Our numerical ice flow model solves the field equations for conservation of mass, linear momentum (equilibrium equations) and angular momentum: DρDt+ρvi,i=0σij,j+fi=0σij-σji=0, where D/Dt is the material time derivative, ρ is density, νi are the components of the velocity vector, σij are the components of the Cauchy stress tensor and fi are the components of the gravity force per volume. We use the comma to donate partial derivatives and the summation convention, in line with notation commonly used in continuum mechanics. None of the terms in the equilibrium equations are omitted. In glaciology such models are commonly referred to as full-Stokes models.

We use an upper-convected Maxwell rheological model that relates deviatoric stresses τij and deviatoric strains eij with e˙ij=12Gτ▿ij+Aτn-1τij, where A is the rate factor, the superscript ▿ denotes the upper-convected time derivative, n is the constant in Glen's flow law (a nonlinear relation with n=3 is used throughout), G is the shear modulus G=E2(1+ν), ν is the Poisson's ratio and E is the Young's Modulus. The upper-convected Maxwell model allows for calculation of large strain under rotation which, although not essential for the strains present in our model, we have chosen to use for completeness. More details of this rheological model can be found in . The deviatoric stresses are defined as τij=σij-13δijσpp, and the deviatoric strains as eij=ϵij-13δijϵpp, where σij and ϵij are the stresses and strains, respectively. This rheological model approximates the visco-elastic behaviour of ice at tidal timescales, and can be thought of as a spring and dashpot in series such that the resulting strain is the sum of the elastic and viscous components and the stresses are equal.

These equations are solved using the commercial finite-element software package MSC.Marc . The ice stream and the underlaying till are treated as two separate deformable bodies. In a previous study we have calculated the migration of the grounding line in response to ocean tides, and accounted for the resulting effect on ice flow upstream from the grounding line in a flow-line setting . Due to computational considerations we have here, however, not allowed the grounding line to migrate over tidal cycles.

Basal velocity is given by a commonly used empirical form that includes effects of hydrology e.g.: ub=cτbmNq, where τb is the tangential component of the basal traction, N is the effective pressure (kept constant for the initial parameter study and subsequently perturbed due to the tide), c is basal slipperiness and both m and q are exponents. The slipperiness, tangential basal traction and effective pressure are all spatially variable.

The effective subglacial water pressure N at the ice-till interface is defined as the difference between the normal component of the basal traction (σnn, with a positive stress acting upwards) and the subglacial water pressure (pw), i.e. N=-σnn-pw, where a positive value for N indicates grounded ice where the downwards pressure of ice exceeds water pressure (as is the case everywhere upstream of the grounding line in this model).

Our approach to including subglacial hydrology within the finite element model framework described in is to reduce the problem to the simplest possible set of equations. Rather than attempt to model a complex system of connected channels and distributed flow, we treat the drainage system as a homogenous porous medium with a characteristic 'conductivity' that, once coupled to the ice-flow model, can be tuned so that the velocity response matches observations. This approach to modelling subglacial hydrology has been used successfully in previous coupled studies eg. .

As a starting point we must lay out how the tide perturbs the subglacial water pressure. We write the subglacial water pressure (pw) at any location upstream from the grounding line as pw(x,t)=ρwgh(x,t)+ρwg(S¯-b(x)), where h(x,t) is the tidally induced perturbation in the hydrological head, ρw is the ocean density, g the gravitational acceleration, b(x) the bed elevation, and the ocean surface elevation S(t) is given by S(t)=S¯+ΔS(t), where S¯ is the mean ocean surface elevation, and ΔS(t) the ocean tide. We incorporate the effects of the tides on subglacial water pressure through the grounding-line boundary condition for the perturbation in the hydrological head h. We assume that at the grounding line the subglacial water system is in direct contact with the ocean, and the subglacial water pressure at that location is therefore equal to the ocean pressure, or pw(x,t)=ρwg(S(t)-b(x))=ρwg(S¯+ΔS(t)-b(x)), at x=xgl, and hence h(xgl,t)=ΔS(t).

