The transient hydroelastic response of an ice shelf under long wave excitation is analysed by means of the finite element method. The simple model, presented in this work, is used for the simulation of the generated kinematic and stress fields in an ice shelf, when the latter interacts with a tsunami wave. The ice shelf, being of large length compared to its thickness, is modelled as an elastic Euler-Bernoulli beam, constrained at the grounding line. The hydrodynamic field is represented by the linearised shallow water equations. The numerical solution is based on the development of a special hydroelastic finite element for the system of governing of equations. Motivated by the 2011 Sulzberger Ice Shelf (SIS) calving event and its correlation with the Honshu Tsunami, the SIS stable configuration is studied. The extreme values of the bending moment distribution in both space and time are examined. Finally, the location of these extrema is investigated for different values of ice shelf thickness and tsunami wave length.

The catastrophic impact of climate change on the Antarctic Peninsula is examined in the works of Scambos et al. (2003) and Skvarca et al. (1999), where attempts to identify the mechanisms of climate-induced, ice shelf disintegration are made. Ice shelf stability is being re-evaluated as wave trains are becoming rougher and elevated temperatures lead to the further weakening of ice formations (Young et al., 2011). In fact, the question of whether ocean wave forcing acts as a collapse triggering mechanism is thoroughly explored in the literature. In particular, gravity wave forcing is depicted as a major cause of rift propagation within an ice shelf, preceding breakup events (Bromirski and Stephen, 2012). Additionally, the effects of infra-gravity waves and intense storm activity are also considered crucial for ice shelf stability (Bromirski et al., 2010).

The present contribution is motivated by the calving event triggered by the Honshu earthquake-generated tsunami, in March 2011. Observational data showed that the Tsunami generated by the aforementioned earthquake in Japan reached the Sulzberger Ice Shelf in Antarctica and caused the formation of two icebergs, the largest being the size of Manhattan island (Brunt et al., 2011). It is evident that the oscillatory flexural bending, induced by wave excitation, is a primary mechanism for ice shelf and ice tongue calving. Ice-tsunami wave interaction is also manifested in the run-up stage, when drifting ice formations are swept by the incoming long wave. The Tohoku Tsunami exhibited the rare feature of transporting large ice masses, causing significant disruptions on the Kuril Islands shoreline as documented in Kaistrenko et al. (2013).

Due to their structural characteristics, namely their negligible bending rigidity and large horizontal dimension, the dynamic response of ice shelves when interacting with the ocean wave field can be effectively modelled as an initial–boundary value problem of hydroelasticity. Hydroelastic analysis is also applied for the study of ice floes subjected to ocean forcing (see Squire, 2007). Under the above considerations, ice shelves can be modelled as constrained semi-infinite plates floating over a water region with either zero or non-zero draft (see Sergienko, 2010). Related to ice shelf modelling, a recent work of Bhattacharjee and Guedes Soares (2012) focuses on the frequency domain problem of a floating semi-infinite plate in the vicinity of a vertical wall. A variety of plate edge conditions are examined, including a free, a fixed and a pinned condition at the vertical wall interface. Brocklehurst et al. (2010) present an analytical solution to the problem of a clamped semi-infinite, homogeneous, elastic plate over a constant bathymetry region. Tkacheva (2013) employs an eigenmode expansion for the solution of a fixed plate on a vertical wall under regular wave loading.

The majority of studies consider the case of harmonic wave excitation, which enables the calculation of the floating body response in the frequency domain. In this case, a common line of work is the modal expansion technique, where the elastic deformation is deduced by the superposition of distinct modes of motion (Belibassakis and Athanassoulis, 2005). The hydrodynamic forces are treated primarily through the employment of the Green function method or the eigenfunction expansion matching method. A number of studies have focused on transient analysis of elastic floating bodies, allowing for non-harmonic wave forcing and time-dependant loads on the body. These attempts incorporate direct time integration schemes, Fourier transforms, modal expansion techniques and other methods (Meylan and Sturova, 2009; Sturova, 2009; Watanabe et al., 1998). For a non-uniform elastic plate floating on shallow waters of variable depth, Papathanasiou et al. (2015) developed a higher-order finite element for the time domain solution of the hydroelastic problem composed of a freely floating or semi-fixed body, while the non-linear transient response is examined in Sturova et al. (2010) by means of a spectral–finite difference method.

In the present contribution, the previous work of the authors on higher order FE schemes (i.e. Papathanasiou et al., 2015) will be applied in the hydroelastic analysis of ice-shelves under long-wave excitation. In Sect. 2 the physical domain and the governing equations are presented. The variational formulation of the previously defined initial boundary value problem is discussed in Sect. 3. In Sect. 4 a case study, with parameters resembling that of the Sulzberger Ice shelf, is analyzed by means of the proposed methodology. The temporal distributions of the maximum and minimum bending moment values, along with their corresponding location along the semi-fixed floating body are given. Finally, a parametric analysis regarding the location of the occurred extreme bending moment values is performed for different ice shelf thickness and initial disturbance wavelength values.

