Brief Historical Tour of Glacier Ice on Earth and its Role in Climate Dynamics1
Kolumban Hutter1*, Dietmar Gross1
Department of Mechanics, Darmstadt University of Technology, Darmstadt, Germany
*Corresponding author: Kolumban Hutter, Department of Mechanics, Darmstadt University of Technology, Darmstadt, Germany. Email: hutter@vaw.baug.ethz.ch
Received Date: 18 June, 2019; Accepted Date:
17 July, 2019; Published Date: 25 July,
2019
Citation: Hutter K and Gross D
(2019) Brief Historical Tour of Glacier Ice on Earth and its Role in Climate
Dynamics. J Earth Environ Sci 7: 174. DOI: 10.29011/2577-0640.100174
Abstract
The science of the physics of ice crystals started almost 500 years ago with Cardano, followed by Kepler, Hooke, Dalton, the two Braggs (father and son) and Pauling. Here, we are concerned with naturally formed ice of glaciers, ice sheets and ice shelves. We do not discuss lake, river, sea and atmospheric ices, even though they equally play a role in today’s General Circulation Models (GCMs) for the Earth.
The study of glacier ice started with the observation of the behavior of the dynamics of Alpine glaciers, specifically that they are not rigid, but moving bodies that deform. A ladder, left in 1788 at the icefall of the Col de Géant by de Saussure was 44 years later found at the three-glacier merge of the Mer de Glace, corresponding to a mean velocity of 375 feet/year. Hugi and Agassiz measured the motion of rocks on the middle moraine of the Unteraar-gletscher and found similar values. However, at early times their location of origin was an enigma; so they were named erratic. Forbes and Tyndall found by position measurements with theodolites that velocities of glacier surface objects are (i) considerably larger in summer than in winter; (ii) larger on the glacier surface than at depth and (iii) larger in the middle of the glacier-width than close to the boundaries, as reported by Helmholtz (1865), [77]. On this basis Rendu and Forbes were the first to identify similarities of glacier flows with streams of very viscous fluids. It lasted until the 1950s when physicists had postulated the constitutive equation for isotropic ice as a non-Newtonian power law fluid (Nye 1952), [134], that was experimentally verified by Glen (1952, 1953), [55,56], and Steinemann (1954, 1956, 1958), [157-159]. Orowan (1949), [136], likely prompted the British contributions. This law is now known as Glen’s flow law. Various second grade fluid alternatives (e.g. Man and others (1985, 1987, 1992 2010), [116-118], provide extensions to capture primary and secondary creep. Comparing results of creep tests from distinct experimentalists disclose unexplained disparities of stress relations.
All large-scale mass Initial Boundary Value Problems (IBVP)s are viscous, heat conducting incompressible Stokes problems, using the Glen-Steinemann flow law, which may involve cold ice regions with the ice temperature below melting, separated (close to the basal boundary) by a Class-I mixture of temperate ice, i.e. ice with water inclusions. The dynamics of the cold-temperate transition surface between the two regions is governed by a (simplified) Clausius-Clapeyron equation (Fowler & Larson (1978, 1980), [49-50] Hutter (1980, 1983), [84-86], Hutter, et al. (1981, 1988), [82, 88], [Blatter & Hutter (1991), [15]).
As for boundary conditions
at the free surface, snow accumulation and radiative heat must be parameterized
as functions of position and time or these quantities must be taken over from
concurrent large-scale flow models of the atmosphere. One must proceed in a
similar way with the ice ocean interface on floating portions of ice shelves.
Here, general ocean-circulation models are operative. Moreover, at the
ice-shelf front, mass loss of ice by calving must be parameterized, which is
significant for marine glaciers, ice shelves, and their dynamic response in
such scenarios.
Motivation
To begin, let us briefly present those articles that treat some aspects of the historical development of the physics of glacier flow. We collect articles in which the focus is devoted to the history of glacier and ice sheet flows approximately during the past 3 centuries but also including the past 30 years. Our focus is the Earth bound slow movement of ice masses and how this creeping flow was, in earlier centuries, intellectually developed and principally rationalized, eventually as a nonlinear viscous heat-conducting fluid. Of similar significance as the early material science properties of ice was the recognition of the concept of ice age(s). For the 19th century scientists, the enigmatic appearance of erratic boulders led to the recognition of the concept of ice age(s). This concept, first spelled out around 1830 by the German poet Goethe, see Cameron, D. (1965), [24], was only formally rationalized as the planetological cause of the ice ages that is coupled with Earth internal processes affecting the time evolution of the Earth’s climate.
The most significant memoir, in which the physics of glaciers is reviewed, is Garry Clarke’s (1987) ‘A short history of scientific investigations on glaciers’, [27]. The author gives a tour of ‘all’ glacier related processes, starting in the first quarter of the 19th century, but primarily focusing on the years ~1940-1987 and describing mostly research performed in the past 50 years related to the physics of glaciers. This work is much geared to accumulate the varied knowledge separated by subjects; it is excellently summarized. Its 19th century coverage is similar to ours, but with a broader coverage of subtopics and has perhaps less of a natural-philosophical emphasis than ours.
The paper ‘Decent of glaciers: some early speculations on glacier flow and ice physics by John Walker and E.D. Waddington (1988), [171], concentrates on the early work of the motion of glaciers as observed in the 19th century. They cover the literature of the period from ~1820 to 1870, notably listing the memoirs of Agassiz (1837, 1840, 1842), [3-5] de Charpentier (1841), [26] Forbes (1841, 1842, 1843), [ 45-47], covering essentially what was stated by Clarke (1987), [27].
The memoir ‘Life, death and afterlife of the extrusion flow theory’ by E.D. Waddington (2010), [170], is devoted to a specialized subject, namely extrusion, which was a fallacious model of ice flow. In our manuscript, we do not deal with such phenomena, i.e. processes that failed to be recognized as realistic.
Two additional memoirs in the Journal of Glaciology are devoted to the hydraulic component of Alpine glacier flow. Gwenn Flowers (2010), [43], devotes her paper to such related works by Almut Iken (1933-2018), [93-95], Similarly, Christina Hulbe, et al. (2010), [81], devoted their paper to Women in glaciology. Both these articles are at best tangibly related to our present concern of glacial flows.
Thus, it transpires that our focus is, first, the birth of the description of glacier ice flows as the motion of a heat-conducting fluid and, second, the presentation of the necessity of extending the flow law of ice from its nascent postulation beyond the Glen-Steinemann flow law as a general nonlinear viscous fluid. We stress circumstances where we believe that present research activities do not strictly enough draw the inferences that should be drawn on the basis of the findings.
To understand how the large ice masses and glaciers on Earth contribute
to the climate scenarios, one must first securely know to what extent the
climate driving of the solar system interacted, and will interact, with the
Earth-interior processes, and how these affect the various components: the
atmosphere, the solid Earth, the oceans and hydrosphere, as snow, ice, water
and vapor, not to forget the biosphere, which also has a significant
anthropogenic component. Unquestionably, there is also an anthropogenic
component, as demonstrated by industrial activities since approximately the
year 1780 and expressed by the equivalent
This brief account of the most significant components of the Earth’s climate system only touches those parts of the entire picture, as they chiefly developed in the last 90 years or so, beginning with Milankowitch’s thesis (1930), [124]. It consists of three distinct but concurrent processes: eccentricity, obliquity, and precession. Eccentricity, the elliptical cycle of variation in shape of the Earth’s orbit around the Sun is about a 100.000-year cycle. Obliquity is the cycle of axial tilt of the Earth’s rotation axis toward or away from the Sun and varies from about 22 to 24.5 degrees, usually taken as 23.5 degrees with about a 26.000-year cycle. As these parameters change, so does the amount of sunlight that hits different latitudes on the Earth; for a NASA illustration of these, see Gustovich (2018), [74].
As a scientific focus, climate dynamics on Earth and its anthropogenic coupling came last. In the period from the mid-17th to the mid-19th century, glaciers were first thought to be rigid objects. That they somehow move was mainly recognized in the 18th century, in parts by conjecturing that the volume expansion of water in the freezing process would push the ice downward in their valleys. Later in the 18th century this belief was replaced by the hypothesis that glaciers would slide over their beds. Moreover, edgy rocks and dirt on the glacier surface, which were seen to move, brought the impetus for the assumption that glaciers move and deform. Careful geodetic measurement with theodolites in the 18th and 19th century, finally, disclosed the creeping character of this motion as deformable like a dough. A further enigma was still the inexplicable presence of erratic boulders within an environment of unmatchable petrography. The solution of this mystery led finally to the hypothesis of earlier ice ages, a fundamental topic of climate dynamics, postulated in the 18th and 19th century and rationally explained in the 20th century.
To be able to describe the deforming motion of glaciers calls for
material science of the ice. The determination of the material properties of
polycrystalline ice as a nonlinear viscous, heat-conducting body - at early
times often denoted as plastic - is vital for the quantification of its melting
processes. This concerns the substance and its appearances as ice, water and vapor. These
are described by the thermodynamic principles (first and second law). It so
happens that the Earth’s climate conditions through the past few million years
have driven the phase changes of
This brief overview of how the phenomena ‘glaciers and ice sheets’
triggered the curiosity of mankind to eventually become a respected part of the
natural sciences of even high significance for the humans’ living on this
planet. This picture must be complemented by the material sciences. These ought
to describe the
where
A(T) is called rate factor, and n (generally chosen as n = 3) determines
the exponent of the power law. In the
glaciological community, the law (1) is called Glen’s flow law (we call
it the Glen-Steinemann flow law, because the law was approximately
simultaneously and independently proposed in the 50s of the last century by
both researchers in the UK and Switzerland, respectively [1]). However, in material science, depending on the field of science, it
is attributed to different distinguished scientists, in rheology to Graham
(1850), Guoy (1910). Ostwald-de Waele [2] brought it forward in (1929) in the context of colloidal fluids and
Reiner already before (1929) in rheological problems. In metallurgy, it is
called Norton’s flow law (1929), and in plasticity theory Orowan had used it.
In Glaciology John Nye (1953), [134], had demonstrated to the applied
glaciologists, how it could be ‘derived’ from simple material postulates of
viscous fluid behavior. It is likely that Orowan, who had also worked on
constitutive behavior of plastic materials at that time, had directed Glen and
Nye into such a nonlinear law; all three were then working at the Cavendish
Laboratory at Cambridge University, UK, where Glen and Nye were junior
collaborators.
So, the Glen-Steinemann flow law has precursors in not too distant related fields. We scrutinize its performance in comparison to laboratory and in-situ experiments and will also report on a number of attempts to the parameterization of the proposed laws. The likely best form of this viscous thermo-mechanical material description will then at last be incorporated in full-scale computations of glaciers and large ice masses in regional and completely Earth-embracing physical-mathematical-numerical models. This is in order to try to understand the climate relevant processes on our Globe for computational reproduction of the past climate variation due to the thermo-mechanical input data from extra-Earth processes, as well as to predict of future reaction.
