Assessment of a porous viscoelastic model for wave attenuation in ice-covered seas

In recent decades, the polar regions have experienced major transformations due to global warming. For example, the rapid decline of summer ice extent in the Arctic Ocean has caught a lot of attention. While there is no doubt that warmer temperatures have been a major factor in transforming the polar seascape, evidence has also shown that ocean waves and their increased activity play an aggravating role, and in turn the presence of sea ice affects the wave dynamics. By breaking up the sea ice, waves cause it to become more fragmented, which in turn increases their capacity to further penetrate and damage the ice cover. A typical setting in the ocean where wave-ice interactions prevail is the marginal ice zone (MIZ) which is the fragmented part of the ice cover closest to the open ocean. It is a highly heterogeneous region comprising various types of sea ice that result from the incessant assault of incoming waves.

Of particular interest to oceanographers is the modeling of wave attenuation in sea ice, a process that has been poorly represented in large-scale wave forecasting models for the polar regions. There are two principal mechanisms for the attenuation of wave energy propagating into an ice field: (i) scattering by ice floes or other inhomogeneities of the ice cover, which is a conservative process that redistributes energy in all directions, and (ii) dissipative processes which are related to various sources, e.g. friction due to the presence of sea ice, inelastic collisions and breakup of ice floes. The relative importance of scattering and dissipation is still unclear, and uncertainties still exist about the actual mechanisms for wave dissipation in sea ice. This has led to a surge of research activity on this topic in recent years. Studies have suggested that dissipative processes are dominant in frazil and pancake ice fields [14, 33], while scattering seems to be the main mechanism for wave attenuation in broken floe fields [22, 40]. Even for a denser ice field, the problem remains complex: e.g. Ardhuin et al. [1] found that dissipation dominates over scattering for long swells in the Arctic ice pack.

With a view to describing wave attenuation in the MIZ, two different approaches have been pursued based on linear theory: (i) discrete-floe models where individual floes with possibly distinct characteristics are resolved assuming an idealized geometry [2, 30, 43], and (ii) continuum models where the heterogeneous ice field is viewed as a uniform material with effective rheological properties including viscosity or viscoelasticity [10, 21, 41]. In case (i), the analysis focuses on wave scattering and typically requires solving a boundary value problem with multiple regions in the horizontal hyperplane. By contrast, case (ii) enables the derivation of an exact algebraic expression for the dispersion relation governing traveling plane waves in the effective medium. Wave attenuation (possibly from scattering and dissipation combined) is encoded in the complex roots of this dispersion relation, and various physical effects are controlled by constant parameters. Recent reviews on this theoretical work can be found in [36, 37].

Models of type (i) have been applied to various floe configurations and have reached a high degree of sophistication. There is now a consensus that the process of wave scattering in sea ice is well understood, and associated parameterizations have been tested for operational wave forecasting [13, 34, 44]. It is however not the case for dissipative processes and, partly for this reason, there has been an increasing effort in recent years at developing and calibrating models of type (ii) [5, 11, 14, 35]. In this framework, details of the attenuation processes are not accounted for and it lies on the calibration to ensure that the effective parameters are assigned suitable empirical values for practical applications.

While continuum models have been employed for some time now, based mostly on thin-plate theory, to describe wave propagation in pack ice [15, 27], their extension to the setting of a more compliant or fragmented ice cover for application to the MIZ is more recent [50]. Earlier versions include the two-layer viscous model of Keller [21] which treats the ice cover as a viscous layer lying on top of an ideal fluid (the ocean). The viscous layer is meant to represent a suspension of ice particles in water. Interaction among these particles and the associated friction leads to wave energy dissipation, which is modeled as a viscous effect. Good agreement has been found in comparison to laboratory data on wave attenuation in grease ice [32, 33]. Keller’s model was extended by de Carolis and Desiderio [10] to allow for a viscous fluid in the lower layer as well. Validation was provided to some extent against laboratory and field measurements. Wang and Shen [41] refined Keller’s model by adding elasticity to the upper layer as this property may be of relevance to broken floe fields. Their viscoelastic model has been tested against laboratory experiments under various ice conditions, and has been calibrated and used in parameterization of wave hindcasts for the Arctic and Antarctic. Building upon this idea, Zhao and Shen [49] developed a three-layer version which features a turbulent boundary layer between the viscoelastic ice cover and the inviscid ocean. Dissipation due to turbulence in the middle layer is associated with some eddy viscosity.

