This text is the english adaptation of the contribution by Olivier Gagliardini to the French book on Louis Lliboutry "Le Champolliion des Glaces", by Marc Turrel and edited by UGA Editions.

Glacier sliding: a contemporary topic in glaciology for which Louis Lliboutry was precursor.

Olivier Gagliardini, Professor UGA, August 2017 (english translation, June 2019 - special thanks to Nathan Maier for smoothing the text)

It is said that working to understand the processes that control the basal sliding of glaciers occupied the majority Louis Lliboutry’s life. This work began in 1958, the same year the Grenoble Glaciology Laboratory was created, when Lliboutry published three short papers in the famous French Comptes Rendus des séances de l’Académie des Sciences. Louis Lliboutry would go on to publish more than 20 articles on this subject, with the last being in 2002. I had the chance to attend one of his last, if not his last, scientific presentation which was still about the sliding of glaciers. The presentation was given at the annual meeting Société Hydrotechnique de France (SHF) held on the campus of Grenoble on March 9 and 10, 2005. I remember the contents of his presentation were far too complex and varied to really be explained within the 15 minutes of allotted time. However, he started his presentation with a scathing remark: “we remain captive to the crude, incomplete, and erroneous theories developed in the 60s by Weertman, myself, Nye, Röthlisberger, and Budd, which should only be of a historical interest today." [0] He then told the audience he was going to present a new view on glacier sliding disseminated within three articles to be submitted to the Journal of Glaciology. In 2007, the year of his death, I published my first contribution on the subject, however, I still wondered what had become of the three articles Lliboutry promised to publish. As more time passed, I eventually forgot about them. Recently I checked to see if any of these three articles had been submitted to the Journal of Glaciology: none have.

As suggested by Andrew Fowler in his 2010 article titled "Weertman, Lliboutry and the development of sliding theory" [1], Louis Lliboutry’s first articles on the subject in 1958 [2,3,4] were surely written in response to an article on glacier sliding published in 1957 [5] authored by the American theoretician Johannes Weertman. In this article, Johannes Weertman proposes that sliding at the base of glaciers results from two phenomena. The first is the melting and refreezing of the ice around the bed roughness (bumps on the bed), and the second is the viscous deformation of ice around the bed roughness. Because of the existence of a lubricating water film between the ice and the bed, these processes allow the ice to slip past its base and slide. For these two processes, Johannes Weertman derived a mathematical expression relating the sliding speed and the shear stress at the base of the glacier, which depended only on the geometric parameters of the bedrock roughness, the thermal parameters of the ice and the rock, and the rheology of ice.

Thanks to the observations in his possession at this time, Louis Lliboutry emphatically noted that such a relationship could not explain the strong variations in glacier flow velocity. The shear stresses at the base of the glacier, which are also important to sliding speeds, varied too little, and compared to observed sliding speeds, the sliding speed values given by the Weertman model were much too low. He then presented a visionary view of the role of glacial water in sliding, where high water pressures at the glacier base has the ability to open of cavities on the downstream side of bedrock bumps. He explained that because the stresses exerted by the ice on the bedrock bumps are lower than on downstream side compared to upstream side, when the pressure of the basal water increases, water cavities can open on the downstream side of the bedrock bumps and "erase" a part of the bedrock roughness. The higher the water pressure, the larger the cavities. Cavitation lowers the resistance to basal sliding and changes sliding speeds, and as such, Lliboutry’s theory also explained the acceleration of glaciers observed at the beginning of the summer season on a seasonal scale or at the beginning of the afternoon at the daily scale. Therefore Lliboutry found that Johannes Weertman’s proposed relationship between slip velocity and basal shear stress was too simple because it didn’t include effective pressure, defined as the difference between ice pressure and water pressure at the base of the glacier.

The problem of glacier sliding is unfortunately much more complex than Lliboutry stated since the geometry of the water cavities cannot be known a prior and must be determined for each effective pressure. Nevertheless in 1958, Louis Lliboutry already showed that in the presence of water cavities, the stress at the base must decrease if the sliding speed and / or the water pressure increase. He found the relationship between basal stress and sliding speeds is no longer direct, and the basal stress can correspond to several sliding speeds given the degree of cavitation: before the opening of cavities, the relation between the velocities and the basal stress is increasing, then when the cavities open, it is decreasing. In 1981 the Swiss glaciologist Almut Iken [6] found the limit for the top of the shear stress-sliding curve. She showed that the ratio of the basal shear stress to the effective pressure must be limited and not exceed the maximum value of the slope of the roughness of the bedrock. It was then necessary to wait until 2005 for theoretical work of Christian Schoof, of the University of British Columbia in Vancouver, to obtain a mathematical form for this law in the particular case of a linear ice rheology [7]. In 2007, thanks to the finite element method, we were able to generalize this law for nonlinear rheologies [8]. Nevertheless, all these formulations remain incomplete since they assume that the water pressure is uniform and constant at the base of the glacier. This point was already noted by Louis Lliboutry while he was rejecting previous sliding theories in his 2005 presentation to the SHF: "They [these theories] are particularly unaware that the subglacial water pressure varies continuously and is not uniform under all of the glacier." To this day and to my knowledge, no article takes into account the spatial and temporal variability of the water pressure, except those of Louis Lliboutry written in the last years of his life. Unfortunately, he will never be able to submit these articles to the Journal of Glaciology.