Early Deformation and Sliding Models of Alpine Glaciers
The proposals for the enigmatic large ice masses by Johann
Jakob Scheuchzer [3] and others in the 18th century
can only be understood, if we accept that prior to the mid-19th century
no quantification of constitutive relations was available to the material
scientists interested in the physics of large ice masses. Nevertheless,
glaciers as natural phenomena at high altitudes of mountainous regions were
attractive, if simply for their mystic appearance. Prior to the 17th century,
not even the deformability of glacier ice was recognized, certainly not
admitted. Glaciers were postulated to be rigid objects, see e.g., Moraltus,
Johann (1669), [105]. Only valley inhabitants close to them accepted some
deformability, because e.g. they observed their snouts to advance or retreat,
but could hardly rationalize a cause, and if they had been able to, they were
too isolated in their valleys to get in contact with musing intellectual
hikers. These modest people often knew it better.
So, with this background in the year 1705, Johann Jakob
Scheuchzer visited Swiss Glaciers and proposed a theory on their motion. He
knew from physics that water is expanding in the freezing process to ice, and
that the ‘Force’ of expansion is so large that cartridge rounds which are
filled with water that freezes, are blasted into pieces. Scheuchzer assumed
that the water in glacier fissures and crevasses that freezes would extend with
such excessive power that its force will unquestionably push the glacier
downward. This concept, later often called ‘Dilatation Theory’, was also
adopted by Jean de Charpentier [4], his brother Toussaint de Charpentier [5] and Louis Agassiz [6] and others. Basic thought of this
concept was the belief that glaciers are permanent storehouses of coldness,
capable to freeze all water that percolates through them. Interesting and
strange in this model is that the volume expansion in the freezing process of
ice is the only cause of the downward motion of the ice of glaciers. As a
question at hindsight one might ask whether these scientists had ever looked at
the water outlets of Alpine glacier termini with their sub-glacier rivers and
generally substantial water discharge that is often rather large, certainly not
zero during summer, as they required by their early deformation postulate.
About in the year 1760, Altmann [7] & Gruner [8] brought forward their opinion that
glaciers would move by means of sliding along their beds. Both published their
contributions independently, Altmann in 1751 and Gruner in 1760. Almost 40
years later, Horace Bénédict de Saussure revived this concept, which became the
‘de Saussure Theory’, even though, according to Tyndall (1878, p 185), he was
never chiefly involved in it.
Cautious Birth of the ‘Plasticity Theory’
The 18th century also brought the first ideas
about the deformability of ice sitting on solid ground. The concept of sliding
should simply have brought the science mountaineers to the postulation of the
deformability of glacier ice. Indeed, glaciers generally mostly move on
non-planar beds; this fact, together with the existing forward motion requires
deformability to maintain the sliding hypothesis. Yet, none of the above
mentioned scientists attributes the terms ‘viscosity’ and ‘kneadability’ to
characterize the deformability of the ice, even though, according to Tyndall
(1878, p. 185), [164], the appearances of many glaciers suggest these
terminologies, if it were not so contradictory to any daily experience with ice.
Needless to say that the concept of viscosity was known since Newton had
introduced it in the Principia, (first edition 1687, third edition 1726),
[132].
In spite of this, these kinds of plastic concepts found their
defenders. In the year 1773, André César Bordier from Geneva published a small
booklet ‘Voyage pittoresque aux Glacieres de Savoyes’, Genève, 1773
(‘Pituresque journey into the Glaciers of Savoy’). He advocates for an over-all
view of the ice mass which moves as a whole from high to lower altitudes in a
manner as seen with other fluids. ‘Do not let us view the ice as an immobile
and stiff material, similar to mollified wax, which, to a certain degree, is
flexible and extensible’. This is likely the first occasion where the
kneadability of glacier ice is mentioned.
However, according to Tyndall, André César Bordier’s [9] concepts were unheard in the natural
scientific community in the 1770s. They were reborn more than 60 years later by
a successor of better scientific training. This person was a catholic priest
and later bishop of Annecy, Louis Rendu [10]. In the year 1840 he submitted ‘Théorie
des Glaciers de la Savoie’, [146], to the Royal Academy of Savoy. Tyndall, was
fascinated by Rendu’s writings and states that it is not known, whether he ever
saw Bordier’s works, probably not, because he never refers to it. Tyndall
quotes a few of Rendu’s statements to illustrate the preciseness of his
expressions:
· Between Mer de Glace
and a river, there exists such a perfect similarity that it is impossible to
find a circumstance in a glacier, which would not equally occur in a river.
· In
flows of water, the movement is not uniform, neither with regard to the width
nor to the depth.
· The friction at the bed and at the sides, paired with the effect of
obstructions, makes the motion to differ from position to position; only toward
the middle of the free surface, one reaches the full motion.
· There
exists a large set of facts, which seem to force us to believe that glacier ice
possesses some sort of extensibility, which allows it to adjust to the local
circumstances, to thin, to swell and to contract as if it were a soft dough.
To corroborate these inferences, Rendu requested careful measurements of the glacier motion; he did not perform these himself, but asked his mountain guide to observe the motion of a particular boulder at the glacier side, which, during 5 years moved 40 feet/year. Other boulder position measurements corroborated the motion. This is a first indication to search for motion patterns of glaciers, which we shall subject to a detailed analysis later on.
To summarize
First descriptions of the motion and deformation of glaciers are due to Bordier in 1773, but the flexibility and extensibility with a characterization of viscous or plastic behavior was recognized by Rendu as late as 1840, [143], and afterwards, but could not yet be phrased in terms of a constitutive relation such as the Glen-Steinemann law.
Erratic Boulders
Since the 19th century, the edged boulders,
apparently incoherently distributed in mountainous territory are named glacial
erratic, which explicitly suggests for us that they were once moved by glaciers
and deposited at their present positions when the carrier ice retreated. These
rocks suggested to the mountaineers misleading or inexplicable or false
behaving, briefly err-behaving. The causal connection had first to be
recognized, and it influenced at least subconsciously the early understanding
of the existence of the Ice Ages. According to Krüger (2009), [105], the oldest
written statement on erratic boulders dates from 1301 and is reported by
‘Henricus dictus von dem Steine’ [11]. The first assured written statement was
found in Johannes Guler’s ‘Raetia’ [12] In his script Krüger (2009), [105]
tells that Guler von Wyneck mentions a colossal rock in Veltlin, in the Italian
Alps, for which he could not see, where it may have broken off. With the
growing development of modern geology in the mid-18th century
the general interest in the erratic boulders grew. In 1727/28 Moritz-Anton
Capeller [13], physician of the town Lucerne,
recognized the alpine origin of the erratic boulders, deposited in the
mountainous foreland. However, he only published his recognition in 1767 in his
‘Pilatii montis historia’, shortly before his death.
Initially, the contemporary scientists as e.g. the notary
Abraham Schellhammer [14] (1675-1735) and Albrecht von Haller [15] both from Berne assigned the
ultimate cause of the positional spreading of the erratic boulders to the
biblical deluge or (later) to a number of exceptional floods. Tobias Krüger
(2009), (2013), [106, 107], reports a number of exceptional and extreme
interpretations, even by scientists of the status of Horace Bénédict de
Saussure (1740-1799) and many others.
One interpretation by the Prussian geologist Christian Leopold
von Buch [16], why an erratic boulder found in the
Jura Eastern slopes, but of likely origin in the upper Valais is as follows: He
states that originally far in the past, today’s Rhone Valley was blocked at
Saint Maurice by a rocky Rigel (bedrock bar) between the Dents du Midi and the
Dents des Morcles. Behind, a gigantic lake was formed up to the mountain peaks.
At the breakdown of this natural Rigel, the water masses were set free with
hardly imaginable power; they threw huge rocks as far as to the Jura (Krüger
(2009), [106], notes that this information was only published after von Buch’s
death in 1867). We will refrain here from reporting further erroneous
propositions of the displaced positions of erratic boulders. Interesting,
however, is that in the mid-18th century, more than 100 years
earlier, in 1742, the engineer and geographer Pierre Martel (1701-1767) from
Geneva, reports of large displaced rocks found during his journey to the Valley
of Chamonix. Its inhabitants had apparently explained to him that the ‘Glacier
du Dois’ once carried these rock pieces down the valley. Krüger (2009), [106],
concludes: For this reason, at the present state of knowledge [meaning 2009],
the inhabitants of the Savoy Alps were the first, who established a causal
relation between glaciers and position-alien rocks.
In the last decade of the 18th century, James
Hutton [17] devoted his working time to geology
and authored the treatise ‘Theory of the Earth’, originally planned as four
volumes. He adored Saussure’s imagination in his theory of erratic boulders,
but rejected Saussure’s concept that erratic boulders reached their places of
final position before the erosion had formed today’s valleys. He writes ‘there
would then have been immense valleys of ice sliding down in all directions
towards the lower country and carrying large blocks of granite to a great
distance where they would be an object of admiration after ages, conjecturing
from whence or how they came.’ As primary cause (be aware, existence of earlier
ice ages was not known at this time) for the extended glaciers of the Alps,
Hutton assumed that the latter had been substantially larger at earlier times.
Therefore, it then must have been colder there, which favored an intensified
glaciation [because atmospheric temperature decreases with height; this was
known at that time]. Damaging of Hutton’s concept was that it required a
tremendous erosion rate to excavate the suspected high-level plane and to form
the valleys from the initially much higher mountains. This would last an
excessively long time. Positive is that he is one of the first, who suggested a
connection between the phenomenon of erratic boulders and the spacious earlier
glaciation.
After Hutton’s death, the theologian and mathematician John
Playfair continued his work [18]. He summarized this work in his book of
the year 1802, [142]. His message is the same as that of Hutton. Krüger, [106],
quotes him as follows:
‘For the moving of large masses of rock, the most powerful
engines which nature employs are without doubt the glaciers, those lakes or
rivers of ice which are formed in the highest valleys of the Alps, and other
mountains of first order. These great masses are in perpetual motion,
undermined by the influx of heat from the Earth, and impelled down the declivities
on which they rest by their own enormous weight, together with that of the
innumerable fragments of rock, which are loaded. These fragments they gradually
transport to their utmost boundaries, where a formidable wall ascertains the
magnitude, and attests the force, of the great engine by which it was erected’,
from ‘John Play fair, Illustration of the Huttonian theory of the Earth’,
Edinburgh 1802, p 388ff §348, [142].
