the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Tidal bending of ice shelves as a mechanism for large-scale temporal variations in ice flow

### Sebastian H. R. Rosier

### G. Hilmar Gudmundsson

GPS measurements reveal strong modulation of horizontal
ice shelf and ice stream flow at a variety of tidal frequencies, most notably
a fortnightly (*M*_{sf}) frequency not present in the vertical tides
themselves. Current theories largely fail to explain the strength and
prevalence of this signal over floating ice shelves. We show how well-known
non-linear aspects of ice rheology can give rise to widespread, long-periodic
tidal modulation in ice shelf flow, generated within ice shelves themselves
through tidal flexure acting at diurnal and semidiurnal frequencies. Using
full-Stokes viscoelastic modelling, we show that inclusion of tidal bending
within the model accounts for much of the observed tidal modulation of
ice shelf flow. Furthermore, our model shows that, in the absence of vertical
tidal forcing, the mean flow of the ice shelf is reduced by almost 30 % for
the geometry that we consider.

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Ocean tides are known to greatly affect the horizontal flow of both ice
shelves and adjoining ice streams, even far upstream of grounding lines (GLs)
(Alley, 1997; Anandakrishnan et al., 2003; Bindschadler et al., 2003a, b; Brunt et al., 2010; Doake et al., 2002; Gudmundsson, 2006; King et al., 2011; Legresy et al., 2004; Makinson et al., 2012; Marsh et al., 2013; Minchew et al., 2016; Rosier et al., 2017a). In some cases the horizontal ice flow
responds at a different frequency to the tidal forcing; for example on the
Rutford Ice Stream (RIS) the primary response is at a fortnightly
(*M*_{sf}) frequency that is not measurable in the vertical tidal
motion (Gudmundsson, 2006). More recent observations have shown that the
*M*_{sf} signal actually increases in strength on the adjoining ice
shelf (Minchew et al., 2016; Rosier et al., 2017a) and also exists on isolated ice shelves
which do not have large ice streams feeding into them
(Gudmundsson et al., 2017; King et al., 2011).

A multitude of mechanisms have been proposed which could lead to a fortnightly modulation in ice flow: a non-linear basal sliding law (Gudmundsson, 2007, 2011; Rosier et al., 2014), tidal perturbations in subglacial water pressure (Rosier et al., 2015; Thompson et al., 2014), grounding line migration (Robel et al., 2017; Rosier et al., 2014) and changes in the effective ice shelf width (Minchew et al., 2016). Identifying the mechanism whereby ocean tides generate the observed tidal modulation in ice flow is important for several reasons. The amplitude of these perturbations is often a significant fraction of mean flow speed, and the perturbations are widespread, impacting ice flow on a large number of ice streams and several ice shelves. Not knowing the root cause of these tidal modulations therefore implies a significant lack in our understanding of the forces controlling the large-scale ice flow of the Antarctic Ice Sheet. Furthermore, there are good reasons to believe that the tidal response is significantly affected by the rheology of ice or mechanical conditions at the base of ice streams, or possibly both in combination. Hence, once the mechanism has been fully identified, one can expect to be able to make inferences about ice rheology and/or basal conditions from observations of tidal modulations in ice flow. The Filchner–Ronne Ice Shelf (FRIS) is a particularly good natural laboratory for obtaining these insights because of the considerable tidal range, which can be as large as 9 m (Padman et al., 2002).

Previous modelling studies have focused almost exclusively on tidal
modulation of ice stream flow
(Gudmundsson, 2007, 2011; Rosier and Gudmundsson, 2016; Rosier et al., 2014, 2015; Sergienko et al., 2009; Thompson et al., 2014; Walker et al., 2012, 2016),
whereas tidal modulation of the flow of ice shelves has received much less
attention. This is possibly because it has often been assumed that the
*M*_{sf} signal observed on ice shelves is driven by processes
occurring on neighbouring ice streams; indeed these make up the bulk of the
proposed mechanisms listed above. Now that new observations show the
*M*_{sf} signal strengthening downstream of GLs
(Minchew et al., 2016; Rosier et al., 2017a), it has become clear that an alternative
mechanism is needed which can generate this signal, independent of anything
occurring on grounded ice (Minchew et al., 2016; Robel et al., 2017; Rosier et al., 2017a).

Here, we will show how the observed widespread tidal modulation in ice flow can be generated within ice shelves themselves through tidal flexure. We begin with a description of this simple mechanism, which results directly from the well-known non-linear aspect of the flow law of glacier ice and hence does not require an ice stream to act as a source of the observed tidal signals. Then in Sect. 3, using elastic beam theory, we derive a simple mathematical description of this mechanism that yields some insights into its importance for various ice shelf configurations. Finally in Sect. 6, we present results from a 3-D full-Stokes viscoelastic model of a confined ice shelf, with a similar geometry to the RIS, that incorporates the new mechanism and is capable of replicating many of the observed characteristics of the tidal response of the Ronne Ice Shelf. These results will show that this mechanism has important implications for both the time-varying and mean flow of ice shelves subjected to strong vertical ocean tides.

The Filchner–Ronne, Larsen and to a lesser extent Ross ice shelves are situated in tidally energetic regions and thereby subjected to large vertical motion at tidal frequencies. By far the largest tidal amplitudes are in the Weddell Sea region, particularly at the grounding line of large ice streams such as Rutford and Evans (Padman et al., 2002). In the grounding zone (here defined as a band along the grounding lines that extends several kilometres into the main shelf) the ice bends to accommodate these large vertical tidal motions. This bending generates longitudinal and shear stresses within the ice which contribute to the effective stress and are strongest near the grounding line during high and low tide. Since ice is a non-Newtonian shear thinning fluid, its effective viscosity will be altered by these tidal stresses. A schematic showing how vertical tidal motion can lead to a reduction in effective viscosity of ice shelf shear margins is shown in Fig. 1. This effect, which we will call “flexural ice softening”, leads to an increase in ice velocity during high and low tide. We will show that this is a direct consequence of the non-linearity of Glen's flow law.

