NHESSNatural Hazards and Earth System ScienceNHESSNat. Hazards Earth Syst. Sci.1684-9981Copernicus GmbHGöttingen, Germany10.5194/nhess-15-1851-2015Hydroelastic analysis of ice shelves under long wave excitationPapathanasiouT. K.KarperakiA. E.karperaki.ang@gmail.comTheotokoglouE. E.BelibassakisK. A.Department of Civil Engineering and Surveying & Geoinformatics Engineering, Technological Educational Institute of Athens, Athens, GreeceSchool of Naval Architecture and Marine Engineering, National Technical University of Athens, Zografou Campus, Athens, GreeceDepartment of Mechanics, School of Applied Mathematical and Physical Science, National Technical University of Athens, Zografou Campus, Athens, Greecenow at: DICAM, University of Trento, via Mesiano 77, 38123 Trento, ItalyA. E. Karperaki (karperaki.ang@gmail.com)19August20151581851185710March20155May201529July20155August2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://nhess.copernicus.org/articles/15/1851/2015/nhess-15-1851-2015.htmlThe full text article is available as a PDF file from https://nhess.copernicus.org/articles/15/1851/2015/nhess-15-1851-2015.pdf
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.
Introduction
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.
Physical domain geometry and governing equations
The ice shelf is represented by an elastic, heterogeneous, thin plate with a
fixed edge, extending infinitely at the y direction (vertical to the
page). The plate of horizontal length L, rests on a layer of inviscid,
incompressible fluid over an impermeable bottom. Assuming shallow water
conditions, the long wave approximation (i.e. wavelength
much greater than water depth) can be employed. The last assumption allows
for dimensionality reduction, resulting in a 1-D system of equations, since
now the z component of the fluid velocity is considered negligible. The
domain is divided into regions S0≡ (0, L)
and S1≡ (L, ∞), with the hydroelastic coupling taking place at the former (Papathanasiou
et al., 2015). In S0, the plate deflection coincides with the water upper surface elevation
η(x, t). The fluid velocity potential in the two regions is denoted as φ0 and φ1
respectively. In order to account for the draft of the plate, the variable
bathymetry function b(x)=H(x)-d(x),
where B(x) is the water depth and d(x)=τ(x)ρi/ρw the draft of the plate,
τ(x) being the plate thickness, is defined. The ice and water density are
ρi and ρw, respectively. The flexural rigidity of the plate is given by
D(x)=Eτ3/12(1-v2), with E being the Young's Modulus of ice and v the Poisson's ratio. The
mass per unit length of the plate is denoted by m(x)=ρiτ.
After introducing the non-dimensional variables x̃=x/L,
η̃=η/L, t̃=tgL-1,
φ0̃=φ0g-1/2L-3/2,
φ1̃=φ1g-1/2L-3/2
and dropping tildes, the governing system of differential equations is
reduced to (see also Sturova, 2009),
Mη¨+Kηxxxx+η+φ0˙=0,x∈S0η˙+Bφ0xx=0,x∈S0,φ1¨-Bφ1xx=0,x∈S1,
where a superimposed dot denotes differentiation with respect to time while
an index following the a function denotes differentiation with respect to
the spatial variable. In addition, the coefficients appearing in the above
equations are defined as
K(x)=D(x)/ρwgL4,M(x)=m(x)/ρwL,B(x)=b(x)/L.
The bending moment and shear force along the shelf are given by the
expressions Mb=Kηxx and V= (Kηxx)x, respectively.
Stress distribution within the ice beam
In agreement with the Euler–Bernoulli beam model, the normal stress varies
linearly along the z direction. The maximum normal stress value at any
given cross section is
σxxmax=MbK|z|max=|z|maxηxx.
The shear stress distribution, as derived from equilibrium relations, varies
quadratically along the vertical direction. Maximum shear stress, located at
the neutral fibre is,
σxzmax=32Vτ/L=3L2Kxηxx+Kηxxxτ.
The above system of equations is supplemented with boundary, interface and
initial conditions. At the fixed end, simulating the ice shelf grounding
line, the deflection and slope are set to zero. At the free edge of the
plate, representing the ice shelf tip facing the ocean, zero bending moment
and shear force is imposed. These conditions read
η(0,t)=ηx(0,t)=Mb(1,t)=V(1,t)=0.