Similar observations and interpretations were also made by
natural scientists from other countries. In Bavaria the geologist, mineralogist
and physicist Mathias von Flurl [19] recognized the position-alien
character of the erratic boulders in Bavaria. In the year 1809, the Bavarian
physician and astronomer Franz von Paula Gruithuisen [20] devoted some of his research to
erratic boulders in the Alps. He knew of the significance of such boulders, and
rejected the hypothesis of large floods to be the cause, but kept the transport
hypothesis of the apparently unmotivated spreading by water; so, the heritage
of the deluge as the principle cause was still behind his thinking. Krüger
(2009), [104], cites von Paula Gruithuisen with arguments which today must be
interpreted as abstruse geological and hydrological thinking. He, further muses
that von Paula Gruithuisen did not seem to know that glaciers do slowly flow
down their valleys. They must be immobile like frozen lakes, and a geological
force must exist, which pushes the glaciers, including their edgy rocks,
upwards to transport the latter large distances. Evidently, such a concept
seems today completely unrealistic. The biblical deluge can best serve in this
situation as a ‘deus ex machina’ to resolve this intellectual dilemma.
About at the same time, during the early 19th century,
Jean-Pierre Perraudin [21] and Siméon Gilliéron [22] came to the conclusion that “The
glaciers of our mountain chains […] had at earlier times a considerably larger
extent than today [sic: at Perraudin’s and Gilliéron’s time]. Our entire valley
was up to a high altitude above the Dranse, a river in the Valley de Bagnes,
occupied by a single huge glacier that extended far down to Martigny, as the
rock boulder evidenced, which were found in the vicinity of this town and which
are too big to ever have been transported by water” (translated form Krüger
(2009), [106], by KH). Apparently, Perraudin’s thesis of 1815 was only based
upon positions where erratic boulders were found. This was an indirect
recognition that positions of erratic boulders were determined by the earlier
glaciation of the valley area, in which they were found.
To summarize:
In the 18th century the contemporary scientists
had abandoned the biblical deluge hypothesis in favor of a series of flood
events as the causes of the enigmatic positioning of the erratic boulders. Only
after the 1740s, first suppositions of the transport of erratic boulders by
glaciers appeared, but the principal postulated causes were largely based upon
rather unrealistic theoretical concepts, which even were contradictory. The
question, how glaciers would move - even the problem whether they would move at
all - remained controversial. This remained so in the 19th century.
Reflected on from the 21st century, this cannot be a surprise
as the climate variations were not yet understood and a continuum mechanical
formulation for material behavior had not yet been established.
Systematic Measurement of the Deformation and Motion of Glaciers
Erratic boulders were of help in searching for the deformation
and motion of glaciers as a whole. Direct observation of objects on the glacial
surface and at certain depths below it, in crevasses and fissures, on the other
hand, brought a breakthrough. The detailed observations and measurements were
done by James David Forbes [23,44-46], and John Tyndall [24] and are expertly described by
Tyndall (1860, 1878) [163] and -- in lecture notes ‘Eis und Gletscher’ (‘Ice
and Glaciers’), held by Herrmann von Helmholt [77], in February 1865 in
Frankfurt am Main and Heidelberg. On pp.108, 109 of these notes he writes. ‘We
have so far compared glaciers according to their manifestation with streams;
this similarity is, however, not just an external one; on the contrary, the ice
of glaciers moves in fact forward, similar to water in a stream, only slower
[…]. Since, namely, the ice at its lower end is incessantly lessened by
melting, it would soon have disappeared, if new mass would not continuously
move forward from above, which would by snowfalls in the firn fields repeatedly
be renewed’ (translation KH). Von Helmholtz reports on a number of such
convincing situations which we itemize below.
Isolated rock boulders on the surface of a glacier can be used
as identifiers of material surface points of glaciers. This observational
procedure can be used for any object positioned on the ice and left there, but
repeatedly observed for its position.
· In the year 1827 Hugi [25] erected a hut on the middle moraine
of the Unteraargletscher, to perform there some measurements. The position of
this hut was determined by him and by Agassiz 14 years later, in the year 1841;
it stood then 4884 feet (~1602m) down-glacier. A somewhat smaller velocity was
found by Agassiz for his own hut of the same glacier, [3-5].
· Measuring positions of the surface boulders by theodolites, made by Forbes and Tyndall, from day to day disclosed for the Mer de Glace that during summer the velocities were of the order of 20-35 inch/day (54.7-96.7 cm/day); during winter, they were about half as large; at the ice surface they were larger than at depth, and at the sides they were considerably smaller than in-between’. (von Helmholtz 1865, [77], pp 110-11, translation by (KH))
· The observation of the kind on motion of glaciers allows also an interpretation, in which orientation crevasses must be formed; since, namely, the forward motion of the ice at different positions is not the same, the relative distances of these mass points change with time. Now, because the ice between any two such points cannot be arbitrarily stretched, it will be fractured and form fissures or crevasses’ (Helmholtz 1865, [77], p 112, translation KH). Von Helmholtz, quoting Tyndall’s conviction that the glacier is hardly able to resist tension but rather ruptures under the influence of such tensions, illustrates this in (Figure 2).
· Both, Tyndall and von Helmholtz, see (Figure 3), beautifully explain how moraines and dust stripes illustrate the deforming motion of the glacier system ‘Mer de Glace’.
The three contributing ice-arms are ‘Glacier du Géant’, ‘Glacier de Léchaut’ and ‘Glacier du Talèfre’, which unite as ‘Mer de Glace’. Side moraines transporting boulders, fallen from the side flanks of the mountains onto the moving ice, in the three upper glacier arms become in the ‘Mer de Glace’ interior moraines, except for the outermost two side moraines to the far orientation crevasses must be formed; since, namely, the forward motion of the ice at the far left and far right. The dust stripes in the ‘Glacier du Géant’ and further down in the ‘Mer de Glace’ all come from the icefall fed by the ‘Col de Géant’ (g in panel (b) of (Figure 3)). The material falling down this icefall -- varying from winter to summer -- forms the ice-dust mingling at the base of the icefall, from which the stripes emerge as displayed in (Figure 3), panels (a) and (b). These stripes have in the downstream stretch of the ‘Glacier de Géant’ a curved appearance across the glacier width that amplifies further down in the ‘Mer de Glace’. These bands of dust and the spread of rocks and stones appear as alternating grey and whitish stripes, likely indicating annual repetitions of the ice, as first noted by Forbes. All this is impressive manifestation that velocities at the sides are smaller when compared with the free surface motion in the interior, as so described by Tyndall in the year 1842, [163].
The dust stripes, indicating the displacement distribution of the free surface are in the ‘Mer de Glace’, limited to the Western part of the glacier; the Eastern part is covered by interior moraines but no curved dusty stripes are seen - at most an onset of spreading that grows in the down-glacier direction [26]. Corroboration of this interpretation is given in detail by Tyndall (1873, pp 93-123). He and his collaborators demonstrated this for the ‘Mer de Glace’, the Grindelwald-, Aletsch- and Morteratsch-Gletscher. The method was to insert wooden sticks at several points along an initially straight line into the snowy surface across the glacier (generally 6 to 10 per transect and to determine by theodolite their positions at consecutive times (generally one to a few days apart), see [Figures 4,5].
This allowed evaluating the travelled distance for each wooden stick at consecutive times. This meticulous procedure was the verification of the bold assertion that ‘the ice of a glacier flows slowly, and similarly to a stream of a very viscous substance, as e.g. honey, coal-tar, or a thick tonic pulp’. The ice does not simply slide over the bed, just as a rigid body does, which slides down a slope; it rather bends and displaces itself in itself and, even though it glides over the base of the valley, the parts, which are in contact with the bottom surface and the side walls of the valley, are effectively slowed down, caused by the significant friction. By contrast, the middle of the free surface of the glacier, which is farthest from the bottom and the walls of the valley, move faster. Louis Rendu and James David Forbes were amongst the first to emphasize the similarity of glacier flows with the flow of a fluid substance’ (Helmholtz, 1865, [77], p 116, translation KH).
Thus, by the 1860s the scientists focusing on the physics of the glaciers had acquired convincing knowledge that glaciers were slowly moving and deforming similarly as water in rivers, creeping soil down mountain slopes, hot lava as rivers in volcanic eruptions, honey or polymeric fluid substances. However, a mathematical formulation of such motions as continuous media was not available at that time. Neither was there any mention that inertial terms in Newton’s law in river motions have a sizeable influence, whilst they can safely be ignored in the determination of glacier motions.
To summarize
Systematic measurements of the motion of the ice on the glacier
surface at selected points on the surface, at its middle line, close to its
side boundaries in fissures and crevasses below the surface were consistently
done by Forbes and Tyndall using theodolitic positioning at consecutive days.
The evaluation of these measurements led to the interpretation that glacier
motion fields are analogous to the motion of terrestrial surface river flows, a
fact explicitly spelled out by Rendu in the first half of the 19th century,
see [143].
Birth of the Concept of an Ice Age
According to Rowlinson [27] (1971), [145] the years from 1840
to 1870 were amongst the most fruitful ones in the history of physics. As for
the solution of the problem of th position of the erratic boulders the
systematic measurements of the motion and deformation of glaciers primarily
done by Agassiz, Forbes and Tyndall were important as was Rendu’s ‘Théeorie des
Gaciers de la Savoie’ (1840), [143]. Forbes had formulated a model of viscous
deformation, which was criticized, because it conflicted with the sliding
model, but Forbes successfully showed that the sliding model was in conflict
with the (experimental) facts, which he had discovered (Rowlinson 1971, [130]).
By 1851 Forbes’ measurements were summarized in his article of 1855 for the
eighth edition of the encyclopedia Britannica. Abbreviated as evident, this
statement reads
“Each portion of a glacier moves…in a continuous manner…The ice in the middle part of the glacier moves much faster than that near the sides or banks; also the surface moves faster than the bottom…. The variation of velocity (as in a river) is most rapid near the sides…. The glacier, like a stream, has its pools and rapids. Where it is embayed by rocks it accumulates, its declivity decreases, and its velocity at the same time…and the increased temperature of the air favor the motion of the ice…. The velocity does not, however, descend to nothing even in the depth of winter…. [These] circumstances of motion … appear to be reconcilable with the assumption of what may be called the Viscous or Plastic Theory of glacier motion, and with that alone … the veined or ribbon structure of the ice is the result of internal forces, by which one portion of the ice is dragged past another…. The veined structure is unquestionably the result of the struggle between the rigidity of the ice and the quasi-fluid character of the motion impressed upon it. That it is so evident not only from the direction of the laminae, but from their becoming distinct exactly in proportion to their nearness to the point where the bruise is necessarily strongest”.
