FariaEA2013 PartII accepted JournalStructGeol

FariaEA2013 PartII accepted JournalStructGeol
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The Microstructure of Polar Ice. Part II: State of the
ArtI
Sérgio H. Fariaa,b,∗, Ilka Weikusatc , Nobuhiko Azumad
a
Basque Centre for Climate Change (BC3), Alameda Urquijo 4-4, 48008 Bilbao, Spain
IKERBASQUE, Basque Foundation for Science, Alameda Urquijo 36-5, 48011 Bilbao, Spain
c
Alfred Wegener Institute for Polar and Marine Research, Columbusstrasse, 27568 Bremerhaven,
Germany
d
Department of Mechanical Engineering, Nagaoka University of Technology,
1603-1 Kamitomioka, Nagaoka 940-2188, Niigata, Japan
b
Abstract
Besides the obvious relevance of glaciers and ice sheets for climate-related issues,
another important feature of natural ice is its ability to creep on geological time
scales and low deviatoric stresses at temperatures very close to its melting point,
without losing its polycrystalline character. This fact, together with its strong mechanical anisotropy and other notable properties, makes natural ice an interesting
model material for studying the high-temperature creep and recrystallization of
rocks in Earth’s interior. After having reviewed the major contributions of deep
ice coring to the research on natural ice microstructures in Part I of this work
(Faria et al., this issue), here in Part II we present an up-to-date view of the modern understanding of natural ice microstructures and the deformation processes
that may produce them. In particular, we analyse a large body of evidence that
reveals fundamental flaws in the widely accepted tripartite paradigm of polar ice
I
Dedicated to Sepp Kipfstuhl on occasion of his 60th birthday.
Corresponding author. Tel.: +34-94-4014690.
Email addresses: [email protected] (Sérgio H. Faria),
[email protected] (Ilka Weikusat), [email protected] (Nobuhiko Azuma)
∗
Preprint submitted to Journal of Structural Geology
September 23, 2013
microstructure (also known as the “three-stage model,” cf. Part I). These results
prove that grain growth in ice sheets is dynamic, in the sense that it occurs during deformation and is seriously affected by the stored strain energy, as well as
by air inclusions and other impurities. The strong plastic anisotropy of the ice
lattice gives rise to high internal stresses and concentrated strain heterogeneities
in the polycrystal, which demand large amounts of strain accommodation. From
the microstructural analyses of ice cores, we conclude that the formation of many
and diverse subgrain boundaries and the splitting of grains by rotation recrystallization are the most fundamental mechanisms of dynamic recovery and strain
accommodation in polar ice. Additionally, in fine-grained, high-impurity ice layers (e.g. cloudy bands), strain may sometimes be accommodated by diffusional
flow (at low temperatures and stresses) or microscopic grain boundary sliding via
microshear (in anisotropic ice sheared at high temperatures). Grain boundaries
bulged by migration recrystallization and subgrain boundaries are endemic and
very frequent at almost all depths in ice sheets. Evidence of the nucleation of new
grains is also observed at various depths, provided that the local concentration
of strain energy is high enough (which is not seldom the case). As a substitute
for the tripartite paradigm, we propose a novel dynamic recrystallization diagram in the three-dimensional state space of strain rate, temperature, and mean
grain size, which summarizes the various competing recrystallization processes
that contribute to the evolution of the polar ice microstructure.
Keywords: ice, glacier, ice sheet, mechanics, creep, recrystallization, grain
growth, microstructure, fabric, texture
2
1
1. Introduction
2
An essential feature of Earth’s dynamics is the hot deformation of large rock
3
masses in a slow and continuous flow regime called creep. The study of creeping
4
rocks is complicated by various factors; among them diversity and inaccessibility.
5
The former means that rocks are seldom monomineral; rather, they are usually
6
made of complex and variable compositions of minerals with distinct properties.
7
The latter expresses the fact that field observations of creeping rocks are often
8
very difficult or even impossible to perform, because most high-temperature de-
9
formation processes occur in Earth’s interior.
10
For these reasons (not to mention other well-known reasons stemming from
11
climatology; Lemke et al., 2007), the creep of ice turns out to be very interest-
12
ing for geologists and geoscientists (Hudleston, 1977; Wilson, 1979, 1982; Burg
13
et al., 1986; Kirby et al., 1991; Zhang and Wilson, 1997; for a deeper discus-
14
sion see Wilson et al., this issue). The abundance, purity, and low melting point
15
of natural ice make the field study of creeping glaciers and ice sheets a feasible
16
task. Polar ice sheets over Greenland and Antarctica are particularly appealing
17
in these respects, because of their immense mass (2.7 and 22.6 × 1018 kg, respec-
18
tively; Lemke et al., 2007) and purity (polar ice typically has an impurity content
19
in the ppb range; Legrand and Mayewski, 1997), as well as their relatively simple
20
and steady flow, when compared to smaller ice bodies like glaciers and ice caps
21
(Paterson, 1994).
22
Evidently, the investigation of creep and recrystallization of polar ice sheets
23
has also its shortcomings, mainly related to the complex logistics and drilling tech-
24
nology necessary for retrieving old ice samples from several kilometres of depth.
25
A brief review of the difficulties and advances in deep ice core drilling in Antarc3
26
tica and Greenland has been presented in the first part of this work (Faria et al.,
27
this issue) —from now on called Part I— together with the major contributions
28
of deep ice coring to the research on natural ice microstructures. Through that
29
historical synopsis we could appreciate how the current paradigm of natural ice
30
microstructures has emerged, and also how it started being challenged in recent
31
times.
32
Here in Part II we discuss in detail these recent challenges and show how they
33
may reveal to us a new perspective of the mechanics and microstructure of natural
34
ice. To achieve this aim, we carefully reconsider several aspects of our current
35
understanding about natural ice microstructures and the deformation processes
36
that may have produced them, including strain-induced anisotropy, grain growth,
37
and dynamic recrystallization, among others. The whole review ends with a new
38
paradigm for the microstructure evolution of natural ice. For convenience, the key
39
concepts invoked in this work are summarized in a glossary in Appendix A.
40
As it will become evident in the next pages, in spite of many insightful stud-
41
ies of natural ice microstructures and deformation mechanisms, our knowledge
42
about this subject is still imperfect and incomplete. On the other hand, we do
43
have enough information to propose novel plausible models, which together with
44
modern technologies are helping to make this field of research more promising
45
and exciting than ever.
