I_{\textit{GF}}|^2\approx |A^I_b|^2$ in the bulk of the ice sheet away from the edge. Since $|A^I|^2 = |A^I_{\textit{GF}}|^2$ everywhere, this claim is supported by figure 3(b) where $|A^I|^2\approx |A^I_b|^2$ except near the ice edge.

We remark that the frequency-averaging provided by the narrow-banded spectrum is necessary for $|A^I|^2$ (and $I^I$ ) to be uniform within the bulk of the ice sheet. For monochromatic forcing, $|A^I|^2$ oscillates in the radial direction around $|A^I_b|^2$ with wavelength $\lambda ^I/2$ due to the plan form-dependent wave interference pattern. The oscillation amplitude depends on $k^I\ell _D$ . The interference pattern is sensitive to frequency such that the oscillations are suppressed for sufficiently large $\Delta k^{I}R$ . Therefore, a smaller bandwidth $\Delta k^I/k^I_c$ is necessary for effective frequency averaging as $k^I_cR$ increases.

Although there are few observations of the directional wave field in the MIZ, the conditions reported by Wadhams et al. (Reference Wadhams, Squire, Goodman, Cowan and Moore1988) are in line with the short-wavelength theoretical results. For instance, the wave field is observed to be isotropic, reaching isotropy within 1.2 km of the ice edge in Run C and within 0.6 km for waves propagating through a band of ice floes. The description of Run C reports a narrow-banded wind sea with peak frequency at approximately 0.3 Hz, corresponding to an open-water wavelength $\lambda ^O=17$ m, propagating through an ice field with typical floe diameter $\ell =80$ m. Although further data about the ice is not available to determine $\lambda ^I$ , this corresponds to $k^O\ell \approx 30$ , indicating that the short-wavelength regime can be applicable in a realistic MIZ. Lastly, the ice concentration is reported to be 70 % during Run C. Using phase-resolved simulations of the MIZ, Montiel et al. (Reference Montiel, Squire and Bennetts2016) show that the wide-spacing approximation is valid for an ice concentration of 70 %.

5.2. Edge correction to theoretical results in the short wavelength regime

We develop an edge correction to the theoretical results to approximate the increase in squared amplitude near the ice edge based on the insight that the edge effects are (mostly) agnostic to plan form for $k^I_c\ell \gg 1$ . We consider the squared amplitude $|A^I|^2$ as a function of edge-normal distance into the ice sheet $x_n$ . Previous sections have established that $|A^I|^2\approx |A^I_b|^2$ for $x_n\gtrsim \lambda ^I_c$ where $|A^I_b|^2$ is a function of $k^I_c\ell _D$ and $k^O_c\gamma$ , but not ice sheet plan form, for $k^I_c\ell \gg 1$ . Through direct computations of circular and rectangular (see § 6) ice sheets, we find that, generally, $|A^I|^2$ for $0\leqslant x_n\lesssim \lambda ^I_c$ is a function of $k^I_c\ell _D$ and $k^O_c\gamma$ , but not ice sheet plan form, for $k^I_c\ell \gg 1$ . Note that this is not the case in regions where the edge has high curvature, such as at sharp corners. For the rectangular ice sheet cases (§ 6) we consider, the corner behaviour affects pointwise quantities, such as maximum bending strain, but generally has a small effect on the spatial mean of squared amplitude, important for determining in-ice dissipation.

We derive an edge correction to the theoretical result $|A^I_e(x_n)|^2$ based on this insight. We assume that regions of the ice edge without high curvature can be approximated as a straight edge for $k^I_c\ell \gg 1$ . We then determine $|A^I_e(x_n)|^2$ using the eigenfunction matching method of Fox & Squire (Reference Fox and Squire1994) to solve for the deflection of a semi-infinite ice sheet subjected to isotropic incident waves. For a given $k^I_c\ell _D$ and $k^O_c \gamma$ , we first determine the free surface elevation in the semi-infinite ice sheet $\eta ^{SI}(x_n;\psi ^O)$ for an incident wave propagating from open water into ice with unit amplitude and incident angles $\psi ^O\in [-\pi /2,\pi /2]$ . We also calculate $\eta ^{SI}(x_n;\theta ^I)$ for an incident wave propagating from ice into open water with amplitude $A^I_b$ and incident angles $\theta ^I\in [-\pi /2,\pi /2]$ . This requires a straightforward modification to the method of Fox & Squire (Reference Fox and Squire1994). We then angle and frequency average using

(5.2) \begin{equation} \langle |A^I_e(x_n)|^2 \rangle _\omega = \Biggl \lt \frac {1}{\pi }\left (\int _{-\pi /2}^{\pi /2}|\eta ^{SI}(x_n;\psi ^O)|^2 {\rm d}\psi ^O + \int _{-\pi /2}^{\pi /2} |\eta ^{SI}(x_n;\theta ^I)|^2{\rm d}\theta ^I\right ) \Biggr \gt _\omega . \end{equation}

Note that $\langle |A^I_e(x_n)|^2 \rangle _\omega \approx \langle |A^I_b|^2 \rangle _\omega$ for $x_n\gtrsim \lambda ^I_c$ . $I^I_e$ is obtained from (5.2) using the definition of intensity in ice (3.3b ).