Rowlinson’s paper is in large parts a detailed historical
account of the dispute and disagreement over how the motion of glaciers had to
be physically interpreted. “Tyndall’s criticism centered on the word ‘viscous’,
which Forbes never defined clearly. He apparently understood it to mean
resistance to shear stress, but used the terms viscous, plastic and semi-fluid
indifferently…Tyndall interpreted it differently.” By viscosity he understands
that property of a semi-fluid body, which permits of its being drawn out when
subjected to a force of tension. He denied that ice could be permanently
stretched without braking and so dismissed Forbes’ viscosity as apparent, not
real. Instead he proposed his theory of ‘fracture and regelation’…” (Rowlinson,
1971, [145]). He describes in detail this somewhat bitter fight that lasted for
years; it also bore elements of jealousy, because Forbes feared that Tyndall
(the younger of the two) might get the Copley Medal of the Royal Society in
1859. William Thomson 1824-1907 (later (1892) Lord Kelvin), apparently was also
involved in this maneuver.
Thermodynamicists of the 19th century played
equally a role in the process of creating a realistic physical basis for the
motion of glaciers. At that time, Julius Robert Mayer (1814 - 1878) [119] (a
physician in Heilbronn, Germany) had formulated the First Law of
Thermodynamics - then called the mechanical law of heat – and
Rudolf Clausius had introduced the Second Law of Thermodynamics [28,29], as
did Lord Kelvin [161], who introduced in 1848, at the age of 24, the absolute
temperature. This was known already earlier, since ice floats on water. So one
knew of the expansion of water in the freezing process to ice, and it is, thus,
not so surprising that Johann Jakob Scheuchzer and later Jean de Charpentier,
[26] and Louis Agassiz, [4,5] at least for a while assumed that only freezing
of intra-glacial water would cause the downward motion of glacier ice.
Nothing of this provides a hint as to the concept of ice ages;
neither Agassiz nor de Charpentier nor any other of the glacier scientists of
that time proposed the concept of very cold climates in the past. Today the
creation of the idea ‘ice age’ is largely still attributed to Louis Agassiz,
but this is erroneous: Surprisingly, it is Johann Wolfgang von Goethe (1749-1832),
one of the world’s greatest poets, who was equally a respected scientist.
Dorothy Cameron (1965), [24], in Vol. 5 of the Journal of Glaciology tells the
story. Goethe as a scientist was the first to attribute the transport of
erratic blocks to glaciers; he spoke out that an ice sheet covered Northern
Germany; furthermore, he is given the credit to first having believed in an ice
age. Cameron (1965), [24], states that Louis Agassiz (1837) and Jean de
Charpentier (1841) were correct when they assigned to Goethe the credit for
being the first to have conceived the concept [28] of the ice sheet. There are very
few written attributes to this fact. We have only found Reinhard Hederich
(1898), [76], being mentioned in a footnote of Alfred Krehbiel’s dissertation
of (1902), [104]. The comment is: ‘…neither should Goethe be forgotten, who
built himself noteworthy correct views of glaciers and erratic bolders…’
Similarly, also Forbes (who was not directly addressing erratic boulders or the
concept of ice ages) was well aware of Goethe’s scientific significance. On the
title page of his ‘Travels through the Alpes’ he cites Geothe’s sentence: ‘Sage
mir was du an diesen kalten und starren Liebhabereyen gefunden hast’ (tell me
what you believe to have found in these cold, rigid fondnesses).
The reference to Goethe is also missing in the reviews by Clarke
(1987), [27], Walker & Waddington (1988), [171] and Waddington (2010),
[170].
Juvenile Struggle with a Creep Law for Ice
As explained in the last section, it was in the 19th century that natural scientists recognized that glaciers are slowly moving ice masses. Helmholtz (1865), [77] and Tyndall (1860, 1878), [163,164], summarized the knowledge as of ~1870; they beautifully describe the slow birth of this new understanding of glacier flows as movements of a very viscous material. For the recognition of this property of deformability a large number of scientists were involved, besides Tyndall, at least Forbes and Rendu ought to be mentioned. At the mid 19-hundreds, further progress in developing a physically based theory for the motion of large ice masses was hampered by two facts,
· The nonexistence of a formal observer-invariant material theory, coupled with the basic physical laws,
· The
primitive state of experimental techniques for the determination of the
constitutive relations (here primarily for the Cauchy stress tensor).
The second half of the 19th and the early 20th century
were needed to develop these two physical, mathematical and engineering-type
specialties. As a prelude to the glaciological activities starting shortly
before 1950, it is important to note that the material sciences of continuous
media -- fluids and solids -- as they developed in the second quarter of the 20th century
were active in a nascent field, devoted to the description of the mainly
continuous deformations of bodies to external driving elements such as forces,
temperature, etc. This new science was coined rheology and became quickly
fashionable among chemists, material scientists, applied mathematicians and
physicists. The important features in the field of material creep to be solved
were how e.g. the stress tensor can be expressed by deformation measures, such
as strain and/or rate of strain tensors, temperature, etc. in a materially
objective manner, i.e. observer invariant form. Markus Reiner (1886 – 1976), an
Austrian-Israeli (civil) engineer, was an early protagonist of the description of
the creep behavior of fluids and developed prior to 1929 together with his
scientific associate Ms. R. Riwlin the Reiner-Riwlin constitutive relation for
fluid creep. Expressed in a nutshell: Let t be the stress
tensor and t´ its deviator and D the
strain rate tensor [29]. These are symmetric 3x3-tensors, and in
a volume-preserving (i.e. incompressible) material, D is a
deviator (its trace or first invariant vanishes). So, a possible constitutive
relation must be expressible as, where the function f, when
evaluated in terms of and T, must equally be a
deviator. The explicit form of f, satisfying these requirements can
be constructed with relatively simple methods of 3x3-matrix algebra and is
given by
in which both sides of this relation are deviators,
and are the second and third invariants of
of this relation are deviators,
The utmost majority of scientific works in glaciology has been done for
the case that
The remainder of this historical review will be devoted to the test of adequacy of the Reiner-Riwlin fluid as a constitutive model for the creep of polycrystalline isotropic ice. Naturally, this historical episode only dates back to the last 60-70 years with important contributions done since ~1980. We do this in this historical article against the recommendation of two reviewers and the handling Scientific Editor of the J. Glaciology. Referee 1 of that version of this paper stated
“There is an important borderline between
and I [sic: the referee] believe that the paper tends to cross the border from history toward the side of expressing a scientific opinion. I view this to be a point where the paper is no longer about history but is about new science being advocated…”
We as authors disagree with this request that the two parts ought to be
separated. Quite contrary: Historical facts ought to be used for the consequences,
which they imply. Trivially, likely all facts of our manuscript have happened
in the past (with the credits given to others) and are therefore part of the
history.
Applied glaciologists have almost exclusively employed the Reiner-Riwlin
fluid model in its reduced form
in which
The scene began in the 1950s with the recognition that creep of ice in large ice masses must be treated as an isotropic nonlinear viscous or plastic body. Creep tests on polycrystalline ice were done first by Perutz (1950), [141]; Glen (1952, ‘53, ‘54, ’55, ‘58, ‘74), [55-61] and Steinemann (1954, ‘56, ‘58), [157-159], followed by Gold (1958), [61-63], Mellor & Smith (1967), [120], Mellor & Testa (1969a,b), [121,122], Jacka (1984), [97], with reviews given by Kuo (1972), [108], Shumskiy (1974), [154] and Mellor (1979), [123]. The commonly used stress systems are tension, compression and simple shear, although combined states of stress had also been looked at. In addition, the attitude of material scientists were, that experimental results obtained with such simple stress systems could be extrapolated to 2- and 3-dimensional stress systems.
Typical creep curves for polycrystalline ice under simple stress states (shear, tension/compression) are as shown in (Figure 6) with primary, secondary and tertiary creep, where
· Primary creep is
decelerating,
· Secondary creep is
steady
· and tertiary creep is accelerating, merging into another steady state or leading to rupture.
Experiments by Steinemann (1958), [153], see (Figure 6a,b) show how
creep curves look like in these ranges and (Figure 7a,b) shows analogous curves
in doubly logarithmic representation. Since in these latter plots
against ln(
Note, we write for the one-dimensional strain rate now
requesting that this law is restricted to a finite stress range
The above representations (2) and (3) are restricted to uniaxial normal
stress and simple shear, but the structure of these formulas was wished to be
also applicable to 3-dimensional stress states. For density preserving
materials one may write an objective strain-rate stress relation for
polycrystalline isotropic ice as a Reiner-Riwlin fluid; i.e., if t is the Cauchy stress and
in which both sides of this relation are deviators and
(ii) the strain rate tensor and stress
deviator are co-axial.
These postulates require that
valid as the general constitutive relation for density preserving
polycrystalline ice.
As his example, Steinemann chose a combined unilateral compression plus
shear test as follows
which, under ideal experimental performance of a volume preserving
isotropic material, yields
Steinemann did perform such creep tests and measured
for which case, according to
in which (11) follows by use of (6). Moreover, at
The combined compression plus shear test with the expressions (7) and
(8) provides a possibility to study the relevance of the third invariant(s) as
independent constitutive variables. Equations
• in simple shear with
• in uni-axial
compression/tension without shear,
It follows that independence of the constitutive relation for the Cauchy stress deviator from the third invariant implies that creep laws obtained for compression/extension and simple shear, respectively, must be brought to coincidence. If this coincidence cannot be verified by experiments to within acceptable errors, then a dependence of the stress strain-rate relation on the third invariant is likely.
To summarize
Multi-axial deformation tests (shear plus compression) show that
isotropic ice does not satisfy Nye’s assumption that the creep law for the
Cauchy stress is independent of the third invariant, see (Figure 8).
Questionable Consolidation of the Co-axiality Hypothesis in the 1960s to 1980s
· The period after the formulation
of the power law by Glen and Steinemann was not characterized by searching for,
and extension of, the flow law to a general constitutive relation of a
nonlinear fluid. Experimentalists rather seemed to be trapped in the simplicity
and beauty of Nye’s two postulates, [135]. The power law was replaced by
functions better matching the experiments (e.g. sinh (•) to the
and write the constitutive law (6) for isothermal processes as
where
The coefficients
(Figure 9) collects measured
To summarize:
· Comparison of creep data of polycrystalline isotropic
ice with co-axial strain-rate stress relations and independence of these
relations of the third invariants, showed that the accelerated monotonic
increase of
· The power law description of the
· The rate factor a(T) of the constitutive relation
The above findings suggest that in addition to the temperature
dependence at least a further variable, of which the cause may be physical, instrumental
or experimental, must also influence the strain-rate stress relation, viz.,
Many experiments are generally made with grains of pure ice mingled with
pure water and then frozen, the rate factor E(?) in the last formula remains
presently a mystery, because it does not even tell the variable. Incidentally, eq.