The water velocity is assumed zero underneath the grounding line and thus
the velocity potential gradient vanishes,
φ0x(0,t)=0.
At the interface between S0 and S1, assuming energy and mass
conservation, the following matching conditions are derived (Stoker, 1957;
Sturova, 2009):
B(1-)φ0x1-,t=B1+φ1x1+,tandφ0˙1-,t=φ1˙1+,t.
The ice shelf is assumed to be initially at rest, while an incoming long
wave transverses region S1 and reaches the free edge of the shelf. The
initial boundary value problem formulation is thus completed with the
following conditions,
η(x,0)=η˙(x,0)=φ0x(x,0)=0,x∈S0andφ1x(x,0)=0,φ1˙(x,0)=-F(x),x∈S1.
In the last of Eq. (9b), F(x) denotes the free surface elevation caused
by the Tsunami wave at an initial time, at an area distant to the ice shelf edge.
Finite elements – variational formulation of the governing equations
In order to derive the variational formulation of the above problem, Eqs. (1)–(3)
are multiplied by the weight functions ν, -w0 and w1, respectively.
Integration by parts yields
∫01Mνη¨dx+∫01Kνxxηxxdx+∫01νηdx+∫01νφ0˙dx+νKηxxx01+νxKηxx01=0,-∫01w0η˙dx-Bw0φ0x01+∫01Bw0xφ0xdx=0,∫1∞w1φ1¨dx-Bw1φ1x1∞+∫1∞Bw1xφ1xdx=0.
Using the conditions described in Eqs. (6)–(8), and adding Eqs. (10)–(12)
the equivalent semi-discrete variational problem is formulated as Papathanasiou et al. (2015).
Find η, φ0 and φ1, such that for every ν, w0
and w1 at any given moment in time it holds
∫01Mη¨hνhdx+∫01νhφ0˙hdx-∫01w0hη˙hdx+∫1∞w1hφ1¨hdx+aηh,νh+b0w0h,φ0h+b1w1h,φ1h=0,
where
aηh,νh=∫01Kνxxhηxxh+νhηhdx,b0w0h,φ0h=∫01Bw0xhφ0xhdx,
and
b1w0h,φ0h=∫1∞Bw1xhφ1xhdx,
while superscript h denotes spatially discrete quantities.
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,
η(xt) and φ0(x, t)
are approximated by ηh=∑i=16ηi(t)Hi(x),
φ0h=∑i=15φ0i(t)Li(x).
Domain S1 is discretized only in region [1, R], where
the positive constant R≫ 1 is selected large enough so that any
disturbance propagating inside S1 does not reach point R within the
time interval of interest. Fourth order Lagrange shape functions are used
for the interpolation of the velocity potential φ1. By substituting
the above approximate solutions into Eq. (13) and letting
the weight functions ν assume the form of the Hermite C1
shape functions while w0 and w1 are substituted by the Lagrange C0
shape functions, the resulting system is derived in the form,
Mu¨+Cu˙+Ku= 0,
with the vector of unknowns being u= [ηηxφ0φ1]T.
Results
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 x= 1 to 800 m at x= 2 (Brunt
et al., 2011). The initial, bell-shaped free-surface elevation considered in
the following examples is
η0=Aexp-x-μ(x0+w)2-x-μ(x-x0+w)(x-x0-w),
where A represents the amplitude, x0 the origin location, w the
half-wavelength and μ a smoothness parameter controlling the steepness
of the initial pulse. Finally, the material constants selected are as follows: ice
shelf density ρi= 922.5 kg m-3, water density ρw= 1025 kg m-3,
Young's modulus E= 5 × 109 Pa and Poisson's ratio
v= 0.3 (see also Sturova, 2009). The acceleration of gravity is
g= 10 m s-2. In the following analysis 400 hydroelastic elements have been
used, while the number of elements in region S1 is selected such that
the element size is the same for both regions S0 and S1. Numerical
experiments have shown that this discretization ensures convergence, as any
further refinement induces virtually no variation of the results. Finally,
the Newmark method has been employed for time integration. The
non-dimensional time interval T= 70 (corresponding to 7000 s) is considered.
