A Collection of Wet Beam Models for WaveIce Interaction
 ^{1}Department of Infrastructure Engineering, The University of Melbourne, Parkville, 3051, VIC, Australia
 ^{2}Department of Mechanical Engineering, Aalto University, Espoo, Finland
 ^{1}Department of Infrastructure Engineering, The University of Melbourne, Parkville, 3051, VIC, Australia
 ^{2}Department of Mechanical Engineering, Aalto University, Espoo, Finland
Abstract. Theoretical models for the prediction of decay rate and dispersion process of gravity waves traveling into an integrated ice cover are introduced. The term “wet beam” is chosen to refer to these models as they are developed by incorporating waterbased radiation forces, including heave damping and added mass, which are absent in most conventional models. Presented wet beam models differ from each other according to the rheological behavior considered for the ice cover. Twoparameter viscoelastic solid models accommodating KelvinVoigt (KV) and Maxwell mechanisms along with a oneparameter 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, threeparameter solid models, termed Standard Linear Solid (SLS) mechanisms, are suggested to be used to reformulate 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 twoparameters wet beam dispersion relationships presented in this research are observed to predict decay rates and dispersion process of waves travelling into actual ice covers, though threeparameter wet beam models were seen to reconstruct the those of freshwater ice formed in a wave flume.
Sasan Tavakoli and Alexander Babanin
Status: closed

RC1: 'Comment on tc202275', Anonymous Referee #1, 16 Jul 2022
Thank you for the opportunity to give a peer review of this interesting article, “A Collection of Wet Beam Models for WaveIce Interaction”.
Summary:
The article contributes to the waveice interaction, especially modeling the wave decay and dispersion when surface water waves propagate through an ice cover. The authors assumed the sources of wave energy dissipation from two mechanisms: one is water wave forces, and the other is the mechanical behavior of the ice layer, denoted as the fluidbased and solidbased energy damping mechanisms, respectively. They present “wetbeam” models that introduce the wave radiation term (heave direction only) in the EulerBernoulli beam theory and different rheologies for ice. The considered rheologies contain Kelvin Vogit (KV) model and Maxwell model and use pure elastic material as reference. Relevant dispersion relations are deduced.
The decay rates and wavenumbers are calculated using the dispersion relations with tuned rheological parameters to fit measurements from fields and lab flumes. The measurements cover landfast ice, broken ice from fields, and two lab flumes experiment with viscoelastic material and freshwater ice. The wet beam models using viscoelastic materials can agree with the measured wave decay rates in the landfast ice and broken ice fields. However, for freshwater ice, the models cannot give a well fit for decay rate and dispersion at the same time. The discrepancy is solved by introducing threeparameter viscoelastic rheologies into their dispersion relations.
The study found that the fluidbased energy damping mechanism is dominant for long waves, and the solidbased mechanism is important for short waves. The damping term in the wave radiation plays a more important role in decay rate than the added mass term. The heave added mass term can affect the wavenumber. It is also interesting to find that the equivalent Young Modulus of an SLStype material using Maxwell approach is close to what is measured in dry tests.
The proposed idea of considering wave radiation in modeling waves propagating through ice cover will be of interest to the readership of the journal. Please see my reports below:
General Comments:
 A few typos need to be corrected, which are listed in the specific comments.
 Do the dispersion relations Eqs. (1315) have multiple roots features like the models mentioned in Mosig(2015)? For example, Figure 2 of Mosig (2015) shows a root distribution in the wavenumber and attenuation domain. In other words, are there multiple roots solved from Eqs. (1315) satisfying ki>0 in this work? If so, what are the criteria for choosing the dominant root?
 What is the reason for using different dimensionless viscosities for KV model and Maxwell model in the last row of figure 2?
 It is unclear what value of the added mass coefficient A is used except in Figure 4 of this manuscript.
 Is there a comparison of wavenumber corresponding to the wave decay rate comparison with Wadhams et al. (1988) and Meylan et al. (2014) in figure 6? It would be comprehensible to have such a comparison.
