Theoretical models for the prediction of decay rate and dispersion process of gravity waves traveling into an integrated ice cover expanded over a long way are introduced. The term “wet beam” is chosen to refer to these models as they are developed by incorporating water-based damping and added mass forces. Presented wet beam models differ from each other according to the rheological behavior considered for the ice cover. Two-parameter viscoelastic solid models accommodating Kelvin–Voigt (KV) and Maxwell mechanisms along with a one-parameter elastic solid model are used to describe the rheological behavior of the ice layer. Quantitative comparison between the landfast ice field data and model predictions suggests that wet beam models, adopted with both KV and Maxwell mechanisms, predict the decay rate more accurately compared to a dry beam model. Furthermore, the wet beam models, adopted with both KV and Maxwell mechanisms, are found to construct decay rates of disintegrated ice fields, though they are built for a continuous ice field. Finally, it is found that wet beam models can accurately construct decay rate curves of freshwater ice, though they cannot predict the dispersion process of waves accurately. To overcome this limitation, three-parameter solid models, termed standard linear solid (SLS) mechanisms, are suggested to be used to re-formulate the dispersion relationship of wet beam models, which were seen to construct decay rates and dispersion curves of freshwater ice with an acceptable level of accuracy. Overall, the two-parameter wet beam dispersion relationships presented in this research are observed to predict decay rates and dispersion process of waves traveling into actual ice covers, though three-parameter wet beam models were seen to reconstruct the those of freshwater ice formed in a wave flume. The wet beam models presented in this research can be implemented in spectral models on a large geophysical scale.

Mutual interaction between water waves and ice is a multi-physical problem, frequently occurring in polar seas where waves can penetrate ice covers, traveling over kilometers until they die out. The phase and group speeds of the resulting gravity-flexural waves advancing through the ice can be different from that of an open-water sea owing to the effects of forces caused by solid motions. There is a pressing need to understand the mutual effects of ice and gravity waves on each other due to the recent retreat of the sea ice in the Arctic

Mathematical modeling of the wave–ice interaction firstly received the attentions of researchers in the 19th century. The first model was developed by

The Greenhill study is the kernel of the next generation of models established for the prediction of mutual effects of water waves and ice. Researchers looking into the wave–ice interaction developed models by modifying the original model of Greenhill. Early developments of models dates back to years between the 1950s and 1970s when scientists had become able to reach polar seas, recording the wave climate. The energy of waves traveling through the ice was observed to be reduced by sea ice

Following the Greenhill's model, various linear mathematical models of wave–ice interaction have been developed. In some of them, interactions between water waves and finite length ice floes have been modeled. Using such an assumption, scattering and radiation problems can be addressed, and motions of a flexible ice floe can be found (e.g.,

Some other models highlight mutual interaction of water waves with an infinite length ice cover. Simply stated, ice cover is viewed as a continuum medium. When this approach is embarked, the ice-induced energy decay is found through finding the root of the dispersion equation. To address the energy decay rates of continuum models, scholars have mostly prescribed viscoelastic (e.g.,

Models developed for the prediction of decay rates and the dispersion process of continuously integrated ice have been mostly developed by assuming small displacement for the cover. Thus, they solve the solid dynamic problem by using an Euler–Bernoulli beam theory. Their applications have been seen to be limited. This can be due to the reason that they have mostly developed by simplifying the problem, neglecting some aspects of the fluid and solid motions. Studies performed in the recent decade provide a clear picture of this fact. In some studies concerned with the dispersion process of waves advancing in ice (or elastic) covers, formulated dispersion relationships were reported to reconstruct the dispersion plot with an effective value of rigidity (or Young's modulus), which is much smaller than what was measured in dry tests

To overcome the limitations of available continuum models, the role of different mechanisms in energy dissipation and the parameters influencing the dispersion process of waves propagating in ice should be understood well. Reviewing the structure of all developed models (example of review papers:

