Auke Barnhoorn
email: auke.barnhoorn@geo.uu.nl

Wouter Van Der Wal

Martyn R Drury

Upper mantle viscosity and lithospheric thickness under Iceland

Keywords: Rheology, deformation mechanism, creep, water content, melt content, mid-ocean ridge, plume

Deglaciation during the Holocene on Iceland caused uplift due to glacial isostatic adjustment. Relatively low estimates for the upper mantle viscosity and lithospheric thickness result in rapid uplift responses to the deglaciation cycles on Iceland. The relatively high temperatures of the upper mantle under the newly formed mid-ocean ridge under Iceland are responsible for the low upper mantle viscosity values. In this study, estimates for lithospheric thickness and upper mantle viscosity under Iceland from glacial isostatic adjustment studies are complemented by a microphysical modelling approach using the theoretical temperature distribution under mid-ocean ridges combined with olivine diffusion and dislocation creep flow laws. The lithospheric thickness (27-40 km) and upper mantle viscosity (2x10 18 -10 19 Pas) outcomes for the upper mantle recent glaciation under the Vatnajökull glacier are consistent with previous reports of viscosity and lithospheric thickness from glacial isostatic adjustment studies. A combination of a 40 km thick elastic lithosphere and an average upper mantle viscosity of 5x10 18 Pas would suggest that the upper mantle under Iceland is most likely dry. The earlier and larger Weichselian glaciation event (~10000 BP) on Iceland is predicted to have had a slightly larger upper mantle viscosity ~10 19 Pas and a lithospheric

A c c e p t e d M a n u s c r i p t

Introduction

The Vatnajökull glacier, located in the south-east of Iceland is the largest ice cap of Iceland having a mean radius of 50.7 km covering an area of ~8100 km 2 (Björnsson, 1998). Its estimated mass loss since 1890 is 435 km 3 (Árnadóttir et al., 2009), much more than the earlier estimate by [START_REF] Sigmundsson | Glacio-isostatic crustal movement caused by historical volume change of the Vatnajökull ice cap, Iceland[END_REF] where the mass loss between 1890 and 1978 is estimated at only 182 km 3 . The Vatnajökull glacier is situated directly on top of the spreading axis in the eastern volcanic zone (EVZ) of the Icelandic mid-ocean ridge and near the inferred centre of the Icelandic hotspot (e.g. Wolfe et al., 1997). Due to the vicinity of the glacier to the active tectonic area, the response of the solid earth to melting of the ice cap is strongly controlled by the properties of the newly formed upper mantle underneath the midocean ridge. The relatively high temperatures in the mantle during rifting result in relatively low upper mantle viscosities and fast relaxation times in comparison with tectonically stable areas such as Fennoscandia. Upper mantle viscosity estimates from glacial isostatic adjustment (GIA) studies for Iceland range between 5x10 17 Pas and 5x10 19 Pas (e.g. [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF][START_REF] Sigmundsson | Glacio-isostatic crustal movement caused by historical volume change of the Vatnajökull ice cap, Iceland[END_REF]Thoma and Wolf, 2001;[START_REF] Sjöberg | Land uplift near Vatnajökull, Iceland, as observed by GPS in 1992, 1996 and 1999[END_REF][START_REF] Fleming | Glacial-isostatic adjustment and the viscosity structure underlying the Vatnajökull ice cap, Iceland[END_REF][START_REF] Pagli | Glacio-isostatic deformation around the Vatnajökull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling[END_REF][START_REF] Biessy | Fast and partitioned postglacial rebound of southwestern Iceland[END_REF][START_REF] Árnadóttir | Glacial rebound and plate spreading: results from the first countrywide GPS observations in Iceland[END_REF][START_REF] Breton | Post-glacial rebound of Iceland during the Holocene[END_REF] compared to 10 20 -10 21 Pas in Fennoscandia (e.g. [START_REF] Lambeck | Sea level change, glacial rebound and mantle viscosity for northern Europe[END_REF][START_REF] Milne | Space-geodetic constraints on glacial isostatic adjustment in Fennoscandia[END_REF]. Elastic thickness estimates in those studies including estimates from A c c e p t e d M a n u s c r i p t 3 seismological studies (e.g. [START_REF] Kaban | Nature of the crust-mantle transition zone and the thermal state of the upper mantle beneath Iceland from gravity modeling[END_REF][START_REF] Kumar | The lithosphere-asthenosphere boundary in the North-West Atlantic region[END_REF] range between a very thin lithospheric thickness of 10 km up to 80 km. Recent glacial isostatic adjustment studies by [START_REF] Pagli | Glacio-isostatic deformation around the Vatnajökull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling[END_REF] and Árnadóttir et al. (2009) provides estimates for both the lithospheric thickness and the upper mantle viscosity. It resulted in a lithospheric thickness of 10-20 km and a viscosity of 4-10x10 18 Pas for [START_REF] Pagli | Glacio-isostatic deformation around the Vatnajökull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling[END_REF] and a lithospheric thickness of 40 km and a viscosity of 10 19 Pas for Árnadóttir et al. (2009).