46
2. Crystalline structure and dislocations
47
Under natural conditions on Earth’s surface, ice occurs in the ordinary hexagonal
48
form of ice Ih. This should not be confused with its closely related cubic variant,
49
ice Ic, which presents a similar tetrahedral coordination of oxygen atoms, but
4
50
is metastable at all temperatures (Bartels-Rausch et al., 2012). Ordinary ice Ih
51
has a rather open lattice, with an atomic packing factor of less than 34%, which
52
accounts not only for its abnormally low density compared to liquid water, but
53
also for the pressure-induced reduction of its melting point at high temperatures
54
(Schulson and Duval, 2009).
55
Oxygen ions build the essence of the ice lattice (from now on the term “ice”
56
refers to ordinary hexagonal ice Ih, except when explicitly mentioned otherwise).
57
They are arranged in a structure which resembles that of wurtzite or high-tridymite
58
(Hobbs, 1974; Evans, 1976; Poirier, 1985), viz. layers of puckered hexagonal
59
rings piled in an alternate sequence of mirror images normal to the c-axis (Fig. C.1).
60
Hydrogen nuclei (protons) remain statistically distributed in the oxygen lattice,
61
building covalent and hydrogen bonds along the lines joining pairs of oxygens
62
(Pauling, 1935). This proton disorder is however not completely arbitrary: it
63
must conform with the Bernal–Fowler rules (also called “ice rules”), which re-
64
quire that two protons should be close to any oxygen, with only one proton per
65
bond (Bernal and Fowler, 1933). Hence, each oxygen is involved in two covalent
66
and two hydrogen bonds.
67
The violation of the ice rules, either by an excess or a deficiency or protons,
68
gives rise to particular point defects in the crystalline structure, known as ioniza-
69
tion and Bjerrum defects. These point defects, together with more conventional
70
molecular defects (vacancies and interstitials) play a fundamental role in the me-
71
chanics of ice, as they influence the motion of the main agents of deformation in
72
ice: dislocations (Glen, 1968; Goodman et al., 1981; Okada et al., 1999; Petrenko
73
and Whitworth, 1999; Louchet, 2004).
5
74
2.1. Slip systems and plastic anisotropy
75
According to the fundamentals of dislocation theory (Hirth and Lothe, 1992;
76
Weertman and Weertman, 1992), possible slip systems in ice can in principle be
77
found on the basal, prismatic, and pyramidal planes, as described in Table D.1 and
78
Fig. C.2.
79
Experience shows, however, that the plasticity of monocrystalline ice is strongly
80
anisotropic (Duval et al., 1983): single crystals of ice deform very readily when
81
the shear stress acts on the basal plane, as epitomized more than a century ago
82
by McConnel’s (1890) “deck of cards” metaphor. This phenomenon was later
83
beautifully illustrated in Nakaya’s (1958) experiments, through the use of shadow
84
photography for revealing slip bands (Appendix A) in deformed monocrystalline
85
ice bars. Not long after, Bryant and Mason (1960) found grouped etch pits and
86
channels along slip bands in formvar replicas of deformed ice monocrystals, cor-
87
roborating the hypothesis that slip bands consist of a high density of dislocations.
88
In polar ice, the optical observation of slip bands turns out to be much more dif-
89
ficult, because of the very low strain rates characteristic of ice sheet flow. Nev-
90
ertheless, advanced digital methods of optical microscopy could show (Fig. C.3)
91
that slip bands are also a common feature of polar ice (Wang et al., 2003; Faria
92
and Kipfstuhl, 2004; Kipfstuhl et al., 2006).
93
The modern explanation for the strong plastic anisotropy of hexagonal ice
94
is that the energy of a stacking fault on the basal plane is so low that perfect
95
basal dislocations may dissociate into Shockley partial dislocations separated by
96
a stacking fault (Fukuda et al., 1987; Hondoh, 2000). Thus, recalling that the
97
self-energy of a dislocation is proportional to the square of its Burgers vector, it
98
follows that a perfect basal dislocation in ice with Burgers vector b is expected
6
99
to stabilize into a ribbon-like structure (Fig. C.4) consisting of a stacking fault
100
delimited by two partial dislocations with Burgers vectors b1 and b2 = b − b1 ,
101
provided that
b2 > b21 + b22 ,
with b2i := bi · bi
(i = 1, 2, ∅) ,
(1)
102
and the energy of the stacking fault created by this dissociation is sufficiently low
103
to preserve the inequality (1).
104
The reason for the low stacking fault energy of ordinary ice is the small energy
105
difference between hexagonal ice Ih and cubic ice Ic (Bartels-Rausch et al., 2012).
106
This leads to the conclusion that the stacking fault between the two partial dislo-
107
cations should possess cubic structure (Hondoh, 2000). Actually, the width of the
108
resulting stacking fault is expected to be rather large, ranging from one to two
109
orders of magnitude larger than the lattice spacing (Fukuda et al., 1987). As a re-
110
sult, cross-slip and climb of such widely extended dislocations should be strongly
111
suppressed, seeing that the stress required to constrict extended dislocations, al-
112
lowing them to move on non-basal planes, is considerably large (Gilra, 1974; the
113
need of full constriction for cross-slip has been objected by Duesbery, 1998, pro-
114
vided that the driving stress on the cross-slip plane is large enough). Another
115
consequence of the dissociation of basal dislocations is that a dislocation with an
116
initially arbitrary shape soon evolves into a combination of long basal and short
117
non-basal segments (Fig. C.4a), owing to the strong tendency of basal segments to
118
elongate (Hondoh, 2000). In fact, theory and experiments suggest that non-basal
119
segments should be one to two orders of magnitude shorter than basal segments
120
(Fukuda et al., 1987; Ahmad and Whitworth, 1988; Hondoh, 2000). Therefore,
121
non-basal dislocation segments are generally too short to significantly contribute
122
to macroscopic deformation (Petrenko and Whitworth, 1999).
7
123
To sum up, the dissociation of basal dislocations into partials and its many
124
consequences are essential for explaining the extreme plastic anisotropy of ice.
125
2.2. Heterogeneous strain and non-basal slip
126
Non-basal slip in high-quality ice single crystals has often been observed by X-
127
ray topography (Fukuda et al., 1987; Ahmad and Whitworth, 1988; Higashi et al.,
128
1988; Hondoh et al., 1990; Shearwood and Whitworth, 1991). These studies re-
129
vealed an interesting feature of ice plasticity, namely the rapid motion of short
130
edge dislocation segments on non-basal planes. While such fast-moving short
131
segments are not expected to significantly contribute to macroscopic deformation,
132
they provide mechanisms for the multiplication of basal dislocations (e.g. as mov-
133
ing Frank–Read sources; Petrenko and Whitworth, 1999) and for accommodation
134
of heterogeneous strain.