The present paper aims to develop continuum wave–ice interaction models by hypothesizing that fluid forces and solid forces emerge simultaneously. Distinguishing the developed models from the previous studies, fluid forces, including damping and added mass, are hypothesized to emerge under a viscoelastic ice beam covering water, vibrating due to the wave forces. To understand the role of rheological behavior of ice, two different two-parameter solid mechanisms, linking stress and strains arising in the solid beam, are employed to establish these models. Accordingly, two new viscoelastic models with incorporation of fluid damping force are introduced. It is also aimed to provide a better understanding of how different rheological patterns can be implemented in the prediction of decay rate of the ice-covered water and seas. This paper is structured as follows. Models are developed in Sect. 2. In Sect. 3, results including wave height decay rates and dispersion curves are presented. At the first step, sensitivities of models to different parameters, shear modulus and viscosity, are analyzed. At the next step, predictions of models are compared against those of field and flume measurements. Afterward, two three-parameter solid models are also used to formulate dispersion relationships to investigate the way consideration of more complex rheological behavior can affect the results. Finally, a discussion on the ability of models in the prediction of decay rates and dispersion curves is presented. In Sect. 4, concluding remarks and suggestions are presented. Appendix A presents an example of dimensional data.

Consider a two-dimensional fluid domain containing water. The domain extent is stretched over an infinite length and has a depth of

Let the fluid be ideal and irrotational. Hence, the fluid motion is represented by the potential field, which is indicated with

The vertical component of velocity is zero at the seabed, which signifies that

It is assumed that fluid forces emerge per unit area when the beam interacts with the water, which are given as

Equation (

For an elastic solid body,

For a viscoelastic material, two different two-parameter models can be used. The first one is Kelvin–Voigt (KV) and the second one is Maxwell. For the former, displacements of both elements (i.e., spring and dashpot) are similar, but the resulting forces are different. For a Maxwell material, however, forces emerging in elements are similar, and displacements are different.

A pictograph of the problem and the different patterns of rheological behavior considered for the ice layer. Panels

For a KV material, the dynamic shear modulus can be written as

For a Maxwell material, the dynamic shear modulus is given by

Dispersion relationships can be established for different materials. The first dispersion relationships is developed for water waves traveling into a pure elastic material as

The second dispersion relationship describes the link between frequency and wavenumber for a KV material, which is found to be

To analyze results and to use any of the models more easily, all parameters are normalized. Parameters are normalized using the Buckingham Pi theorem, enabling us to perform scaling law. For the first dispersion relationship, eight parameters are involved, but for the second and third relationships, nine parameters are involved. Therefore, five non-dimensional numbers are identified for the pure elastic model, and six non-dimensional numbers are identified for models developed for viscoelastic materials.

Non-dimensional numbers have been previously presented by different authors, concerted with the field wave–mud or wave–ice interaction

Results of wet beam models are presented in five separate sub-sections. The first sub-section presents a discussion on the effects of different parameters on the dispersion and decay rate plots. The primary aim is to provide a better understanding of the sensitivity of models to mechanical behavior of material and fluid forces. The second and third sub-sections discuss the ability of models in reconstruction of the decay rate and dispersion process through comparing their results against field and flume measurements. A brief summary of these measurements is documented in Table

Cases studied in the present paper.

Figure

As seen, for a viscoelastic material with a larger flexural rigidity,

Effects of the dynamic viscosity

To provide a clear picture of the effects of the fluid damping on the decay rates,

Now we discuss the decay rates of covers with a lower flexural rigidity (second row). No critical

The last row of Fig.

Effects of fluid damping coefficient on

Figure

The left panel of Fig.

The decay rates of a Maxwell material considering different damping coefficients are plotted in the right panel. Trends of the presented curves are consistent with the ones observed in the left panel of Fig.

Log–log plots of

To understand the role of fluid-based damping and the effects of viscosity on the decay rate more deeply,

The

The dispersion relationships are also employed to calculate wavenumbers of waves propagating into viscoelastic covers. Results are depicted in Fig.

Effects of the added mass force

As apparent, when the elasticity number of a KV material is greater (left) and the added mass is nil, the dimensionless wavenumber is below 1.0 at small values of

For a KV material with low density and a zero added mass coefficient, the wavenumber is below 1.0 when the dimensionless open-water wavelength is small. Compared to a KV material with a greater elasticity number (left panel),

The added mass force can affect the dispersion process of waves propagating into cover with lower elasticity number by increasing the wavenumber, though its influences on the waves advancing in the cover with lower elasticity number are more noticeable compared to cover with larger elasticity number.