Since it is the magnitude of stresses and not their sign that contributes to
the effective viscosity, there is no difference in the flexural ice-softening
effect between high and low tide. The only time that the effective viscosity
of an ice shelf subjected to large tides will increase to that of an ice
shelf without tides is when the vertical deflection is small, i.e. between
high and low tide or during neap tides. As a consequence there are two other
important repercussions for the ice shelf flow that arise from this
mechanism, aside from the direct increase in velocity at high and low tide.
Firstly, the mean flow of an ice shelf is greater in the presence of large
tides because, even at its slowest, it will be flowing at least as fast as an
ice shelf without tides. Secondly, because the change in velocity (due to
flexural ice softening) during spring tide is larger than during neap tide,
the ice shelf flow will be modulated at an *M*_{sf} period (provided
the rheology is non-linear, as is the case for glacier ice). Since many large
ice shelves are confined on three sides by grounded ice, the bending stresses
are generated along their entire length. This mechanism could therefore
explain how the *M*_{sf} signal increases in strength downstream of
ice stream grounding lines, as evidenced by recent GPS and satellite
observations (Minchew et al., 2016; Rosier et al., 2017a).

Elastic beam theory provides a useful starting point for evaluating the
magnitude of these tidal bending stresses on an ice shelf and their impact on
its effective viscosity. We start from a simple confined ice shelf whose
geometry is invariant across flow (in the *y* direction) and with a constant
ice thickness gradient in the down-flow *x* direction. The ice shelf is
symmetrical about the centreline, which is distance *W* from the two
sidewalls at *y*=0 and *y*=2*W* (Fig. 1). For this
analytical solution we assume that the portion of the ice shelf that we
investigate is sufficiently far from the GL such that the only bending occurs
across-flow. The situation near the main GL of a narrow confined shelf will
be a complex combination of along- and across-flow stresses that we shall
ignore for now. Deviatoric stresses are defined as

where *σ*_{ij} are the components of the Cauchy stress tensor,
*δ*_{ij} is the Kronecker delta and $p=-{\mathit{\sigma}}_{kk}/\mathrm{3}$ is the
isotropic pressure. We use the comma to denote partial derivatives and the
summation convention, in line with standard tensor notation.

We immediately make the simplifying assumptions (motivated by full-Stokes
calculations presented below) that ${\mathit{\tau}}_{xx}={\mathit{\tau}}_{xz}=\mathrm{0}$; hence ${\mathit{\tau}}_{yy}=-{\mathit{\tau}}_{zz}$, ${\mathit{\sigma}}_{zz}=-p-{\mathit{\tau}}_{yy}$ and ${\mathit{\sigma}}_{xx}=-p$.
Furthermore, we assume that the only important contributions to *τ*_{yy}
and *τ*_{yz} are due to tidal bending. The force balance equations in *x*
and *z* reduce to the following form:

Note that in this system *σ*_{zz} is not cryostatic, unlike in the
shallow-shelf and shallow-ice approximations. We are interested in finding an
expression for the across-flow variation in downstream velocity, *u*(*y*), for
which we need an expression for *τ*_{xy}. As we show in
Appendix A, *τ*_{xy} is essentially independent of the tidal
stresses (as well as *x* and *z*) and can be approximated by

where ${F}_{d}=\mathit{\rho}g{\partial}_{x}s$.

Linear elastic beam theory gives us an expression for the elastic stresses that will arise due to tidal bending (Robin, 1958). Although strictly derived for an infinitely long ice shelf (or, in the orientation of bending that we consider, infinitely wide), we show in Appendix B that the equations in Robin (1958) provide a good approximation for the geometry that we are interested in. The two contributing stresses, related to the bending moment and its derivative, are the across-flow longitudinal bending stress,

and the across-flow shear bending stress,

where

*w*_{a} is the vertical tidal motion, *E* is Young's modulus of ice, *ν*
is Poisson's ratio, *h* is local ice thickness and *ρ*_{w} is
the density of seawater. The vertical coordinate, *z*, is defined as the
vertical distance above the neutral axis of the ice shelf, which we assume to
be halfway through its thickness.

At this stage we employ a Maxwell rheological model consisting of a linear elastic spring and a non-linear viscous dashpot, whose behaviour is modelled by Glen's law (Glen, 1955), connected in series. With this viscoelastic model the total strain is the sum of the viscous and elastic strains, and the stress is equal in the two components. In this way, we can express the horizontal shear strain rate as

where

and, based on the assumptions given above,

Motivated both by our findings in the Appendix that ${\dot{\mathit{\tau}}}_{xy}\approx \mathrm{0}$ and by the fact that this elastic term can only ever yield a linear
response to the tidal forcing, we discard it and focus only on the non-linear
viscous response. We are concentrating on the non-linear response because only
this can explain modulation of horizontal ice shelf flow at an
*M*_{sf} frequency, given that the *M*_{sf} constituent is
absent in the vertical tidal forcing.

By assuming that *n*=3, we can separate the velocity into unperturbed and
time-varying components. Integrating with respect to *z* and *y* then gives
the depth-averaged velocity $\stackrel{\mathrm{\u203e}}{u}$ as

where s is the surface elevation, b is the bed elevation and
$\stackrel{\mathrm{\u203e}}{{\mathit{\tau}}_{xy}}$ is the depth-averaged shear stress. We have split this
into the three components, denoted as the unperturbed (*u*_{0}), long(itudinal)
bending stress and shear bending stress contributions to ice flow. Evaluating
the integrals for each term and neglecting the overbar since everything is
now depth-averaged yields

where $\mathit{\xi}=\mathit{\lambda}W-\frac{\mathit{\gamma}}{\mathrm{2}}$ and *γ*=2*λ**y*;

and

The shear and across-flow longitudinal components can be combined, such that
the total (time-varying) velocity $u={u}_{\mathrm{0}}+\mathrm{\Delta}u$. Along the
centreline at *y*=*W*, the change in velocity due to tides (Δ*u*) is

where

To illustrate the consequences of a typical tidal action for the ice shelf
flow, we assume that the time-varying sea level *w*_{a}(*t*) can be written as
the sum of two cosines of amplitude ${a}_{{M}_{\mathrm{2}}}$ and ${a}_{{S}_{\mathrm{2}}}$ and angular
frequency ${\mathit{\omega}}_{{M}_{\mathrm{2}}}$ and ${\mathit{\omega}}_{{S}_{\mathrm{2}}}$, i.e.