In this study, we present a microphysical approach to determine the lithospheric thickness (elastic layer) and upper mantle viscosity under the Icelandic mid-ocean ridge and specifically under the Vatnajökull glacier. Information on the rheological behaviour of the main upper mantle mineral olivine is combined with information on the temperature distribution under mid-ocean ridges, information on the microstructural state of the upper mantle under mid-ocean ridges (grain size, water content) and stress levels within the upper mantle due to glaciation events to determine the viscosity of the upper mantle. This approach provides detailed estimates on the lateral and radial variations in upper mantle viscosity and lithospheric thickness that can be tested in geodynamical models and compared with glacial isostatic adjustment studies that so far neglected lateral variability in viscosity and lithospheric thickness. In this study, we have based estimates of viscosity and elastic thickness for Iceland on data from the temperature distribution of the mantle under oceanic lithosphere calculated from a half-space cooling model for oceanic lithosphere (Turcotte and Schubert, 2002) and the olivine flow laws from [START_REF] Hirth | Rheology of the upper mantle and the mantle wedge: A view from experimentalists[END_REF] for linear diffusion creep and non-linear dislocation creep in a wet and dry upper mantle. The half-space cooling model simulates a normal mid-ocean ridge setting and a simplified approximation of plume activity under Iceland. Temperatures of the plume at the center of the ridge axis are 100 °C and 200 °C higher than temperatures at the ridge in a mid-ocean ridge setting, based on estimates of excess temperatures of 75-200 °C due to plume activity under Iceland (e.g. Ito [START_REF] Ito | Dynamics of mantle flow and melting at a ridge-centered hotspot: Iceland and the Mid-Atlantic Ridge[END_REF][START_REF] Ito | Mantle flow, melting, and dehydration of the Iceland mantle plume[END_REF][START_REF] Ruedas | Temperature and melting of a ridge-centred plume with application to Iceland. Part I: Dynamics and crust production[END_REF][START_REF] Schmeling | Crustal accretion and dynamic feedback on mantle melting of a ridge centred plume: The Iceland case[END_REF]. The static temperature distribution in the upper mantle, based on the half-space cooling model, used here (Fig. 1) is less complicated than dynamic models for plume activity and mantle melt generation (e.g. [START_REF] Ito | Dynamics of mantle flow and melting at a ridge-centered hotspot: Iceland and the Mid-Atlantic Ridge[END_REF][START_REF] Ito | Mantle flow, melting, and dehydration of the Iceland mantle plume[END_REF][START_REF] Ruedas | Temperature and melting of a ridge-centred plume with application to Iceland. Part I: Dynamics and crust production[END_REF][START_REF] Marquart | On the geometry of mantle flow beneath drifting lithospheric plates[END_REF][START_REF] Schmeling | Crustal accretion and dynamic feedback on mantle melting of a ridge centred plume: The Iceland case[END_REF]. However, those models only contain a relatively simple dependency of viscosity on pressure and temperature, whereas the viscosity dependence of this study includes also the influence of water, melt fraction and grain size. By using the flow laws of [START_REF] Hirth | Rheology of the upper mantle and the mantle wedge: A view from experimentalists[END_REF] for olivine rheology, solid-state viscosity variations (both radially and laterally) under an active mid-ocean ridge are calculated. The study shows that large variations in elastic thickness and viscosity are expected under the mid-ocean ridge setting of Iceland due to cooling-related temperature variations.

Temperature dataset and olivine viscosity modeling

Geotherms for the oceanic lithosphere under Iceland are constructed using a time-dependent half-space cooling model (Turcotte and Schubert, 2002):

) 2 ( ) ( ) ( c s m s z erf T T T z T     (1)
where T s is the surface temperature, T m the mantle temperature at a depth of 410 km (in this study we used T m = 1400, 1500 or 1600 °C),  c the age of the oceanic lithosphere and  the thermal diffusivity (taken as 1 mm 2 s -1 ; Turcotte and Schubert, 2002). Estimates of upper mantle temperatures from [START_REF] Goes | Shallow mantle temperatures under Europe from P and S wave tomography[END_REF] using P-and S-wave tomography data for Iceland (Fig. 1A) give temperatures in accordance with a time-dependent cooling model of the oceanic lithosphere. The oldest volcanic rocks outcropping on Iceland have an age of approximately 16 Ma [START_REF] Foulger | Older crust underlies Iceland[END_REF]. The steady-state spreading rate of Iceland in plate motion models is 19-20 mm/year (e.g. Árnadóttir et al., 2009). Average growth rate of each plate on both sides of the spreading ridge is thus around 1 cm/year = 10 km/Ma. Based on the [START_REF] Ito | Dynamics of mantle flow and melting at a ridge-centered hotspot: Iceland and the Mid-Atlantic Ridge[END_REF][START_REF] Ito | Mantle flow, melting, and dehydration of the Iceland mantle plume[END_REF][START_REF] Maclennan | Plume-driven upwelling under central Iceland[END_REF][START_REF] Ruedas | Temperature and melting of a ridge-centred plume with application to Iceland. Part I: Dynamics and crust production[END_REF][START_REF] Putirka | Mantle potential temperatures at Hawaii, Iceland, and the mid-ocean ridge system, as inferred from olivine phenocrysts: Evidence for thermally driven mantle plumes[END_REF][START_REF] Schmeling | Crustal accretion and dynamic feedback on mantle melting of a ridge centred plume: The Iceland case[END_REF] is imposed on a mid-ocean ridge setting (T m = 1500 and 1600 °C). The olivine-wadsleyite transition at ~410 km is taken as the lower boundary of the vertical temperature distribution in the upper mantle (Fig. 1B). The dry solidus of for an anhydrous mantle and the solidus at 100 ppm H 2 O for a wet upper mantle of [START_REF] Hirschmann | Dehydration melting of nominally anhydrous mantle: The primacy of partitioning[END_REF] are taken as melting curves. When the temperatures along the geotherms of Fig. 1A exceed the solidus temperature a partially molten upper mantle is assumed to be present (Fig. 1C).

Olivine viscosity modelling

Diffusion and dislocation flow laws for olivine from [START_REF] Hirth | Rheology of the upper mantle and the mantle wedge: A view from experimentalists[END_REF] are used to calculate the upper mantle viscosity distribution for a dry/wet and melt-free upper mantle and a dry/wet and melt-bearing upper mantle. The diffusion and dislocation creep flow laws describe the dependence of the strain rate (  ) on temperature (T), grain size (d), deviatoric stress (), pressure (P), water content ( O fH 2 ) and melt content ( ):