(15), also states that the assumption of the thermo-rheological simplicity
assumption of the constitutive relation for the stress tensor is likely
invalid, see Morland & Lee (1960), [127]. We conclude:
• For simple shear and uniaxial compression Nye’s
postulates give exactly the same class of constitutive functions, but if the
ψ-functions determined by the experiments differ from one another; this
difference is an indication that
• Steinemann performed this kind of combined experiment,
measured
On the other hand, equation (11) can for isothermal processes be written
as
Glen constructed (Figure 8) more than 60 years ago and demonstrated that
Nye’s postulate ought to be abandoned.
It is time to do this!
Recent climate relevant research that employs constitutive relations of
polycrystalline isotropic ice as a viscous material generally assumes
co-axiality of the strain-rate tensor and the Cauchy stress deviator that
hardly deviates from the simple power law of the Glen-Steinemann type. This
essentially remained so, even though facts to the contrary are well known to
the experimental specialists. There is a necessity for a more general form of
the constitutive relation with non-coaxiality of the strain rate tensor and the
Cauchy stress deviator (.) The simplest such constitutive
relations are the Reiner-Riwlin fluids, which for density preserving fluids
have one of the forms
In these formulae, different from eq. (4), the stretching tensor
and
Morland and Staroszczyk (2019) [127] – whom we follow here - have
performed a Gedanken experiment; it is a combination of compression and shear
deformation, and it serves as a suggestion to catch the interest of
experimentalists. So, let X and x be position vectors of ice particles
in the reference and present configurations, respectively; then, compressing or
extending and shear deformations can be described as
Here,
from which
and
Li & Jacka (1996), [121], Warner et. al. (1999), [172], Treverrow,
et al. (2012), [162] and Budd, et al. (2013), [17], conducted
experiments that fit the above conditions and performed these with the
assumption
The requirement
With the above expressions (19)-(21), restricted by relation (22), and
the stress expressions (we write
and, upon using
in which
Moreover, eliminating between these two expressions
Finally, substituting these results into
This equation delivers
To summarize: The above results, due to
Morland & Staroszczyk (2019), [130], are important for the following
reasons:
1.
In the experiments the strain-rates
which are equally point-wise known for these experiments.
2. So, provided the experiment can be conducted as intended and described above,
we have pointwise knowledge of
The parameter j=1,2… J counts the
number of experimental points. Kriging is popular method to determine the mathematical
expressions of
3. Equation (29) is independent of
the response functions
4. Prerequisites of the results of the
solution are that
It is evident that in principle combined compression-shear tests can be
conducted, in which the scalar functions
cannot be maintained. So, (Figure 11) supports the Reiner-Riwlin structure
of the constitutive relation
The result also confirms for the Budd, et al. (2013) experiments, [17], that the deviation from co-axiality is not small. In almost all cases ice is treated as volume preserving ( ) and the momentum equation is applied in the Stokes approximation, i.e., the acceleration terms are ignored.
Applications to Climate Dynamics
Field equations for cold and temperate ice regions
Glaciers, ice sheets and ice shelves in climate relevant analyses are mostly treated as polycrystalline isotropic power law fluids, but restricted to constitutive relations of the form ‘ice plus inclusion of water’. This is so, even though careful experiments have shown that a complete Reiner-Riwlin structure with non-coaxiality of the stress-strain rate relation is likely better modeling the creep behavior of isotropic ice. The concentration of the water in the ice varies according to the thermal regime that prevails as a result of input of geothermal heat from the interior of the Earth, the heat flow exchange at the ice-atmosphere interface and the water production per unit mixture volume due to viscous heat of the mixture. Cold ice is ice with vanishing water content and temperature below the freezing point, while temperate ice is at the pressure melting point, in which the temperature is related to the pressure.
Furthermore, the contact surface of the cold and temperate ice – this surface is called the Cold Temperate Transition surface (CTS) at which the Clausius-Clapeyron equation must hold, is an internal singular surface. The thermomechanical conditions that are described by the jump conditions of the field equations in the cold and temperate subsets of the polythermal ice mass determine locally, how the melting and freezing processes evolve along the CTS. On the other hand, within the temperate ice the production of moisture equals the production of water mass per unit volume.
The governing field equations of ice in the cold ice region are the
balances of mass, momentum (in the Stokes approximation) and internal energy.
The constitutive relations describing the creeping flow are here given by the
Reiner-Riwlin strain-rate--stress relation, eqs. (4) & (5) (i.e. not the
usual Glen-Steinemann law), and the heat flux vector q is given by Fourier’s law of heat conduction, whereas the stress
evolution is governed by the Stokes equations, and the temperature evolution
follows from the internal energy balance. These equations are for a density
preserving fluid of the form
and
Correspondingly, in temperate ice regions, the physical laws are a
class-I binary mixture; i.e. the equations comprise of the balances of mass,
momentum and energy for the mixture as a whole plus a balance for the water
mass. Interpreting the equations on the left-hand side of equations (32) as the
mass, momentum and energy balances of the mixture as a whole, then these
equations must be complemented by the balance of the water mass,
in which w is the moisture content per unit mass, j is the moisture flux and C the moisture production rate per unit
volume of the mixture. For this mixture model constitutive relations are needed
for j, C and the
where
in which eq. (4) has been substituted, here adequate for a Reiner-Riwlin
fluid. Moreover, employing the Caley-Hamilton theorem for the stress deviator
Substituting this into the above formula yields
It follows that
If Nye’s co-linearity assumption of
D and
as stated e.g. by Hutter (1983), [86]. Furthermore, according to
Lliboutry (1979), [113], the presence of melt-water can significantly affect
the constitutive relation of the stress deviator. He concludes that the
constitutive equation for the creep law remains formally valid but coefficients
depend now on the moisture content w and not on T, which is related to the
pressure via the Clausius-Clapeyron equation. For the Reiner-Riwlin, fluid this
means that
Analyses of the constitutive functions for the creep laws for temperate
ice have also been given with more details than here by Blatter & Hutter
(1991), [15], Fowler & Larson (1978), (1980), [49, 50],, Greve (1995, ’97, 2000),
[67-70], Hutter (1982, ‘83), 2019), [83,86,90], Hutter, et al. (1988), [88], however
only for the situation that the strain-rate is collinear to the stress deviator.
For the Reiner-Riwlin fluid, the equation for the moisture content takes the
form
Note that C depends also on the third stress deviator invariant, if stress and strain-rate are not co-axial. In any concrete situation, the union of these statements defines the initial boundary value problems (IBVP) that must mathematically be solved to arrive at a set of quantifications of the variables as functions of space and time that will provide information on the climate relevant questions for which the IBVPs were formulated.
Phase change properties of a viscous-heat conducting fluid are well known, see e.g. Hutter (1983 or 2019), [86,90], but need carefully be introduced for materials, which are kinematically handled as density preserving. The difficulty is that there cannot be a pressure melting formula and neither a Clausius-Clapeyron formula in a density preserving material. We, thus, will treat here compressible viscous fluids and present phase change properties of such materials and then will come back to the case of a density preserving material.
A phase change surface of a continuous material is defined as a special
singular surface at which the temperature and the velocity component that is
tangential to this surface are continuous. Thus,
where
(41)
in which
Hutter (2019), [90] gives proof of these relations. In this proof, the
fact that the components of v and u that are tangential to
In thermostatic equilibrium one has
Next, eliminating
When both phases are in thermostatic equilibrium, then
The pressures on both sides of
an equation that is usually written as
in which
It consists of the jump of the internal energy across
The interpretation of this equation is facilitated, if the relations
are applied. Both,
Differentiating this expression with respect to the temperature yields
(see Hutter 1983), [79],
The Clausius-Clapeyron equation is the inverse of this expression,
Since the temperature variations in problems involving phase changes are usually small, c in (50) is generally regarded as a constant.
Notice that in a density preserving material
In non-equilibrium the chemical potential must be defined by
as suggested by (40), and relations
The exact exploitation of (52) has so far not been achieved. This is the
reason why one assumes near equilibrium behavior and restricts
Hence
Since
Hence, in contrast to the Clausius-Clapeyron equation in thermodynamic
equilibrium, it is not the thermodynamic pressure entering formula (55) but the
normal traction,
Shallow flow approximations and Stokes models
The above analysis for the description of the thermo-mechanical response across a phase change surface addresses this behavior on an interior surface where cold and temperate ice touch each other. Such ice masses are also bounded by other surfaces, e.g. the free surface separating ice from the atmosphere, the interfaces between ice & solid ground and ice & water for ice shelves floating on ocean or lake water, see (Figure 12). In a transition of an ice sheet into an ice shelf, a singular line, called grounding line, separates the grounded sheet from the floating shelf. The case, where an ice sheet ‘sits’ primarily on solid ground, but also has floating portions (on sub-glacial lake(s)), also occurs; Antarctica e.g. is an ice continent with a substantial portion resting on sub-Antarctic lakes.
Each glacier, ice sheet and ice shelf has its own set-up and boundary conditions
that together with the field equations define its initial boundary value
problem. Two limiting situations are
sketched in (Figure 12). An ice sheet (left part) rests on solid ground, and
the flow is dominantly horizontal with smaller vertical velocity components.
This corresponds to deformations, principally governed by shearing. Length
scales of field variations are about 10 times the ice sheet depth or larger and
normal stress effects can to first order be ignored in comparison to the shear stresses
The above described properties almost trivially suggest that the
transition behavior from an ice sheet to an ice shelf must necessarily be
treated by a combination of the SOSIA & SOSSA [35], for a graphical sketch, see (Figure 13).
At the outset one ordinarily does not have sufficient information,
whether the ice mass is cold or temperate or polythermal [36]. More specifically, even if one should have access to the base of such
an ice mass and can reasonably estimate where the solid base is cold and where
it is temperate, one still has generally no initial knowledge of its
thermodynamic state -- cold, temperate or polythermal with unknown location of
the CTS. Extensive initial computations by trial and error will have to be
conducted just to estimate the initial conditions for a reliable evolution
scenario. So far such computations have only been performed by employing the
power flow law, even though our review has evidenced that the law (31) requires
amendments by replacing the Glen-Steinemann power law by a functional relation
reproducing a finite viscosity at zero stress deviator, or even better to use
the Reiner-Riwlin fluid law. The standard procedure is to initially assume an
ice mass to be wholly cold. This likely corresponds to polar glaciers & ice
sheets for which the entire ice mass has temperatures below the melting point
except perhaps at, and very close to, the basal surface. If computations should
generate small basal-near boundary layers with temperatures above the melting
point, then the temperature at these points is set equal to the melting point
in this sub-region and the no-slip boundary condition is replaced at the base
by a perfect sliding law. A few iterations may suffice to reach convergence.