These two cosines represent the principal lunar (*M*_{2}) and solar (*S*_{2})
semidiurnal tides, which dominate in the area of interest. Crucially, because
the velocity is a function of tidal deflection squared, new frequencies
emerge which, if we assume it takes the form of Eq. (16), expands
as follows:

where ${\mathit{\omega}}_{{M}_{\mathrm{sf}}}={\mathit{\omega}}_{{S}_{\mathrm{2}}}-{\mathit{\omega}}_{{M}_{\mathrm{2}}}$ and
${\mathit{\omega}}_{{M}_{S\mathrm{4}}}={\mathit{\omega}}_{{M}_{\mathrm{2}}}+{\mathit{\omega}}_{{S}_{\mathrm{2}}}$. The four emergent frequencies
that we expect to see are labelled according to their respective tidal
constituent names. Depending on the relative size of the *M*_{2} and *S*_{2}
vertical tidal forcing, different frequencies will dominate in the horizontal
ice flow response. In the case of the Filchner–Ronne ice streams, the
amplitude of the *S*_{2} constituent is typically about half that of the *M*_{2}
constituent. As a result, the *S*_{4} frequency will be much smaller than the
other three. In terms of velocities, the amplitudes of the *M*_{sf}
and *M**S*_{4} components will be equal, and larger than the *M*_{4} component
as long as ${a}_{{S}_{\mathrm{2}}}>{a}_{{M}_{\mathrm{2}}}/\mathrm{2}$.

Several useful results are now easily obtained with Eqs. (17) and
(14); for example the amplitude of the *M*_{sf} component in
ice shelf velocity is simply $\left(B{a}_{{M}_{\mathrm{2}}}{a}_{{S}_{\mathrm{2}}}\right)/\mathrm{2}$. Integrating with time
gives an expression for displacements, which are more readily measured with
in situ GPS. Once again, the amplitude of the *M*_{sf} component in
displacements in this case becomes
$\left(B{a}_{{M}_{\mathrm{2}}}{a}_{{S}_{\mathrm{2}}}\right)/\mathrm{2}({\mathit{\omega}}_{{S}_{\mathrm{2}}}-{\mathit{\omega}}_{{M}_{\mathrm{2}}})$. Even more interesting is
the result of the first term of Eq. (17), which acts to increase
the time-averaged ice shelf velocity (*u*_{mean}). The size of this
effect, which we call the *n*_{shift}, is given by

such that ${u}_{\text{mean}}={u}_{\mathrm{0}}+{n}_{\text{shift}}$. Interestingly, within
this framework all tidal energy at the original (vertical) semidiurnal
forcing frequencies disappears (as can be seen by squaring the tidal forcing,
Eqs. 16–17). In reality linear elastic effects and
changes in damming stresses would be expected to produce some response at
these frequencies, and these terms are included in the 3-D model described in
Sect. 4. Note that from Eq. (10) onwards these
results have been derived under the assumption that *n*=3. For *n*=1 bending
stresses have no impact on the ice shelf viscosity, and so the *M*_{sf}
flow modulation and *n*_{shift} would be identically equal to zero.

Using the simple set of equations outlined above, we can easily explore the
parameter space to see how the strength of the tidal response changes. Of
particular interest is how the *n*_{shift} leads to an increase in the
mean speed of the ice shelf. In Fig. 2 we show speed-up along
the ice shelf medial line (solid black contour) as a percent of the baseline
speed with no tides, i.e. *u*_{mean}∕*u*_{0} (the parameters chosen are
shown in Table 1). This shows that, for a given tidal amplitude, the
*n*_{shift} effect will be most strongly felt on a narrow, thin ice
shelf. Conversely, the amplitude of the *M*_{sf} signal in ice shelf
displacements (dashed contour) is strongest for wide, thick ice shelves. The
apparent discrepancy is because, with all other parameters held constant, a
wider ice shelf will flow much faster, and so the increase in speed as a
percent of the baseline is much less.

Note that we use a different value of *E* to obtain bending stresses for
analytical solution than in our full-Stokes model. Using the instantaneous
Young's modulus of 9 GPa (suggested by laboratory experiments) would result
in bending stresses that are too large. This is because ice behaves
viscoelastically at tidal frequencies and *E* is frequency-dependent. This
behaviour is captured by our full-Stokes model, but, since the much simpler
elastic beam model does not include this complexity, instead we treat
this value as a tuning parameter and pick a value of *E* that best matches
our modelled bending stresses, which turns out to be 800 kPa.

In order to explore the idea of flexural ice softening in more detail, we
undertook modelling experiments on an idealised ice stream/shelf domain using
the commercial finite-element software MSC.Marc, which has been used
extensively in the past to explore the tidal response of ice streams
(Gudmundsson, 2011; Rosier and Gudmundsson, 2016; Rosier et al., 2014, 2015). The idealised ice
stream is 28 km wide (to match the approximate average width of the RIS) and
consists of a 150 km floating shelf and 80 km grounded ice
(Fig. 3). Although data now exist showing tidal modulation on
other ice streams, the RIS lends itself well to an idealised study of this
kind because of its relatively simple geometry and because its flow has
remained largely unchanged over the measurement period
(Gudmundsson and Jenkins, 2009). Surface and bed slopes of the ice stream and
ice shelf portions of the model are approximate averages of the slopes found
on RIS, and ice thickness at the downstream limit of the domain is 1420 m.
The model is run forward in time for 60 days in order to resolve the
*M*_{sf} signal. The grounding line position is fixed and cannot
migrate at tidal frequencies, since our focus is only on the effects of tidal
bending stresses. We investigate several test cases (Sect. 5), some
of which require a slightly different model set-up, which we describe in the
relevant sections.