135
Although the study of individual dislocations in carefully prepared ice single
136
crystals, deformed under precisely controlled conditions, yields invaluable infor-
137
mation about the fundamental properties of dislocations in ice, it is evident that the
138
deformation processes naturally occurring in polycrystalline ice are much more
139
complex. Hondoh and Higashi (1983) and Liu et al. (1993, 1995) used X-ray to-
140
pography to study the interactions between dislocations and grain boundaries in
141
ice bicrystals and polycrystalline ice, respectively. They could demonstrate that
142
the regions surrounding grain boundaries (viz. the “mantle” of the grain, after
143
Gifkins, 1976) generally deform before the grain interiors (viz. the “core” of the
144
grain). Dislocations are emitted from stress concentrations at grain boundaries,
145
caused by strain misfits and/or grain boundary sliding, and this process completely
146
overwhelms any lattice dislocation generation mechanism. Depending on the rel-
147
ative configuration of grain boundaries and applied stress, not only basal disloca8
148
tions but also fast non-basal edge segments can be emitted by grain boundaries,
149
trailing screw segments behind them.
150
These findings are in close agreement with the results from microscopic obser-
151
vations of natural ice microstructures in fresh ice core samples (Wang et al., 2003;
152
Faria and Kipfstuhl, 2004, 2005; Kipfstuhl et al., 2006, 2009; Weikusat et al.,
153
2009a,b), where abundant evidences of heterogeneous strain and internal stresses
154
can be found in form of multiple subgrain boundaries and dislocation walls, bent
155
slip bands, pinned and bulged grain boundaries (cf. Sect. 4). In particular, the
156
large amount of subgrain boundaries and dislocation walls in regions surrounding
157
grain boundaries clearly indicates the tendency of polar ice grains to develop in-
158
tracrystalline strain gradients and high internal stresses in their “mantle” region,
159
while preserving their “cores.” Additionally, it is not uncommon to observe the
160
manifestation of internal stress concentrations through bulged or cuspidate grain
161
boundaries with radiating subgrain boundaries and dislocation walls (examples
162
can be found in almost all micrographs shown here, e.g. Fig. C.5; see also Kipfs-
163
tuhl et al., 2006; Faria et al., 2009, 2010; Weikusat et al., 2009b). In fact, accord-
164
ing to recent statistical studies on subgrain boundaries in polar ice (Weikusat et al.,
165
2010, 2011; see Sect. 4.1), internal stresses are high enough to produce a consid-
166
erable amount of non-basal dislocations, as revealed by the significant fraction
167
of tilt boundaries on basal planes, which are formed by geometrically necessary
168
non-basal edge dislocations.
169
Recalling the fact that the strong plastic anisotropy of ice has been known for
170
more than a century (McConnel and Kidd, 1888; McConnel, 1890), the findings
171
described above should seem unsurprising: large internal stresses and heteroge-
172
neous strains that vary in space with a wavelength comparable to the grain size are
9
173
actually expected in a polycrstalline material made of such remarkably anisotropic
174
grains (Remark 1).
175
Remark 1. The homogeneous deformation by dislocation glide of an incom-
176
pressible polycrystal into an arbitrary shape requires the activity of at least five in-
177
dependent slip systems, in order to avoid geometric incompatibilities between the
178
grains (Taylor, 1938). If the condition of homogeneous strain is waived, then only
179
four independent systems are necessary, provided that the strain gradients result-
180
ing from geometric incompatibilities are balanced by internal stresses (Hutchin-
181
son, 1976). In the case of ice, the basal plane provides only two independent
182
slip systems: further two systems must be active by slip or climb on prismatic
183
and/or pyramidal planes. Notwithstanding, non-basal deformation of ice requires
184
stresses at least 60 times larger than those for basal slip at the same strain rate, so
185
that large internal stresses are expected in ice undergoing dislocation creep (Duval
186
et al., 1983; Wilson and Zhang, 1996).
187
Despite their fundamental importance for the mechanics of glaciers and ice
188
sheets, internal stresses and heterogeneous strain phenomena have been largely
189
ignored (or treated as a secondary issue) in models of the microstructure evolu-
190
tion of natural ice. For instance, recrystallization models based on an average
191
dislocation density (e.g. De la Chapelle et al., 1998; Montagnat and Duval, 2000)
192
are often invoked in support of the tripartite paradigm of polar ice microstruc-
193
ture (also called “three-stage model”; see Sect. 3.3 of Part I). From the results
194
discussed here, and extended in Sects. 4 and 5, it turns out that such models are
195
not appropriate for describing the microstructure evolution of polar ice, because
196
they seriously underestimate recrystallization processes, which are very sensitive
10
197
to internal stress concentrations and localized values of dislocation density close
198
to grain boundaries.
199
Recently, the small-scale modelling of the effects of internal stresses and het-
200
erogeneous strains on the evolution of ice microstructures has become a very ac-
201
tive research topic, as reviewed in this Issue (Montagnat et al., 2013). On the
202
other hand, on the much larger scale of ice sheet dynamics, this problem becomes
203
particularly difficult, because a multiscale continuum model is needed. To our
204
knowledge, there is only one theory currently capable of dealing simultaneously
205
with large scale ice sheet flow and dynamic recrystallization, taking into account
206
the effects of strain heterogeneities and internal stresses (Faria, 2006a,b; Faria
207
et al., 2006b). It models the polycrystal as a heterogeneous structured medium
208
within the framework of the general theory of Mixtures with Continuous Diversity
209
(MCD; Faria, 2001; Faria et al., 2003). As pointed out by Placidi et al. (2004)
210
and Faria and Kipfstuhl (2004), internal stresses are modeled by the orientational
211
couple-stress tensor $∗ (sometimes also called “polygonization tensor”), which
212
describes the action of localized bending stresses acting on the ice lattice. Het-
213
erogeneous strain is modelled by a set of N scalar-, vector-, or tensor-valued dis-
214
location parameters B∗κ (with κ = 1, 2, . . . , N), which characterize the spatial
215
arrangement of dislocations in the polycrystal (Faria et al., 2006b).
216
At this point it should be clear that, in order to improve large-scale glacier and
217
ice sheet models, we have first to find out realistic, explicit expressions for abstract
218
concepts like the “orientational couple-stress tensor” and the set of “dislocation
219
arrangement parameters,” which require information from detailed investigations
220
of the type described in this section, as well as results from models on the small
221
polycrystalline scale, as those reviewed elsewhere in this Issue (Montagnat et al.,
11
222
2013).