Generally, authors only dispute about a temperature dependent rate
factor and restrict their focus on cold ice. They write
For details, see Cuffey & Paterson (2010), [32]. However, the
arguments in which way this parameterization is justified is not very
convincing to us. For instance, the value of the parameter
As an example, it is known, that ice that was formed during an Ice Age
contains a larger amount of dust than Holocene ice; this affects its
deformability. This effect is incorporated by making
Thus, Pleistocene ice is approximately three times softer than Holocene
ice [37].
The above approach can be used with the Stokes equations [38] or the simplified equations of the shallow ice approximation (SIA,
Hutter 1983, [79]) or the Parallel Ice Sheet Model (PISM, see Bueler &
Brown (2009), [18], that combines the shallow ice approximation with the
shallow shelf approximation (Weis, et al. 1999), [159]. This model combines the
horizontal shear deformation of vast ice sheets and small sliding with the
longitudinal straining of ice shelves and ice streams, in which sliding effects
near the basal boundary layer may be dominant. According to Greve (1997), [60],
for the SIA to be a valid approximation the horizontal length scales of the topography
must be at least of the order of 10km. At shorter wave-length resolutions,
Stokes models are generally needed and used [39]. In the same spirit, the PISM hybrid model has been used repeatedly to
simulate dynamics of ice fields [40].
Schoof (2003), [133] has proposed a proposition for applications of PISM
to situations with typical relatively small undulation scales (smaller than ~10
ice depths but larger than 1 ice depth). His scheme introduces a further
multiplication factor θ, extending the rate factor in (31) to
Geometries for ice sheet extents at the Last Glacial Maximum (LGM, ~24.000 years BP) for the Alpine ice-field have been reconstructed on the basis of positions of terminal moraines and erratic boulders [42]. Moreover, trimlines were identified with the maximum ice surface elevation at the LGM. Geomorphological reconstructions evidenced for the Alpine ice field details regarding dominant flow domains, flow directions, and ice-free rock formations (nunataks), constrained by bed topography [43]. Imhof, et al. (2019) report: ’While the reconstructed maximum ice extent can be matched fairly well, the models produce ice thicknesses much greater than suggested by geo-morphological evidence, on average 500 m thicker for the Rhine Glacier (Becker, et al. (2016), [11]), and 800 to 861 m thicker for the Rhone Glacier (Becker, et al. (2017), [12], Seguinot, et al (2018), [152]). Only the modeling study by Cohen, et al. (2018), [29] is able to match the reconstructed ice surface elevation’.
If we understand the theoretical basis of these software applications
properly, the authors, who try to numerically reproduce the ice fields at the
LGM employ software that does not explicitly predict the evolution of the
moisture content in temperate ice regions. Their models nowhere employ or
describe the distribution in space and time of the moisture content in the
temperate ice domain. The models are, thus, not able to describe the growth or
retreat of the evolution of these regions. This important additional complexity
will influence the total melting rate of such ice fields that is likely to gain
significance in today’s climate scenarios. Moreover, Schoof’s (2003), [149],
additional rate factor
So, we must treat temperate ice as a mixture of ice with water inclusions [44] for which the balance laws of mass, momentum and energy are complemented by a balance of mass for the water, formulated as a diffusion equation for this moisture content (defined as the ratio of water mass to mixture mass, Hutter (1982), [83]. The constitutive equation for the mixture stress employs so far the Glen-Steinemann power law whose rate factor depends on the moisture content (that replaces the temperature). The temperature on the other hand stays at the melting point via the Clausius-Clapeyron relation. Moreover, the moisture production rate in the moisture diffusion equation is related to the power of working in the energy equation. The structure of the emerging partial differential equations is parabolic with non-vanishing moisture diffusivity ν, else hyperbolic. The temporal evolution of polythermal ice sheets inferred by using the evolution of the class-I mixture of polythermal ice is today (2019) just at its initial steps, see Greve (1997, 2000), [69,70], Seddik, et al. (2017), [151]. Intensive work on this model in the near future is pressing.
The theoretical parts of the IBVPs have kept mathematically oriented glaciologists and climatologists busy since the 1980s. Yet, to integrate the ‘exact’ Stokes problem computationally is still very difficult, the reason being that integration over a climate cycle (of the order of 100’000 years) or more, to reliably obtain present-day initial conditions for geometry, ice velocity and temperature distributions, is mainly a question of CPU times [45]. To circumvent this, singular perturbation methods were employed. It led to the shallow-ice and shallow-shelf approximations [SIA, SSA] [46] and their second order improvements [SOSIA, SOSSA] [47]. Today, there exist several open source programs that solve the Stokes formulations for small-scale (in space and time) analyses [48], for SIA and SSA problems (Greenland, Antarctica, and the Alps) [49] and Earth-embracing climate studies. To improve this, Schoof’s, (2003) [149], additional rate factor θ operates as a smoothing operator of the basal sliding resistance, yet the suggested smoothing operation may at best be interpreted as the supposition of perfect sliding at the rough bed that is transferred to a viscous sliding law on the smoothed-out bed. Moisture mass production is not involved. Yet, to adequately model the melting rates and, therefore, the moisture evolution in the likely temperate basal boundary layer correctly, high resolution of the rough basal geometry is unavoidable.
The scientific reports and papers coming out annually are large. They are concerned e.g. with the deglaciation of the mountainous glaciers and ice caps [50], the ice loss of Greenland [51] and West Antarctica [52], coupled with sea level rise (of the order of up to 1m at the end of the year 2100 [53]. They affect the global circulation of the ocean currents[54] and the Earth’s climate [55] and likely leading to millions of displaced people (e.g. of Bangladesh) [56].
In computations of ice sheet or ice shelf performances over millennia the various domains bounding the evolving ice mass may not be treated as passive rigid continua. The variation of the weight of an ice sheet due to the mass loss/gain by external climate driving requires considering the lithosphere and asthenosphere as heat conducting nonlinearly viscous (plastic) continua. These exert important effects on the non-steady response of the climate variations in time and space on the Earth’s surface, and this affects the melting and freezing processes of the ice on the Earth surface. Such effects are incorporated in today’s ice sheet models.
To summarize
Today’s (2019) state of the art when employing the Stokes equations of the creeping ice flow dynamics suffers from the following deficits:
· Present programs are not flexible enough to apply them for sufficiently long climate scenarios of the order of several (tren to hundred) thousand years.
· This software does not strictly account for the two different thermodynamic states – cold and temperate – that may exist in glaciers and ice sheets with an internal cold-temperate transition surface where the two regions touch.
· The material response is exclusively based on the Glen-Steinemann flow law that has been shown not to be adequately match able with known laboratory experiments.
The quadratic terms of the Reiner-Riwlin fluid need to be accounted for, if satisfactory agreement with measured data in multiaxial states of stress is to be obtained.
Summary
We have outlined in the first half of this review that it lasted several centuries until a rational physical understanding of the motion of glaciers (and ice sheets/shelves) had developed. According to the early understanding large ice masses on Earth were thought to be rigid and non-movable, then sliding rigid objects and finally bodies that move like very viscous fluids, similar to honey or a dough. This state of knowledge was achieved by the mid 19 hundred, but it took until the mid-1950s until it was recognized by specialists that this behavior could be categorized as a subject of rheology. It needed the detailed laboratory experiments of Glen and Steinemann to understand the creep of cold ice as a substance whose deformation could be subjected to a mathematical model, known as a nonlinear heat conducting fluid with power law rheology. We could have stopped our review at this point and could have reported how most glaciological specialists spent their time by perfecting the numerical-mathematical approaches for a great number of realistic glaciers, ice sheets and ice shelves on the globe; this would probably have made an acceptable review report. The glaciological community would likely have been happy with this, as the model would stay in conformity with Nye’s (1956), assertion [135], (i) that the third invariant does not affect the strain-rate-stress relation and (ii) strain rate and stress deviators are coaxial. We preferred to continue by searching for possible pitfalls that may be contained in this model. However, already in 1980, a committee of ice researchers alerted in a review of experimental studies on the non-negligible effect of the third invariant – a call that has largely been ignored (see Hooke & Mellor and 11 others (1980)).
Already in the late 1950s Steinemann conducted combined shear-compression
tests. Glen took these data and checked whether they would be in conformity
with the co-axiality postulate. His results are summarized in (Figure 8). If
Nye’s conjecture would be correct, all theoretical curves plotting
A further weakness of the power law parameterization of polycrystalline isotropic ice is that a polynomial fit of degree 3 of the experimental curves do better fit the individual experimental data (Smith & Morland). However, a better fit of data from different authors or sites cannot be obtained this way.
In ice sheet computations of ice shields e.g. for the Alps (and other large ice masses) at the LGM, the basic concepts illustrated above have restricted attention to power laws of cold ice with adjustments of the rate factor in a rather ad-hoc manner to local geo-morphologically inferred ice sheet geometry.
Finally, we have not found any substantial application of the mathematical theory of polythermal ice masses. The application of their mathematical theory would need to use the class-I mixture theory, in which the moisture (mass) balance equation would be employed in the temperate ice region, and in which the moisture jump condition would be employed at the Cold-Temperate Transition Surface (CTS). This would give a better knowledge of the water budget in such ice fields.
Furthermore, whereas the mathematical model for polythermal ice masses has been proposed more than 30 years ago (Fowler 1977, [48], Hutter 1982, [83] with improvements by Greve 1999, [7]), its application to realistic situations still awaits its explicit use.
In conclusion, this review has disclosed two extensions of present-day ice sheet models, (i) the replacement of the power law rheology by a full Reiner-Riwlin parameterization and (ii), the explicit use of the Class-I mixture model to properly account for the phase change processes in the temperate ice regions and along the CTS.
[1] The authors were repeatedly critisized by the attribution of Glen’s flow
law to Glen-Steinemann and the arguments of this critique seem to be the facts
that Glen published the power law prior to Steinemann, according to
these’experts’. However, the documented facts do not necessarily transmit the
historical facts. True is that Steinmann’s Ph. D. dissertation was completed
around 1953, i. e. at the same time as Glen’s, and then submitted, but sitting
on the professor’s shelf, collecting dust for about 5 years, when it was
published in 1958 (in the German language): Experimentelle Untersuchungen zur
Plastizität von Eis, Beiträge zur Geologie der Schweiz, Nr. 10, 1958) in a
geological periodical of the Swiss Natural Sciences. Steinemann managed to
publish some excerpts between 1952 and 1958 (in the English language); these
are the papers that are usually referred to in the literature. KH has not seen
the 1958-paper being generally referenced internationally. Steinemann left
glaciology and went to Lausanne as a professor of physics. He might well have
been fed-up with the German-Swiss environment in the 1950s. However, Glen
took-up Steinemann’s experimental findings in shear-compression experiments and
generated the evidences of the failure of the power law in his 1958-paper. In
this context it is worth mentioning that the first paper on the power law of
ice is written by Orowan [136], and Glen &
Perutz, [54], cite Steinemann’s power law
already in 1954. Moreover, also in a series of other papers the (power) flow law
is attributed to both Glen and Steinemann, see e.g. Budd et al. (2013) [17]. We might also add that there is a “Steinemann
Island” close to the Antarctic Island, named by UK Antarctic Place-names
Committee (UK-APC) in 1960 for Samuel Steinemann, Swiss Physicist, who made
laboratory investigations on the flow of single and polycrystalline ice.