## 4.1 Field equations

The full-Stokes solver MSC.Marc uses the finite-element method in a Lagrangian frame of reference to solve the field equations

representing conservation of mass, linear momentum and angular momentum,
respectively. In the above equations, D∕D*t* is the material time
derivative, *v*_{i} are the components of velocity, *σ*_{ij} are the
components of the stress tensor, *ρ* is the ice density and *f*_{i} are the
components of the gravity force.

We use a non-linear Maxwell viscoelastic rheology in a slightly modified form to Eq. (7), which can be written as

where the full stress tensor contributes to the effective stress, i.e.

and the superscript *▿* denotes the upper-convected time derivative:

(Christensen, 1982). We use the same rheological parameters as in
Gudmundsson (2011), which are found to replicate the behaviour of the
more complex Burgers model at tidal frequencies, i.e. *E*=4.8 GPa and
*ν*=0.41, where $E=\mathrm{2}G(\mathrm{1}+\mathit{\nu})$ (Shames and Cozzarelli, 1997).

## 4.2 Boundary conditions

At the downstream limit of the domain we prescribe the ice shelf stresses

and

where *p*_{b} is a buttressing term. A value of 250 kPa was chosen
for *p*_{b}, in order to reproduce ice shelf velocities similar to
those observed at the outlet of the RIS. At the upstream boundary we apply
the cryostatic pressure ${\mathit{\sigma}}_{xx}=\mathit{\rho}g(s-z)$. At the ice surface, a
stress-free boundary condition (BC) of the form *σ*_{ij}*n*_{j}=0 is used, where
*n*_{j} is the outward unit vector normal to the surface.

The ocean pressure normal to the ice–ocean interface (*p*_{w}) is
applied as an elastic foundation (see Gudmundsson, 2011 for details).
This is exactly equivalent to a normal stress of

where *z* is the depth below sea level and *w*_{a}(*t*) is the time-varying vertical tidal motion (Sect. 5.1).

Upstream of the grounding line, along the ice–bed interface (green and orange shaded regions in Fig. 3), we use a Weertman style sliding law of the form

where *c* is basal slipperiness, *τ*_{b} is the along-bed
tangential component of the basal traction and *m* is a stress exponent. In
all of our experiments we use a non-linear sliding law with *m*=3. Similarly,
slipperiness values beneath the ice stream are kept fixed in all experiments
to a value that approximately matches the mean flow velocity of the RIS.
Beneath the margin, slipperiness is made several orders of magnitude smaller
to restrict ice flow in this portion of the model.

We treat one side of the model ice stream as the medial line, since the problem is symmetrical (${\partial}_{y}h=\mathrm{0}$), meaning we only need to model half of the ice stream with no lateral flow as the appropriate BC. The other side is treated as a grounded sidewall with no slip, such that $u=v=w=\mathrm{0}$ (referred to hereafter as the clamped BC). In one of the experiments (n3xy) the constraint on vertical velocity is removed, as explained in Sect. 5.

## 4.3 Discretisation

The model uses 20-node isoparametric hexahedral (brick) elements with a
27-point Gaussian integration scheme. These quadratic elements allow accurate
representation of stresses and strains with far fewer numbers of elements
than would otherwise be needed when using linear elements. Element size
varies from a maximum horizontal dimension of ∼2 km to a minimum of
∼300 m around the grounding line and in the shear margins. The
finite-element mesh is unstructured, with a GL that curves to avoid an unnatural
grounding zone corner. The ice is three elements thick vertically, resulting in
nine integration points through its depth. The model mesh is shown in
Fig. 3. The n3xyz simulation (Sect. 5) was repeated
with double the horizontal resolution to check if this affected results.
*M*_{sf} amplitude changed by a maximum of 3 %, and ice velocity by
a maximum of 2.5 %, and so the default resolution was deemed sufficient.

We conduct three simple model experiments to investigate the effects of flexural ice softening within our model. Model runs are named such that n1 or n3 denotes whether we use a linear or non-linear ice rheology and xy or xyz signifies which degrees of freedom are clamped on the sidewall boundary.

- n3xyz
In the first experiment we run the model with non-linear ice rheology and sidewalls clamped in

*x*,*y*and*z*. This is designed to simulate the “Rutford” case whereby the margins are essentially stagnant and flexure occurs all along the GL, both where the main body of the ice stream meets the ocean and downstream of this point along the sides. In order to approximately match the observed 1 m d^{−1}flow velocities of the floating portion of RIS, we adjust the ice rate factor (*A*) uniformly. - n3xy
For the second experiment we run the model as in n3xyz, but the sidewalls downstream of the GL are not clamped vertically (

*z*direction). With this set-up there is no bending along the sidewalls downstream of the GL, so flexural stresses are only generated in the grounding zone around*x*=0. This experiment is akin to a fast-flowing ice shelf bounded by stagnant floating ice, as can be found on the floating portion of some fast-flowing outlet glaciers. - n1xyz
The third experiment uses the same set-up and boundary conditions as in n3xyz except that ice rheology is made linear, such that

*n*=1 in Eq. (22). This experiment is done to demonstrate the difference in response due only to changing*n*from one to three. In this experiment therefore, the ice viscosity is not stress-dependent, such that the bending stresses do not cause a reduction in the effective viscosity of ice. As such, it is not a “realistic” situation (since ice is known to have a non-linear rheology) but serves to emphasise that this non-linearity is the important one at play in our model. In order to produce sensible ice shelf velocities, the rate factor*A*is adjusted uniformly so that the background flow speed (denoted*u*_{mean}in the previous analysis) is approximately the same as the other experiments.