223
3. Creep of glacier ice
224
Section 2 of Part I warned about the potential injustice of naming milestones
225
for defining decisive moments in scientific research. In the case of ice mechan-
226
ics, however, the period 1947–1952 is widely acknowledged for establishing a
227
paradigm shift that irreversibly changed the glaciologists’ attitude to the mechan-
228
ics of glaciers and ice sheets (Sharp, 1954; Waddington, 2010). Its milestone is
229
Glen’s (1952) article on mechanical tests showing that the secondary creep of
230
ice could be described by a power law (of the type proposed by Norton, 1929,
231
in metallurgy), therefore confirming a conjecture about the non-Newtonian creep
232
behavior of ice (Perutz, 1949, 1950; cf. Sect. 2.1 of Part I). Glen’s (1952) pre-
233
liminary study was soon complemented by Glen and Perutz (1954), Steinemann
234
(1954), Glen (1955) and others, including the corroboration of the suitability of
235
such a power law for modeling glacier flow (Nye, 1953, 1957).
236
3.1. The creep curve
237
Isotropic polycrystalline ice (viz. homogeneous polycrystalline ice with no lat-
238
tice preferred orientation; cf. Appendix A) exhibits a creep curve typical of many
239
polycrystalline materials undergoing high-temperature creep (Fig. C.6). It is char-
240
acterized by a preliminary “instantaneous” Hookean elastic strain (cf. Remark 2),
241
followed by three creep stages. Natural ice in glaciers and ice sheets is expected
242
to undergo all these creep stages in situ, even when subjected to polar conditions
243
(viz. stresses lower than 0.1 MPa, temperatures down to −50◦ C, strain rates about
244
10−12 s−1 , and total shear strains exceeding 1000%).
12
245
Remark 2. Budd and Jacka (1989) report that the Hookean elastic strain of isotropic
246
polycrystalline ice reaches 0.024% at 0.2 MPa octahedral stress, and has little de-
247
pendence on temperature. Indeed, according to Gammon et al. (1983), the vari-
248
ation in the elastic properties of isotropic polycrystalline ice in the temperature
249
range between −50◦ C and close to the melting point should lie below 10%, al-
250
though they may vary considerably with the impurity content of ice.
251
The achievement of all three creep stages in laboratory tests simulating polar
252
conditions is clearly impossible, since this would require thousands of years of
253
uninterrupted straining under carefully controlled conditions. Therefore, the creep
254
behavior of natural ice is usually extrapolated from mechanical tests performed at
255
higher temperatures or stresses (e.g. Steinemann, 1954; Glen, 1955; Lile, 1978;
256
Jacka, 1984; Jacka and Li, 2000), and then compared with field measurements of
257
glacier flow or the deformation of glacial tunnels and deep boreholes (e.g. Nye,
258
1953; Paterson, 1977; Fischer and Koerner, 1986; Talalay and Hooke, 2007).
259
During the first creep stage, usually called transient or primary creep, the
260
strain rate decreases rapidly. This deceleration is due to work hardening mainly
261
produced by the load transfer from easy-glide to hard-glide systems and the in-
262
creasing strain incompatibilities between the grains, which build up internal stresses
263
and localized heterogeneous strains (Wilson, 1986; Petrenko and Whitworth, 1999;
264
Schulson and Duval, 2009; cf. Sect. 2.2), both clearly identified by the forma-
265
tion of the first dislocation walls and subgrain boundaries (Hamann et al., 2007;
266
Sect. 4.1). Primary creep in ice extends to about 1% of strain, irrespective of
267
temperature or stress (Budd and Jacka, 1989), and a considerable fraction of it
268
consists of a recoverable “delayed-elastic” strain (sometimes also called “anelas-
269
tic” strain), implying that part of the deformation is recovered after the load is
13
270
removed, in a relaxation process that can take several hours (Duval, 1978). Budd
271
and Jacka (1989) report primary recoverable strains of 0.15% and 0.30% for
272
isotropic polycrystalline ice at −10◦ C compressed at 0.2 MPa and 1.0 MPa oc-
273
tahedral stress, respectively. It is believed that the delayed elasticity of ice is
274
mainly caused by the relaxation of internal stresses by dislocation back-gliding
275
(Glen, 1975; Cole, 2004; Schulson and Duval, 2009).
276
The primary creep of ice ends with the inception of secondary creep. In con-
277
trast to other materials, a steady-state regime has not been observed in the sec-
278
ondary creep of ice at any temperature down to −50◦ C, or at stresses as low as
279
22 kPa octahedral (Budd and Jacka, 1989; Remark 3).
280
Remark 3. We emphasized above the conjunction “or” in order to make clear that
281
the minimum strain rate could not be achieved so far in any single test combining
282
the lowest temperature and stress just mentioned. Jacka and Li (2000) report
283
minimum strain rates attained in some extreme compression tests, including one
284
ran during more than five years at −45◦ C and 550 kPa octahedral stress, as well as
285
another one executed at −19◦ C and 100 kPa octahedral. Russell-Head and Budd
286
(1979) describe a sequence of strain rate minima attained in a shear test performed
287
at 22 kPa octahedral stress and an initial temperature of −2◦ C, with subsequent
288
temperature drops to −5◦ C and −10◦ C after each strain rate minimum.
289
Instead of reaching a steady state, the secondary creep of ice seems to be es-
290
sentially a transition zone between 0.5% and 2% strain that connects the deceler-
291
ating primary creep to the accelerating tertiary creep. Its main characteristic is the
292
inflection point in the creep curve, which occurs at about 1% strain, irrespective
293
of temperature or stress, and defines the minimum strain rate for the whole creep
14
294
process. As demonstrated by Jacka (1984), this minimum is best visualized in a
295
log–log plot of strain rate versus strain (Fig. C.6), which has since then become a
296
standard in the ice mechanics literature.
297
In spite of not being identified as a true steady state, the secondary creep of
298
ice has a fundamental physical meaning: its minimum strain rate defines the point
299
where hardening caused by evolving internal stresses is counterbalanced by the
300
softening produced by dynamic recovery and recrystallization, e.g. through the
301
re-arrangement of geometrically necessary dislocations into low-energy structures
302
(subgrain boundaries, dislocation walls, etc.) and the obliteration of localized
303
internal stresses by strain-induced grain boundary migration (SIBM), among other
304
processes (Remark 4 and Sects. 4 and 5).
