Observations show that the flow of Rutford Ice Stream (RIS) is strongly
modulated by the ocean tides, with the strongest tidal response at the 14.77-day tidal period (

The majority of ice streams in Antarctica are forced at their boundary by
ocean tides, either directly or through the motion of an adjoining ice shelf.
Measurements have shown the flow of ice streams to be greatly affected by
ocean tides over large distances upstream from the grounding line

Linearly detrended horizontal displacements on the RIS reproduced by
a tidal fit to the original measured data. Measurements are shown from five
GPS stations at 20

An interesting aspect of the tidal observations on RIS is the long period (

One of the key motivations for studying the impact of tides on ice-stream flow is that modelling work has shown the response to reflect mechanical conditions at the glacier bed. Hence, observing and modelling tidally induced modulations in ice-stream motion provides a window into the mechanisms that influence basal sliding.

As initially suggested by

The motivation for this work are recent modelling studies that suggest that
any models using time-invariant sliding-law parameters, while ignoring the
effects of tidally induced sub-glacial pressure variations on sliding, will
fail to reproduce the RIS observations in quantitative terms. Recent work by

The first measurements of this effect made by

Here we use a 3-D nonlinear visco-elastic model with a geometry closely matching that of RIS to investigate the causes for the observed tidal response. We couple our ice-mechanical model to a model describing the changes in basal water pressure due to ocean tides, by allowing basal velocity to change in response to changes in effective basal water pressure.

The paper is organised as follows. We first describe our nonlinear
visco-elastic model and present the basic governing equations. We then perform
a full-Stokes surface-to-bed inversion of medial line surface velocities to
determine the time-averaged spatial distribution of basal slipperiness. We
then establish in a thorough parameter study that the model of

Our numerical ice flow model solves the field equations for conservation of
mass, linear momentum (equilibrium equations) and angular momentum:

We use an upper-convected Maxwell rheological model that relates deviatoric stresses

These equations are solved using the commercial
finite-element software package MSC.Marc

Basal velocity is given by a commonly used empirical form that includes
effects of hydrology

The effective subglacial water pressure

Our approach to including subglacial hydrology within the finite element
model framework described in

As a starting point we must lay out how the tide perturbs the subglacial
water pressure. We write the subglacial water pressure (

The tidally induced perturbation in hydrological head is then modelled as a
diffusion process, i.e.

Thus, our approach is to solve for tidal perturbations in hydraulic head (rather than water pressure) which
is known at the grounding line and transmitted upstream through a simple diffusion process controlled by the conductivity

This is coupled to our ice-stream model through the sliding law
(Eq.

The hydrological coupling leads to six constants:

Our model geometry is based on the RIS, however, we have not attempted to
reproduce its geometry exactly and our thickness distribution in along-flow
direction corresponds to the mean ice thickness across the ice stream. The
3-D model domain (Fig.

3-D model domain, showing the boundary forces (black arrows) and
flow constraints (red arrows). The subglacial drainage system extends a
further

A no-slip condition is applied along one of the lateral boundaries and a
free-slip condition along the other. The latter represents the ice stream
medial line, giving an overall width of

Two boundary conditions are necessary to solve for the diffusion of hydraulic
head upstream from the grounding line. As mentioned earlier, at

Ocean pressure is applied to the base of the floating ice shelf as a spring
foundation

Schematic showing the various mechanisms by which tides can influence ice-stream flow. Note that grounding line migration, crevassing and tidal currents are not included in the model.

Preliminary experiments were conducted in which the stress exponent of the
flow law (

A Bayesian inversion approach was used to empirically calculate the

Since the model response to a change in slipperiness is nonlinear, the
inversion will not converge to an optimum solution in a single iteration and
so a Newton-Gauss iterative approach is used of the form

Our treatment of the prior covariance matrix is the same as

We reduce the number of calculations needed by only taking into account
along-flow variations in slipperiness. This simplification is justified due to the simple geometry and because we only seek to match the medial line ice-stream velocity. Buttressing (

Although this brute force approach to inverting for basal slipperiness is
computationally more expensive than others such as the adjoint method, there
are a number of advantages to this method such as giving an explicit estimate
of the inversion error. Furthermore, because each element of the

Modelled

As discussed above, to date no model has been presented that can reproduce
the tidally induced horizontal velocity variation observed on the RIS.
Admittedly, most models have focused on trying to identify the mechanism
responsible for the rather striking observation that the response of the
ice stream is concentrated at tidal frequencies absent in the forcing.
However, it would be expected that if the mechanism has been correctly
identified, and is the primary cause for the velocity fluctuations, modeled
amplitudes would be close to those measured. In fact modelling work presented
so far has always produced too small a response at the

To address the open question of whether RIS observations can be replicated
through stress transmission alone, our first modelling aim is to establish an
upper bound on the possible

We performed an extensive parameter study, with the stress exponent

The results of the parameter study are summarized in
Fig.

Both the decay length scale (Fig

The amount of buttressing needed to match observed velocities increases as

Decay of the

We now couple our hydrological model (Sect.

Modelled detrended horizontal surface
displacements taken along the ice-stream medial line at 20

Coupled model results obtained through optimization of hydrological
parameters are shown in Fig.

The only feature of these results that is arguably not in agreement with
observations is the amplitude of the semidiurnal tidal constituent detrended
displacements. Comparison between Figs.

We perform a sensitivity analysis to determine whether the

Comparison in Fig.

Sensitivity analysis of model parameters (

Response of

A reduction in

The large difference in

The four characteristics of the model's tidal response are plotted against
exponents

We find that stress transmission alone cannot fully explain the observed

This nonlinearity arises largely in two of the parameters:

Spatial variations in

In order to understand the interaction between the hydrology and stress
transmission mechanisms it is important to consider the relative timing with
which they act on the ice stream. As explained previously, an exponent

Results from Fig.

None of the other parameters within the model had such a large effect on the
length scale and the implication is that a nonlinear sliding law is required
in addition to any nonlinear response to subglacial pressure variations.
Matching the observed long period modulation of ice-stream flow requires a
balance between large

An explanation for this increase in length scale with

The requirement of high conductivity in order to transmit the tidal signal far enough upstream to match observations suggests that there must be a channelized drainage system beneath the RIS. This could consist of a few large channels that transmit the tidal pressure wave far upstream which then permeates through the till on either side of the channel, leading to changes in effective pressure over large portions of the ice-stream base.

Observations of surface motion of the RIS show a strong, nonlinear response
that propagates a long way upstream from the grounding line. The nonlinear
response of this ice stream and others in the region is striking both in its
amplitude and extent and matching observations is not possible through stress
transmission considerations alone. Coupling with a hydrological model that
sends tidally induced subglacial pressure variations far upstream is required
to explain these observations. Furthermore, three other requirements must be
met; low effective pressure across the entire ice-stream bed, a highly
conductive subglacial drainage system and a nonlinear sliding law such that

Hydrological and basal sliding model parameters that produced a best fit to
observations were

We wish to thank the editor and the two anonymous reviewers for their insightful comments that have helped greatly improve this paper. We also thank J. Kingslake, Jan De Rydt and C. Martín for their comments on the manuscript and many helpful suggestions during the course of this work. The original GPS data was collected with the support of NERC GEF loan 785. This study is part of the British Antarctic Survey Polar Science for Planet Earth Programme. Funding was provided by the UK Natural Environment Research Council through grants NE/J/500203/1 (SHRR PhD studentship) and NE/F014821/1 (JAMG Advanced Fellowship). Edited by: F. Pattyn