## 5.1 Tidal forcing

The time-varying vertical tidal forcing is implemented as a stress acting
normal to the ice shelf base (Eq. 27). For all the
experiments described above the model is forced with the principal
semidiurnal (*M*_{2}, *S*_{2}) and diurnal (*O*_{1}, *K*_{1}) tidal
constituents, i.e. the four tidal constituents which are generally largest
beneath the Ronne Ice Shelf. Their amplitudes are derived from GPS
measurements of vertical ice shelf motion 20 km downstream from RIS GL
(Gudmundsson, 2006). The tidal forcing is kept intentionally simple to
avoid complicating any interpretation of our full-Stokes model results.

We now present results from our viscoelastic 3-D full-Stokes model of an
idealised ice stream/shelf system. We begin by examining the modelled
response at *M*_{sf} frequency, since previous models do not reproduce
observations of this non-linear effect on floating ice shelves.
*M*_{sf} amplitude in horizontal surface ice displacements is shown in
plan view for the three experiments in Fig. 4. For the
n3xyz experiment, which can be thought of as the typical situation for a
confined ice shelf subjected to large vertical tides, *M*_{sf}
amplitude increases continuously downstream of the GL
(Fig. 4a). In the across flow (*y*) direction the amplitude
increases towards the medial line. Also shown are contours of ice shelf
velocity (*u*), which increase from 1 m d^{−1} upstream of the GL to more
than 3 m d^{−1} on the shelf.

In the n3xy experiment the only change with respect to the n3xyz
experiment is to remove the vertical clamp BC acting along the sidewall of
the floating portion of the model. With this change in sidewall BC the
*M*_{sf} amplitude is similar at the *x*=0 GL where bending stresses
are still generated. Downstream of this region however the *M*_{sf}
amplitude decays rapidly to zero with distance (Fig. 4b),
whereas in the n3xyz experiment the amplitude continues to increase with
distance. Ice velocities on the floating shelf are lower than in the n3xyz
experiment, and across-flow shear is less pronounced, such that the ice
velocity contours are further apart.

For the n1xyz experiment, (Fig. 4c), where the only change
compared to the n3xyz experiment is to change the value of *n* from one to
three, the *M*_{sf} response is even more localised to the GL region
and the amplitude is close to zero.

Other tidal frequencies in the n3xyz experiment that emerge from the
frequency doubling (Eq. 17), such as *M**S*_{4} , show very
similar spatial patterns to the *M*_{sf} responses shown in
Fig. 4a. In the n1xyz experiment, these frequencies are
completely absent.

Running the standard n3xyz experiment with and without tides reveals how the mean ice shelf flow is affected by tidal bending stresses. Averaging over the entire floating portion of the shelf, mean velocity is increased by ∼ 35 % when the experiment is run with a vertical tidal forcing equivalent to that experienced near the RIS GL, as opposed to no tidal forcing.

To explore the role of flexural stresses in more detail, we plot across-flow
profiles for each component of the deviatoric stress tensor
(Fig. 5). Stresses are taken from the n3xyz experiment at
*x*=100 km, to avoid the 2-D bending stresses at *x*=0, and for a positive
vertical tidal deflection of 2 m. The stress is normalised by the
depth-averaged horizontal shear stress at the margin *ρ**g**W*∂_{x}*s*, as
predicted by the analysis in Sect. 3 (for the ice shelf
surface slope in the model of $\mathrm{5.4}\times {\mathrm{10}}^{-\mathrm{4}}$ the stress scale is
67.5 kPa). Distance from the margin is normalised by the ice shelf
half-width (*W*=14 km). Surface and bed across-flow bending stresses
(*τ*_{yy}) are equal in amplitude but opposite in sign, and so all the
stresses are plotted as the depth averages of their absolute values. This is
more relevant for our purposes, since it is the absolute amplitudes of these
stresses, and not their signs, that impact the effective stress.

Our numerical results show that the contributions of across-flow and shear
bending stresses to the effective stress, and therefore their relative
impacts on effective ice viscosity, change significantly with increasing
distance away from the ice shelf margins. At the margins, both across-flow
and shear bending stresses contribute about equally to the total effective
stress. With increasing distance away from the margins, both bending stress
terms behave as damped cosine waves (Eqs. 4 and 5);
however the resulting “waveforms” are phase-shifted with respect to one
another. This can be seen in Fig. 5, where *τ*_{yy} shows a
clear minimum at a distance of $y/W\approx \mathrm{0.2}$ before increasing again,
whereas the minimum for *τ*_{yz} is discernible at $y/W\approx \mathrm{0.4}$. As a
consequence of this damped behaviour, bending stresses are largest near the
grounding line but, for this geometry, have very little impact on effective
viscosity along the ice shelf medial line, where they have decayed to almost
zero (the fact that the *τ*_{yy} term is relatively large at the medial line
is a result of ice shelf spreading, not bending in the grounding zone). Note
that, since *λ* is a function of ice thickness, the location of the
bending stress minima will shift as the thickness changes.

At this stage we can briefly evaluate the validity of the assumptions made in
Sect. 3. The expression for the across-flow variation in
*τ*_{xy}, given by Eq. (3), varies from the value
calculated by our full-Stokes model by a maximum of 5 %. The assumption
that ${\mathit{\tau}}_{yy}\approx -{\mathit{\tau}}_{zz}$ holds near the margin – as shown in
Fig. 5, where the modelled absolute values of these two
stresses are approximately equal – but begins to break down at a distance of
*W*∕2, where the *τ*_{xx} becomes increasingly large due to ice shelf
spreading. Finally, the vertical shear stress (*τ*_{xz}) is approximately
zero everywhere apart from within one ice thickness of the GL, where the
effects of neighbouring ice shearing vertically in the grounded margin are
felt. Nevertheless, even in this region *τ*_{xz} contributes less than
2 % of the total effective stress.