In the spirit of this continuum approach, Chen et al. [3] recently proposed a more elaborate two-layer model where the ice cover is viewed as a homogeneous isotropic poroelastic material according to Biot’s theory [4]. More specifically, the heterogeneous ice field is described as a mixed layer with a solid phase and a fluid phase as the two limiting configurations. Each phase is assumed to be slightly compressible. Dissipative effects are included via two different mechanisms: viscosity within each phase of the ice layer, and friction caused by the relative motion between its fluid and solid constituents. A parameter of interest in this model is the ice porosity which may serve to provide a measure of ice concentration. Despite the complicated nature of this formulation, an exact linear dispersion relation can be derived and numerical estimates of physically relevant solutions can be found using relatively simple selection criteria. Preliminary tests were conducted in [3] to verify consistency with predictions from simpler models (e.g. open water, mass loading, purely elastic) in their respective limits [6, 46]. A more detailed review of this dispersion relation together with those from other viscoelastic representations will be presented in the next section.

The main goal of this paper is to further assess the porous viscoelastic model of Chen et al. [3] by testing it against both laboratory experiments and field observations of wave attenuation in sea ice. These are taken from the literature, and allow us to probe a wide range of ice conditions and wave frequencies. This is accomplished by fitting the theoretical predictions to data on attenuation rate via error minimization over a set of rheological parameters. As a result, numerical estimates for both the attenuation rate and the set of effective parameters are obtained from the fitting process. This model’s performance is also checked by comparing it to other existing viscoelastic formulations under the same various conditions. The purpose of such a comparison is two-fold. First, it helps examine in detail the parametric dependence in viscoelastic theories, and the extent to which common rheological parameters may differ in their range of values. Indeed, this difference may be of several orders of magnitude for such effective parameters. Second, it helps validate our data fitting method as we can check with previous independent calibration results from the literature. Given the relatively large parameter space in this poroelastic setting, a sensitivity analysis is also performed to gauge the individual contributions of rheological parameters to the fitting process. A notable finding from our study is that Chen et al.’s model can reproduce to some degree the roll-over of attenuation rate as observed in field measurements from the Arctic MIZ. This intriguing phenomenon has generally eluded linear scattering or viscoelastic models and, while various possible causes have been suggested, it is still not well understood [25, 39].

The remainder of this paper is organized as follows. Section 2 recalls the linear dispersion relation obtained from the porous viscoelastic model of Chen et al. [3] and describes the data fitting procedure. Other existing viscoelastic formulations are also briefly reviewed. Section 3 presents the corresponding fits to data on attenuation rate from a selection of laboratory experiments and field observations. Section 4 discusses the estimation of shear modulus and kinematic viscosity, and compares results among three different viscoelastic models. Section 5 shows sensitivity tests on a set of rheological parameters that are relevant to the poroelastic system. Finally, concluding remarks are provided in Section 6.

The dispersion relations reviewed in this section are derived from continuum models for linear traveling waves in the two-dimensional case (one horizontal direction and one vertical direction).

