Ice-shelf forced vibration modelling is performed using a full 3-D finite-difference elastic model, which also takes into account sub-ice seawater flow. The ocean flow in the cavity is described by the wave equation; therefore, ice-shelf flexures result from hydrostatic pressure perturbations in sub-ice seawater layer. Numerical experiments have been carried out for idealized rectangular and trapezoidal ice-shelf geometries. The ice-plate vibrations are modelled for harmonic ingoing pressure perturbations and for high-frequency spectra of the ocean swells. The spectra show distinct resonance peaks, which demonstrate the ability to model a resonant-like motion in the suitable conditions of forcing. The spectra and ice-shelf deformations obtained by the developed full 3-D model are compared with the spectra and the deformations modelled by the thin-plate Holdsworth and Glynn model (1978). The main resonance peaks and ice-shelf deformations in the corresponding modes, derived by the full 3-D model, are in agreement with the peaks and deformations obtained by the Holdsworth and Glynn model. The relative deviation between the eigenvalues (periodicities) in the two compared models is about 10 %. In addition, the full model allows observation of 3-D effects, for instance, the vertical distribution of the stress components in the plate. In particular, the full model reveals an increase in shear stress, which is neglected in the thin-plate approximation, from the terminus towards the grounding zone with a maximum at the grounding line in the case of the considered high-frequency forcing. Thus, the high-frequency forcing can reinforce the tidal impact on the ice-shelf grounding zone causing an ice fracture therein.

Tides and ocean swells produce ice-shelf bends and, thus, they can initiate break-up of ice in the marginal zone (Holdsworth and Glynn, 1978; Goodman et al., 1980; Wadhams, 1986; Squire et al., 1995; Meylan et al., 1997; Turcotte and Schubert, 2002) and also excite ice-shelf rift propagation. No strong correlation between rift propagation rate and ocean swell impact have been revealed so far (Bassis et.al., 2008), and it is not clear to what degree the rift propagation can potentially be triggered by tides and ocean swells. Nevertheless, the impact of tides and ocean swells is a fraction of the total force (Bassis et al., 2008) that produces ice calving in ice shelves (MacAyeal et al., 2006). Moreover, a resonant-like motion in suitable conditions of long-term swell forcing (several periods of the swell impact) can cause a fracture in the ice-shelf (Holdsworth and Glynn, 1978). Thus, new knowledge on the vibration process in ice shelves is important for the investigation of ice-sheet–ocean interactions and sea level change due to alterations in the ice calving rate.

Several models of ice-shelf bends and ice-shelf vibrations have been proposed by several researchers, e.g. Holdsworth (1977), Hughes (1977), Holdsworth and Glynn (1978), Goodman et al. (1980), Lingle et al. (1981), Wadhams (1986), Smith (1991), Vaughan (1995), Schmeltz et al. (2001), Turcotte and Schubert (2002), on the basis of elastic thin plate/elastic beam approximations. These models allow simulations of ice-shelf deformations, calculate the bending stresses emerging due to the processes of vibrations, and assess possible effects of tides and ocean swell impacts on the calving process. Further development of elastic-beam models for description of ice-shelf flexures implies application of viscoelastic rheological models. In particular, tidal flexures of ice-shelf are obtained using the linear viscoelastic Burgers model by Reeh et al. (2003) and Walker et al. (2013), and using the nonlinear 3-D viscoelastic full Stokes model by Rosier et al. (2014).

Ice-stream response to ocean tides has been described by full Stokes 2-D finite-element employing a non-linear viscoelastic Maxwell rheological model by Gudmundsson (2011). This model revealed that tidally induced ice-stream motion is strongly sensitive to the parameters of the sliding law. In particular, a non-linear sliding law allows for the explanation of an ice stream response to ocean forcing at long-tidal periods (MSf) through a nonlinear interaction between the main semi-diurnal tidal components (Gudmundsson, 2011).

A 2-D finite-element flow-line model with an elastic rheology was developed by O. V. Sergienko (Bromirski et al., 2010; Sergienko, 2010), and was used to estimate mechanical impact of a high-frequency tidal action on the stress regime of ice shelves. In this model (Sergienko, 2010), seawater was considered as an incompressible, inviscid fluid, and was described by the velocity potential.

In this work, the modelling of forced vibrations of a buoyant, uniform, elastic ice-shelf, floating in shallow water of variable depth, was developed. The simulations of the bends of the ice-shelf were performed by a full 3-D finite-difference elastic model. The main objectives of the study were as follows: firstly, to introduce a method that provides stability to the numerical solution in the full finite-difference elastic model based on the coupling of the fundamental momentum equations with the wave equation for the water layer; secondly, to compare the results – the amplitude spectra and the ice-shelve deformations – obtained by the full 3-D model developed here with the spectra and the deformations modelled by the thin-plate Holdsworth and Glynn model (1978) (Appendix A) with the intention of revealing the principal distinctions, if any, and specifications of the full model.

The 3-D elastic model is based on well-known momentum equations (e.g.
Lamb, 1994; Landau and Lifshitz, 1986):

The sub-ice water is considered as an incompressible non-viscous fluid of
uniform density. Another assumption is that the water depth changes
gradually horizontally. Under these assumptions, the sub-ice
water flows uniformly in a vertical column, and the manipulation with the
continuity equation and the Euler equation yields the wave equation
(Holdsworth and Glynn, 1978)

For harmonic vibrations, the method of separation of variables yields the
same equations, in which only the operator

The boundary conditions are (i) stress free ice surface, (ii) normal stress
exerted by seawater at the ice-shelf free edges and at the ice-shelf base,
and (iii) rigidly fixed edge at the origin of the ice-shelf (i.e. in the
glacier along the grounding line). In detail, the well-known form of the
boundary conditions, for example, at the ice-shelf base is expressed as

In this developed model, we considered an approach where the known
boundary conditions (Eq.