[2] As an example of the many fundamental articles on this subject we
restrict here mentioning that of W. Oswald in 1929 [137].
[3] Johann Jakob Scheuchzer
(02.08.1672 – 23.06.1733) physician and natural scientist from (and in) Zürich,
Switzerland. He studied medicine in Altdorf, near Nürnberg and in Utrecht,
where he received his medical doctorate. In the same year, he took his first
journey to the Alps. In the year 1695, when one of the four physicians of the
town Zurich had passed away, the Zurich government appointed him as his
successor. Scheuchzer’s merits are wide embracing: he replaced the
trigonometric determination of the heights of positions on the Earth by much
more accurate barometric instruments; with his ‘Herbarium divularium’ he became
the founder of paleo-botany. In the year 1713 he drew with his ‘Nova Helvetiae
Tabula Geographica’ the best and most accurate map of Switzerland at that time.
In several journeys through Switzerland he also visited in 1705 the ‘Rhone
Glacier’, and wrote a report, which also contains the other Swiss glaciers as
they were known at that time.
[4] Jean de
Charpentier (07.12.1786, Freiberg (Saxony)-12.09.1855, Bex, Switzerland) was a
Swiss geologist; he was first mining engineer in Silesia, worked then in the
Pyrenees where he wrote several geological memoirs. In the year 1813 the
government of the Canton Vaud, Switzerland, offered him the directorship of the
salt mine in Bex where he stayed until his death.
[5] The older brother of Jean, Toussaint de Charpentier (1779-1847),
equally a geologist, mining engineer and entomologist also adopted this phase
change argument as cause for the forward glacier motion.
[6] Louis Agassiz (28.05.1807, Neuchatel
-14.12.1873, Cambridge, Mass, USA), a descendent of a clergyman, was a leading
paleontologist and natural scientist of the 19th
century focusing on fish-fossils and glaciology. He received his higher
education from the universities of Zurich, Heidelberg and Munich in medicine
and natural sciences and received his doctorate in philosophy from the
University of Erlangen and his medical doctorate from the University of Munich.
He was professor at the University of Neuchâtel and (later at Harvard
University, Boston, Mass. USA, where he stayed until his death). Since 1836,
Agassiz was a leading glaciologist. He, with his research group, performed in
1840, see [4], measurements on the
Aare-Gletscher, where they built a hut (‘Hôtel des Neuchâtelois’), focusing on
the properties of glacier ice, its temperature and the water circulation as
well as the motion and mobility of the ice. He was one of the first to
postulate the existence of the Ice Age, but interpreted it as a sudden event
that preceded the rise of the Alps. The German poet Johann Wolfgang von Goethe
anticipated the postulation of the Ice Age in 1831 (see Cameron, D., (1965), [24])
[7] Johann Georg Altmann (21.04.1695, Zofingen – 18. 03.1758, Ins) studied
at the Theological School in Berne, Switzerland, before he worked during
1725/26 and again 1734/35 as protestant clergyman in Wahlern. From 1734 to 1757
he taught in Berne as Professor of eloquence and ancient Greek and ethics. His
‘Versuch einer historischen und physischen Beschreibung der helvetischen
Eisbergen’ (‘Attempt of a historical and physical description of the Helvetic
icebergs’), edited by Heidegger and Compagnie, Zurich 1751, brings forward his
sliding postulate of glacial movement. (Available as a pdf at ETHZ library)
[8] Gottlieb Sigmund Gruner (20.07. 1717, Trachselwald, Berne--10.04. 1778,
Utzensdorf) was born into a Bernese patrician family and grew up in the town of
Burgdorf in the Canton of Berne. After studying law he was employed as
archivist for the Landgrave of Anhalt-Schaumburg and travelled as such through
Prussia and Silesia. In 1779 he permanently returned to Switzerland, where he
was employed as an attorney in the Canton Berne. Gruner had broad interests
that also included mineralogy and geology. He wrote a three-volume treatise
‘Die Eisberge des Schweizerlandes’ (‘The ice mountains of Switzerland’) that
appeared in 1760/62. In this treatise, he presented his theory of sliding of
glaciers. (This is an excerpt from Tobias Krüger, 2009), [105]).
[9] André César Bordier (24.12.1746, Geneva – 18.03.1802, Céligny) studied
protestant theology prior to 1770 and was a clergyman from 1770 - 1774. In 1775
he became a member of the ‘Council of the 200’ in Geneva, in 1777 he became the
first judge at the court martial, 1790 councilor and in 1791 syndic trustee of
Geneva. In 1794 he was (in absence) sentenced to death by the revolutionary
court of justice of Geneva.
[10] Louis Rendu (09.12.1789, Meyrin – 08.08.1859 Annecy)
was a French roman-catholic priest and later bishop of Annecy and a scientist.
He was the author of ‘Theorie des
glaciers de la Savoie’, an important book on the mechanisms of glacial motion. The Rendu
Glacier, Alaska, U.S. and Mount Rendu, Antarctica are named
for him, see also Hollier, J. & Hollier, A.
(2016), [79].
[11] State Archive Solothurn (Switzerland) :ISIL:
CH-000043-5, document collection, document of 5. Oct, 1301, according to [105].
[12] Guler von Wyneck, Johannes (31.10.1562, Davos –
03.02.1637, Chur, both in the SE of Switzerland) received his first education
in the Latin school of Chur and studied then in Zurich, Basel and Geneva.
Because he was in 1582 elected as local governmental secretary
(‘Landschreiber’, in German), he could not complete his studies. In the year
1587 he became ruler (‘Landeshauptmann’, in German) of the Veltlin. In his
second marriage, he was married to Elisabeth von Salis, a descendent of the
very influential family in Graubünden, which brought to him the noble attribute
‘von Wyneck’. He was politically very influential, and furthered the formation
of an alliance of the Drei-Bünden (three confederations), but left to Zurich to
escape local political disturbances. He then withdrew from political activities
for more than a decade. His merits as a historian and mapmaker are exceptional,
evidenced by his chronograph ‘Raetia’.
[13] Moritz-Anton Capeller (09.06.1685, Willisau–16.09
1769, Beromünster, both in Switzerland) grew up in Lucerne, where, after
passing the Latin school, he completed his education in the Jesuit college.
From 1700-1704 he studied mathematics and philosophy at the Collegium
Helveticum in Milano; the medical studies were completed in the year 1706 in
the Lothringian Academy at Pont à Mousson. During the Spanish war, he served as
physician and engineer in Naples. He returned 1710 to Lucerne to follow his
father as a private physician of the town. He served as member of the Lucerne
Town Parliament and also had some assignments as an engineer (e.g. correction
of the torrents) and acted as a teacher of mathematics and geometry at the
local school of artillery. He earned scientific recognition and fame by his
crystallographic-mineralogical works (‘Prodromus crystallo-graphiae’) that
brought him the membership in the Royal Society of London. His principal work
is on the history of the mountain Pilatus (‘Pilatii montis historia’,
1723-1728). His interests were wide, and so were his correspondences e.g., with
Jakob Scheuchzer and Albert von Haller.
[14] Abraham Schellhammer assigned the distribution of such
orphane rocks to the biblican flood in his ‘Topo-graphia’ (1732). He was aware
of their Alpine origins, because he wrote that they came from the ‘destruction
of the moundains’.
[15] Albrecht von
Haller (16.10.1708, Berne – 12.12.1777, Berne) was a Swiss anatomist,
physiologist, naturalist,
encyclopedist, bibliographer and poet. As an infant prodigy, he had
mastered Greek, Hebrew and Latin at the age of 15 when he had already acted as
author of numerous translations from Ovid, Horace and Virgil as well written
original lyrics, dramas and an epic. He studied medicine in Tübingen and
Leiden. His professional activity was started as physician in Berne in 1729,
where aside of this, he also studied mathematics and botany for which he
produced the basis of his great work on the flora of Switzerland. While there,
he conducted many journeys through the Alps. As a result he created in 1729 the
poem ‘Die Alpen’ (‘The Alps’) that appeared in 1732 as the first edition of his
embracing work ‘Gedichte’ (Poems’), demonstrating his appreciation of the
mountains. In 1736, Haller was called by King Georg II as professor of
medicine, anatomy, botany and surgery to the newly founded University in
Göttingen, where he became famous through his work in medicine and botany. He
left there in 1753 to return to Berne.
[16] Christian Leopold von Buch (26.04.1774 - 04.03.1853) was a German geologist and paleontologist born in Stolpe an der Oder (now a part of Angermünde, Brandenburg) and is
remembered as one of the most important contributors to geology in the first
half of the nineteenth century. His scientific interest was devoted to a broad
spectrum of geological topics: volcanism, petrology, fossils, stratigraphy and mountain formation. His most remembered accomplishment is the scientific definition of the
Jurassic system.
[17] Scott James Hutton (03.06.1726 - 26.03.1797, both in Edinburgh), was an
independent scholar and claimed founder of geology, also worked as a physician,
and was earlier founder of a chemical factory. He was also active as a
multi-year farmer.
[18] John Playfair (03.10.1748, Benvie (near Dundee) – 07.20.1819, Burntisland, five,
Scotland) was a Church of Scotland minister, remembered as a scientist and
mathematician, and a professor of natural philosophy at the University of
Edinburgh. He is best known for his book Illustrations of the Huttonian Theory of the
Earth (1802), which summarized the work of James Hutton. It was
through this book that Hutton's principle of uniformitarianism, first reached a wide audience. Playfair's textbook Elements of Geometry made a brief
presentation of Euclid's parallel postulate known now as Playfair's axiom. In 1783, he was a co-founder of the Royal Society of
Edinburgh. He served as General Secretary to the Society during
1798-1819.
[19] Mathias von Flurl (05.02.1756, Straubing – 27.07.1823, Kissingen) is
the founder of the Bavarian mineralogy and geology. In 1781, he was called as
professor to the University in Munich. From 1788 to 1806, he modernized as
factory commissioner the china factory Nymphenburg. 1792 he published the first
geological overview of Bavaria in his ‘Beschreibung der Gebirge von Baiern und der oberen Pfalz’ (‘Description
of the mountains of Bavaria and the upper Pfalz’) including the first
geological maps of Bavaria. He thus established the groundwork of geology and
mineralogy of Bavaria. Starting in 1822, he alluded the Elector Max IV. Joseph
to acquire various collections of minerals and laid the foundation for the
‘State of Bavaria’s Mineralogical Collection’ in Munich. He was also the
official inspector of the Bavarian salt-works.