The dispersion relation associated with the porous viscoelastic model proposed in [3] (hereafter referred to as CGG) is given by

with

where $g$ is the acceleration due to gravity, $H$ is the water depth and $c_{f}$ is the speed of sound in water. The reader is directed to [3] for a detailed derivation of this model and to the Appendix where the expressions of coefficients $T_{1}$, $T_{2}$, $T_{3}$ are recalled for convenience. These coefficients are functions of various wave parameters and rheological parameters. Wave parameters include the angular frequency $\omega$ and complex mode $\kappa=k+{\rm i}\,q$ where $k$ is the wavenumber and $q$ is the attenuation rate. Rheological parameters include the water density $\rho_{f}$ as well as the ice density $\rho_{s}$, porosity $\beta$, shear modulus $\mu$, Poisson’s ratio $\nu$, kinematic viscosity $\eta$ and thickness $h$. Ice porosity is represented by a dimensionless parameter whose range is $0\leq\beta\leq 1$, with the limiting values $\beta=0$ (solid phase) and $\beta=1$ (fluid phase) corresponding to pack ice and near-open water, respectively. This parameter may be related to ice concentration $C$ via the relation $\beta=1-C$, namely $\beta$ is the complement of $C$. It should be pointed out that, in this continuum framework, the elasticity, porosity and viscosity parameters do not necessarily correspond to intrinsic properties of sea ice but rather they are meant to represent effective properties of the heterogeneous ice field under consideration, similar to e.g. homogenization modeling of wave propagation in complex media [7, 9]. These parameters may thus vary over a wider range than the typical values for sea ice. In view are potential applications to large-scale wave forecasting in the MIZ where various types of sea ice coexist. Whenever the information is available, $\beta$ and $h$ will be assigned values corresponding to mean ice concentration and mean thickness of the ice cover (e.g. field studies often report an estimate of the fraction of ice-covered surface that may be used for $C$). Aside from bulk viscosity which is typically regulated by $\eta$, this model also describes friction due to the relative motion between fluid and solid parts of the ice field. In the equations, the coefficient controlling this mechanism is defined by

where $a$ denotes the fluid pore size in the porous medium. From the viewpoint of effective medium theory for wave propagation in the MIZ, this parameter $a$ may be related to some characteristic horizontal size of open-water areas in the fragmented ice cover.

In the next section, predictions from (1) will be tested against a selection of laboratory experiments and field observations. For each set of experimental data, comparison with other existing models will be provided as well. These include recent viscoelastic models by Wang and Shen [41] and Mosig et al. [31], which we find convenient to present in detail below because they share common rheological parameters, and this will be of relevance to the subsequent discussion. These two models are simpler than the present one in the sense that they do not take into account ice porosity, accordingly their dispersion relations are simpler. The dispersion relation resulting from Wang and Shen’s model [41] (hereafter referred to as WS) can be written as

where

and

On the other hand, the dispersion relation produced by Mosig et al.’s model [31] (hereafter referred to as EFS) takes the form

where

Note that the EFS model also makes use of the thin-plate approximation and thus is significantly simpler than both CGG and WS models which instead consider the ice cover as a distinct layer with an actual thickness. Preliminary comparison between these three models can be found in [3].

For a given value of $\omega$ and other parameter values, the dispersion relation is solved numerically for $\kappa$ using the root-finding routine fsolve in Matlab. More specifically, because $\kappa$ is complex, Eq. (1) is split up into its real and imaginary parts. This leads to a system of two independent equations that are solved simultaneously for the two unknowns $k$ and $q$. The fsolve algorithm is essentially a quasi-Newton method with a numerical approximation of the Jacobian matrix. We have successfully used this Matlab routine in previous work [16, 18] to compute traveling wave solutions of nonlinear partial differential equations. Considering that multiple roots for $k$ and $q$ may exist here [31, 48], we apply the selection criteria proposed in [41] to find a dominant pair $(k,q)$ associated with a physically relevant solution. We choose the converged values $(k,q)\in\mathbb{R}_{+}^{2}$ such that $k$ is closest to the open-water wavenumber $k_{0}$ which solves

and $q$ is the lowest attenuation rate possible. Accordingly, we run the root finder fsolve for a range of initial guesses around $(k,q)=(k_{0},0)$ and select the converged values for which the error $|\kappa-(k_{0}+{\rm i}\,0)|$ is minimum among all the roots found. In doing so, we were able to get acceptable solutions in all the cases we considered.