305
Remark 4. The above explanation of the physical meaning of the secondary creep
306
of ice holds for the ductile regime only, which is the focus of this review. At high
307
stresses and/or low temperatures, ice becomes brittle and the characteristic soft-
308
ening of secondary and tertiary creep regimes (if they can be achieved prior to
309
material failure) is mainly caused by crack formation, which eventually leads to
310
the fracture of the ice specimen (Petrenko and Whitworth, 1999; Schulson and
311
Duval, 2009).
312
The creep response of ice following the minimum strain rate is somewhat more
313
complicate. In most mechanical tests, performed at temperatures above −15◦ C
314
and stresses higher than 0.3 MPa (corresponding to minimum strain rates about
315
10−8 s−1 ), the secondary creep gives way to accelerating tertiary creep after 1–
316
2% of strain, which eventually reaches a stable, steady-state regime after ca. 10%
317
strain (Budd and Jacka, 1989). The accelerating part of tertiary creep is accompa-
318
nied by the development of lattice preferred orientations (LPOs) and an increase
15
319
in the mean grain size. The latter eventually reaches a tertiary steady-state size,
320
which can be roughly predicted by the relation (Jacka and Li, 1994)
D2ss =
ϕ
,
σ3
(2)
321
where Dss is the linear dimension of the mean grain size in the tertiary steady-
322
state stage, σ is the applied stress, and ϕ is a dimensional factor with negligible
323
temperature dependence. It should be noticed that the rapid LPO formation in
324
such “fast” experiments is not caused by slip-driven lattice rotation, since strains
325
of only a few percent are not sufficient to produce noticeable LPOs by lattice rota-
326
tion alone (Azuma and Higashi, 1985; Jacka and Li, 2000). Rather, this early LPO
327
formation must be related to the nucleation of new grains (SIBM-N; Appendix A).
328
Steinemann (1958) was the first to suggest that, for a given temperature and
329
stress regime, the ratio between the tertiary maximum and the secondary minimum
330
strain rates (nowadays called strain-rate enhancement) could be expressed as a
331
function of the minimum strain rate, that is
ε̇max
= f (ε̇min ) ,
ε̇min
(T = const.)
(3)
332
where ε̇max and ε̇min denote the tertiary maximum and the secondary minimum
333
strain rates, respectively, while f is an increasing function of the minimum strain
334
rate. Indeed, at lower temperatures and stresses (corresponding to minimum strain
335
rates of about 10−9 s−1 ), the strain-rate enhancement abates and the LPO devel-
336
opment slows down. As remarked by Steinemann (1958), this reflects the fact
337
that nucleation recrystallization (SIBM-N) is no longer effective, being gradually
338
replaced by migration recrystallization (SIBM-O) and rotation recrystallization
339
(RRX; cf. Sects. 4, 5, and Appendix A).
16
340
At even lower temperatures and stresses (e.g. 0.1 MPa at −20◦ C, or any equiv-
341
alent stress–temperature combination resulting in minimum strain rates about 10−10 s−1 ),
342
observations are inconclusive. Secondary minimum strain rates could be achieved
343
at 1% strain in a few tests after several years of continual deformation (e.g. Jacka
344
and Li, 2000), but many more years would be necessary in order to investigate
345
tertiary creep under such slow conditions.
346
3.2. Creep laws
347
Glen (1955) and Barnes et al. (1971) have shown that the creep of ice up to the
348
minimum strain rate (that is, including the primary and early stages of secondary
349
creep, prior to acceleration), is reasonably well fitted with Andrade’s Law (An-
350
drade, 1910) in the form (from now on, the creep regimes in which a given equa-
351
tion is valid will be expressed by the acronyms PC, SC and TC within square
352
brackets, denoting primary, secondary and tertiary creep, respectively)
ε = ε0 + ln 1+ βt1/m + κt
≈ ε0 + βt
1/m
+ κt ,
[PC, SC] (4)
353
with m = 3, where the approximation is valid for small strains, such that βt1/m 1
354
and ε . 1%. In (4), ε and ε0 are the true (logarithmic) and instantaneous elastic
355
strains, respectively, t denotes time, while β and κ are parameters depending on the
356
applied stress and temperature. It is not difficult to recognize that β describes the
357
material response at the onset of primary creep, while κ represents the secondary
358
asymptotic “steady-state” strain rate, which would be reached if the accelerating
359
tertiary creep had not occurred. Consequently, βt1/m is sometimes called the tran-
360
sient creep term, while κt is the secondary “steady-state” creep term.
361
For temperatures and stresses usually considered in ice creep tests, experience
17
362
shows that the early stage of transient creep (ε . 0.01%; Budd and Jacka, 1989)
363
is characterized by a roughly linear relation between stress σ and strain ε within
364
a fixed time interval, therefore implying that β ∝ σ. On the other hand, Glen
365
(1955) attempted to use (4) for deriving the stress dependence of the asymptotic
366
secondary minimum strain rate κ from creep tests, but the accuracy of the method
367
was impaired by the onset of recrystallization and the difficulty to identify the end
368
of the transient creep. From tests performed at −0.02◦ C between 0.15–0.90 MPa,
369
he found κ ∝ σn with n = 4.2.
370
An independent determination of the secondary minimum strain rate was pur-
371
sued by Glen (1952, 1955), by determining a power-law relation between the min-
372
imum strain rate actually observed in experiments and the stress required to pro-
373
duce it. In its most popular version (due to Nye, 1953), the power law that would
374
soon be known as Glen’s Flow Law takes the form
ε̇ = Aσn
[SC] (5)
375
(cf. Remark 5), or in tensorial formulation (cf. Hutter, 1983; Paterson, 1994;
376
Hooke, 2005)
ε̇ = Aσn−1 σ ,
[SC] (6)
with
σ = σT , ε̇ = ε̇T , tr (σ) = tr (ε̇) = 0 ,
q q 2
1
ε̇ := 2 tr ε̇
and
σ := 12 tr σ2 .
(7)
(8)
377
Remark 5. Power-law relations similar to (5) were introduced in fluid dynamics
378
in 1923 by de Weale and Ostwald (cf. Ostwald, 1929) and some years later in
379
metallurgy by Norton (1929).
18
380
In the above equations, (·)T denotes the transpose and tr(·) the trace of the re-
381
spective tensor. The tensors σ and ε̇ describe the deviatoric (traceless) Cauchy
382
stress and the strain rate, respectively. The non-negative scalars σ and ε̇ are the
384
square roots of the deviatoric second invariants of σ and ε̇, and consequently cor√
respond to 3/2 times the octahedral shear stress and strain rate. At temperatures
385
below circa −10◦ C, the flow parameter A is assumed to depend on temperature T
386
and hydrostatic pressure p according to an Arrhenius-like equation (Remark 6)
383
A = α e−(Q+pV)/kB T ≈ α e−Q/kB ϑ ≈ α e−Q/kB T ,
(9)
387
where Q and V are the activation energy and volume for creep, kB is the Boltzmann
388
constant, and the parameter α is usually regarded as a constant, although it may
389
also depend on such factors as grain size, impurity and/or water content (Alley,
390
1992; Paterson, 1994).