) exp( ) exp( 2 RT PV E O fH d A r p n        (2)
where A and  are constants, n, p and r are the stress, grain size and water fugacity exponents, E the activation energy and V the activation volume. For the modeling of dry/wet, melt-A c c e p t e d M a n u s c r i p t 6 bearing upper mantle a melt fraction ( ) of  = 0.04 is used. A melt content of 4% is just above the upper limit for estimates for melt that can remain in a peridotite (up to 2-3%; Faul (1997), 1-3%; [START_REF] Ruedas | Temperature and melting of a ridge-centred plume with application to Iceland. Part I: Dynamics and crust production[END_REF]). We have used a higher value of 4% to illustrate that the presence of melt has only a relatively small reducing effect on the overall viscosity within the upper mantle under Iceland. A study of the Oman ophiolite [START_REF] Dijkstra | Microstructures and lattice fabrics in the Hilti mantle section (Oman Ophiolite): Evidence for shear localization and melt weakening in the crust-mantle transition zone[END_REF] shows that olivine grain sizes in mantle rocks of mid-ocean ridges have grain sizes between 0.8 and 3.3 mm. An average grain size of 2 mm is used in the modelling as a representative grain size for the upper mantle in mid-ocean ridge settings. Stress profiles within the upper mantle due to glacial loading have been calculated for Iceland using a three-dimensional finite element model for glacial isostatic adjustment based on the commercial program Abaqus. We use the geometry of [START_REF] Schotman | Constraints on shallow low viscosity zones in Northern Europe from future GOCE gravity data[END_REF] who studied Scandinavia, and scaled down the spatial dimensions by a factor of 10 because the Icelandic glaciations cover a much smaller area. The model has a total of 97x97 elements in the horizontal direction and 7 element layers in the vertical direction. Elements in the inner surface area of 292 x 292 km are 8 x 8 km wide.

Density and Young's modulus parameters are obtained by volume averaging from PREM.

The rheology is a combination of dislocation and diffusion creep [START_REF] Barnhoorn | Lateral radial and temporal variations in upper mantle viscosity and rheology under Scandinavia[END_REF] with parameters taken from [START_REF] Hirth | Rheology of the upper mantle and the mantle wedge: A view from experimentalists[END_REF] and a temperature-depth profile for a cooling oceanic lithosphere of 4 million years old (Turcotte and Schubert, 2002). The top 15 kilometer is taken to be fully elastic, which is reasonably close to the 10 km elastic thickness of [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF]. Note however, that the real elastic thickness in the model for glacial loading depends on the effective viscosity which is a function of stress.

The model is loaded separately by two glaciations. (i) the Weichselian (or Younger Dryas) glaciation is modelled following [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF], approximating the ice cap by four discontinuities in height. The surface extent of the ice cap is 160 km and the maximum height is 1800 m. We have included a 100000 year growth phase of the ice cap ; unloading A c c e p t e d M a n u s c r i p t 7 takes place at different time steps between 10000 and 9500 years before present according to [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF]. (ii) A model for the recent growing and melting of the Vatnajökull ice cap is taken from [START_REF] Fleming | Glacial-isostatic adjustment and the viscosity structure underlying the Vatnajökull ice cap, Iceland[END_REF]. A perfectly plastic ice cap (discretised on our 8 x 8 km elements) grows from a radius of 45 km and central thickness of 800 m at 900 AD to a radius of 52 km and a central thickness of 915 m at 1890 AD. After that melting takes place in two different rates to end up with a radius of 51 km and 904 m in 2001 AD. The GIA model provides the von Mises equivalent stress as output:

3 2 ij ij q   (3)
where ij  is an element of the stress tensor. Von Mises equivalent stresses are calculated at the center of the layers (3.75, 11.25, 20, 37.5, 70, 120, 185) and averaged (Fig. 2) over a horizontal area of 200 km radius and also averaged over time (from 20 ka BP up to 9 ka B.P.

for the Weichselian ice cap; from 10 ka B.P. up to 2001 for the recent melting of the Vatnajökull ice cap). The stresses are interpolated for every kilometre in the upper mantle to produce Fig. 2. Using a faster melt rate between 1890 and 2003 (Árnadóttir et al. , 2009) than reported by [START_REF] Fleming | Glacial-isostatic adjustment and the viscosity structure underlying the Vatnajökull ice cap, Iceland[END_REF] resulted in average stress values that differed by less than 7% in areas where the stress is high enough to influence the effective viscosity. The resulting difference in stress using both melt rate estimates is too small to significantly influence the outcomes of this microphy modelling study. The Von Mises stress is a measure of deviatoric stress and can be used directly in flow laws derived from uni-axial experiments [START_REF] Ranalli | Rheology of the Earth[END_REF]. We assume that the equivalent stress can be used as the stress parameter in Eq. ( 2).

The von Mises stresses is the equivalent stress multiplied by 3. The maximum equivalent stress value of 1 MPa at shallow depths for the Weichselian glaciation (Fig. 2) can be compared to estimates of an average stress in the lithosphere and upper mantle due to a simple model of glacial unloading in Scandinavia which are in the range of ~0.5 to at maximum 30 A c c e p t e d M a n u s c r i p t 8 MPa (Barnhoorn et al., in press). The smaller ice cap of Iceland in comparison to Scandinavia causes the average stresses due to glacial loading to be lower for this study. A stress of 1 MPa is also the lower stress estimate for convective processes in the upper mantle [START_REF] Ranalli | Rheology of the Earth[END_REF].

Pressure (P) is calculated as

km GPa P / 033 . 0  .
A composite flow law (e.g. [START_REF] Ranalli | Rheology of the Earth[END_REF] in which both competing mechanisms dislocation creep and diffusion creep contribute to the deformation is used to calculate the total viscosity where:

disl diff total         (4)
and

total total diff diff disl disl             2 ; 2 ; 2    (5)
Two melt contents have been modelled (0% and 4%) for a dry (C OH = 0 H/10 6 Si) and wet upper mantle (C OH = 100 H/10 6 Si). The lateral extent of the melting zone in the 4% melt scenario is controlled by the crossing of the geotherm with the solidus of the upper mantle of [START_REF] Hirschmann | Dehydration melting of nominally anhydrous mantle: The primacy of partitioning[END_REF]. Melt-controlled rheology is in this scenario concentrated under the present-day ridge axis and extends ~120 km for T m = 1400 °C away from the ridge axis.