The two lower panels of Fig. 4 show the calculated dimensionless wavenumbers of gravity waves propagating into a solid cover hypothesized to behave in the same way Maxwell materials do. Left and right panels respectively show the curves corresponding to covers with relatively large and low elasticity numbers. The trends of curves plotted in the left panel confirm that the dynamic viscosity can slightly affect the wavenumber of waves advancing in the cover with a large elasticity number. The influence of dynamic viscosity on wavenumber is not significant and emerges over a relatively narrow range, (the close-up view provides the evidence). Dimensionless wavenumber is seen to be insensitive to the dynamic viscosity of Maxwell materials when Elasticity number is low (right panel).

This section presents comparisons between predictions of the models and decay rates found through field and flume measurements. In all runs, the

Viscoelastic models cannot follow the field data when fluid damping is set to be zero, and tails of curves diverge from those of the field data. This can be seen in both upper and lower rows of Fig.

For the KV material, a dynamic viscosity of

Comparisons between

The decay rates of two different broken ice fields are calculated using the wet beam models and are compared against measurements. Field data and calculated decay rates are plotted in Fig.

Wet beam models established for viscoelastic materials can predict decay rate curves of a broken ice field with a good level of accuracy when fluid damping is incorporated into the calculations. Similar to curves plotted in Fig.

Comparisons between

Wet beam models are employed to calculate the decay rates of waves traveling into viscoelastic covers, with an aim to understand whether they can be used to predict attenuation rates measured in flume tests or not. The reconstructed

The curves reconstructed by viscoelastic models cannot follow the experimental data if the fluid damping is not considered. Dashed and dashed-dotted curves plotted in the first and second columns provide shreds of evidence. Their results only match with experimental data at very short dimensionless open-water wavelengths. Viscoelastic-based models can accurately predict decay rates when the fluid damping is set to be non-zero. The elastic model can calculate decay rates with an acceptable level of accuracy. The fluid damping that gives the best fitting for pure elastic material is slightly greater than those of models built for viscoelastic materials.

Comparison between

At the final stage, the decay rates of the freshwater ice formed in the wave flume of the University of Melbourne are predicted by using the presented models. Figure

The KV model can accurately predict the decay rates of 1 cm thick ice when dynamic viscosity is set to be

The KV model under-predicts the decay rate curve of 1.5 cm thick ice cover regardless of the values of the dynamic viscosity and the fluid damping coefficients. The Maxwell model, however, can construct

Comparison between

Comparison between

The accuracy of models in the calculation of the dispersion process of waves traveling through viscoelastic covers is also evaluated. First, the dispersion curves of waves propagating into landfast ice are constructed, and then the dispersion curves of waves advancing in freshwater ice are plotted.

Figure

Comparison between

Figure

Note that wavenumbers of field tests documented in

In Sect. 3.2 and 3.3, it was observed that proposed models cannot accurately calculate the dispersion process and decay rates of waves advancing into the freshwater ice. Artificial effects may have contributed to the dispersion and dissipation process, though the large difference between inputs giving the best fit for the decay rate and dispersion plots still leaves us with a big question mark about the main reason for such a difference.

As discussed, other researchers have hypothesized that the effective elasticity or dynamic viscosity of the wet ice interacting with water waves can be different from what have been measured in dry tests. While the whole paper looks into ability of KV and Maxwell in the prediction of decay rates and dispersion process and discusses how accurate their results can be, the idea of developing other models by considering various viscoelastic materials arises. We can use the other solid models to evaluate whether the Young's modulus giving the best result for decay rates and dispersion process matches with what was found in dry tests or not.

Standard linear solid models. Panels

To provide an answer to the above question, the linear combination of KV and Maxwell materials, known as the standard linear solid (SLS) model, is used to describe the mechanical behavior of ice. Two spring elements (

Comparison between predictions of SLS models and values measured by

The decay rates are seen to be well predicted by both models. The interesting point is that the KV model was found not to be able to construct the decay rates of 1.5 cm thick. But as seen here, the results of the SLS KV model fairly agree with experimental measurements. The SLS Maxwell model can predict the decay rates very well. The trend of the decay rates given by SLS Maxwell and SLS KV are very similar.

Note that adding more elements may make the dispersion relationship more complex, leaving us with more options (i.e., more inputs), though it can also lead to over-fitting as more parameters are needed to be tuned. In the future, more studies can be carried out to understand the mechanical behavior of different types of viscoelastic ice, and it can be investigated whether it is required to use solid models with more than two elements or not.