391
Remark 6. Above −10◦ C the increase of the minimum strain rate with tempera-
392
ture is enhanced and the Arrhenius law breaks down (Glen, 1955, 1975; Hooke,
393
1981; Budd and Jacka, 1989). It is believed that grain boundary sliding and the
394
presence of water within the grain boundaries may be the main causes of this creep
395
enhancement (Barnes et al., 1971). Due to the lack of a more realistic alternative,
396
an empirical Arrhenius-like equation similar to (9) is frequently used to model the
397
temperature dependence of ice creep above −10◦ C, including an apparent (and in
398
fact temperature-dependent) activation energy with no physical meaning (Mellor
399
and Testa, 1969b; Budd and Jacka, 1989; Paterson, 1994).
400
Rigsby (1958a) asserted that the effect of the activation volume of ice is in
401
most cases negligibly small (−55 . V . 32 cm3 /mol, according to Jones and
19
402
Chew, 1983) and can be accounted for in (9) by using the pressure-dependent
403
temperature relative to the melting point
ϑ := T + Bp ,
(10)
404
with B = 98 K/GPa (Lliboutry, 1976; Remark 7).
405
Remark 7. It should be noticed that the value of the constant B, which is appro-
406
priate for natural ice, does not coincide with the theoretical value of the relation
407
between pressure and melting temperature of pure ice (Clausius–Clapeyron rela-
408
tion) −dT m /dp = 74 K/GPa. As explained by Glen (1974) and Lliboutry (1976),
409
this discrepancy is mainly due to the natural saturation of air in water.
410
Values of the exponent n in (5) and (6) derived from experiments and field
411
measurements range from 1 to 4, with a general consensus for using n = 3 (Hobbs,
412
1974; Hooke, 1981; Weertman, 1983; Budd and Jacka, 1989; Alley, 1992; Pater-
413
son, 1994; Petrenko and Whitworth, 1999; Schulson and Duval, 2009). In his
414
pioneering work, Glen (1952) found n = 4. After extending his preliminary re-
415
sults, he came to n = 3.2 (Glen, 1955). in a later review, Glen (1975) eventually
416
suggested n = 3.5 for stresses above about 0.1 MPa, with its value falling off with
417
decreasing stress towards (but not necessarily reaching) unity. A similar fall-off of
418
the exponent n at sufficiently low stresses has been observed and/or suggested by
419
a number of authors, based on field and laboratory results (e.g. Mellor and Testa,
420
1969a; Hooke, 1973; Goodman et al., 1981; Doake and Wolff, 1985; Pimienta
421
and Duval, 1987; Goldsby and Kohlstedt, 1997; Azuma et al., 2000; Peltier et al.,
422
2000; Cole and Durell, 2001; Durham et al., 2001; Goldsby and Kohlstedt, 2001,
423
2002; Marshall et al., 2002; Song, 2008). The case n ≈ 2 is usually associated
20
424
to grain boundary sliding, while n → 1 is believed to be caused by diffusional
425
flow or Harper–Dorn creep (Goodman et al., 1981; Duval et al., 1983; Weertman,
426
1983; Alley, 1992; Goldsby and Kohlstedt, 2001).
427
From the mathematical point of view, a power-law exponent n → 1 at van-
428
ishing stresses would also be welcomed by modelers (see e.g. Thompson, 1979;
429
Hutter, 1982, 1983; Fowler, 2001). The case n > 1 when σ → 0 leads to an infinite
430
effective viscosity dσ/dε̇, and consequently to some pathological singularities in
431
the modeling of ice-sheet flow (e.g. an infinite surface curvature on the ice divide
432
and infinite slope at the ice-sheet margin). Owing to this, simple generalizations
433
of (5) have been proposed, like
ε̇ = AI σ + AII σn
[SC]
(11)
434
with n non-integer, or alternatively the polynomial form
ε̇ =
N
X
Ai σi
[SC]
i=1
(12)
435
with i integer (e.g. Meier, 1958, 1960; Lliboutry, 1969; Colbeck and Evans, 1973;
436
Thompson, 1979; Hutter, 1980, 1981; Hutter et al., 1981; Smith and Morland,
437
1981; Pettit and Waddington, 2003). The parameters AI , AII and Ai are usually as-
438
sumed to be functions of temperature, and possibly also of other factors, like grain
439
size, water/impurity content, etc. (Remark 8). More sophisticated generalizations
440
of (5), based e.g. on the Garofalo or the Prandtl–Eyring models, are discussed by
441
Barnes et al. (1971) and Hutter (1983).
442
Remark 8. Flow law generalizations like (11) or (12) are not necessarily mathe-
443
matical artifices to overcome numerical singularities: they may in fact represent
21
444
the competition of several deformation mechanisms. For instance, Azuma et al.
445
(1999, 2000) proposed a combination of dislocation creep (n = 3) and diffusional
446
flow (n = 1) to explain the weaker c-axis clustering observed in fine-grained,
447
high-impurity ice layers (viz. cloudy bands) at low temperatures and stresses in
448
the Dome Fuji deep ice core.
449
Compared to secondary creep, the tertiary creep of ice has been much less
450
studied, in spite of its widespread occurrence in nature. The reason is, as already
451
mentioned in Sect. 3.1, the extremely long period necessary to reach tertiary creep
452
in deformation tests under the low temperatures and stresses typically found in
453
glaciers and ice sheets.
454
From a series of tests at −11.5◦ C, −4.8◦ C and −1.9◦ C, with stresses ranging
455
from 0.3 to 1.6 MPa (corresponding to strain rates between 10−8 and 10−5 s−1 ),
456
Steinemann (1958) derived the following power law, valid for the secondary and
457
tertiary regimes
ε̇ = Aσn ,
n = n0 + P(σ, T ) ,
[SC, TC]
(13)
458
where A(T ) is still given by (9), n0 = const., and P is a polynomial function of
459
σ and T , such that n = n0 during secondary creep. During tertiary creep, n may
460
reach quite large values, depending on the applied stress and temperature, e.g.
461
n ≥ 10 for σ = 1.6 MPa and T = −1.9◦ C.