The contributions of Johannes Weertman in 1957 and Louis Lliboutry in 1958 have initiated many works on glacier sliding, including experimental, theoretical, and numerical approaches. However, no new processes have actually been proposed to explain the sliding of a glacier on hard bedrock. In fact, it is astounding that the conceptual basis for glacier sliding on bedrock was laid out more than 60 years ago, but we still know so little about glacier basal friction. Overall, the work over the last six decades which followed these two contributions focused "only" on refining the mathematical formulations linking sliding velocities, basal ice stresses, and water pressures. The same is true for Louis Lliboutry’s work, where he spent the rest of his career proposing a long series of formulas, more or less restricted to particular cases of glacier sliding, in his many papers that followed those of 1958.

Even today where the form for the law linking sliding velocity, basal ice stresses and water pressure has emerged, the physical parameters entering into this law remain poorly known. This lack of knowledge comes from the fact that observing the base of glaciers remains a very difficult task. Nevertheless, very recently a series of numerical and observational tools have emerged which suggest imminent discoveries in the study of basal friction are coming. On the numerical side, the inverse methods make it possible to reconstruct the basal friction using only measurements of glacier surface velocities, obtained for example, by satellite measurements. In addition, our understanding of glacial hydrology and recently developed models provide an estimate of the distribution of water pressures at the base of the glacier. On the observation side, recent work using seismometer networks deployed on and around glaciers has shown that glacier basal slip is not limited to the creep processes described by Johannes Weertman and Louis Lliboutry, but is also related to sudden sliding events (called "stick slip") similar to those observed on faults of the Earth’s crust. These same seismic observations also make it possible to "listen" to the noise generated by the hydrological network at its base, and thus to determine the geometric and pressure characteristics of the subglacial drainage network.

In 2005, Louis Lliboutry concluded his speech at the SHF, noting, "There is still a lot of numerical work to be done. I leave it to the younger researchers, who are not afraid by the fact that the theme is not fashionable." On this last point he was wrong: the societal issues associated with rising sea levels driven by mass loss from glaciers and ice sheets has brought this subject "back to fashion" and has also contributed to the emergence of the new theoretical, numerical, and experimental approaches which have been developed over the last ten years. Our understanding of friction at the base of glaciers is far from complete, but it rests on solid foundations whose first stones were laid by Louis Lliboutry in 1958 and continued throughout his life.

[0] All citations in blue italics are from the 6-page summary that Louis Lliboutry distributed on March 10, 2005 for his presentation at the SHF conference in Grenoble, and whose title is "Glissement et hydraulique sous-glaciaires" (Subglacial sliding and Hydraulics). The english translation of this 6-page manuscript is given in the Appendix of Roldan Balsco PhD thesis.

[1] Fowler, A. 2010. Weertman, Lliboutry and the development of sliding theory. J. Glaciol., 56(200), 965–972.

[2] Lliboutry, L. 1958a. Ondes cinématiques sur un glacier et glissement sur le lit. C. R. Hebd. Séances Acad. Sci., 247(2), 114–116.

[3] Lliboutry, L. 1958b. Frottement sur le lit et mouvement par saccades d’un glacier. C. R. Hebd. Séances Acad. Sci., 247(2), 228–230.

[4] Lliboutry, L. 1958c. Contribution à la théorie du frottement du glacier sur son lit. C. R. Hebd. Séances Acad. Sci., 247(3), 318– 320.

[5] Weertman, J. 1957. On the sliding of glaciers. J. Glaciol., 3(21), 33–38.

[6] Iken, A. 1981. The effect of the subglacial water pressure on the sliding velocity of a glacier in an idealized numerical model. J. Glaciol., 27(97), 407–421.

[7] Schoof, C. 2005. The effect of cavitation on glacier sliding. Proc. R. Soc. London, Ser. A, 461(2055), 609–627.

[8] Gagliardini O., D. Cohen, P. Råback and T. Zwinger, 2007. Finite-Element Modeling of Subglacial Cavities and Related Friction Law. J. of Geophys. Res., Earth Surface, 112, F02027.