For the following tests, we prescribe a number of physical parameters such as $g=9.81$, $\rho_{s}=917$, $\rho_{f}=1025$ and $c_{f}=1449$ (in SI units), and fit the model predictions to the experimental data by optimizing with respect to other parameters. The water depth $H$ is also specified, since this is either known information from laboratory experiments or a representative value may be used for a specific oceanic region when comparing to field observations. Furthermore, because the range of Poisson’s ratio is typically small ($0<\nu<1/2$), we set it to be $\nu=0.4$, after checking that it does indeed not play a major role (see Section 5). This reduction in the parameter space helps simplify the analysis, which is especially relevant for the CGG model considering that it involves many physical parameters.

We will focus our attention on the subset $(\beta,\mu,\eta)$ when fitting the model predictions to the experimental data. Throughout this study, we will only consider data on the attenuation rate, a reason being that data on the wavenumber were not reported by field observations and our focus here is on the attenuation process as in [22, 31, 34, 38]. Our fitting procedure is basically a direct search approach. For a given triplet $(\beta,\mu,\eta)$ and a range of values of $\omega$, we apply the above-mentioned root-finding scheme to find a set of pairs $(k,q)$. We repeatedly run this procedure over a specified region of parameter space $(\beta,\mu,\eta)$, and select the set $\{q_{j}\}$ for which the $L^{2}$ error

between numerical estimates $\{q_{j}\}$ and experimental data $\{\widehat{q}_{j}\}$, is minimum among all the solutions calculated. The best fit so obtained returns a set $\{q_{j}\}$ for the attenuation rate, as well as a triplet $(\beta,\mu,\eta)$ for these rheological parameters. For the problem at hand, we use a straightforward definition (6) of the error as in [11], which is readily applicable to all the cases explored and which allows for a direct comparison among the various models involved.

It turns out that information on ice concentration was also reported in some of these studies (which we used to determine the ice porosity), and thus only the pair of parameters $(\mu,\eta)$ is to be found from the data fitting. In this process, the ranges of values for $(\mu,\eta)$ and their resolutions are chosen in a heuristic manner based on extensive trials, considering previous work [31, 33, 47] and our own experience [3, 4]. The larger the region of parameter space to be covered, the higher the computational cost. Typically, we conduct a preliminary search over an extended rough region of parameter space and then refine the search over smaller better-resolved sectors. Aside from testing the performance of the CGG model, such an analysis also helps calibrate it by estimating rheological parameters for potential applications in realistic conditions. As discussed below, while the CGG, EFS and WS models share common physical parameters such as $\mu$ and $\eta$, their respective numerical values according to the data fitting may differ significantly. Note that we use the same procedure to obtain fitting curves from the EFS and WS models.

Although it is somewhat different in character from the CGG, EFS and WS models, we will also include a comparison with the two-layer viscous model recently proposed by Sutherland et al. [38] (hereafter referred to as SRCJ), which estimates wave attenuation in sea ice by

This formula is partly heuristic because it was derived based on scaling arguments and dimensional analysis. It is nonetheless appealing due to its stunning simplicity and has been shown to produce satisfactory results in comparison with experimental data. Considering that viscous models have been successful at describing wave attenuation in such ice covers as grease ice, predictions from (7) may serve as a suitable independent reference, especially for the tests involving laboratory experiments. The coefficients $\Delta_{0}$ and $\epsilon$ are dimensionless empirical parameters whose range is $0<\Delta_{0}<1$, $0<\epsilon<1$. The same $L^{2}$ error (6) is used to optimize (7) with respect to the pair $(\Delta_{0},\epsilon)$ when fitting to measurements. The parameter $\Delta_{0}$ is a measure of the relative motion between ice and water at the bottom boundary of the ice layer. To first approximation, $\Delta_{0}\simeq 1$ corresponding to a no-slip boundary condition. The parameter $\epsilon$ is thought to be a function of ice porosity, exhibiting a similar behavior. In particular, the limit $\epsilon\rightarrow 0$ is analogous to $\beta\rightarrow 0$ (pack ice) for which, according to (7), there should be no wave dissipation within the ice layer. Sutherland et al. [38] pointed out that such a parameterization is consistent with observations of wave attenuation through the MIZ, being several orders of magnitude greater in frazil and pancake ice than in a broken floe field [14]. This is somewhat counterintuitive considering that the latter represents a more rigid ice cover than the former. Note however that the dominant mechanism for wave attenuation in broken floe fields is believed to be scattering and is of different nature from the viscous-type dissipation taking place in pancake ice fields. Preliminary tests of the CGG model in [3] are also consistent with these observations and indicate a tendency for $q$ to decrease as $\beta\rightarrow 0$ over all frequencies.