Outside of this melting zone, the upper mantle deforms by a melt-free rheology ( = 0). The lithospheric thickness is defined as the layer in which the viscosity is high enough so that no measurable influence on the relaxation process is exerted. Analyses of the viscosity cut-off value for Scandinavia (Barnhoorn et al., in press) shows that viscous deformation starts to affect predicted sea-level changes by around one meter for an average viscosity in the [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF] indicates that roughly the first vertical 100 km of the upper mantle is affected by recent glaciation, whereas the radius of ~160 km present during the Weichselian glaciation [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF] indicates that in that period the first ~300 km of the upper mantle is affected by the Weichselian glaciations. In addition, our stress distribution profiles calculated for both glaciation events show that at depths of 100 km for the recent glaciation event and at 300 km for the Weichselian glaciation event the upper mantle is still subjected to some loading induced stresses. At larger depths for both events the stresses decrease towards zero.

We therefore have calculated average viscosities from depths below the lithospheric thickness to a depth of 100 km for recent glacial isostatic adjustments on Iceland and from depths below the lithospheric thickness to a depth of 300 km for the glacial isostatic adjustment during the Weichselian period. Maximum depths of the upper mantle affected by the glaciation events are approximations. Establishing the exact depth-sensitivity requires more detailed computations using for example Frechet kernels [START_REF] Mitrovica | A complete formalism for the inversion of post-glacial rebound data: Resolving power analysis[END_REF]. Horizontally averaged viscosity profiles and vertically averaged viscosity trends have been produced for both glaciation events. Single values representing the average viscosities and lithospheric thicknesses under both icecaps have been determined for comparison with glacial isostatic

Results

The rheological modelling for the Iceland mid-ocean ridge shows that diffusion creep dominates the deformation in the upper mantle (Fig. 3). Only at shallow depths under the mid-ocean ridge a small region can occur where dislocation creep dominates the deformation. This is only the case for the Weichselian glaciation event. All deformation during the recent glaciation event is controlled by diffusion creep according to our model. As a result of the dominance of diffusion creep under Iceland most of the upper mantle experiences stressindependent viscosities and therefore the Weichselian and recent glaciation events produced almost identical viscosity fields within the upper mantle (Fig. 4). Significant variations in upper mantle viscosity occur during both glaciation events both with depth and with increasing distance from the ridge axis (Fig. 4). The radial variations in viscosity are characterised by a decrease in viscosity up to depths of ~70 km caused by the rapid increase in temperature with depth and the presence of a partially molten region to depths of ~70 km (Fig. 2b). Below that the radially averaged viscosity (Fig. 5) increases from depths >70 km to 410 km caused by the increased importance of the pressure dependence of the rheology (Fig. 4 and 5), while temperatures are relatively constant at those depths (Fig. 1).

The microphysical modelling shows a strong, but narrow low-viscosity layer for the Weichselian glaciation at depths of ~30-70 km (Fig. 6) due to the influence of the melt content on viscosity and the increase in temperature close to reaching the adiabatic temperature gradient. The decrease in viscosity in this layer due to the presence of the melt is ~1-2 orders of magnitude in comparison with the viscosities in a melt-free upper mantle. The low viscosity zone for the recent glaciation starts even shallower (~15 km) due to closer A c c e p t e d M a n u s c r i p t 11 proximity of the Vatnajökull glacier to the mid-ocean ridge axis where higher temperatures prevail (Fig. 6). A wet upper mantle (C OH = 100 H/10 6 Si) results in a reduction of the viscosity of around 1 order of magnitude in comparison with a dry upper mantle (Fig. 5B and5D). The melting temperatures for a wet upper mantle solidus are lower than those for a dry upper mantle (Fig. 1) causing the partially-molten region, and hence melt-induced lowering of the viscosity, to extend to larger distances away from the mid-ocean ridge axis within a wet upper mantle (Fig. 1 and Fig. 5). Plume activity within the upper mantle under Iceland may have resulted in lower upper mantle viscosities than in a normal mid-ocean ridge scenario.

Higher temperatures in the range of T of 100-200 °C have been postulated for Iceland (e.g. [START_REF] Ito | Dynamics of mantle flow and melting at a ridge-centered hotspot: Iceland and the Mid-Atlantic Ridge[END_REF][START_REF] Ito | Mantle flow, melting, and dehydration of the Iceland mantle plume[END_REF][START_REF] Maclennan | Plume-driven upwelling under central Iceland[END_REF][START_REF] Ruedas | Temperature and melting of a ridge-centred plume with application to Iceland. Part I: Dynamics and crust production[END_REF][START_REF] Putirka | Mantle potential temperatures at Hawaii, Iceland, and the mid-ocean ridge system, as inferred from olivine phenocrysts: Evidence for thermally driven mantle plumes[END_REF][START_REF] Schmeling | Crustal accretion and dynamic feedback on mantle melting of a ridge centred plume: The Iceland case[END_REF]. We have simulated the effect of additional plume activity on Iceland by increasing the mantle temperatures at the bottom of time-dependent half-space cooling model (Turcotte and Schubert, 2002) with 100 and 200 °C to T m = 1500 °C or 1600 °C (Fig. 7). The higher temperatures with the mantle due to the plume lower the viscosities by ~0.6 orders of magnitude per 100 °C excess temperatures (Fig. 6 and8).

The vertical temperature gradients and resulting viscosity gradients underneath a midocean ridge result in a large increase in lithospheric thickness (Fig. 5A (Fig. 5). In the case of a wet upper mantle, average upper mantle viscosities range are predicted to be between 3x10 18 Pas and 7x10 18 Pas during the Weichselian glaciation event and 7x10 17 Pas and 7x10 18 Pas for the recent glaciation event (Fig. 5).