At first, the effects of an initial pulse with A= 0.5 m,
μ= 50 × 105 m-2 and w= 8000 m are considered. In Fig. 1, a visual
representation of the given pulse propagation is plotted. The bell-shaped
disturbance is split into two waves travelling in opposite directions. The
pulse propagating to the left (towards the negative x axis) is partially
reflected when reaching the bathymetric variation at x= 2. As the wave
propagates over decreasing depth, its amplitude increases, while the
velocity decreases. The velocity reduction is evident in the curved
trajectory path for 1 ≤x≤ 2, as shown in Fig. 1. At x= 1, the wave
impacts the ice shelf free edge, initiating the propagation of the
hydroelastic wave while it is partially reflected. The hydroelastic wave,
featuring dispersive characteristics, is fully reflected at the grounding
line (fixed end, x= 0), at t≈ 67. The dispersive nature of the
hydroelastic pulse can also be seen in Figs. 2 and 3, manifested as the
formation of smaller disturbances preceding the main elevation wave. These
disturbances reach the grounding line at earlier times than the main pulse
and lead to an increase of the bending moment locally. This phenomenon is
displayed in the maximum and minimum bending moment time profiles (Figs. 2
and 3) as spikes, located at x= 0 and appearing in the time interval from
t≈ 55 to t≈ 65.
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 L= 100 km.
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, t≈ 30 the maximum
bending moment is seen to increase. The location of the maximum bending
moment value is found to follow the main pulse towards the fixed end. At
t= 34, the entire pulse has passed underneath the floating cantilever,
causing an increase in maximum bending moment. At the same time, the
location of the maximum value for the bending moment is shifted back near
the free edge. This is due to the fact that the entire wavelength of the
initial pulse has passed underneath the floating cantilever, causing the tip
to bend again as it recovers to the initial undeformed state. At t= 44
the location of the maximum bending moment value is shifted towards the
ice shelf tip once again. The above can be attributed to flexural effects
taking place at the right side of the propagating disturbance. As the
hydroelastic wave propagates away from the free edge, the tip is restored to
its original position causing additional flexing in the interior of the
cantilever. Due to the fact that, in the present work, the grounding line is
simplistically modelled as a fixed boundary, the global bending moment
extrema are found at the fixed edge, at the time of reflection t= 67. Prior
to full reflection, a series of spikes in the maximum bending moment
distribution are caused by the dispersed hydroelastic waves reaching the
fixed edge before the main pulse.
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 ξ measures the distance of the point of occurrence of the
extreme value from the free edge.
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 ξ denotes the
distance from the free edge up to the location of the extreme value along
the semi-fixed floating body x= 1. In all cases, the extreme bending moment
values have been considered in a time interval excluding the effects of the
fixed end forcing (t≥ 50). In that manner, Fig. 4 demonstrates the
location of extreme bending moments for the phase during which the main
pulse enters the region of hydroelastic interaction. As can be seen in the
aforementioned figure, the location of the extreme bending moment is
relatively insensitive to variations of the wavelength. For thickness values
of 80 and 100 m, this location is found to be at about 2 % of the ice
shelf length (2 km into the 100 km long ice shelf), calculated from the free
edge. The above results are found in agreement with the work of Squire (1993),
where the breakup of shore fast ice, modelled as a semi-infinite,
thin floating plate, is investigated in the frequency domain. Furthermore,
as thickness increases, the location of extreme values seems to shift
towards the interior of the ice shelf. For a thickness of 120 m, location ξ
is placed at approximately 10 % of the ice shelf length and
features a slight variation with increasing initial disturbance wavelength.
However, this variation is very small when compared to the total length of
the beam. Another interesting feature is that in this last case (120 m
thickness) the maximum absolute value found corresponds to negative values
of the bending moment (see Fig. 5), whereas for thickness values of 80 and
100 m the maximum absolute bending moment values are found to be positive
(sagging moments). This feature explains the different shape of the 120 m
curve in Fig. 4, when compared to the curves corresponding to 80 and 100 m,
which closely resemble one another. The fact that in the case of 120 m the
extreme bending moment values are negative might be attributed to the beam
thickness being very large compared to the water depth under the ice shelf.
Finally, these results are strongly dependent on the form of the incoming
wave, in the sense that if another wave profile instead of an elevation
pulse is chosen, the bending moment fields will be of a different nature.
Plot of maximum and minimum bending moment value distributions for
w= 8000 m and τ/L= 0.0012. Extreme bending moment value is
negative, during the entry phase, for an ice shelf thickness value of 120 m.
Conclusions
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.
Acknowledgements
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
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