 Do you consider the wave excitation force to be another necessary potential source? Because the excitation forces, radiation forces, and static forces are the common forces that need to be considered in hydrodynamics. It could occur in low ice concentration fields of ice floes.
Specific Comments:
Line 117, Eq. (9), shear stress modulus G_E is equal to shear modulus G. Do you mean G is the elastic modulus or Young's modulus?
Line 157, ko is not claimed.
In the bottom row of Figure 2, the Elasticity number corresponding to the dashed gray curve is not specified. By the way, the right column could be removed since the data are already presented in the other columns.
In figure 3, the FS model corresponding to the blue curve is not defined in the left panel. in the right panel, what is the reason for the sudden drop of the blue curve near the nondimensional wavenumber = 580ï¼
Line 230, it seems to be a typo, change the word ‘travailing’ to ‘traveling’
Line 243, I feel the paragraph is confusing, except “The heave added mass coefficient is seen to affect the dispersion process of waves propagating into the cover with lower
Rigidity”, which can be read from Figure 2(right). It is acceptable to continue with “ the heave added mass coefficient can …”. But I don’t see why it ‘matches with’ large rigidity.
Line 276 typo, correct the word ‘viscoelastic’.
Figure 6’s caption, a typo, move a ‘by’ from '... data measured by by Wadhams et al. (1988), upper row, and Meylan et al. (2014) …'.
The fluid damping coefficient B of red solid curves in the legends in the top row of Figure 8 is partially missed technology.
Line 322, change “Left and right panels … Maxwell and KV materials.” to “Left and right panels … KV and Maxwell materials.”
Line 455, a grammar error in “dispersion curves Maxwell model give is sensitive to dynamic viscosity”
 AC1: 'Reply on RC1', Sasan Tavakoli, 13 Oct 2022
 AC3: 'Reply on RC1', Sasan Tavakoli, 13 Oct 2022

RC2: 'Comment on tc202275', Anonymous Referee #2, 26 Jul 2022
My apologies to the authors for getting to this review later than I anticipated when I accepted the job. The delay is especially unfortunate as there seems to be a fundamental error in the theoretical framework of the study that means I cannot recommend revisions that give a pathway to publication.
The authors are proposing a model for wave propagation in ice covered water that includes wave radiation forces (added mass and heave damping), which they say are absent in most models. However, this is not correct as others (e.g. Squire, Meylan and coworkers) have developed many models that include radiation forces (none of which are referenced). Their models of elastic ice floes contain the rigid body modes of heave and pitch (in 2D) as well as elastic modes (see e.g. Meylan & Sturova, 2009, Journal of Fluids and Structures). Here, the authors have attempted to incorporate radiation forces directly into a dispersion relation for the floating ice but its implementation appears to be incorrect. Consider the damping term, which should express the transfer of energy from the body motion to radiating waves, so that no energy is lost from the wave–ice system. It should not, as it does here, induce an imaginary component of the wavenumber and hence wave energy dissipation.
The term in the dispersion relation used to represent heave radiation is identical to that derived from the Robinson–Palmer model, which has been used by many previous authors and shown to be capable of giving reasonable predictions of wave attenuation (again, lots of references missing). Therefore, key findings, such as “decay rates were observed to be poorly predicted if the fluidbased energy damping is not taken into account”, must be reinterpreted in the context of the RP model and lose their novelty.
Aside from the issues with the radiation force, the paper comes across as contributing yet more models of waves in ice covered waters with parameters tuned to particular datasets but without the general predictive capabilities needed for improved understanding of the wave–ice system. It is not surprising that adding more tuning parameters allows for better agreement with observations. Advances require connections between the parameter values and the ice properties associated to the different datasets.