In the ice-shelf forced vibration problem, the boundary conditions for the
water layer are as follows: (i) at the boundaries coinciding with the lateral
free edges:

Numerical solutions are obtained with a finite-difference method, which is
based on the standard coordinate transformation

Numerical experiments with ice flow models and with elastic models
(Konovalov, 2012, 2014) have shown that the method, in which the initial
boundary conditions Eq. (

Constitutive relationships between stress tensor components and
displacements correspond to Hook's law (e.g. Landau and Lifshitz, 1986;
Lurie, 2005):

Substitution of these relationships into Eq. (

The numerical experiments with ice-shelf forced vibrations were carried out for a physically idealized ice plate having rectangular and trapezoidal profiles (Fig. 1). The three experiments that differ in ice-shelf/cavity geometries as shown in Fig. 1 are considered here. A difference in the spectra obtained from among the three experiments implied an impact of the cavity geometry and of the ice-shelf geometry on the eigenfrequencies of the shelf–water system.

In Experiment A, ice-shelf thickness and the water layer depth were kept constant (Fig. 1a).

In Experiment B, an expanding water layer was considered (Fig. 1b). The
expanding water layer is in agreement with the observations (e.g.
Holdsworth and Glynn, 1978) and leads to the change in the spreading velocity of a long gravity wave in the channel (due to changes in

In addition, in Experiment C, a tapering ice-shelf was considered (Fig. 1, c). As in the case of the expanding cavity, firstly, the tapering ice-shelf is in agreement with the observations (e.g. Holdsworth and Glynn, 1978). Secondly, the taper of the ice-shelf should yield changes in the eigenfrequencies of the shelf–water system due to a change in the average ice-plate thickness.

Figures 2–4 show the amplitude spectra obtained for all the three experiments. The amplitude spectrum, shown in Fig. 2, is split into parts for a better visualization of the resonance peaks in the spectrum.

Figures 5 and 6 show the ice-shelf deformations that responded to the eigenfrequencies derived from the amplitude spectra in Experiment A and C, respectively.

The first four eigenvalues can be distinguished easily
in the spectra shown in Fig. 2. They are approximately equal to 37.1, 14.2, 7.1, and 4.21

This experiment reveals the same trend in the difference
between the eigenvalues obtained from both considered models (Fig. 3).
Specifically, as in Experiment A, the maximum difference between the
eigenvalues is observed for the first eigenvalue. The first three
eigenvalues are approximately equal to 43.2, 16.8, and 8.4

There are no new particulars (in comparison with previous experiments) in the
relative position of the resonance peaks obtained from the considered models
(Fig. 4) in Experiment C. The first four eigenvalues are approximately equal
to 66.9, 21.2, 10.3, and 5.9

Figure 7 shows shear stress distributions in the vertical cross-section
along the centre line. Obtained distributions reveal that the maximum shear
stress is next to the grounding line. For instance, in Experiment A, near-resonance forcing, which corresponds to the second eigenvalue (Fig. 5b),
induces a threshold value of

The ice-shelf forced vibration modelling can be performed by the 3-D full elastic model, although the volume of the routine sufficiently increases in comparison with the thin-plate model.

The numerical experiments have shown the impact of shelf/cavity geometry on the amplitude spectrum. The alterations of the geometries excite shifts in the peak positions. Therefore, the prediction ability for resonant-like ice-shelf motion is dependent on (i) detailed ice-shelf surface/base topography (ii) detailed numbers and positions of the crevasses, and (iii) detailed seafloor topography under the ice-shelf.

The complementary shear stress, which can be derived in the full model, in
the case of high-frequency forced vibrations, are an order of magnitude less
in the maximum than the maximal value of the component

In the forced vibration problem, in which the dissipative factors are neglected, amplitudes in the peaks (Fig. 2), in general, are undefined (unlimited). A realistic finite motion in the peaks can be modelled by considering the limitation of the ingoing overall water flux in the model, which is based on the original equations for the water layer (continuity equation and Euler equation). This model includes applicable boundary conditions for ingoing water flux and, hence, yields specific amplitude spectra with limited amplitudes in resonance peaks (Konovalov, 2014).

Thus, the full 3-D model yields quantitatively similar results, which were
obtained by a model based on thin-plate approximation (Holdsworth and Glynn,
1978). The maximum relative deviation for the eigenvalues in the test
experiments does not exceed 11 % and the maximum is observed for the first
eigenvalue. This can be explained by the assessment of the first
eigenfrequency, which is obtained considering the thin-plate approximation.
The assessment is expressed as

Holdsworth and Glynn forced vibration model (1978), which is considered in the test experiments (A, B and C) as the basic model, includes the following equations.

Thin-plate vibration equation (the momentum equation) is

The wave equation for water layer is

The boundary conditions are as follows:

At

At

At

In this work, the method, in which the initial boundary conditions Eq. (

We rewrite, for instance, the first equation from Eq. (

We write the approximation of the derivative

The standard (typical) boundary conditions at the ice-shelf base Eq. (

Thus, at the ice-shelf base, we apply the equation

Therefore, after the coordinate transformation, the applicable equations at
ice-shelf base can be written as

Finally, the same manipulations lead to the following equations at the free
edges:

At

At

At