462
More recently, it became customary in glaciology to follow an alternative ap-
463
proach, in which the power-law exponent is kept constant, e.g. n = n0 = 3 in
464
(13), and all microstructural changes characteristic of tertiary creep are subsumed
465
into the flow parameter A. The usual procedure is to introduce a dimensionless
22
466
enhancement factor E, such that
ε̇ = EAσn ,
n = n0 ,
[SC, TC]
(14)
467
where A(T ) is still given by (9), n0 = const., and the enhancement factor E satisfies
468
the compatibility condition
E |ε̇min = 1 ,
[SC]
(15)
469
which ensures that (14) is equivalent to (5) during the secondary creep of isotropic
470
ice. By extending Steinemann’s (1958) results summarized in (3), Jacka and Li
471
(2000) could show that, for a given stress regime,
max(E) =
ε̇max
= Emax (ε̇min , T ) ,
ε̇min
(16)
472
where Emax is an increasing function of temperature and secondary minimum
473
strain rate. In particular, for uniaxial compression at high stresses and temper-
474
atures, they found the upper bound Emax = 3. Likewise, for simple shear at high
475
temperatures and stresses Budd and Jacka (1989) report the upper bound Emax = 8.
476
These upper-bound values are believed to be the result of the symmetry superposi-
477
tion of the applied stress on fully developed Lattice Preferred Orientations (LPOs)
478
through Curie’s principle (Rosen, 1995, 2005).
479
In the case of natural ice, the enhancement factor E is either derived from
480
direct observation (Shoji and Langway, Jr., 1984; Dahl-Jensen, 1985; Wang et al.,
481
2002) or modeled as a function (or functional) of a suitable set of variables that
482
satisfactorily describe the microstructural evolution of ice during tertiary creep
483
(Lile, 1978; Azuma, 1995; Placidi et al., 2010). It is believed that the main cause
484
of enhancement is the strain-induced anisotropy due to LPOs, but other factors
23
485
may play also an important role, like impurities or grain stereology (i.e. grain
486
sizes, shapes, and arrangement, see Appendix A).
487
Remark 9. It is important to have in mind that only those effects emerging in
488
the tertiary creep should enter in the definition of the enhancement factor E. For
489
instance, the effect of hardening provoked by the interaction of dislocations with
490
dispersed fine particles (Ashby, 1966) is already active during secondary creep
491
and consequently should not be included in E, but rather in the factor α of (9).
492
Unfortunately, it is a formidable task to study the enhancement of tertiary
493
creep by impurities and/or grain stereology in deformation tests at the low temper-
494
atures, stresses, and impurity concentrations typical of glaciers and ice sheets. On
495
the other hand, such an enhancement is frequently observed in the field through
496
ice-core and borehole studies (Gundestrup and Hansen, 1984; Fischer and Ko-
497
erner, 1986; Dahl-Jensen and Gundestrup, 1987; Etheridge, 1989; Paterson, 1991;
498
Cuffey et al., 2000a,b), but in such cases it is very difficult to identify the real
499
agent of the effect because, as explained in detail in Part I, anisotropy, grain size
500
and shape, soluble and insoluble impurity concentrations all correlate generally
501
well with climate signals. Be that as it may, a clear example of tertiary creep en-
502
hancement by impurities and/or grain size and shape is offered by the study of a
503
“soft ice” layer discovered at the EDML drilling site in Antarctica (Faria et al.,
504
2006a, 2009, in preparation; see also Part I): microstructural analyses revealed the
505
occurrence of strain accommodation by microscopic grain boundary sliding via
506
microshear (cf. Drury and Humphreys, 1988; Bons and Jessell, 1999). Evidences
507
suggest that this phenomenon is triggered by a combination of high impurity con-
508
tent and temperature with small grain sizes and a suitable LPO, which facilitates
24
509
the sliding of grain boundaries and leads the microstructure to recrystallize into a
510
characteristic “brick wall” pattern that promotes further microshear.
511
Sophisticated tensorial models that explore the anisotropy of natural ice LPOs
512
have also been proposed (Azuma, 1994; Gödert and Hutter, 1998; Morland and
513
Staroszczyk, 1998; Gillet-Chaulet et al., 2005; Faria, 2006b; Placidi and Hutter,
514
2006), although their use in large scale computer models has been greatly ham-
515
pered by their intrinsic mathematical complexities (Montagnat et al., 2013). They
516
are generally characterized by a fourth-rank tensor-valued fluidity F (or its recip-
517
rocal, the viscosity µ = F−1 ) such that
ε̇ = Fσ .
[SC, TC]
(17)
518
The fluidity tensor F is usually a function or functional of the stress, tempera-
519
ture, and a set of time-dependent vector- and/or tensor-valued variables used to
520
describe the LPO symmetry. In some models the fluidity tensor may also depend
521
on additional factors already mentioned, like grain size, impurity concentration or
522
water content (Faria, 2006b).
523
3.3. Flow–structure interplay and the tripartite paradigm
524
From the discussions in Sects. 3.1 and 3.2 it turns out that the regimes of strain,
525
stress, strain rate and temperature typically found in polar ice sheets cannot be
526
simultaneously achieved in laboratory. Extrapolations of the results of extreme
527
creep tests (e.g. Russell-Head and Budd, 1979; Pimienta and Duval, 1987; Jacka
528
and Li, 2000; Goldsby and Kohlstedt, 2001) do not converge to a unified con-
529
clusion, leaving open the possibility that several mechanisms of deformation, re-
530
crystallization and recovery may be coincidently active in polar ice. Therefore,
25
531
in order to acquire a better understanding of the interplay between flow and mi-
532
crostructure in ice sheets, we must resort to indirect approaches. The most ef-
533
fective of them is undoubtedly the microstructural analysis of ice core samples,
534
which is reviewed in the ensuing sections. Before embarking on such a review,
535
however, it may be interesting to approach the interplay issue from the standpoint
536
of large-scale ice-sheet mechanics.
537
For several decades, the tripartite paradigm (also called “three-stage model”;
538
cf. Sect. 3.3 of Part I) has defined the status quo in regard to our general under-
539
standing of polar ice microstructures. It has set the framework for interpreting
540
the evolution of grain sizes (Stephenson, 1967; Gow, 1969; Alley et al., 1986a,b;
541
Durand et al., 2006) and lattice preferred orientations (Alley, 1992; Alley et al.,
542
1995; Thorsteinsson et al., 1997), as well as the onset of dynamic recrystallization
543
(Duval and Castelnau, 1995). It has also established the basis for polycrystalline
544
ice models (De la Chapelle et al., 1998; Montagnat and Duval, 2000; Faria et al.,
545
2002; Ktitarev et al., 2002) and provided arguments in disputes about deforma-
546
tion mechanisms in polar ice (Pimienta and Duval, 1987; Duval and Montagnat,
547
2002).