Figure 6 shows the phasing of velocity, effective stress
and strain heating rates in the model shear margin relative to vertical tidal
motion (vertical motion is taken along the medial line to show the undamped
tidal amplitude). Strain heating rate is calculated as ${\dot{e}}_{E}{\mathit{\tau}}_{E}/\mathit{\rho}{C}_{p}$, using a specific heat capacity of 1955.4 J K^{−1} (equivalent to an
ice temperature of −20 ^{∘}C; Cuffey and Paterson, 2010). This shows that
modelled ice velocity, effective stress and strain heating are greatest just
before high and low tide, as would be expected from a viscoelastic rheology.
Effective stress in the shear margin is increased by over 50 % during the
highest tides of the spring cycle. Strain heating rate in the shear margin is
enhanced by vertical tidal motion, and so this mechanism could enhance the
shear heating effect which has been invoked to explain the inferred softness
of Ronne Ice Shelf shear margins (Larour et al., 2005).

The analysis of Sect. 3, together with full-Stokes
viscoelastic modelling, suggests that flexural ice softening could play
an important role in the generation of the *M*_{sf} signal that is
readily observed across the entire Ronne Ice Shelf (Rosier et al., 2017a).
Flexural stresses due to vertical tidal motion can generate a fortnightly
modulation in ice flow along any GL based only on the fact that ice is
non-Newtonian. This mechanism is felt most strongly for a confined ice shelf,
where bending occurs in the margins along the entire length of the shelf. New
observations reveal that the *M*_{sf} signal is generally larger on
ice shelves than on the adjoining ice streams and tends to increase in
amplitude in the downstream direction towards the ice front
(Minchew et al., 2016; Rosier et al., 2017a). Furthermore, the *M*_{sf} signal has
now been observed to lead in phase on the ice shelf, casting some doubt on
previous mechanisms that acted only on grounded ice (Minchew et al., 2016). Our
modelling work shows that flexural ice softening can replicate this phasing
and amplification of the *M*_{sf} signal downstream of ice stream GLs.
Furthermore, these tidal bending stresses will lead to a net speed-up of the
ice shelf.

Two alternative mechanisms have been proposed to explain the *M*_{sf}
amplification on ice shelves, both reliant on GL migration.
Minchew et al. (2016) argues that, if the sidewall GL migrates over a tidal
cycle, this will lead to a change in the effective width of the ice shelf as
proportionally more of it ungrounds. Observed changes in the distance between
the two maxima of lateral shear strain rate between high and low tide are
interpreted as being caused by grounding line migration (Minchew et al., 2016).
An alternative explanation is that flexural ice softening in the shear
margins leads to a steepening of the across-flow velocity profile at the
boundary, thereby shifting the apparent margin as defined above. Calculating
lateral shear strain rate 100 km downstream of the n3xyz simulation shows
that each peak can shift by ∼ 500 m over a tidal cycle, leading to an
apparent widening of 1 km even though there is no grounding line migration
in the model. Alternative evidence of GL migration does exist in other parts
of the FRIS (Brunt et al., 2011), and this mechanism could be locally important;
however, it seems unlikely that it could explain the pervasiveness of the
*M*_{sf} signal across the entire shelf, since it is so reliant on
local bedrock topography.

A previous modelling study has shown that GL migration is itself a strong
non-linearity which can generate an *M*_{sf} response in ice flow
(Rosier et al., 2014). Robel et al. (2017) explored this idea in more detail and
suggested that temporal variability in the area in which an ice shelf contacts
the bed (due to GL and pinning point migration) is the dominant non-linearity on RIS
leading to the observed *M*_{sf} response. Within their framework,
flexural stresses are ignored and the tidally varying ice shelf strain is a
function of competing hydrostatic and buttressing stresses. The
Robel et al. (2017) model was flexible enough to allow for many of the observed
aspects of the tidal modulation to be replicated. However, in the absence of
a physically motivated model of GL migration, knowledge of the sub-shelf
bathymetry or even strong evidence for GL migration in the area,
the extent to which this mechanism plays an important role remains an open question.

The flexural ice-softening mechanism produces a frequency doubling in the
response of the ice shelf, since the marginal ice will be softest just
preceding high and low tide. This is evident in the analysis of
Sect. 3, which reveals that ice shelf velocity modulation
will be dominantly at *M*_{4} and *M**S*_{4} frequencies in contrast to the
*M*_{sf} frequency, which dominates the displacements. In order to
check that our 3-D viscoelastic model reproduces this behaviour, we performed
a tidal analysis on modelled displacement and velocity at the ice stream
medial line, 100 km downstream from the GL. Figure 7
shows the results of this tidal analysis as a frequency power spectrum,
showing only constituents with a high signal-to-noise ratio. Surface
horizontal displacements show a dominantly *M*_{sf} response, with
almost no clear response at other frequencies (Fig. 7a).
In the horizontal ice velocity (Fig. 7b) the *M*_{4} and
*M**S*_{4} frequencies emerge, with similar amplitudes to the *M*_{sf}
in agreement with Eq. (17). Other non-linear frequencies such as
*M*_{f} , arising from interaction of the two diurnal tidal
constituents, should be present but are not resolvable with a simulation time
of 60 days.

As stated above, alternative mechanisms for generating an *M*_{sf}
signal on floating ice assume that GL migration is the dominant process. Ice
shelf velocities from the viscoelastic model proposed by Robel et al. (2017)
(using the parameters selected to match observations on RIS) are dominated by
*M*_{2} and *S*_{2} frequencies. Since the mechanism is non-linear, higher
frequencies such as *M*_{4} and *M**S*_{4} are also generated, but in that model
they are of a lower amplitude than the semidiurnal frequencies. In order to
determine which mechanism is most likely responsible for observations on the
RIS, therefore, we can look at whether short-term ice shelf velocity
modulation is dominantly *M*_{4} and *M**S*_{4} or *M*_{2} and *S*_{2}.