In this section, we test the CGG model against three different sets of laboratory experiments and two different sets of field observations. Altogether, these span a wide range of wave frequencies and ice-cover types. In each case, the CGG model is tested by best fitting its predictions to experimental data on the attenuation rate, based on the numerical scheme described earlier. Predictions by other existing models are also shown for comparison and numerical values of their rheological parameters as determined by the data fitting are discussed.

We further check the performance of the CGG model by comparing its estimates of shear modulus $\mu$ and kinematic viscosity $\eta$ with predictions by the EFS and WS models. These two parameters are important measures of effective viscoelastic properties of the ice cover and are common to all three continuum formulations. Such an assessment would be suitable as part of the calibration of these models in view of potential applications to large-scale wave forecasting in the polar regions. Similar parametric calibration of the EFS and WS models has been conducted in [5, 31, 47], although these studies used different methods to fit the theoretical predictions to experimental data.

Table 1 lists values of $\mu$ and $\eta$ as determined from the CGG, EFS and WS fits to the laboratory experiments and field observations that we discussed in the previous sections. To highlight the effective character of these models as applied to wave propagation in various ice-cover types, these estimates are presented in such a way that they are normalized relative to typical values $\mu\simeq\mu_{0}=10^{9}$ Pa for pack ice [31, 45] and $\eta\simeq\eta_{0}=10^{-2}$ m² s^-1 for grease or pancake ice [14, 33]. As alluded to in Section 2, Table 1 confirms that both parameters can take a wide range of values depending on the particular model and ice conditions. Among all the cases considered, the highest values of $\mu$ and $\eta$ are both achieved by the EFS fit to the Greenland Sea observations [40]. The lowest value of $\mu$ is given by the WS fit to the laboratory Test 1 of Newyear and Martin [32]. The lowest value of $\eta$ is returned by the CGG fit to the Tank 2 experiment of Wang and Shen [42]. These extreme values are highlighted in blue (highest) and red (lowest) in Table 1. As expected, for all three models, the shear modulus $\mu$ (which is a measure of the ice-cover’s elasticity) is found to be smallest for grease ice (Test 1 in [32]) and largest for a broken floe field (Arctic MIZ [40]). Their estimates of $\mu$ for both Antarctic and Arctic MIZ remain overall within two orders of magnitude from the typical value $\mu_{0}$ for pack ice. By contrast, the kinematic viscosity $\eta$ (which may represent a combination of various attenuating effects in this continuum framework) exhibits a more complicated behavior depending on the particular model and ice conditions. We point out however that all three models predict $\eta$ to be on the order of $\eta_{0}$ for the grease-ice experiments [32], which is consistent with values $\eta=2$–$3\,\eta_{0}$ inferred by Newyear and Martin [33] who fitted data from [32] to Keller’s two-layer viscous model [21]. For a broken floe field, the viscosity estimates from both EFS and WS models are found to be larger by several orders of magnitude than their counterparts for grease ice. On the other hand, the corresponding predictions from the CGG model remain comparable between these two types of ice cover.