[20] Franz von Paula Gruithuisen (19.03.1774, Burg Haltenberg am Lech – 21.06.1852,
Munich) was a Bavarian physician and astronomer. He taught medical students before becoming a professor of
astronomy at the University of
Munich in 1826. He believed that the Earth's moon was habitable and
was the first to suggest that craters on the Moon were
caused by meteorite impacts.
[21] Jean-Pierre Perraudin (25.04.1767, Lourtier – 03.01.1858, Lourtier)
from the Canton of Valais, Switzeland was first member of the local council of
the village Bagnes and later member of the council of the Canton of Valais. He
concluded, based on observations that originally much larger glaciers
transported erratic boulders and that the ordered scratches at rock walls must
have been caused by rocky boulders that were moved and rotated at the glacier
beds. When in 1818 the ice dammed lake of the Glacier Giétroz erupted, he
informed the engineer Ignaz Venetz, who communicated this idea that later was
taken up by Jean Charpentier and Louis Agassiz.
[22] Gilliéron Jean-Siméon-Henri (1779-1838), was a deacon in Vevey, at Lake
Geneva.
[23] James David Forbes (20.04.1809, Edingurgh, Scotland -- 31.12.1868
Bristol, England) was a Scottish physicist and glaciologist, who worked
extensively on the conduction of heat and on seismology. He was professor at
Edinburgh University from 1833-1859 when he became Principal of the United
College of St. Andrews. His principal glaciological research is summarized in
‘Travels through the Alps of Savoy’ (1843), [46].
[24] John Tyndall (02.08.1820, Leighlingbridge Ireland – 04.12.1893, Haslemere, England)
was a prominent 19th-century Irish physicist. His initial scientific fame arose
in the 1850s from his study of diamagnetism. Later he
made discoveries in the realms of infrared radiation and the physical properties of air. Tyndall also published more than a
dozen science books, which brought the state-of-the-art 19th century experimental
physics to a wide audience. From 1853 to 1887, he was
professor of physics at the Royal
Institution of Great Britain in London. As an
enthusiastic mountaineer in the Valais Alps of Switzerland he developed a
strong interest in glaciology to which he contributed enormously by his books. The Pic Tyndall in the Alpes, the Tyndall Mountains in
Antarctica, the Tyndall Lake and Glacier in Chile and the Tyndall Moon Crater
are named after him.
[25] Franz Joseph Hugi (23.01.1791,
Grenchen, 25.03.1855, Solothurn, both in Switzerland) was a Swiss geologist and
researcher of the Alps. He started as a catholic priest and teacher at the
orphanage in Solothurn, Switzerland. There, he founded the Society of Natural
Sciences of the Canton Solothurn, which he handed over to the Canton in 1836.
Having served in the orphanage and as a teacher in the ‘Realschule’ (upper
level public school) he received in 1833 in the newly opened high school of
Solothurn the professorship in physics, but was discharged 1837 because of his
transfer to the protestant faith. In the year 1844 he received an honorary
doctoral degree from the University of Berne. His research on glaciers is
summarized in the scripts ‘About the nature of glaciers and winter journey into
the polar sea’ (1842) and ‘The glaciers and erratic boulders’ (1843). A
detailed evaluation of Hugi’s significance as a glaciologist is given in Albert
Krehbiel’s doctoral dissertation (1902), [104].
[26] The difference in these dust distributions is caused
by the fact that the ice in the glacier de Léchaud does not fall through an ice
fall as at g, as seen in panel (b) of (Figure 3); here, the
discharge difference in summer and winter generates annual repetitions of the
amount of dust in the Mer de Glace that causes the curved dust stripes.
[27] Sir John Shipley Rowlinson (12.05.1926 – 15.08.2018) was a British
chemist, who earned his doctorate at the University of Oxford and had research
and teaching positions at the University of Wisconsin, the University of
Manchester, the Imperial College London and Oxford University, where he retired
as professor in 1993. His research was on capillary and cohesion, but he also
wrote about the history of science. He was a Fellow of the Royal Society of
London and the Academy of Engineering. He routinely climbed the Swiss Alps and
the Himalaya.
[28] Cameron writes ’theory’ instead of ‘concept’, but a
theory was established only later by Milankowich (1930), [124].
[29] All bold faced quantities are second rank tensors or 3x3-matrices:
[30] We take here the position that the reader knows that a graph, in which
log(y) is plotted against log(x) as a straight line corresponds to a power law
of y against x.
[31] This fluid is called after M. Reiner and R. Riwlin, but is by far
mostly referred to as ‘Reiner Rivlin fluid’. Before 1929, Reiner had worked on
a number of projects, now considered as rheology, as said by Scott-Blair
(1976), [150]!, “he developed the ‘Reiner-Riwlin
equation’ – his colleague, Miss R. Riwlin was killed in a road accident. Her
nephew, Roland S. Rivlin (now spelled with a ‘V’) is a famous
[British]-American rheologist”. The Reiner-Riwlin equation does not reproduce
primary, secondary and tertiary creep for that purpose, the constitutive stress
must also depend on higher order Rivlin-Ericksen tensors (see e..g. Man, et al.
(1985, ‘87, ‘92, 2010), [115-119], and Hutter
(2019), [90]).
[32] In this process a further
redundant equation,
arises and is used. This equation is useful in
the derivation of the results (28).
[33] This quantity agrees with the free enthalpy of which another denotation
is Gibbs free energy.
[34] To avoid this numerically rather difficult problem of the determination
of sharp interfaces one could perhaps employ here with advantage the phase
field theory.
[35] Baral (1999), [8],
Baral et al. (2001), [9] Schoof & Hindmarsh
(2010), [149], Kirchner et al. (2011), [103]. Weis (1999), [176],
Weis et al. (2001), [174].
[36] About 50 years ago, Swiss glaciologists said that Swiss glaciers are
wholly cold. However, already in the 1980s there were sufficient indications
that they safely had to be assumed to be polythermal. An early paper is
Haeberli (1976), [75].
[37] A possibility for determining
[38] A prominent software, integrating the Stokes equations by the finite
element method, is called’ Elmer/Ice’, see Gagliardini et al. (2013), [52].
[39]
Leysinger-Vieli & Gudmundsson (2004)[98], Le
Meur et al. (2004), [97], Adhiraki &
Marshall (2013), [2], Braedstrup et al. (2016), [16], Seddik et al. (2017), [135].
[40]
Golledge et al (2012), [59], Yan et al. (2018), [161].
[41] The value for θ follows from a
smoothing procedure of the basal topography, see Schoof (2003), [133]
[42]
Benz-Meier (2003),[13], Bini et al. (2009), [14], Coutterand (2010), [31],
[43] Florineth &
Schlüchter, (1998), [41], Florineth (1998), [42], Benz-Meier (2003), [13],
Kelly et al. (2004), [102], Bini et
al. (2009),[14], Coutterand (2010), [31].
[44] Incidentally, a distribution of the water production could also be
estimated with the models not explicitly using a moisture mass balance equation
by evaluating a posteriori in the temperate region the moisture production rate
C according to eq. (37).
[45] Numerical solutions of the Stokes equations are today (2019) only
possible for realistic territories for integrations over small times into the
future, e.g.,
[46] Fowler (1977), [48],
and Weis, Greve & Hutter (1999], [176], and
Calov, (1994), [20], Calow & Hutter (1996), [21] and (1997), [22].
[47] Baral (1999), [8],, Baral et al. (2001), [9],
Schoof & Hindmarsh (2010), [149], Kirchner
et al. (2011), [103], Weis (2001), [174], Weis, et al. (1999), [176].
[48] Approximately 1000m element lengths or somewhat less.
[49] Becker et al. (2016), (2017), [11,12], Calov (1994), [20],
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[50] e.g.
Jouvet et al. (2009, 2011), [99,100],,
Adalgeirsdóttir et al. (2011), [1], Zekollari et
al. (2014,
2017), [179,180],
Réveillet et al. (2015), [144], Gilbert et al.
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[51] e.g.
Fürst et al. (2015), [51], Vizcaino et al.
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[23], Rückamp
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[52] Favier
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[53] De
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[54] e.g.
Swingedouw et al. (2008), [160], Gierz et al. (2015), [53], Becker et al. (2016), [11], Jackson and Wood
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[55] e.g.
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[56] Neumann
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Figure 5: Dirt stripes (left) of the ‘Mer de Glace’. Its middle moraine separates
the ice of ‘Glacier du Géant’ from that of the ‘Glacier de Léchaud’ and Glacier
du Talèfre’, which shows only traces of moraines; these diffuse to the sides.
Tyndall suspects and concludes that the processes in the ice fall produce an annually
repeated formation of an ice-dust mingling (via quasi-periodic ice rock
slides). (right) End of the dust stripe when the glacier curbs into the snout
area. From Tyndall (1860, 1898), [163, 164], differently arranged.
Figure 6: (a) Creep curves for
polycrystalline ice at -4.8°C under various stresses at first and secondary
creep, from Steinemann (1958). (b)
Principal sketch, showing first, secondary and tertiary creep, principal sketch,
from Hutter (2018), [90].
Figure 7 (left) Strain rate during
secondary creep for polycrystalline ice as a function of stress. (right) Strain rate for tertiary creep
of polycrystalline ice as a function of stress, from Steinemann (1958), [158] as
drawn by Hutter (1982), [90].
Figure 8: A graph of
Figure 9: (a) Various strain-rate
stress data from Glen, [59], Mellor & Testa (1969 a,b), [122], and Steinemann
1956) [159], identified in the inset. (b) Same as in panel (a), but now
normalized to the same temperature at 273.13 K. The panel also shows two
polynomial laws due to Butkovich & Landauer (1960), [19]. The strain-rate
unit is given in
Figure 10: Same as (Figure 9). The
separate data sets are correlated with 3-term polynomials (J = 2) with
polynomials as given in
Figure 11: Data points in the
invariants plane defined by the axes
Figure 12 Sketch of a coupled ice
sheet--ice shelf system. Upstream (downstream) of the grounding line the ice flow
is governed by the SIA-equations (SSA equations). In a transition region across
the grounding line, both sheet flow and shelf flow features prevail. So, to
match the sheet and shelf flows, the SIA and SSA have to be extended to the
SOSIA and SOSSA to properly transfer the flow through the transition zone, from
Kirchner, et al. (2011), [103].).
Figure 13: Velocity profiles and
stress states for ice-sheets (A) and ice-shelves (B), showing at the top the
characteristic vertical velocity profiles that mimic sliding and gliding for
SIA and only sliding for SSA; in the bottom row the infinitesimal cubes are
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