The tidally induced perturbation in hydrological head is then modelled as a diffusion process, i.e. ∂th=K∂xx2h, where K is the hydraulic conductivity. In the context of Darcy groundwater flow, K can be expressed as K=ρwgkμSs, where k is the permeability, μ the viscosity of water, and Ss the specific storage capacity. In reality this parameter combination is poorly constrained and here treated as an unknown.

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 K. When modelling the spatial and the temporal variations of the subglacial drainage system water, we only attempt to describe the perturbations in effective pressure due to tides. This avoids the complications of calculating the temporally averaged pressure field, which is unnecessary as the effects of the mean pressure on basal flow are already accounted for in the temporally averaged value of the basal slipperiness which we derive in our inversion (see below).

This is coupled to our ice-stream model through the sliding law (Eq. ) which we expand to consider perturbations in N: ub=cτbm(N¯+ΔN)q, where ΔN=-ρwgh(x,t) and N¯ is mean effective pressure such that N=N¯+ΔN. Re-arranging this gives ub=c′τbm(1+ξ)q, where c′=cN¯-q and ξ=ΔN/N¯. This now puts slipperiness and mean effective pressure into a new c′ term which is a function of x but not a function of t. In this way the baseline effective pressure and slipperiness conditions that affect the mean velocity of the glacier are separated from the perturbed terms. The c′ term is what is inverted for, as described later, to match observed medial line flow. Re-arranging the equation in this way means that N¯ only affects the relative size of the non-dimensionalised perturbation ξ and not the mean flow which is constrained by observations.

The hydrological coupling leads to six constants: N¯, K, ν, E, m and q which are treated as unknowns. The rheological parameters E (Young's modulus) and ν (Poisson's ratio) are constrained to some extent from previous visco-elastic modelling efforts on tidally induced motion, with values of E expected to be ∼ 4.8 GPa and ν of ∼ 0.41 . The sliding law exponents m and q are treated here as tunable parameters. Note that once c′ has been determined, through the inversion procedure outlined below, K and N¯ only affect modeled flow through their combined effect on ξ. Sensitivity of the model to the choice of these parameters is presented later.

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. ) has zero bed slope, a surface slope of 0.0036 and ice thickness at the grounding line of 2040 m. This simple geometry is derived from average bed and surface profiles along the RIS medial line from BEDMAP2 data . While using constant slopes is a simplification and in reality the bed undulates considerably over the 100 km length being considered, there is no obvious overall shallowing or deepening, and the surface slope is relatively uniform. The width and length of the model domain are 16 and 120 km, respectively. The model width does not vary alongflow and the value chosen is an approximate average width for the region of interest. The hydrological component of the model extends a further 100 km upstream.

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 32km for the symmetrical problem approximately matching that of the RIS . Along the upper in-flow boundary, a surface traction is prescribed based on the analytical solution for the flow of a uniformly-inclined slab of ice. At the downstream boundary, a surface traction is prescribed based on the analytical solution for the flow of an ice shelf in one horizontal dimension , i.e. σxx=-ρig(s-z)+ρigH21-ρiρw-pb, where ρi and ρw are ice and water density respectively (910kgm-3 and 1030kgm-3), H is the ice thickness, s is the ice surface, z is the depth, g is gravitational acceleration and pb is a back pressure term that we treat as unknown and include in the inversion. In the case of the RIS, a non-zero value of pb could, for example, be expected to result from lateral resistance to ice-shelf flow.

Two boundary conditions are necessary to solve for the diffusion of hydraulic head upstream from the grounding line. As mentioned earlier, at x=xgl subglacial water pressure and ocean pressure are assumed to be equal, leading to the boundary condition given in Eq. . At the upstream boundary the condition h→0 as x→∞ is strictly correct for this form of diffusion equation. Since this is not possible to implement in our model we use h=0 at x=200km, assuming that h is very small at the upstream boundary. This can be justified analytically by solving Eq.  to give a decay length scale, for some periodic change in hydraulic head, of 2K/ω where ω is the tidal angular frequency being considered. For the range of conductivity values and tidal frequencies considered here, the model domain of 200km is far larger than this length scale, thus this boundary condition can be safely applied without influencing the model results.