To compare the results of this microphysical modelling with estimates of constant viscosity and constant lithospheric thickness from glacial isostatic adjustment studies for Iceland we have produced a single upper mantle viscosity value and lithospheric thickness value for each of the two glaciation events (averaging out all lateral variations in viscosity and thickness). We have used the range in lateral extent for the Weichselian glaciation event underneath all of the land mass of Iceland [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF] viscosities to be around 10 18 Pas or lower (Fig. 9). The average lithospheric thicknesses for both glaciations differ depending on which method is used to define the lithospheric thickness. For the Weichselian glaciation, the average lithospheric thickness in a dry upper mantle are either 100 km or ~ 30 km depending on the definition used and ~40 km or ~10 km for the recent glaciation event (Fig. 9). Additional plume activity or a wet upper mantle scenario reduce the lithospheric thickness estimates considerably. Pas) in comparison with studies of continental areas such as Fennoscandia. This is due to the relatively high temperature conditions under the newly formed oceanic lithosphere of Iceland.

Discussion

There is a significant spread in upper mantle viscosity estimates for Iceland between 10 17 Pas and 10 20 Pas and in elastic lithospheric thickness estimates between 10 and 80 km amongst the different studies (Fig. 10) over the past two decades [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF][START_REF] Sigmundsson | Glacio-isostatic crustal movement caused by historical volume change of the Vatnajökull ice cap, Iceland[END_REF]Thoma and Wolf, 2001;[START_REF] Kaban | Nature of the crust-mantle transition zone and the thermal state of the upper mantle beneath Iceland from gravity modeling[END_REF][START_REF] Sjöberg | Land uplift near Vatnajökull, Iceland, as observed by GPS in 1992, 1996 and 1999[END_REF][START_REF] Kumar | The lithosphere-asthenosphere boundary in the North-West Atlantic region[END_REF][START_REF] Pagli | Glacio-isostatic deformation around the Vatnajökull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling[END_REF][START_REF] Biessy | Fast and partitioned postglacial rebound of southwestern Iceland[END_REF][START_REF] Árnadóttir | Glacial rebound and plate spreading: results from the first countrywide GPS observations in Iceland[END_REF][START_REF] Breton | Post-glacial rebound of Iceland during the Holocene[END_REF]. The most recent glacial isostatic adjustment studies that provide estimates of both upper mantle viscosity and lithospheric thickness for Iceland [START_REF] Pagli | Glacio-isostatic deformation around the Vatnajökull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling[END_REF][START_REF] Árnadóttir | Glacial rebound and plate spreading: results from the first countrywide GPS observations in Iceland[END_REF] are compared with estimates from our study. The best correspondence for the recent glaciation event of both the elastic thickness and average upper mantle viscosity estimates of this study is with the study of Árnadóttir et al. (2009). An average viscosity in the range of 6x10 18 Pas to ~2x10 19 Pas and an elastic thickness of 40 km (Árnadóttir et al., 2009) is only approached by the results of this study for a dry upper mantle scenario without any additional plume activity (average viscosity ~5x10 18 Pas, Fig. 9). All other scenarios produce too low average viscosity values for the recent glaciation event (Fig. 9). Using the depth at which the viscosity falls below 10 20 Pas as the definition for the lithospheric thickness produces a thickness of ~10 km (Fig. 9), which is considerably below the thickness preferred by Árnadóttir et al. (2009). This suggests that using the definition of the depth at which the average viscosity is lower than 10 23 Pas results in more realistic estimates of lithospheric glaciation event (Fig. 9). Our viscosity estimates are in line with estimates of glacial isostatic adjustment studies for the Weichselian deglaciation [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF][START_REF] Biessy | Fast and partitioned postglacial rebound of southwestern Iceland[END_REF][START_REF] Breton | Post-glacial rebound of Iceland during the Holocene[END_REF], although here significantly larger lithospheric thickness values (~100 km) are predicted for the Weichselian glaciation event than the ~15 km lithospheric thickness predicted by [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF]. A correspondence of the upper mantle viscosities estimates of this study can also be achieved with the study for the recent glaciation event of [START_REF] Pagli | Glacio-isostatic deformation around the Vatnajökull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling[END_REF]. A mean upper mantle viscosity of ~10 18 Pas of [START_REF] Pagli | Glacio-isostatic deformation around the Vatnajökull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling[END_REF] would imply that the upper mantle under Iceland is most likely wet (Fig. 9), as a dry upper mantle produces a higher average viscosity of ~5x10 18 Pas. However, the lithospheric thickness estimates of 10-20 km of [START_REF] Pagli | Glacio-isostatic deformation around the Vatnajökull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling[END_REF] do not correspond to our estimates of elastic thickness in a wet upper mantle of ~30 km. Upper mantle viscosity values in the range of 10 18 Pas can also be achieved by the presence of a plume under Iceland affecting the viscosities of the mantle. This possibility can not be excluded in the comparison with [START_REF] Pagli | Glacio-isostatic deformation around the Vatnajökull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling[END_REF], but a plume can not explain the viscosity estimates of Árnadóttir et al. (2009). Upper mantle viscosity estimates in the range of 10 17 Pas (e.g. [START_REF] Sjöberg | Land uplift near Vatnajökull, Iceland, as observed by GPS in 1992, 1996 and 1999[END_REF] or as high as 10 20 Pas are not preferred by Árnadóttir et al (2009) since they do not fit with recent GPS uplift data for Iceland. Using our modeling approach, very low viscosities values around 10 17 Pas can only be produced with either a very wet upper mantle (C OH ≥ 1000 H/10 6 Si) or with very high deep mantle temperatures due to additional plume activity under Iceland (T m > 1600 °C). Upper mantle viscosity estimates around 10 20 Pas or even higher can on the other hand only be achieved by having very large olivine grain sizes around 5-10 mm in a dry upper mantle, which is much larger than the average grain size present at mid-ocean ridges (e.g. [START_REF] Dijkstra | Microstructures and lattice fabrics in the Hilti mantle section (Oman Ophiolite): Evidence for shear localization and melt weakening in the crust-mantle transition zone[END_REF]. Since By comparing the glacial isostatic adjustments studies of [START_REF] Pagli | Glacio-isostatic deformation around the Vatnajökull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling[END_REF] and Árnadóttir et al. (2009) with this study using a different approach, similar and relatively low average upper mantle viscosity estimates for the mid-ocean ridge under Iceland can be obtained. However, this study shows that lateral variations in upper mantle viscosity and lithospheric thickness are likely to occur under an active tectonic setting such as the Icelandic mid-ocean ridge. A large heterogeneity in temperature distribution within the upper mantle causes large lateral and radial variability in viscosity. So far, glacial isostatic adjustment studies have neglected any lateral variability in their geophysical models of the upper mantle (e.g. [START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF][START_REF] Sigmundsson | Glacio-isostatic crustal movement caused by historical volume change of the Vatnajökull ice cap, Iceland[END_REF][START_REF] Sjöberg | Land uplift near Vatnajökull, Iceland, as observed by GPS in 1992, 1996 and 1999[END_REF][START_REF] Fleming | Glacial-isostatic adjustment and the viscosity structure underlying the Vatnajökull ice cap, Iceland[END_REF][START_REF] Árnadóttir | Glacial rebound and plate spreading: results from the first countrywide GPS observations in Iceland[END_REF]). This study shows that glacial isostatic adjustment studies of Iceland, particularly concerning the Weichselian glaciation event, should in the future incorporate such lateral and radial variability in viscosity and lithospheric thickness.