The ice shelf is represented by an elastic, heterogeneous, thin plate with a
fixed edge, extending infinitely at the

In agreement with the Euler–Bernoulli beam model, the normal stress varies
linearly along the

In order to derive the variational formulation of the above problem, Eqs. (1)–(3)
are multiplied by the weight functions

Find

A special hydroelastic element is developed and employed for the solution of
Eq. (13). The reader is directed to the previous work by Papathanasiou et
al. (2015) for more details concerning the proposed finite element scheme.
The interpolation degree selected features 5th order Hermite
polynomials for the beam deflection/upper surface elevation in the
hydroelastic region and 4th order Lagrange polynomials for the
approximation of the water velocity potential. Hence, within each element,

In the present section, the simplistic, mechanical model described above
will be employed for the calculation of the hydroelastic response of the
Sulzberger Ice shelf under long wave forcing. The SIS is simulated by a
semi-fixed plate of 100 km in length. For the employed bathymetric profile,
mean depth values were used. In Brunt et al. (2011) it is mentioned that the
water column depth in front of the ice-shelf is 150 m, while it increases to
800 m within 100 km from the ice shelf front. Thus, the ocean depth under the
ice shelf is assumed to be 150 m, while a mildly sloping bottom is
considered over a distance of 100 km from the edge of the ice shelf (see
Fig. 1). The water depth increases from 150 m, at

At first, the effects of an initial pulse with

Space-Time plot of the bell-shaped pulse propagation. The
bathymetric profile is shown in a schematic below. All dimensions are
normalized with respect to the plate length

The present analysis aims to provide some first and simple means for the estimation of long wave impact on floating, slender formations and their response, as a first step towards the hydroelastic modelling of ice shelves. As illustrated in Fig. 1, phenomena, such as wave reflection, hydroelastic dispersion, bending moment variation are well reproduced. In order to investigate the generated stress field within the floating body, the bending moment distribution is examined. Bending moment distributions are directly linked to maximum normal stress values. In particular, for notched or pre-cracked specimens it is usually those normal stresses that mostly influence crack initiation and propagation. The latter phenomena are crucial when a pre-existing crack happens to be inside a tensile zone of large magnitude.

Maximum bending moment temporal profile (red thick line) and location of corresponding values along the floating cantilever (black line). A detailed figure of the profile after the wave impact is presented, along with representative snapshots of the deformed ice shelf.

Minimum bending moment temporal profile (blue thick line) and location of corresponding values along the floating cantilever (black line). A detailed figure of the profile after the wave impact is presented, along with representative snapshots of the deformed ice shelf.

Typically, for the bending of thin beams, the normal stresses due to bending
are dominant and as a first approach, shear stresses may be neglected. In
Figs. 2 and 3 the maximum and minimum bending moment temporal and spatial
distributions are shown. For the maximum bending moment, the temporal
distribution is shown in a thick red line, while the location of the
corresponding values along the ice shelf is given by the thin black line
(Fig. 2). When at rest, the maximum bending moment is zero in absence of
flexural effects. Immediately after impact,

As shown in Fig. 3, the minimum bending moment intensifies until the entire pulse wavelength has passed under the floating cantilever, at which point the minimum bending moment value remains virtually constant up to the arrival of the dispersed wave train at the fixed edge.

Location of extreme bending moment along the semi-fixed floating
body for various values of thickness and initial disturbance wavelength.
Variable

Notably, the notion that the pulse will reach the fixed end is rather
unrealistic. The induced flexural effects will cause the bending failure of
the semi-fixed floating body long before the hydroelastic pulse arrives at
the grounding line. As seen in Figs. 2 and 3, the maximum and minimum
bending moment values reach a plateau approximately after the full
disturbance passes underneath the ice shelf. Considering the effects before
the hydroelastic wave train reaches the grounding line, namely for short
times after the long wave impact, the corresponding location of the given
extreme bending moment value along the ice shelf may be linked to both ice
shelf thickness and initial disturbance form. Figure 4 displays a parametric
study of the extreme bending moment value location for different ice shelf
thickness and tsunami wavelength values. Variable

Plot of maximum and minimum bending moment value distributions for

In the present work, the transient hydroelastic response of a semi-fixed floating cantilever, resembling an ice shelf, is studied by means of a higher order finite element scheme. The simple model derived above is able to provide valuable information regarding the kinematic and stress fields induced by long wave forcing on an ice shelf. An illustrative case study is presented with parameters selected so as to approximately simulate the Sulzberger Ice Shelf topology and the relevant calving event conditions, initiated by the 2011 Honshu Tsunami. Bending moment profiles, as generated by a long wavelength elevation pulse, are studied and critical points of the induced stress field are located. During the wave entry in the hydroelasticity dominated region, the locations of extreme bending moments is found to be relatively insensitive to the excitation wavelength for given ice shelf thickness values. Important extensions of the present study include 3-D hydroelastic interaction, as well as the investigation nonlinearity effects of both in the hydrodynamic model and in the elastic subregion. Finally, the study of tsunami-ice interaction in the run-up stage constitutes another possible future research direction in the context of the present model applications.

The present work has been supported by the project HYDELFS funded by the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF 2007-2013) -Research Funding Program ARHIMEDES-III: Investing in knowledge society through the European Social Fund. In particular, author Theodosios K. Papathanasiou acknowledges support from the aforementioned program for the period from 6 September 2012 to 30 September 2014. Edited by: I. Didenkulova Reviewed by: E. Pelinovsky and another anonymous referee