Ocean pressure is applied to the base of the floating ice shelf as a spring foundation a more detailed description can be found in and the tidal forcing is introduced into the model as a perturbation in mean sea level. The tidal forcing is taken from the CATS2008 tidal model output , using the largest six tidal constituents at the RIS grounding line (M2, S2, O1, K1, K2 and N2). This model performs particularly well in this region since it is constrained by previous GPS measurements in this area and comparison with the vertical GPS record of shows very close agreement. Tidal currents beneath the ice shelf are not included in the model since the effect on basal drag is negligible and effects on basal melt are too slow to affect velocities at daily timescales. A schematic showing the various tidal processes, including some not included in the model, is shown in Fig. .

Preliminary experiments were conducted in which the stress exponent of the flow law (m) was changed to examine the effect on Msf response. Changing this parameter alters the mean flow in a non-trivial way that cannot be simply accounted for by altering slipperiness over the entire domain. Since the Msf response is sensitive to mean velocity it is important when comparing results to keep the mean velocity as close to observations as possible. To reproduce the general pattern of observed surface velocities on RIS, and in particular the general increase in velocities towards the grounding line, we invert for slipperiness (c′) using the medial line velocities obtained from the MEaSUREs InSAR velocity data set (note the term slipperiness here encompasses bed slipperiness and mean effective pressure). Although these InSAR-derived velocities are potentially flawed in regions with long period tidal modulation in flow , we address this by increasing the a priori error estimate (discussed later) to be larger than the errors provided in the data set. In general a comparison of the InSAR velocities with in situ GPS measurements does show some differences but the only large discrepancy is on the ice shelf where we are not concerned with matching the velocities.

A Bayesian inversion approach was used to empirically calculate the i×j sensitivity matrix K describing the sensitivity of surface velocities to basal slipperiness. The method and equations are broadly similar to those presented in except that, rather than using analytical expressions for the sensitivity matrix, it is computed as the partial derivative of the forward model with respect to the state vector. The sensitivity matrix is given by [K]pq=∂up∂cq′, where p and q are nodal numbers along the upper and lower surfaces of the finite element mesh. Here the measurement vector u has i elements and is the surface velocity, and the state vector c′ has j elements and is the slipperiness at the bed. Thus we calculate, for each element of the state vector, the change in measurement vector, giving one entire column of K. This is repeated for every element of the state vector to build up a complete sensitivity matrix.

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 ci+1′=ci′+S^-1(KiTSe-1[u-F(ci′)]-Sa-1[ci′-ca′]), where S^-1=KiTSe-1Ki+Sa-1 is the Fisher information matrix, Se is the covariance of measurement errors, Sa is the covariance of a priori errors and F(c′) is the forward model . Measurement errors (σe) are assumed to be uncorrelated and have a normal distribution, such that the measurement error covariance matrix is proportional to the identity matrix, in the form Se=σe2Im. We choose a large value of 0.2 md-1 for σe to account for errors arising from undersampling of tidal effects in this area.

Our treatment of the prior covariance matrix is the same as , based on the assumption that basal slipperiness is spatially correlated, whereby each prior estimate of c′ at location i is related to a neighbouring location i-1 by ci′=ϕcci-1′+ϵc, where ϵc has variance σ2. The elements of Sa-1 can then be given by [Sa]pq=σa2e-|p-q|/λ, where λ is a decay length scale, related to ϕc by λ=-1/ln⁡ϕc and the variance is σc2=σ21-ϕc2. This results in a covariance matrix which has σa2 along the diagonal and non-zero off-diagonal elements.