Figure 3(b) compares the edge correction $|A^I_e|^2$ with direct computation of $|A^I|^2$ . We observe that $|A^I_e|^2$ recovers $|A^I_b|^2$ in the bulk of the ice sheet, as expected. At the ice edge $x_n=0$ , $|A^I_e|^2$ is within 0.05 of $|A^I|^2$ . Overall, $|A^I_e|^2$ is a good approximation of $|A^I|^2$ .

Note that the edge correction applies to ice sheets of general plan form. We show its applicability for rectangular ice sheets in § 6.

5.3. Range of validity of theoretical results with respect to the short-wavelength parameter $k^I\ell$

We consider the spatial mean of squared amplitude $\langle |A^I|^2\rangle _{\boldsymbol{x}}=(1/\mathcal{A})\int _{S_I}|A^I|^2{\rm d}S$ as a simple surrogate to evaluate the range of validity of the theoretical results and the edge correction with respect to $k^I_cR$ . Figure 4 displays $\langle |A^I|^2\rangle _{\boldsymbol{x}}$ as a function of $k^I_cR$ for three $(k^I_c\ell _D,k^O_c\gamma )$ pairs. These $(k^I_c\ell _D,k^O_c\gamma )$ pairs correspond to ice thicknesses $h=0.5$ , $1$ and $3.5$ m with the rest of the typical wave and ice parameters outlined at the start of § 5 held constant. We observe that $\langle |A^I|^2\rangle _{\boldsymbol{x}}$ approaches $|A^I_b|^2$ as $k^I_cR$ increases with a difference of less than 0.01 at $k^I_cR = 25$ for each ice thickness in figure 4(a).

Figure 4.

Spatial mean of squared amplitude in circular ice sheet, normalised by (a) $|A^O|^2$ or (b) $|A^I_b|^2$ , for $(k^I_c \ell _D,k^O_c \gamma )=(0.56,0.028)$ (red), $(0.86,0.074)$ (black) and $(1.19, 0.26)$ (blue). We display direct computational results ( ), $\langle |A^I_e|^2\rangle _{\boldsymbol{x}}$ ( ) and $|A^I_b|^2$ ( ) (in (a) only). Here corresponds to figure 3.

We can understand $\langle |A^I|^2\rangle _{\boldsymbol{x}}$ as the area-weighted average of $|A^I_b|^2$ and edge effects contained to $\mathcal{A}_e = \lambda ^I_cP$ , the area of $S_e^I$ , where $P$ is the perimeter of the ice sheet. From § 5.2, we know that the amplitude of edge effects remains constant as $k^I_cR$ varies which we denote as $\mathbb{C}$ . The area-weighted average can then be expressed as $\langle |A^I|^2\rangle _{\boldsymbol{x}} = (|A^I_b|^2 (\mathcal{A} - \mathcal{A}_e) +\mathbb{C}\mathcal{A}_e)/\mathcal{A} = |A^I_b|^2 + \tilde {\mathbb{C}}/(k^I_cR)$ , where $\tilde {\mathbb{C}}$ absorbs additional constants.

While this simple argument reveals that the edge effects decay with the inverse of $k^I_cR$ , we can use the edge correction to capture the leading-order effect, i.e. determine $\tilde {\mathbb{C}}$ . We make a theoretical prediction of $\langle |A^I|^2\rangle _{\boldsymbol{x}}$ by using the one-dimensional function $|A^I_e(x_n)|^2$ to form a two-dimensional surface of revolution $|A^I_e(\boldsymbol{x})|^2$ with $x_n=0$ along the circumference. We compare the theoretical prediction $\langle |A^I_e(\boldsymbol{x})|^2\rangle _{\boldsymbol{x}}$ with the direct computational result in figure 4 and find that $\langle |A^I_e(\boldsymbol{x})|^2\rangle _{\boldsymbol{x}}$ is a good approximation of $\langle |A^I|^2\rangle _{\boldsymbol{x}}$ for $k^I_cR\gtrsim 4\pi$ . At $k^I_cR=4\pi$ , $\mathcal{A}_e=3\mathcal{A}/4$ , so edge effects are important over most of the area. As a result, we expect the assumptions underlying the edge correction to no longer apply for $k^I_cR\lt 4\pi$ . The surprisingly good performance of the theoretical prediction also suggests that approximating the Green function for general plan form with the infinite-ice Green function remains valid even when $\mathcal{A}_e$ is comparable to $\mathcal{A}$ .