On a related note, we see that $\mu$ and $\eta$ as determined by the CGG fit only vary over $6$ and $3$ orders of magnitude respectively, among all the cases considered. By contrast, $\mu$ and $\eta$ as predicted by the WS fit vary over $10$ and $6$ orders of magnitude respectively, while both parameters in the EFS model vary over $10$ orders of magnitude. This suggests that both $\mu$ and $\eta$ in the CGG model may only require moderate tuning in view of potential applications to operational wave forecasting. Our estimates of $\mu$ ($4.2\times 10^{11}$ Pa) and $\eta$ ($4.2\times 10^{6}$ m² s^-1) from the EFS fit to the Antarctic MIZ data are consistent with those reported in [31] for the same model ($\mu=4.9\times 10^{12}$ Pa, $\eta=5.0\times 10^{7}$ m² s^-1). Both of them are several orders of magnitude larger than the reference values $\mu_{0}$ and $\eta_{0}$ (especially for $\eta$). Regardless of how close the fit is, the EFS model tends to require very large values of these parameters in order to reproduce wave attenuation in broken floe fields of the Antarctic and Arctic MIZ. This may be interpreted as a way to make up for the thin-plate approximation so that elastic properties of the ice cover would be sufficiently well captured in these situations.

Our estimates of $\mu$ ($\{4.2,2.5\times 10^{5},8.3\times 10^{5}\}$ Pa) and $\eta$ ($\{1.5\times 10^{-2},45.0,131.6\}$ m² s^-1) from the WS fit to the laboratory measurements of Zhao and Shen [47] are in good agreement with their own findings ($\mu=\{21,5\times 10^{5},1\times 10^{6}\}$ Pa, $\eta=\{1.4\times 10^{-2},61,140\}$ m² s^-1) for Test 1, 2, 3 respectively (see their Table 2). These authors also fitted the WS model to a data set from [42] (it was not clearly stated which experiment was considered) and obtained $\mu=48$ Pa, $\eta=4\times 10^{-2}$ m² s^-1 which again are fairly close in terms of order of magnitude to our own results ($\mu=\{1.1\times 10^{3},4.0\times 10^{2}\}$ Pa, $\eta=\{9.6\times 10^{-2},5.1\times 10^{-2}\}$ m² s^-1) for Tank 2 and 3 respectively. When examining the CGG and WS models against the field observations, we see that their predictions of $\mu$ are comparable to each other on the order of $10^{7}$ Pa, which contrasts with the much higher values from the EFS fit, as noted above. This similarity however does not extend to $\eta$ since the WS model yields values that are higher than the CGG predictions by several orders of magnitude. Again, the range of estimated $\eta$ from the CGG fit is strikingly narrow among all the cases considered, in comparison to the other two models.

It is also worth mentioning that the estimates $\mu=3.3\times 10^{7}$ Pa and $\eta=1.2\times 10^{-2}$ m² s^-1 from the CGG fit to the Bering Sea measurements are consistent with those ($\mu=2.3\times 10^{9}$ Pa, $\eta=1.5\times 10^{-2}$ m² s^-1) reported in [26] for the same data set. As stated earlier, these authors used a thin-plate viscoelastic model and were able to emulate the roll-over phenomenon to some extent (see their Fig. 13). In that study, $\mu$ was assigned a typical value $\sim\mu_{0}$ for pack ice while $\eta$ was deduced from the data fitting. Interestingly, the fitting curve shown in Fig. 13 of [26] bears a resemblance to the CGG curve in our Fig. 5 (modulo the switch between wave frequency and period for the horizontal axis). Lastly, we remark that the CGG predictions of $\eta\sim 10^{-2}$–$10^{0}\,\eta_{0}$ for the data sets from the Antarctic and Arctic MIZ are encouraging in view of earlier measurements that reported values of eddy viscosity under large ice floes, ranging from $2.4\times 10^{-3}$ m² s^-1 in the central Arctic Ocean [20] to $2.1\times 10^{-2}$ m² s^-1 in the Weddell Sea (Antarctic MIZ) [28].