The modelling of a mid-ocean ridge setting assumes a relatively thin crust of <10 km under Iceland. Different observations of the thickness of the crust under Iceland are reported with thickness varying between a thin crust scenario of 10-15 km and a thick crust scenario of 20-40 km (summarised by [START_REF] Björnsson | Temperature of the Icelandic crust: Inferred from electrical conductivity, temperature surface gradient, and maximum depth of earthquakes[END_REF]. If, in the thick crust scenario, the temperature distribution of the thickened crust is the same as the temperature distribution would be in a time-dependent cooling lithosphere, then the mechanically defined elastic lithosphere values reported here would still be valid estimates. If, on the other hand, the thickened crust under

Iceland is also significantly cooler, then the lithospheric thicknesses of 25-40 km obtained in this study will be too small. The thin crust scenario, also favoured by [START_REF] Björnsson | Temperature of the Icelandic crust: Inferred from electrical conductivity, temperature surface gradient, and maximum depth of earthquakes[END_REF] for Iceland, is in line with the mid-ocean ridge model used in this study.

In conclusion, based on the closest match of upper mantle viscosity and lithospheric thickness between Árnadóttir et al. (2009) and this study, upper mantle viscosity estimates around 5x10 18 Pas of this study for the recent glaciation event are preferred, implying that a dry upper mantle is most likely present under Iceland. A dry upper mantle scenario is in contrast with the conclusion drawn from a similar rheological modelling study for Scandinavia and where a wet upper mantle was concluded to be most likely present (Barnhoorn et al., in press;[START_REF] Schotman | Constraints on shallow low viscosity zones in Northern Europe from future GOCE gravity data[END_REF]. The different outcomes may point to lateral variations in water content within the upper mantle e.g. between continental lithosphere and oceanic lithosphere.

Conclusions

Figure captions

  the spreading rate, geotherms for the oceanic lithosphere between 0 and 20 Ma (Fig.1A) and a two-dimensional cross-section of the temperature distribution under the Icelandic mid-ocean ridge (Fig.1B) are constructed. The cross-section encompasses the first 200 km on either side of the ridge axis corresponding to the first 20 Ma of oceanic lithosphere generation. Mantle temperatures at the base of the modelled range are set at 1400, 1500 and 1600 °C to reflect the temperature distribution in a normal mid-ocean ridge setting (T m = 1400 °C;Turcotte and Schubert, 2002) and settings where a plume with excess temperatures for Iceland of 100-200 °C (e.g.

  lithosphere below 10 25 Pas. Since relaxation times for the Icelandic ice caps are much smaller than the relaxation times for Scandinavia (hundreds of years compared to tens of thousands of years), we have selected a viscosity cut-off value of 10 23 Pas to represent the maximum viscosity value at which viscous deformation contributes to glacial isostatic adjustment due to recent melting on Iceland. We have determined the lithospheric thickness for any distance A c c e p t e d M a n u s c r i p t 9 away from the ridge axis in two ways: 1) as the depth at which the average viscosity from the top surface of the Earth to that depth is less than 10 23 Pas, 2) as the depth at which the viscosity falls below 10 20 Pas. We have chosen a maximum viscosity value of 10 20 Pas sinceÁrnadóttir et al. (2009) state that in their GIA model the lithosphere behaves essentially elastic above 10 20 Pas. The lithospheric thickness varies in both cases perpendicular to the ridge axis. Below the lithosphere, the upper mantle is deforming viscously and the average viscosity values reported here are the averages of the viscosity values below that lithospheric thickness. The radius of ~50 km for the Vatnajökull glacier present nowadays

  Iceland. Those studies usually assume that the upper mantle under Iceland only has one viscosity and lithospheric thickness without any lateral variations.

  Fig.5shows that defining the lithosphere as the depth at which the average viscosity is lower than 10 23 Pas produces thicknesses significantly larger than defining the lithosphere as the depth at which the viscosity is lower than 10 20 Pas (~130-150 km at a distance of 200 km

  and the lateral extent of the Vatnajökull glacier for the recent glaciation event to determine the average viscosities and lithospheric thicknesses. Maximum viscosity values in a dry upper mantle under the recent Vatnajökull glacier are around 5x10 18 Pas and would have been around 10 19 Pas during the Weichselian glaciation. Plume activity and/or a wet upper mantle scenario reduce average

  in this study predicts the lithospheric thickness and upper mantle viscosities under Iceland using a theoretical temperature distribution under mid-ocean ridges combined with the rheological behaviour of olivine. It results in estimates that are in line with estimates from glacial isostatic adjustment studies and seismological studies. All studies show smaller elastic thicknesses (<100 km) and relatively low viscosity values (< 10 20

  microphysical study. Using the preferred dry upper mantle scenario with a lithospheric thickness definition of an average viscosity smaller than 10 23 Pas results in an average viscosity of 10 19 Pas and an lithospheric thickness of ~100 km for the Weichselian

  or very high viscosity estimates do not fit with the most recent uplift rate scenarios(Árnadóttir et al., 2009), a wet and/or hot upper mantle scenario or a very large grain size in combination with a dry upper mantle scenario are not likely to be present in the upper mantle under Iceland.