We include results for multiple $k^I_c\ell _D$ where $\ell _D = (D/(\rho g))^{1/4}$ in figure 4 to show that the theoretical result as well as the edge correction are applicable over a wide range of ice stiffnesses. Figure 4(a) shows that $|A^I_b|^2$ decreases for increasing ice thickness $h$ or elastic modulus $E$ , and that the highest $k^I_c\ell _D$ result is closest to $|A^I_b|^2$ for the entire range of $k^I_cR$ . This is because the amplitude of the edge effects $(|A^I(r=R)|^2-|A^I_b|^2)$ generally decreases as $k^I_c\ell _D$ increases. In figure 4(b), we normalize the results of figure 4(a) with respect to $|A^I_b|^2$ to enable comparison between different $k^I_c\ell _D$ . The results approach zero (corresponding to $|A^I_b|^2$ ) in order of increasing $k^I_c\ell _D$ , opposite the trend in figure 4(a). This is because the amplitude of the edge effects, normalised by $|A^I_b|^2$ , generally increases as $k^I_c\ell _D$ increases. Note that the oscillations in the highest $k^I_c\ell _D$ result are due to the plan form-specific interference pattern mentioned in § 5.1 which are not totally suppressed for the selected bandwidth.

Figure 4(b) shows that both $k^I_cR$ and $k^I_c\ell _D$ are important to the range of validity of the theoretical results. This is indicative of the three relevant length scales of the problem: $k^I_c$ , $\ell _D$ and $\ell$ . We explore the interplay between these length scales in more depth in § 5.4.

5.4. Role of ice stiffness in predicting wave intensity in ice

Up to this point, we have mainly considered the short-wavelength regime ( $k^I_cR\gg 1$ ) where both theoretical results are applicable. However, it turns out that $|A^I|^2\approx |A^I_b|^2$ away from the ice edges even for $k^I_c\ell =O(1)$ when $\ell /\ell _D\gg 1$ . We depict the parameter space of the three dimensionless parameters $k^{I}_{c}R$ , $k_c^I\ell _D$ and $R/\ell _D$ in figure 5(a) for a circular ice sheet with radius $R$ as an example. We also indicate how the various figures in this work cover different areas of the regime map. With $R/\ell _D$ on the abscissa and $k^I_cR$ on the ordinate, contours of constant $k_c^I\ell _D$ extend radially from the origin and increase in the anticlockwise direction.

We reiterate that the theoretical results apply when $k^I_c R\gg 1$ as evident in figure 4. Conversely, it is also clear in figure 4 that the theoretical results do not apply when both $k^I_cR$ and $R/\ell _D$ are $O(1)$ , corresponding to the lower left-hand quadrant of figure 5(a).

In figures 5(b) and 5(c), we plot $\langle |A^I|^2\rangle _{\boldsymbol{x}}$ as a function of $k^I_cR$ for constant $R/\ell _D$ to demonstrate the role of $R/\ell _D$ . In these figures, ice stiffness $k^I_c\ell _D$ is no longer constant but varies with $k^I_cR$ . At the y-axis, $k^I_cR = k^I_c\ell _D=0$ and we recover the long-wave limit where $\langle |A^I|^2\rangle _{\boldsymbol{x}}=|A^O|^2$ . In the other limit where $k^I_c\ell _D\gg 1$ , $\langle |A^I|^2\rangle _{\boldsymbol{x}}$ approaches zero as the ice sheet behaves as a rigid body. Figure 5(b) further supports that the theoretical results do not apply when $k^I_cR$ and $R/\ell _D$ are $O(1)$ since the computational results depart significantly from $|A^I_b|^2$ for $0.5\lesssim k^I_cR\lesssim 2.5$ . However, in figure 5(c) for $R/\ell _D = 20$ , the ice response is found to agree with $|A^I_b|^2$ for $k^I_cR = O(1)$ , i.e. the lower right-hand quadrant of figure 5(a), as well as for $k^I_cR \gg 1$ . We remark that $\langle |A^I_e|^2\rangle _{\boldsymbol{x}}$ is also plotted in figure 5(c) for $k^I_cR\gt 2\pi$ and is visually indistinguishable from the computed ice response.

We can use the Green function result to explain why $|A^I|^2\approx |A^I_b|^2$ also applies for $k^I_cR=O(1)$ when $R/\ell _D\gg 1$ . Figure 6(a) displays the radial cross-section of $|A^I|^2$ for $k^I_cR = 5.35$ and $R/\ell _D = 20$ . As in § 5.1, we first validate that $|A^I|^2=|A^I_{\textit{GF}}|^2$ everywhere in the ice sheet. We then show that (4.11) is valid in the bulk of the ice sheet by showing that $|A^I_{\textit{GF}}|^2 \approx |A^I_b|^2$ , or, equivalently, $|A^I|^2 \approx |A^I_b|^2$ away from the ice edge in figure 6(a). While the SWA result does not apply to this case, the more general Green function result explains the observed behaviour.