Given the rather large number of rheological parameters associated with the ice cover in the CGG formulation, it is of interest to check their individual relevance to this problem and test the sensitivity of attenuation rate predictions with respect to these parameters. For this purpose, we take the Bering Sea observations as a representative discriminating case because it exhibits unusual features such as the roll-over phenomenon and contains a fair number of data points. We focus our attention on the following parameters: $h$ (thickness), $\beta$ (porosity), $a$ (pore size), $\eta$ (kinematic viscosity), $\mu$ (shear modulus) and $\nu$ (Poisson’s ratio). As alluded to in previous sections, some of them which are related to geometrical features of the ice cover (e.g. thickness, porosity, pore size) may be estimated by in-situ measurement or remote sensing, while others which are related to material properties (e.g. kinematic viscosity, shear modulus) would be more difficult to determine or guess. With this in mind, a sensitivity analysis may help assign predefined values to some of these parameters (as opposed to other parameters that may require more tuning), in order to reduce the parameter space for the CGG model.

For each of these parameters, Fig. 7 displays a set of curves for the attenuation rate as predicted by the CGG model. The reference regime of parameters is given by the corresponding best fit to the Bering Sea data, as discussed in the previous section (see Fig. 5). This set of curves is obtained by varying the parameter under consideration while freezing the other parameters at their original best-fitting values. The objective of such an analysis is to examine how perturbations in individual parameters would affect the original best fit. The range of perturbations for each parameter is chosen to be an interval around its best-fitting value.

Ice thickness is a distinctive feature of the ice cover in the CGG formulation, as opposed to the viewpoint in the thin-plate approximation. Fig. 7(a) reveals that the roll-over tends to shift upward and to higher frequencies as $h$ is decreased. This tendency is quite pronounced and suggests strong sensitivity of $q$ with respect to $h$. A decrease in $h$ by a factor of $3$ shifts the peak outside the frequency range of the experimental data, and moves it out of sight in this figure. The fact that shorter waves (i.e. at higher frequencies) experience more attenuation in thinner nice (i.e. for smaller $h$), which is rather counter-intuitive, is reminiscent of a common feature in models for water waves over seabed composed of a viscous mud layer, where dissipation has a non-monotonic dependence on mud-layer thickness, with thicker layers being less dissipative [3, 6, 8]. Ice porosity and pore size are rheological parameters that are characteristic of the present model. As illustrated in Figs. 7(b) and (c), increasing $\beta$ or decreasing $a$ has the basic effect of raising the attenuation rate and accentuating the roll-over. In either case, the peak remains around $f=0.13$ Hz as $\beta$ or $a$ is varied. Note that the dichotomy in variation between $\beta$ and $a$ is attributed to their contrasting roles in the friction process. Because the parameter $b$ depends linearly on $\beta$ while it is inversely proportional to $a^{2}$ according to (2), friction is thus enhanced (and so is the roll-over) as $\beta$ is increased or $a$ is decreased. A similar behavior occurs as $\eta$ is increased (see Fig. 7d), which is anticipated considering the linear dependence of $b$ on $\eta$. A slightly more complicated picture is observed for the variation with respect to $\mu$. Inspecting Fig. 7(e), the roll-over tends to shift upward and to lower frequencies as $\mu$ is increased. The sensitivity of $q$ to $\nu$ is relatively weak and is confined to the high-frequency region, as suggested by Fig. 7(f). This explains why, for convenience and given that $0<\nu<1/2$, we set $\nu=0.4$ in the previous computations (which is close to the typical value $\nu=1/3$ for pack ice [45]).

Our sensitivity tests indicate that all these parameters have some influence on the roll-over, affecting its amplitude and/or position. Sensitivity of $q$ with respect to $h$ and $\mu$ seems to be most nontrivial, and is particularly strong for $h$. Liu et al. [26] also concluded from their model-data comparison that the frequency at which the roll-over occurs depends on ice conditions, especially ice thickness. In light of this sensitivity analysis and results from Section 3, to help reduce the parameter space, it would also be reasonable to fix the pore size with some predefined value of order $O(10)$ m for potential applications to wave forecasting in the MIZ. This is even more relevant considering that this parameter only appears in the expression (2) of the friction coefficient for the CGG model.