  Microphysical modelling using the temperature distribution under mid-ocean ridges combined with olivine diffusion and dislocation creep flow laws with an olivine grain size of 2 mm and stress distributions for the upper mantle produced lithospheric thickness and upper mantle viscosity estimates for the upper mantle under Iceland during the Weichselian glaciation event and the recent glaciation event which nowadays forms the Vatnajökull glacier. Using the most recent estimates of Árnadóttir et al. (2009) as best estimates for upper mantle viscosity and lithospheric thickness, the results of this study that best match the estimates of Árnadóttir et al. (2009) (viscosity ~5x10 18 Pas and thickness of ~40 km) suggest that the upper mantle under Iceland is most likely dry. Also, the presence of a plume under Iceland can not explain the recent viscosity values reported for Iceland. Glaciation during the Weichselian glaciation event should have produced viscosity values ~10 19 Pas and a lithospheric thickness of ~100 km for Iceland based on this microphysical modelling study. The Weichselian viscosity A c c e p t e d M a n u s c r i p t 18 estimate is in line with GIA studies[START_REF] Sigmundsson | Post-glacial rebound and asthenosphere viscosity in Iceland[END_REF][START_REF] Biessy | Fast and partitioned postglacial rebound of southwestern Iceland[END_REF][START_REF] Breton | Post-glacial rebound of Iceland during the Holocene[END_REF]). However, this study predicts a significantly larger Weichselian lithospheric thickness due to the lateral variability of the lithospheric thickness in the modelling. Because the deviatoric stress level is mostly below 1 MPa, dislocation creep only dominates in a small area underneath the ridge axis. However, viscosity between the two Weichselian and recent glaciations still differs because of the depth-sensitivity which depends on the size of the ice caps. Large lateral variations in upper mantle viscosity and lithospheric thickness are present perpendicular to the ridge axis with a lithospheric thickness in excess of 100 km at 200 km from the ridge axes.

  ., Wolf, D., 2001. Inverting land uplift near Vatnajökull, Iceland, in terms of lithosphere thickness and viscosity stratification. In: Sideris, M.G. (Ed.), Gravity, Geoid and Geodynamics 2000 (Springer, Berlin), IAG Symposia 123, 97-102. Turcotte, D.L., Schubert, G., 2002. Geodynamics 2 nd edition. Cambridge University Press, 456 pp. Van der Wal, W., Wu, P., Wang, H., Sideris, M.G., 2010. Composite rheology in glacial isostatic adjustment modeling, Journal of Geodynamics, 50(1), 38-48, doi:10.1016/j.jog.2010.01.006. Wolfe, C.J., Bjarnason, I.T., VanDecar, J.C., Solomon, S.C., 1997. Seismic structure of the Iceland mantle plume. Nature 385, 245-247.

Figure 1 :

 1 Figure 1: A) Geotherms (black lines) for oceanic lithospheres with an age between 0 and 20

Figure 2 :

 2 Figure 2: Averaged von Mises equivalent stress for the upper mantle under Iceland at seven

Figure 3 :

 3 Figure 3: Dominant deformation mechanism map for the Weichselian glaciation event in a

Figure 4 :

 4 Figure 4: Viscosity distribution under Iceland for a dry and wet (C OH = 100 H/10 6 Si) upper

Figure 5 :

 5 Figure 5: Elastic thickness (A and C) and mean upper mantle viscosity (B and D) for Iceland for a dry and wet (C OH = 100 H/10 6 Si) upper mantle at melt contents of 0 and 4% during the

Figure 6 :

 6 Figure 6: Horizontally averaged upper mantle viscosity profiles of the upper mantle for the

Figure 7 :

 7 Figure 7: Viscosity distribution under Iceland for a dry, melt-free upper mantle at excess

Figure 8 :

 8 Figure 8: Elastic thickness (A and C) and mean upper mantle viscosity (B and D) for Iceland

Figure 9 :

 9 Figure 9: Mean lithospheric thickness (mean of all thicknesses under the glaciated areas) and

Figure 10 :

 10 Figure 10: Compilation of lithospheric thickness and upper mantle viscosity estimates for

  Figure 2

  Figure 3

η

  Figure 4

  Figure 5

  Figure 6

  Figure 7

η

  Figure 8

  Figure 9

Acknowledgements

Financial support from the Netherlands Organisation for Scientific Research (NWO) and TOPO-Europe is acknowledged. Constructive reviews by two reviewers significantly improved the manuscript.


Can theorems help treat cancer?

Jean Clairambault 1 INRIA Paris-Rocquencourt, BANG project-team, BP 105, F78153 Le Chesnay Cedex and Laboratoire Jacques-Louis Lions, UPMC, 4 Place Jussieu, F75005 Paris

Can mathematicians who prove theorems be useful to physicians who treat patients with cancer? Can mathematics have an impact on cancer biology? Is mathematics more than just a tool in the hands of [a few open-minded] physicians willing to understand and treat cancer from an innovative point of view? ...and is mathematical biology more than just a toy in the hands of mathematicians? I imagine that most readers of this journal would join me in answering "yes" to these questions. But let me be more explicit in my questioning: Is there a place for a very abstract theoretical activity, as is proving theorems, in a team of physicians daily committed to treating human diseases (in particular cancer)? Can these physicians see the contribution of mathematicians to their difficult task as more than helping them to "think differently" (as some of them state it), but further as actually taking part in a therapeutic decision? As regards mathematical biology, when it is applied to medical questions, do we mathematicians feel enough committed to go beyond a few publications in the numerous journals dedicated to applications of maths to biology, and to collaborate with medical teams in a long term effort to improve in common medical treatments? Building on recent articles [START_REF] Clairambault | Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments[END_REF][START_REF] Clairambault | Optimising cancer pharmacotherapeutics using mathematical modelling and a systems biology approach[END_REF] in which I have outlined technical ways by which one can achieve a working collaboration between mathematicians and physicians toward such mutual usefulness, I will explain briefly below why the answers to these questions should be "yes".