Most of our observations of the short-term velocity fluctuations on floating
ice come from GPS units. Tidal analysis of these records is typically done on
their measured displacements, rather than the much noisier velocities
calculated from the time derivative of their measured position. By first
fitting a tidal model to GPS measurements of horizontal ice flow downstream
of the RIS and then calculating the velocity from this smooth field, we can
get a better velocity signal with which to do further analysis. A convenient
measure of the importance of each tidal constituent is the percent energy
(PE) (Codiga and Rear, 2004). Tidal analysis with UTide (Codiga, 2011) of the
measured horizontal ice displacements 20 km downstream of RIS GL show that
the *M*_{sf} signal dominates with 87 % of PE, followed by the
diurnal and semidiurnal tidal constituents. Analysis of the velocities,
calculated as described above, reveals that the two largest constituents are
*M**S*_{4} and *M*_{4} with 21 and 11 % of PE, respectively. Based on the
arguments given above, these results provide compelling evidence that the
flexural ice-softening mechanism is responsible for the majority of the
observed *M*_{sf} signal on the RIS.

One consequence of not including GL migration in our model is to generate
artificially large stresses at the GL during high tide, where tidal stresses
are acting to lift the ice from the bed but the clamped boundary condition
prevents this from happening. For comparison, stresses were obtained for a
simulation in which the GL was allowed to migrate, forced by a positive 2 m
tidal deflection. At the GL node, effective stress was 67 % greater
in the pinned case, but this effect is highly localised, and depth-averaged
effective stress at the GL is only 12 % greater. If bed geometry on RIS is
such that the GL can migrate a meaningful distance, our model would slightly
overestimate the reduction in shear margin effective viscosity due to bending
stresses at high tide. Our aim here is to investigate the flexural
ice-softening mechanism in isolation, and including GL migration would
complicate any interpretation, particularly given the unknown bed geometry of
RIS. GL migration could play a role in generating the *M*_{sf} signal
observed across the Ronne Ice Shelf, depending on whether the local bed
geometry permits it. That being said, both the simplicity of the flexural
ice-softening mechanism and the ease with which it explains many
aspects of the observed tidal modulation in ice shelf flow suggest that it
is likely to be the primary mechanism at play.

In all our full-Stokes model experiments the *M*_{sf} signal decays
rapidly upstream of the grounding line, contrary to observations which show
the signal persists at least ∼ 80 km upstream of the Rutford, Evans
and Foundation Ice Stream GLs (Gudmundsson, 2006; Minchew et al., 2016; Rosier et al., 2017a).
Previous studies have proposed that a non-linear basal sliding law could
generate the *M*_{sf} signal on grounded ice
(Gudmundsson, 2007, 2011; King et al., 2011; Rosier et al., 2014). The model
presented in this paper also uses a non-linear sliding law, but when the
flexural softening mechanism is absent and the non-linear sliding law is the
only mechanism at play (experiment n1xyz) it fails to reproduce the
observed *M*_{sf} amplitude and decay length scale
(Fig. 4c). Other mechanisms have been suggested which could
promote propagation of this signal far upstream, for example weakened margins
or tidal pressurisation of the subglacial drainage system
(Rosier et al., 2015; Thompson et al., 2014). Since our focus is on the ice shelf, we do
not include any of these mechanisms in this model.

The flexural softening mechanism which we have described acts in the grounding zone which may often coincide with a shear margin, a portion of the ice sheet that is complex and remains poorly understood. Shear margins are typically heavily crevassed due to the intense shear straining, making them difficult to access and instrument. These crevasses change the effective bulk properties of the ice, altering the flexural profile compared with undamaged ice (Rosier et al., 2017b). Furthermore, repeated straining will alter the ice fabric and make it highly anisotropic (Alley, 1988; Azuma, 1994). In the grounding zone, repeated tidal straining may itself alter the ice fabric, although this has never been investigated to our knowledge. Finally, lateral and tidal straining will cause strain heating (Fig. 6d). A consequence is that ice within floating shear margins subjected to large tides may be warmer as a result of tidal flexure, although the presence of crevasses could lead to a complex depth-dependent temperature profile (Harrison et al., 1998; Perol and Rice, 2015). All of the processes described above will interact with tidal flexure, and further modelling is required to evaluate their effects in detail.

Remote-sensing techniques suggest that the amplitude of the *M*_{sf}
signal shows considerable spatial heterogeneity (Minchew et al., 2016). There
remains some debate about the correct value for the ice rheological exponent
*n* and whether it might vary spatially (Cuffey and Paterson, 2010), although this is often conveniently ignored in
modelling studies. Since the amplitude of the *M*_{sf} signal on the
ice shelf is highly sensitive to the value of *n*, further modelling of this
effect might help to provide new insights into ice rheology. For example, it
might be that the observed spatial pattern and magnitude of the
*M*_{sf} effect on the shelf downstream of RIS can only be reproduced
for certain choices of *n*, although it would be difficult to separate this
from other factors at play. In the context of the flexural ice-softening
mechanism, this heterogeneity could also arise due to variation in ice
properties such as thickness, fabric, and damage.

We present results from both analytical and full-Stokes models, which show
that tidal bending stresses in ice shelf margins can give rise to large-scale
temporal variations in ice flow. The non-linear rheology of ice means that, as
an ice shelf bends to accommodate vertical tidal motion, stresses generated in
the grounding zone reduce the effective viscosity of ice. This leads to
modulation of ice shelf velocity at a number of frequencies, including the
*M*_{sf} frequency, which is readily observed on many Antarctic ice
shelves (Gudmundsson et al., 2017; King et al., 2011; Minchew et al., 2016; Rosier et al., 2017a). In addition,
the non-linear response changes the mean flow of the ice shelf when it is
subjected to vertical tidal motion.

This mechanism relies only on the non-linear rheology of ice and can explain many recent GPS and satellite observations of tidal effects on ice shelf flow. By causing an increase in ice velocity twice during one tidal cycle, it leads to a strong frequency-doubling effect which is potentially diagnosable from careful measurement of ice shelf velocity with high temporal resolution and accuracy. Tentative analysis of GPS measurements from the floating portion of RIS suggests that these characteristic frequencies can be seen in existing data and that their relative amplitudes match those of our model.

The bending stresses investigated in this study are typically ignored and difficult to incorporate into large-scale ice sheet models; however this work shows that these stresses have a role to play in the overall flow regime. Full-Stokes modelling of a tidally energetic region such as the FRIS would lead to further insights into the importance of this mechanism, as well as its relevance for ice flow models and possibly even ice rheology.