 AC2: 'Reply on RC2', Sasan Tavakoli, 13 Oct 2022

RC3: 'Comment on tc202275', Anonymous Referee #3, 21 Oct 2022
Theoretical model
The theoretical model is
 purely elastic ice, or damping in the ice from the imaginary part of the Young's modulus. The specific formulation for the damping comes either from the KelvinVoigt or the Maxwell rheology and gives different frequency dependance in the damping coefficient.
 damping in the fluid from B, the radiation damping coefficient. (This is the same as the RobinsonPalmer model.)
 extra inertia from A, the added mass coefficientThe main novelty to me are the different ice rheologies, but the fluid damping effectively has little novelty (with the exception of A) but only introduces a more complicated
(and physically more dubious) justification for the RobinsonPalmer (RP) model. I would remove the physical justification completely as (a) unnecessary and (b) physically dubious. (Note I am not proposing to remove the RP model itself as applying an old model to new data can still be interesting.)I say it is physically dubious as the added mass and damping are usually derived from solving the hydrodynamic equations (Laplace's equation + sea floor condition +
boundary condition (7)) with A=B=0 when a body is forced to oscillate. So to put them into (7) seems a bit circular. (Incidentally, in equations 5 and 6, $z^4$
should be $z_{xxxx}$.)In the authors' reply to Reviewer 2, they talk about continuum media (I guess effective media). Maybe they are trying to represent the attenuation due to scattering
by a large number of scatterers. Phaseresolving scattering models do predict that wave energy does decay into ice but they also conserve energy. While they would not be the
first authors to represent the attenuation due to scattering with a dissipative model (eg Williams, Bouillon & Rampal, 2017, The Cryosphere)(for lack of a good alternative),
they aim to represent it entirely with RobinsonPalmer dissipation instead of empirically, as most authors do.It should also be noted that scattering models give quite different results to RobinsonPalmer especially at long periods, and since RobinsonPalmer (combined
with the dissipation inside the ice itself) gives realistic results in these case it begs the question of why they are bringing in scattering at all.Results
 right hand columns of fig 2 not needed
 why not just have $k_i$ instead of $\alpha$ since the attenuation is only coming from the dispersion relation?
 ice rheologies give different attenuation behaviours (peaks in attenuation) at high frequencies. This is interesting that peaks can be produced with different rheologies.
However, once you start to introduce more complexity (I am thinking especially of the SLS models) there are more parameters to be tuned and there is a danger of overfitting.
Moreover, the peak in attenuation may not be real as instrument noise and local nonlinear wave generation of highfrequency waves can give the appearance that high frequencies are being attenuated more than they are (Thompson et al, 2021, J. Geophys. Res.), so trying to fit them too accurately may not be wise.
Status: closed

RC1: 'Comment on tc202275', Anonymous Referee #1, 16 Jul 2022
Thank you for the opportunity to give a peer review of this interesting article, “A Collection of Wet Beam Models for WaveIce Interaction”.
Summary:
The article contributes to the waveice interaction, especially modeling the wave decay and dispersion when surface water waves propagate through an ice cover. The authors assumed the sources of wave energy dissipation from two mechanisms: one is water wave forces, and the other is the mechanical behavior of the ice layer, denoted as the fluidbased and solidbased energy damping mechanisms, respectively. They present “wetbeam” models that introduce the wave radiation term (heave direction only) in the EulerBernoulli beam theory and different rheologies for ice. The considered rheologies contain Kelvin Vogit (KV) model and Maxwell model and use pure elastic material as reference. Relevant dispersion relations are deduced.
The decay rates and wavenumbers are calculated using the dispersion relations with tuned rheological parameters to fit measurements from fields and lab flumes. The measurements cover landfast ice, broken ice from fields, and two lab flumes experiment with viscoelastic material and freshwater ice. The wet beam models using viscoelastic materials can agree with the measured wave decay rates in the landfast ice and broken ice fields. However, for freshwater ice, the models cannot give a well fit for decay rate and dispersion at the same time. The discrepancy is solved by introducing threeparameter viscoelastic rheologies into their dispersion relations.