To assess the recently proposed CGG model, we test it against a selection of laboratory experiments and field observations taken from the literature, concerning wave attenuation in sea ice. Altogether, these measurements span a wide range of ice conditions and wave frequencies. We fit the theoretical predictions to data on attenuation rate via error minimization, which in turn yields estimates for effective rheological parameters in addition to estimates for the attenuation rate. Whenever the information is available, the porosity parameter is assigned a value that is the complement of the mean ice concentration. To further check this model’s performance, we also compare it to other existing viscoelastic theories under the same various conditions. Numerical solutions of the dispersion relations can be found using relatively simple selection criteria. As a byproduct, we independently recover (via a different fitting procedure) a number of results that are similar to those reported in previous studies. Special attention is paid to the EFS and WS formulations which share some common features with the CGG system. For such parameters as $\mu$ (shear modulus) and $\eta$ (kinematic viscosity) which control the viscoelastic properties, we find that the range of estimated values (over all the situations considered) may differ significantly from one model to another. Among these three representations, the CGG (resp. EFS) model turns out to be the one for which the predicted range of both $\mu$ and $\eta$ is the narrowest (resp. widest) in orders of magnitude. Even for individual situations, this difference in parameter recovery may be considerable. As expected, for grease ice, all three models predict $\eta$ on the order of $\eta_{0}=10^{-2}$ m² s^-1 and $\mu$ to be essentially negligible compared to the typical value $\mu_{0}=10^{9}$ Pa for pack ice. On the other hand, for broken floe fields, there is more variability in the determination of these parameters, especially for $\eta$. We obtain in this case values of $\mu$ that may be lower (for CGG and WS) or higher (for EFS) than $\mu_{0}$ by a few orders of magnitude. Estimates of $\eta$ from the EFS and WS fits tend to be larger than $\eta_{0}$ by several orders of magnitude, while those from the CGG fit remain around this reference value. Overall, the CGG model provides good fits to the data on attenuation rate for the various cases under consideration. Against the Antarctic MIZ data, the EFS counterpart appears to be a clear favorite, but the corresponding fit is achieved for values of $\mu$ and $\eta$ that are both excessively high, a fact which has also been pointed out in [31]. By contrast, the CGG fit returns markedly lower values for these parameters, with $\mu$ being lower than $\mu_{0}$ by two orders of magnitude and $\eta$ being essentially equal to $\eta_{0}$. In comparison to the Arctic MIZ data (from both the Bering and Greenland Seas), the CGG model is able to reasonably reproduce the roll-over of attenuation rate, unlike both EFS and WS counterparts which only predict a monotonic growth with frequency. According to the poroelastic formulation, this intriguing phenomenon is attributed to friction caused by the relative motion between fluid and solid components of the ice cover, which highlights the role of porosity in the present description of wave-ice interactions, as such friction is directly connected to the porous nature of the ice cover. This dissipative mechanism could possibly be a contributing factor in the roll-over effect as reported in field observations.

In the future, various extensions of this work may be envisioned. Further calibration of the CGG model would be suitable using more data (and larger data sets), which may include e.g. recent data from the Arctic MIZ as in [5]. Using such data requires substantial processing and analysis. It would also be of interest to perform more tests against other data sets exhibiting a roll-over in order to further assess the possible role of friction in this phenomenon. While the CGG system features a larger number of physical parameters than other existing viscoelastic formulations. this study suggests that some of these parameters may be assigned predefined values, or may be estimated by in-situ measurement or remote sensing. Moreover, it is conceivable that the parametric dependence in the CGG model may be mathematically simplified by examining asymptotic regimes (in the limit of vanishing parameters) according to the specific type of ice cover under consideration. Finally, it would be appropriate to extend these results to the three-dimensional setting (for wave propagation in two horizontal directions) as well as to the nonlinear case. Discrepancies that we have observed in comparison to experimental data may partly be attributed to such effects. Nonlinear theory of wave-ice interactions has drawn increasing attention in recent years [12, 17, 18, 19].