Figure 5.

(a) Regime map relating $R/\ell _D$ , $k^I_cR$ and $k^I_c\ell _D$ where regions with solid green border and dashed purple border indicate approximate range of validity for both SWA and Green function result and Green function result only, respectively. Spatial mean of squared amplitude in circular ice sheet for (b) $R/\ell _D = 2$ and (c) $R/\ell _D = 20$ . We display direct computational results ( ) in (b) and (c), $|A^I_b|^2$ ( ) in (b) and (c), and $\langle |A^I_e|^2\rangle _{\boldsymbol{x}}$ ( ) in (c). In (c), denotes the specific geometry of figure 6(a).

Figure 5. Long description

The image contains three panels labeled (a), (b), and (c). Panel A: A regime map graph showing the relationship between kR and R/vD. The x-axis is labeled R/vD and the y-axis is labeled kR. The graph includes regions with solid green borders and dashed purple borders indicating the range of validity for both short-wavelength asymptotic (SWA) and Green function results, and Green function results only, respectively. Several lines are plotted for different values of kc vD. Panel B: A line graph showing the spatial mean of squared amplitude in a circular ice sheet for R/vD = 2. The x-axis is labeled kcR and the y-axis is labeled <|A|^2>x/<|A0|^2>. The graph includes two lines, one solid and one dashed, representing different computational results. Panel C: A line graph showing the spatial mean of squared amplitude in a circular ice sheet for R/vD = 20. The x-axis is labeled kc vD and the y-axis is labeled <|A|^2>x/<|A0|^2>. The graph includes two lines, one solid and one dashed, representing different computational results. The solid line in Panel C denotes a specific geometry.

Figure 6.

(a) Radial distribution of squared amplitude for a circular ice sheet with $k^I_c\ell _D = 0.27$ , $k^O_c\gamma = 0.016$ , $R/\ell _D = 20$ and $k^I_cR = 5.35$ . We display direct computational results ( ) and $|A^I_b|^2$ ( ). (b) Regime map from figure 5(a) where shaded blue region indicates the edge-affected region is much smaller than the total area, $\mathrm{Im}\{k^I_{cc}\}R\gg 1$ .

While the explanation above describes the response away from the edge-affected region, it does not explain why this region is small even for $k^I_cR=O(1)$ when $R/\ell _D\gg 1$ . We can understand this by recognizing that the edge effects are related to the complex-valued roots of the flexural-gravity dispersion relation: $\omega ^2 = (gk^I + D(k^I)^5/\rho )/(1+\gamma k^I)$ . These complex-valued modes exhibit oscillatory decay with distance from the ice edge with the real and imaginary parts determining the wavelength and $e$ -folding length respectively. We find that the imaginary part of the complex-conjugate wavenumber, denoted $\mathrm{Im}\{k^I_{cc}\}$ , generally determines the length scale of the edge layer. Consequently, when $\mathrm{Im}\{k^I_{cc}\}R \gg 1$ , the edge-affected layer is much smaller than the bulk of the ice sheet where $|A^I|^2 \approx |A^I_b|^2$ . We show representative contours of constant $\mathrm{Im}\{k^I_{cc}\}R$ in figure 6(b).

For $k^I_c\ell _D \lt 1$ , $\mathrm{Im}\{k^I_{cc}\}\sim 1/\ell _D$ . This is evident in figure 6(a) where the edge effects extend $O(\ell _D)$ from the ice edge (which is equivalent to $O(0.1\lambda ^I_c)$ ). On the other hand, for $k^I_c\ell _D \gt 1$ , $\mathrm{Im}\{k^I_{cc}\}\sim k^I_c$ . This explains why the theoretical result is valid for $k^I_cR\gg 1$ when $k^I_c\ell _D \gt 1$ and for $R/\ell _D\gg 1$ when $k^I_c\ell _D \lt 1$ . Note that $k^I_cR\gg 1$ and $R/\ell _D\gg 1$ are equivalent when $k^I_c\ell _D = 1$ . This corresponds with what we see in figure 3 where the edge effects extend $O(\lambda ^I_c\approx \ell _D)$ from the edge.

For typical physical parameters and 0.5 m-thick ice, $\ell _D \approx 9$ m, and $\ell$ ranges from $O(1\,\mathrm{m})$ close to the edge of the MIZ to $O(1\,\mathrm{km})$ in the interior. Here $\ell /\ell _D\gg 1$ is therefore likely to apply to an appreciable portion of ice sheets in the MIZ.