No experimental data are used in the paper; the modelling is motivated by data published in a previous data paper (Rosier et al., 2017a).

We start from the simplified *z* momentum given in Eq. (2b),
together with expressions for the bending stresses *τ*_{yy} and
*τ*_{yz} (Eqs. 4 and 5, respectively). Applying the
surface boundary condition $\mathit{\sigma}\widehat{n}=\mathrm{0}$, we find that

Since *τ*_{yz}=0 at the surface, this reveals that *σ*_{zz}(*s*)=0.

Using this result and integrating the *z* momentum (Eq. 2b)
from the surface to arbitrary depth *z*, we arrive at an expression for
$p(x,y,z,t)$:

Inserting this into the *x* momentum of Eq. (2a) gives

where

and $\mathit{\zeta}=z(h+\mathrm{2}z)$. Note that the *x* dependence of Eq. (A2)
is through the ice thickness *h*, which also appears in the expression for
*λ* (Eq. 6). Integrating from the surface to the bed and
dividing by ice thickness yields the depth-averaged across-flow gradient in
horizontal shear stress:

With the boundary condition that $\stackrel{\mathrm{\u203e}}{{\mathit{\tau}}_{xy}}$ is zero at the
centreline, we can integrate along *y* to give an expression for depth-averaged horizontal shear stress, which is

It turns out that the second term on the right-hand side of Eq. (A7) is
much smaller than the other two for any sensible choice in parameters, and so
the horizontal shear stress is balanced by the driving stress term to a very
good approximation. Since the geometry along the *x* direction does not
change with time, the only temporal variation in *τ*_{xy} enters through
the smaller second term. As such, ${\dot{\mathit{\tau}}}_{xy}\approx \mathrm{0}$: a curious
finding given the large changes in centreline velocity but one that is borne
out by examination of the stresses in our full-Stokes model
(Sect. 6).

For a comparison with the idealised system of equations presented above, we
take a 2-D slice through the ice shelf in the full-Stokes model (presented in
Sect. 4) and look at the deviatoric stresses. We take this slice
far away from the GL at *x*=0 to avoid the additional bending stresses in
this region. The lateral shear stress *τ*_{xy} is found to vary linearly
from zero at the medial line to ∼ 70 kPa at the margin and is
approximately constant with depth (see also Fig. 5). Maximum
variation in *τ*_{xy} over a tidal cycle is ∼ 3 %, despite the ice
velocity doubling at the medial line. This matches closely with the profile
predicted by Eq. (A7) using parameters taken from the model. The
main discrepancy in stresses between the full-Stokes model and the simplified
system of Eq. (2a) is that modelled *τ*_{xx} becomes
relatively large near the medial line; however since this is not the case
near the margins, where most of the lateral shearing takes place, the
approximation appears to not be a bad one.

Much of the work on tidal bending of floating ice is based on beam theory, specifically the analysis of elastic beams on elastic foundations first explored by Hetenyi (1946). The classical solution for bending of a floating ice tongue was first derived by Robin (1958) and has since been used extensively in studies of ice flexural process (Holdsworth, 1969, 1977; Hulbe et al., 2016; Lingle et al., 1981; Rignot, 1998; Smith, 1991; Stephenson, 1984; Sykes et al., 2009; Vaughan, 1995). We will call this set of equations the long-beam model (LBM). The set of boundary conditions (BCs) chosen in the LBM are as follows:

where *w*(*y*) is the vertical deflection of the neutral axis and *w*_{a} is the
change in sea level due to tides. The assumption in
Eq. (B1) that ice is freely floating at the far-field
boundary is valid in many circumstances; however the shelf downstream of RIS
is only ∼30 km wide, and so this set of BCs might not be appropriate. A
better set of BCs for a narrow ice shelf consists of a beam clamped at both
ends, such that

Starting from the beam equation for a floating ice shelf,

subject to the BCs in Eq. (B2), we arrive at the solution

where *λ* is given in Eq. (6) and the constants *C*_{1} to
*C*_{4} are

If the product *λ**W* is large (specifically, large in comparison to
*π*), then the hinge zone is narrow compared to the ice shelf width. In this
situation, ${C}_{\mathrm{1}}\approx {C}_{\mathrm{2}}\approx \mathrm{1}$ and ${C}_{\mathrm{3}}\approx {C}_{\mathrm{4}}\approx \mathrm{0}$, such
that Eq. (B4) reduces to the LBM solution (Robin, 1958).
For the RIS where *W*≈ 14 km, this turns out to be the
case, and so the simpler LBM differs only very slightly from the solution
given in Eq. (B4). As a result, we can safely use the LBM to
approximate bending stresses on the RIS.

The authors declare that they have no conflict of interest.

We are grateful to Rob Arthern, Brent Minchew and Teresa Kyrke-Smith for very
helpful discussions and two reviewers for their constructive comments, which
greatly improved the quality of the manuscript. Sebastian H. R. Rosier was
funded by the UK Natural Environment Research Council large grant “Ice
shelves in a warming world: Filchner Ice Shelf System”
(NE/L013770/1).

Edited by: Olivier Gagliardini

Reviewed by: Martin Lüthi and Victor Tsai

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- Abstract
- Introduction
- Flexural ice-softening mechanism
- Analytical solution for flexural ice softening
- Full-Stokes model description
- Model experiments
- Model results
- Discussion
- Conclusions
- Data availability
- Appendix A: Derivation of across-flow shear stress
- Appendix B: Analytical solution for double-clamped elastic beam
- Competing interests
- Acknowledgements
- References

- Abstract
- Introduction
- Flexural ice-softening mechanism
- Analytical solution for flexural ice softening
- Full-Stokes model description
- Model experiments
- Model results
- Discussion
- Conclusions
- Data availability
- Appendix A: Derivation of across-flow shear stress
- Appendix B: Analytical solution for double-clamped elastic beam
- Competing interests
- Acknowledgements
- References