The study found that the fluidbased energy damping mechanism is dominant for long waves, and the solidbased mechanism is important for short waves. The damping term in the wave radiation plays a more important role in decay rate than the added mass term. The heave added mass term can affect the wavenumber. It is also interesting to find that the equivalent Young Modulus of an SLStype material using Maxwell approach is close to what is measured in dry tests.
The proposed idea of considering wave radiation in modeling waves propagating through ice cover will be of interest to the readership of the journal. Please see my reports below:
General Comments:
 A few typos need to be corrected, which are listed in the specific comments.
 Do the dispersion relations Eqs. (1315) have multiple roots features like the models mentioned in Mosig(2015)? For example, Figure 2 of Mosig (2015) shows a root distribution in the wavenumber and attenuation domain. In other words, are there multiple roots solved from Eqs. (1315) satisfying ki>0 in this work? If so, what are the criteria for choosing the dominant root?
 What is the reason for using different dimensionless viscosities for KV model and Maxwell model in the last row of figure 2?
 It is unclear what value of the added mass coefficient A is used except in Figure 4 of this manuscript.
 Is there a comparison of wavenumber corresponding to the wave decay rate comparison with Wadhams et al. (1988) and Meylan et al. (2014) in figure 6? It would be comprehensible to have such a comparison.
 Do you consider the wave excitation force to be another necessary potential source? Because the excitation forces, radiation forces, and static forces are the common forces that need to be considered in hydrodynamics. It could occur in low ice concentration fields of ice floes.
Specific Comments:
Line 117, Eq. (9), shear stress modulus G_E is equal to shear modulus G. Do you mean G is the elastic modulus or Young's modulus?
Line 157, ko is not claimed.
In the bottom row of Figure 2, the Elasticity number corresponding to the dashed gray curve is not specified. By the way, the right column could be removed since the data are already presented in the other columns.
In figure 3, the FS model corresponding to the blue curve is not defined in the left panel. in the right panel, what is the reason for the sudden drop of the blue curve near the nondimensional wavenumber = 580ï¼
Line 230, it seems to be a typo, change the word ‘travailing’ to ‘traveling’
Line 243, I feel the paragraph is confusing, except “The heave added mass coefficient is seen to affect the dispersion process of waves propagating into the cover with lower
Rigidity”, which can be read from Figure 2(right). It is acceptable to continue with “ the heave added mass coefficient can …”. But I don’t see why it ‘matches with’ large rigidity.
Line 276 typo, correct the word ‘viscoelastic’.
Figure 6’s caption, a typo, move a ‘by’ from '... data measured by by Wadhams et al. (1988), upper row, and Meylan et al. (2014) …'.
The fluid damping coefficient B of red solid curves in the legends in the top row of Figure 8 is partially missed technology.
Line 322, change “Left and right panels … Maxwell and KV materials.” to “Left and right panels … KV and Maxwell materials.”
Line 455, a grammar error in “dispersion curves Maxwell model give is sensitive to dynamic viscosity”
 AC1: 'Reply on RC1', Sasan Tavakoli, 13 Oct 2022
 AC3: 'Reply on RC1', Sasan Tavakoli, 13 Oct 2022

RC2: 'Comment on tc202275', Anonymous Referee #2, 26 Jul 2022
My apologies to the authors for getting to this review later than I anticipated when I accepted the job. The delay is especially unfortunate as there seems to be a fundamental error in the theoretical framework of the study that means I cannot recommend revisions that give a pathway to publication.
The authors are proposing a model for wave propagation in ice covered water that includes wave radiation forces (added mass and heave damping), which they say are absent in most models. However, this is not correct as others (e.g. Squire, Meylan and coworkers) have developed many models that include radiation forces (none of which are referenced). Their models of elastic ice floes contain the rigid body modes of heave and pitch (in 2D) as well as elastic modes (see e.g. Meylan & Sturova, 2009, Journal of Fluids and Structures). Here, the authors have attempted to incorporate radiation forces directly into a dispersion relation for the floating ice but its implementation appears to be incorrect. Consider the damping term, which should express the transfer of energy from the body motion to radiating waves, so that no energy is lost from the wave–ice system. It should not, as it does here, induce an imaginary component of the wavenumber and hence wave energy dissipation.