At present, biologists and physicians do not look to mathematicians for theorems. An obvious reason for this is the completely different status of knowledge in mathematics and biology and medicine. Unlike mathematics, biology is a field where little is known for certain, for the very nature of the phenomena under study is often inaccessible, and deductive reasoning about uncertain facts of questionable value. Moreover, theorems as such do not exist in biology, although it is not impossible that theorems proven in a mathematical framework were inspired by problems posed initially as biological questions. The clinical relevance of such theoretical results is, however, seldom clear. More precisely, if our aim is to identify optimal schedules for the treatment of cancers by combining different drugs and taking account of cell cycle phase times, it is possible theoretically to determine optimal schedules for the delivery of drugs (with at least a theorem of existence for such an optimal schedule, and numerically determined schedules, as proposed for instance in [1]). In practice it is not easy to establish whether the assumptions that must hold for the theorems and optimal solutions to be valid are fulfilled -for they rely on the measurement of intracellular quantities that are difficult, or even impossible, to obtain. As a result, the practical application of mathematical results may depend upon biological 'facts' that, as is usual in biology, may be very likely, but are seldom certain.

This does not, however, mean that theorems are of no use to physicians. Firstly measurements that can not be made today may tomorrow become routine (consider, for example, the determination of cell cycle phases in individual proliferating cells using fluorescence markers [START_REF] Sakaue-Sawano | Tracing the silhouette of individual cells in S/G2/M phases with fluorescence[END_REF], which until recently was unimaginable). Secondly, when presented with physiologically based sets of competing hypotheses, mathematicians can use theoretical models to compare scenarios and to identify those which best (in some sense) reproduce the biological system under investigation. For instance, in a therapeutic setting, theoretical exploration of alternative model scenarios may reveal that a given treatment strategy must be forsaken in favour of one which always yields better results.

The impact of mathematics in cancer medicine is at its best when it is based on genuine collaborations between mathematicians and clinicians and typically operates in one of two different ways. On the one hand, we have model-based predictive representations of tumour growth without treatment, or of its molecular counterparts at the intracellular level -this I will term the physicist's, or descriptive, point of view. On the other hand we have theoretical optimisation of cancer treatments -which I will call the physician's, or interventionistic, point of view. These alternative directions reflect the fact that cancer medicine does not deal only with the so-called 'natural history of cancers' (an expression borrowed from the vocabulary of medical schools, which means the description of cancer progression in the absence of medical intervention); the obvious main objective of medicine is to heal and, where possible, to cure patients with cancer.

This alternative is not a philosophical or moral consideration about a right or wrong way to act; it is plainly practical. It would certainly be delusional to hope to describe normal physiology before considering perturbations associated with diseased states, since medical progress has never proceeded in this way: physiology has always made its major advances by exploiting insight from pathology. It is indeed from neuropathological observations that both Broca and Wernicke, long before functional cerebral imagery could even be dreamt of, discovered the cerebral areas of language that now bear their names. In the same way, in my opinion, it would be equally delusional, mutatis mutandis, to hope to describe diseases in an efficient manner, from a medical viewpoint, i.e., with the aim to cure patients, without taking account of previous successes and failures of [mainly pharmacological] treatments -a view already expressed in [START_REF] Bernard | Introduction à l'étude de la médecine expérimentale[END_REF] -, which may be considered as perturbations of 'natural' physiopathology. It is only in the limited field of biology considered independently of medicine (which some biologists even regard as a plain "applied branch" of their discipline, a rather extreme attitude) that the first strictly descriptive viewpoint may, to an equally limited extent, be justified in cancer modelling.

Defining a formalism that will help mathematicians better to understand the phenomena at stake in cancer (cautiously avoiding claiming to 'explain' these phenomena to biologists and physicians, an attitude that is seldom appreciated by them and will very likely spoil a working relationship; this is a personal viewpoint also presented in [START_REF] Clairambault | Commitment of mathematicians in medicine. A personal experience, and generalisations[END_REF]) is thus more efficient when it is firstly thought of as 'what to do?', i.e., modelling in order to control what and by which means on which targets. Most oncologists are sensitive to this argument, at least those who consider mathematicians as possible contributors to therapeutic improvements. Among the others, the "physicist's viewpoint" may be more popular, limiting collaborative work to the descriptive biology of cancer, which may also, of course, have therapeutic consequences, predicted or not, and will certainly not be neglected by physicians. These arguments lead me to answer "yes" to the second question too: either directly through active collaboration with oncologists to optimise cancer treatments, or indirectly, by focussing on cancer biology, mathematicians can certainly have a positive impact on physicians dealing with cancer, making them think differently and possibly opening new therapeutic tracks.

In the same way, I will answer the third question positively: mathematics is not a tool in the hands of physicians, for they deal with mathematicians, rather than mathematics. With mutual interest on both sides, both increased understanding and practical improvements can emerge from such collaborations. As regards the attitude of mathematicians themselves to mathematical biology (fourth question), I would say that it depends primarily upon the attitude of the mathematicians: either they use mathematical biology as a source of mathematical problems and care little about applications, or they actively seek out like-minded physicians in order to address problems of genuine biomedical importance, to cast them in a mathematical framework and to ensure that the impact of theoretical solutions is fully exploited for practical, therapeutic purposes. The latter attitude is the most demanding in terms of commitment and patience, but the one that will bear the richest fruits for both parties. Needless to say also, a third way that involves conjugating where possible, the search for nice mathematical problems with a commitment to therapeutic enterprises, without opposing them, is clearly the most satisfying for the mathematician and for the physician alike.

E-mail: jean.clairambault@inria.fr