Decay rates and dispersion curves of gravity waves propagating into an ice cover can be constructed using the presented wet beam models. With the same setups (e.g., dynamic viscosity and Young's modulus), different models can give different curves for the decay rates and the dispersion process. Thus, the choice of the model for the prediction of the decay rates and the dispersion process is very important. This needs a better understanding of the ice mechanics and the way it is formed. Simply stated, the mechanics of ice is a very complicated field of research, with lots of open questions that have not been answered yet.

Models formulated in the present research follow the basis of the Euler–Bernoulli beam theory, which describes displacements of a solid layer by assuming small motions. To provide a clear picture of this theory, transverse sections of a beam flexed due to an external/internal load should be assumed. If the displacement field follows the Euler–Bernoulli law, the normal vector of any transverse section is always parallel to the axis of the beam. That is, no local rotational motion occurs. This is different from what happens for a solid body following the Timoshenko–Ehrenfest beam theory, where rotational motions are taken into consideration

Using mechanical behavior prescribed for the ice and employing an Euler–Bernoulli equation, dispersion relationships can be formulated. If the ice layer is assumed to be dry and no fluid–solid interaction is taken into account, dispersion relationships for wave motions in dry beams are acquired. If the multi-physical problem is considered, the ice layer is assumed to settle down on the water surface and fluid forces emerge. If the fluid motion is assumed to be linear, a fluid damping force can also be included in the beam motion equation, as was suggested by

In the present research, it is hypothesized that energy dissipation is caused by solid-based energy damping and fluid-based energy damping. The former is dominant over high frequencies (corresponding to short waves in an open-water condition), and the latter is dominant over small frequencies (corresponding to long waves in an open-water condition). The solid-based energy damping is caused by the viscoelastic resistance emerging in the solid and is absent for an elastic material. Regardless of the viscoelastic model used to treat the mechanical behavior of the material, the solid-based energy damping decreases with an increase in the wavelength. With an increase in elasticity number, the wavelength range over which solid-based energy damping is dominant becomes wider. This matches with physics. The body is more rigid, and solid responses can contribute to energy damping of longer waves compared to bodies with low rigidity. The fluid-based energy dissipation is generated by a velocity dependent force, decreasing with the increase in wavelength. The energy damping triggered by a pure elastic cover is only due to the presence of fluid damping force. Thus, to compensate for the lack of solid-based energy damping of an elastic cover, a larger fluid damping coefficient needs to be considered. Energy decay triggered by the landfast ice covers which have large rigidity can be computed using the viscoelastic models if fluid damping is included. If it is not, models cannot work properly, and the predicted decay curves diverge from field measurements. This well confirms that fluid-based energy damping contributes to total energy dissipation. Both viscoelastic models were seen to construct the decay curves and fairly follow field measurements. But the dynamic viscosity values were seen to be much different. A KV material may be a more realistic indicator of ice behavior as the landfast ice is expected to be solid.

Decay rates of broken ice fields were seen to be calculated by both viscoelastic models when the fluid-based energy damping is considered. This provided us with another piece of evidence for the contribution of fluid damping coefficient force to energy dissipation. Compared to landfast ice covers, different values of the dynamic viscosity were observed to give the best fitting for the decay rates of waves advancing in water partially covered with broken ice floes. The coexistence of open water and ice floes on the upper layer of the fluid domain can explain this. Models are formulated for an integrated layer of solid ice. If water is included, a mixed thin layer represents the cover. A volume fraction model may describe the dynamic viscosity of the layer. The dynamic viscosity of the water is much smaller compared to that of ice, but the water entrapped between ice floes may be turbulent, leading to eddy generation (turbulent flow can cause damping of waves, e.g.,

The decay rates of the freshwater ice were also calculated. The pure elastic materials can reconstruct the decay rates only if a very large fluid damping coefficient is used, which may not be realistic. But, viscoelastic models were seen to predict the decay rate plots. Models, however, were not able to capture the dispersion process under freshwater ice covers with the same input observed to give the fitted decay rates. Effective values were seen to construct dispersion plots with an acceptable level of accuracy. This has been observed and reported by other scholars, who measured the wavelength and phase speed of disintegrated elastic/viscoelastic covers. The interesting point is that, when a Maxwell model is used, the dynamic viscosity can affect the dispersion process. This motivates us to build other models which formulate the storage modulus by applying the dynamic viscosity. Two available linear models were introduced. One is SLS KV and the other is SLS Maxwell. Both models include two springs and one dashpot. The equivalent Young's modulus giving the best fitting when SLS KV is used is