The term in the dispersion relation used to represent heave radiation is identical to that derived from the Robinson–Palmer model, which has been used by many previous authors and shown to be capable of giving reasonable predictions of wave attenuation (again, lots of references missing). Therefore, key findings, such as “decay rates were observed to be poorly predicted if the fluidbased energy damping is not taken into account”, must be reinterpreted in the context of the RP model and lose their novelty.
Aside from the issues with the radiation force, the paper comes across as contributing yet more models of waves in ice covered waters with parameters tuned to particular datasets but without the general predictive capabilities needed for improved understanding of the wave–ice system. It is not surprising that adding more tuning parameters allows for better agreement with observations. Advances require connections between the parameter values and the ice properties associated to the different datasets.
 AC2: 'Reply on RC2', Sasan Tavakoli, 13 Oct 2022

RC3: 'Comment on tc202275', Anonymous Referee #3, 21 Oct 2022
Theoretical model
The theoretical model is
 purely elastic ice, or damping in the ice from the imaginary part of the Young's modulus. The specific formulation for the damping comes either from the KelvinVoigt or the Maxwell rheology and gives different frequency dependance in the damping coefficient.
 damping in the fluid from B, the radiation damping coefficient. (This is the same as the RobinsonPalmer model.)
 extra inertia from A, the added mass coefficientThe main novelty to me are the different ice rheologies, but the fluid damping effectively has little novelty (with the exception of A) but only introduces a more complicated
(and physically more dubious) justification for the RobinsonPalmer (RP) model. I would remove the physical justification completely as (a) unnecessary and (b) physically dubious. (Note I am not proposing to remove the RP model itself as applying an old model to new data can still be interesting.)I say it is physically dubious as the added mass and damping are usually derived from solving the hydrodynamic equations (Laplace's equation + sea floor condition +
boundary condition (7)) with A=B=0 when a body is forced to oscillate. So to put them into (7) seems a bit circular. (Incidentally, in equations 5 and 6, $z^4$
should be $z_{xxxx}$.)In the authors' reply to Reviewer 2, they talk about continuum media (I guess effective media). Maybe they are trying to represent the attenuation due to scattering
by a large number of scatterers. Phaseresolving scattering models do predict that wave energy does decay into ice but they also conserve energy. While they would not be the
first authors to represent the attenuation due to scattering with a dissipative model (eg Williams, Bouillon & Rampal, 2017, The Cryosphere)(for lack of a good alternative),
they aim to represent it entirely with RobinsonPalmer dissipation instead of empirically, as most authors do.It should also be noted that scattering models give quite different results to RobinsonPalmer especially at long periods, and since RobinsonPalmer (combined
with the dissipation inside the ice itself) gives realistic results in these case it begs the question of why they are bringing in scattering at all.Results
 right hand columns of fig 2 not needed
 why not just have $k_i$ instead of $\alpha$ since the attenuation is only coming from the dispersion relation?
 ice rheologies give different attenuation behaviours (peaks in attenuation) at high frequencies. This is interesting that peaks can be produced with different rheologies.
However, once you start to introduce more complexity (I am thinking especially of the SLS models) there are more parameters to be tuned and there is a danger of overfitting.
Moreover, the peak in attenuation may not be real as instrument noise and local nonlinear wave generation of highfrequency waves can give the appearance that high frequencies are being attenuated more than they are (Thompson et al, 2021, J. Geophys. Res.), so trying to fit them too accurately may not be wise.
Sasan Tavakoli and Alexander Babanin
Sasan Tavakoli and Alexander Babanin
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