GPS measurements reveal strong modulation of horizontal
ice shelf and ice stream flow at a variety of tidal frequencies, most notably
a fortnightly (

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)

A multitude of mechanisms have been proposed which could lead to a
fortnightly modulation in ice flow: a non-linear basal sliding law

Previous modelling studies have focused almost exclusively on tidal
modulation of ice stream flow

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.

Schematic showing the flexural ice-softening mechanism for a
confined shelf, together with the geometry of the problem described in
Sect.

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

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

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

We immediately make the simplifying assumptions (motivated by full-Stokes
calculations presented below) that

Note that in this system

Linear elastic beam theory gives us an expression for the elastic stresses
that will arise due to tidal bending

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

By assuming that

To illustrate the consequences of a typical tidal action for the ice shelf
flow, we assume that the time-varying sea level

Several useful results are now easily obtained with Eqs. (

Contour plot of ice shelf speed-up due to tides, as a percent of the
baseline speed, predicted by the analytical solution in Eq. (

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

Note that we use a different value of

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

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

We use a non-linear Maxwell viscoelastic rheology in a slightly modified form
to Eq. (

Choice of parameters used in Eq. (

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

The ocean pressure normal to the ice–ocean interface (

Upstream of the grounding line, along the ice–bed interface (green and orange
shaded regions in Fig.

We treat one side of the model ice stream as the medial line, since the
problem is symmetrical (

Finite-element mesh used in the full-Stokes viscoelastic model
(Sect.

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

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.

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

For the second experiment we run the model as in n3xyz, but the sidewalls downstream of the GL are not clamped
vertically (

The third experiment uses the same set-up and boundary conditions as in n3xyz except that ice rheology is made
linear, such that

The time-varying vertical tidal forcing is implemented as a stress acting
normal to the ice shelf base (Eq.

Plan view of

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

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

For the n1xyz experiment, (Fig.

Other tidal frequencies in the n3xyz experiment that emerge from the
frequency doubling (Eq.

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

Across-flow transects of depth-averaged non-dimensional stress from
the full-Stokes viscoelastic model (Sect.

To explore the role of flexural stresses in more detail, we plot across-flow
profiles for each component of the deviatoric stress tensor
(Fig.

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.

At this stage we can briefly evaluate the validity of the assumptions made in
Sect.

Time series of vertical ice displacement at
the medial line

Figure

The analysis of Sect.

Two alternative mechanisms have been proposed to explain the

A previous modelling study has shown that GL migration is itself a strong
non-linearity which can generate an

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.

Tidal analysis of horizontal displacement

As stated above, alternative mechanisms for generating an

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)

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

In all our full-Stokes model experiments the

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

Remote-sensing techniques suggest that the amplitude of the

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

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

We start from the simplified

Using this result and integrating the

It turns out that the second term on the right-hand side of Eq. (

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.

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

Starting from the beam equation for a floating ice shelf,

If the product

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