Based on what was observed when SLS models were used, it can be concluded that a two-parameter models such as Maxwell and KV, with a realistic Young's modulus, cannot reproduce mechanical behavior of the freshwater ice formed in the flume, and its behavior is different from that of the landfast ice. There are still doubts about the behavior of broken sea ice. Both the Maxwell and KV models were found to give the best fitting. One reason is that the wave phase is not measured in most of the field tests that took place in the broken ice field, and researchers were mostly concerned with decay rates. Since the phase speed and dispersion plots are not available, the performance of models in the reconstruction of dispersion plots cannot be evaluated. Especially, this could show whether changes in the dynamic viscosity can modify the accuracy of the Maxwell model in the prediction of dispersion or not (the dispersion curves the Maxwell model gives are sensitive to dynamic viscosity). That may be still an open question for the future.

The common approach used to predict decay rates and the dispersion of waves penetrating an ice cover is to formulate a dispersion relationship of a continuum medium, the roots of which give the decay rate and the relative wavenumber in an ice-covered sea. The majority of models developed for wave–ice interaction have been developed based on two common approaches. First, ice was assumed to be a viscous fluid or a viscoelastic solid layer. Second, ice was assumed to be elastic, and a fluid damping term is used to calculate the decay rate. This paper aimed to present wet beam models by considering both of fluid-based and solid-based energy dissipation mechanisms by accommodating different rheological for the ice layer (KV and Maxwell). Thus, the models were called “wet beam”, which refers to water-based forces that are taken into consideration.

Predictions of viscoelastic models and field measurements were quantitatively compared against each other. KV and Maxwell models were seen to reconstruct the decay rates and dispersion process of landfast ice with a great level of accuracy. Decay rates were observed to be poorly predicted if the fluid-based energy damping is not taken into account, suggesting that this mechanism has a very important role in ice-induced energy decay over the range of long waves. The decay rates of unconsolidated ice fields were also seen to be accurately predicted by KV and Maxwell models by setting a non-zero value for the fluid damping coefficient.

Decay rates and wave dispersion of freshwater ice were predicted. The pure elastic model was seen to predict the decay rate with an unrealistic fluid damping coefficient. The decay rates predicted by KV and Maxwell models were seen to agree with experiments, though the dispersion plots were observed to diverge from the experimental data.

Two standard linear solid models were used and two other dispersion relationships were formulated. These relationships were seen to predict the attenuation rate and the dispersion plots with a good level of accuracy. But, the SLS model which was fundamentally based on the KV material gave the best fitting with an unrealistic Young's modulus.

Overall, the wet beam dispersion relationships developed by accommodating two-parameter solid models are able to predict the decay rates and dispersion process of ice fields. But, for a freshwater ice flume, a three-parameter solid model may increase the accuracy of the predictions. As wet beam models were observed to be capable of fitting decay rates of different field measurements, they can be employed in wave spectral modeling of polar seas in the future and can also be coupled with ice break-up models to simulate the evolution of marginal ice zones. In the future, nonlinear models can also be developed to consider the effects of wave steepness on dispersion, and other beam theories can be employed. In addition, viscosity and turbulence effects can also be incorporated into wet beam models.

The data presented in Sect. 3.2 are dimensionless. As was observed in Fig.

Comparison between

The code used for construction of the curves can be provided upon reasonable request.

All the experimental data were extracted from figures or tables of other references listed in Table 1

ST and AVB conceptualized the research. AVB supervised the research and provided the funding. ST formulated the models and visualized the data. ST and AVB discussed the results. ST wrote the draft of the paper. AVB reviewed the paper.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Sasan Tavakoli was supported by a Melbourne Research Scholarship. Alexander V. Babanin acknowledges support through the US ONR and ONRG (grant no. N62909-20-1-2080).

This paper was edited by Daniel Feltham and reviewed by three anonymous referees.