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 (pb, which is particularly relevant for flow velocities near the grounding line) is inverted for by adding a single non-dimensionalised element to the end of the state vector. This is treated in the same way as the other state vector elements apart from having its own (uncorrelated) prior error estimate.

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 K matrix is independent of all the others, it is possible to easily parallelize its calculation, meaning that run times need not be orders of magnitude greater if sufficient computing resources are available. The sensitivity matrix need not be calculated for each iteration and in fact it is advantageous to iterate a number of times using the same matrix before re-calculating it. The iteration was continued until it converged on the maximum a posteriori solution, in contrast to many other similar studies which stop iterating once the misfit between model output and observations is below a given threshold.

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 Msf amplitude in the absence of any temporal changes in bed conditions, i.e. ōther than those resulting from direct stress transmission through the ice due to the flexing of the ice in response to tides. In the context of our modelling methodology described above this is equal to setting the stress exponent, q, of the effective water pressure in the sliding law (see Eq. ) to zero. In effect we repeat the fully 3-D simulations conducted in but with a broader range of parameters, an ice-stream geometry closer to that of RIS and a basal slipperiness distribution (c(x)) determined through a formal inversion of surface velocities. Our tunable model parameters with no subglacial hydrological coupling are the Poisson's ratio (ν), the Young's modulus (E) and the stress exponent (m). We set the stress exponent (n) in Glen's flow law to n=3, and determine the rate factor A from a static temperature distribution defined in the model using the commonly used temperature relation given by .

We performed an extensive parameter study, with the stress exponent m of 1, 3, 5 and 10, and the Young's modulus of 3, 4.8 and 6 GPa. The Poisson's ratio was varied between 0.3 to 0.45, but was found to have almost no effect on the modelled tidal response and we do not discuss those results further. For every value of the basal sliding-law stress exponent m, we first determined the maximum a posteriori distribution of basal slipperiness (c′) using our inversion approach. In the surface-to-bed inversion the long-term average flow in the absence of tidal forcing was matched to the observed velocity, and a (purely) viscous flow model was therefore used in the forward step. We then forced our visco-elastic time-dependent model by tides. For each given value of m and the associated basal slipperiness distribution, tidal response was calculated for a range of elastic rheology parameters. From modelled horizontal displacements curves, we then calculated tidal amplitudes and phases as a function of distance along the medial line. By fitting an exponential curve to the spatial variation in tidal amplitudes, we then determined decay length scales for each tidal component, as well as phase velocities. The decay length scale we refer to here is defined as the e-folding length scale, or the distance for a given signal (in this case the horizontal tidal signal) to decay by factor e.

The results of the parameter study are summarized in Fig. . In Fig. a the amplitude of the Msf frequency 10km upstream from the grounding line is shown. The modelled Msf amplitudes are never larger than a few centimetres. The largest values are found for high m and high E values. Although somewhat higher Msf amplitudes could be obtained by increasing m even further, the modelled results show that this increase is sub-linear as a function of m. Furthermore, for m>10 other model outputs that must match observations such as phase velocity, decay length scale and notably M2 amplitude, would also increase beyond the range of desired values. The model is, thus, not able to reproduce the observed magnitude of the Msf tidal amplitude.

Both the decay length scale (Fig c) and phase velocity (Fig d) increase with increasing m, in agreement with the analytical solution derived in previous work .

The amount of buttressing needed to match observed velocities increases as m is increased and varied from 650KPa to 850KPa for m=1 to 10. Note that the inversion procedure, in minimising the cost function, tries to find a solution that does not vary significantly from the a priori estimates of slipperiness and buttressing, and therefore this buttressing value may be to some extent artificial if the a priori buttressing estimate and error are poorly chosen. For this reason a large value (1000 kPa) is chosen for the error estimate of buttressing used in the inversion.

Decay of the Msf amplitude upstream of the grounding line for all parameter study simulations is plotted in Fig.  (blue lines) and compared with the observed amplitudes (crosses). This clearly shows the disparity between desired amplitude and the range of possible amplitudes using the mechanism described above. The conclusion from this parameter study, in agreement with , is that stress transmission alone cannot explain the large amplitude of Msf modulation, with maximum amplitudes 10km upstream approaching ∼0.05m, considerably smaller than the desired 0.3m. Clearly an additional nonlinear effect is needed to match observations. Although stress-transmission can reproduce the qualitative aspects of the data, in particular the generation of Msf response, the effects are (at the most) about an order of magnitude smaller than revealed by measurements.

We now couple our hydrological model (Sect. ) to the 3-D full-Stokes model by using values of q>0 in order to see whether this can explain measurements made on the RIS.

Coupled model results obtained through optimization of hydrological parameters are shown in Fig. . This provides a much better agreement with GPS measurements than any previous combination of parameters for the model with no subglacial water pressure coupling. Notably, the Msf amplitude and decay length scale are both large and match very closely with data (Fig. ). The hydrological model used a mean effective pressure (N¯) of 105 kPa, pressure exponent (q) of 10 and conductivity (K) of 7 × 109 m2d-1. Other model parameters were E=4.8GPa, ν=0.41 both in accordance with the optimum Maxwell rheology given by and m=3.

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.  and shows an M2 amplitude (visible in both figures as the higher frequency modulation overlain on the long period Msf signal) that is approximately twice as large at the grounding line as the amplitude determined by tidal analysis of the data. Possible explanations for this are that the M2 amplitude may be too small to be sufficiently resolved by the GPS receivers that originally made the measurements or limitations of the simple Maxwell rheology. Errors in the GPS measurements are of the order of centimetres; more details of the original data set can be found in and a description of similar processing in .

We perform a sensitivity analysis to determine whether the Msf response is robust or highly sensitive to certain parameters. Figure  shows change in Msf amplitude (panel a), M2 amplitude (panel b), Msf decay length scale (panel c) and Msf phase velocity (panel d) compared to the optimized model for a ±10% change in each parameter.

Comparison in Fig. a and b suggests that the calculated Msf and M2 amplitudes are closely correlated and thus, for the parameters tested here, there is no clear modification of the model that would decrease the semidiurnal (M2) amplitude without also reducing the Msf response. Softening the ice by reducing E may be one possible route, since this appears to increase Msf amplitude more than M2 amplitude, however this parameter is more tightly constrained than others since the rheology of ice is not entirely unknown and the sensitivity is too small to solve the issue. Msf amplitude is most sensitive to normalized changes in N¯ and q, as might be expected since it is the nonlinearity here that drives the majority of the long period modulation in flow.

A reduction in m increases the nonlinear response of the modeled ice stream, the reverse of the response with no hydrological coupling, but increases the Msf length scale and phase velocity. Overall, all parameters are most sensitive to the choice in N‾. This is not surprising, since ΔN is small and as N‾ gets large the dimensionless number ξ will drop out and that source of nonlinearity disappears.

The large difference in Msf amplitude between the parameter study simulations and those that include tidally induced subglacial pressure variation poses an important question; is a nonlinear sliding law where m>1 required at all, given that the Msf modulation appears to be largely generated by water pressure changes. Results from the sensitivity analysis suggest that the stress exponent m remains a crucial parameter in altering characteristics of the Msf response. To look at this in more detail, the model was rerun with varying exponents q and m, with the aim of examining the characteristics of the Msf response given changes in the dominance of the two mechanisms.

The four characteristics of the model's tidal response are plotted against exponents q and m, each varying between 1 and 10, in Fig. . These results show that reducing m leads to an increase in amplitude of both tidal frequencies investigated, but a decrease in the length scale and phase velocity. An Msf decay length scale of ∼50km is observed on the RIS but panel c shows that for m=1 the length scale is smaller up to q=10 and in fact appears to have reached an asymptote. Increasing m for any given value of q however leads to a large increase in the length scale. The mechanism by which increasing m reduces Msf amplitude but increases length scale is discussed later but suggests that a flow low with m>1 is still required to reproduce the RIS tidal response.