548
The cornerstone of the tripartite paradigm is the assumption that Normal Grain
549
Growth (NGG) dominates the evolution of the polar ice microstructure in the up-
550
per hundreds of meters of the ice sheet, including the firn layer, according to the
551
parabolic law
D2 − D20 = K t ,
(18)
552
where D2 is the mean grain cross-sectional area at time t, D20 is its extrapolated
553
initial value, and K is the grain growth rate (Stephenson, 1967; Gow, 1969; Alley
554
et al., 1986a; Paterson, 1994; De la Chapelle et al., 1998). This assumption has
26
555
recently been challenged by Kipfstuhl et al. (2006, 2009) through a detailed mi-
556
crostructure study of Antarctic ice and firn from the EDC and EDML sites. These
557
authors found clear evidence of migration and rotation recrystallization (RRX) al-
558
ready at very shallow depths (a few tens of meters at EDML) and identified them
559
as one of the dominant mechanisms of microstructure evolution in deep firn and
560
bubbly ice (Figs. C.7 and C.8). Laboratory experiments and computer simulations
561
of normal grain growth (Roessiger et al., 2011, 2013; Azuma et al., 2012) have
562
also cast doubts on the tripartite paradigm, by showing that the microstructure of
563
shallow polar ice seems to be affected by processes other than NGG.
564
Based on these recent results and the information discussed in the previous
565
sections, we can now investigate the reasons for the failure of the tripartite paradigm.
566
In the pioneering work of Gow (1969), which established the notion of NGG in
567
polar ice, mean grain size was derived from the cross-sectional areas of the 50
568
largest grains in a sample. Clearly, this method is fast and practical, but it ignores
569
(i.e. it cuts off) most of the grain size distribution and is therefore inappropriate
570
(Remark 10).
571
Remark 10. Gow (1969) justified this approach by his observation of a certain
572
uniformity in the size of grains disaggregated from specific snow layers. Such
573
uniformity is however questionable and has not been observed in modern studies.
574
It has possibly been caused by a bias towards larger grains, which is introduced
575
during the process of disaggregation of the fragile snow and firn.
576
As discussed in Part I, despite its shortcomings the 50-largest-grains method
577
has been used for determining the mean grain sizes of several firn and ice cores, in-
578
cluding GISP2. More elaborated methods, like the linear intercept (Dye 3, GRIP,
27
579
GISP2), the counting of grains within a given area (Camp Century, Byrd, Vos-
580
tok) or the modern Automatic Fabric Analysis, AFA (NGRIP, EDC, Dome F)
581
share a common limitation: they are all based on thickness-integrated images of
582
the ice sample, so that the resolution of the method is limited by the thickness
583
of the thin section under analysis (usually around 0.3–0.5 mm). Grains or grain-
584
boundary features smaller than the section thickness cannot be identified, and very
585
inclined boundaries give rise to large experimental errors. This limitation imposes
586
a serious cut-off in the grain size distribution, which handicaps interpretations of
587
microstructure evolution in natural ice.
588
To date, the best solution for improving the resolution of ice microstructure
589
analyses is actually based on the old, pioneering work of Seligman (1949), illus-
590
trated in Fig. C.9: we simply record the the grain-boundary grooves on the ice
591
surface, which are naturally produced by thermal etching. Today it is no longer
592
necessary to cover the ice sample with a sheet of paper and rub it with a pencil,
593
in order to record its microstructure. We can simply photograph the thermally
594
etched ice surface with a high-resolution digital camera. This is the physical prin-
595
ciple of the Microstructure Mapping method (µSM), proposed by Kipfstuhl et al.
596
(2006). If the thermal etching is well done, the resolution of the µSM method is
597
limited mainly by the resolution of the optical equipment and the digital image
598
analysis software. Current set-ups work with resolutions in the range 3–65 µm
599
(Kipfstuhl et al., 2006, 2009). Another promising option, with even higher reso-
600
lution than µSM, is Electron Backscatter Diffraction (EBSD; Iliescu et al., 2004;
601
Piazolo et al., 2008; Weikusat et al., 2010; Prior et al., 2012). The use of EBSD
602
on ice is technically very difficult and is still in its infancy, but rapid technological
603
and methodological developments suggest that it may become a powerful tool for
28
604
future studies of ice microstructure.
605
In the sequel, we investigate the validity of the tripartite paradigm in the
606
EDML site. The reason for selecting this site is twofold: first, it provides the
607
most detailed and up-to-date information about polar firn and ice microstructures;
608
second, it offers one of the best examples of “typical” Antarctic ice, because the
609
EDML drilling site is representative of the Antarctic plateau without being located
610
at such an unusual place like an ice dome (e.g. EDC, Dome F) or above a large
611
subglacial lake (viz. Vostok).
612
The increase of grain size with depth in EDML polar firn was studied by Kipf-
613
stuhl et al. (2009) at three distinct “resolutions,” viz. average grain area of the 100
614
largest grains, of the 500 largest grains, and of all grains in each firn section. These
615
three “resolutions” were chosen in order to investigate how the afore-mentioned
616
cut-off of the grain size distribution affects our perception of grain growth. From
617
the results of that study, we can now calculate the grain growth rate K appear-
618
ing in (18) for each of the three cut-offs. We find K100 = 3.3 × 10−3 mm2 /a for
619
the 100 largest grains, K500 = 2.0 × 10−3 mm2 /a for the 500 largest grains, and
620
Kall = 1.5 × 10−4 mm2 /a when all grains in the sample are taken into account.
621
These values can be compared with Paterson’s empirical curve relating growth
622
rate and temperature, derived from a compilation of field measurements of grain
623
growth rates in firn from various polar locations (Fig. 2.5 of Paterson, 1994). For
624
the EDML site, where the mean temperature in firn and shallow ice is ca. −45◦ C
625
(Table B.1 of Part I), Paterson’s curve predicts a grain growth rate in the range
626
2 (±1) × 10−3 mm2 /a. Clearly, the EDML values of K100 and K500 are compatible
627
with Paterson’s empirical prediction, while the most reliable of them, Kall , is too
628
low by one order of magnitude.
29
629
The cause of this serious discrepancy is related to the different cut-offs of the
630
grain size distributions. The flawed rates K100 and K500 describe solely the kinetics
631
of the larger grains, that is, of truncated grain size distributions. In this manner,
632
they systematically ignore the formation, existence, and kinetics of smaller grains.
633
It is evident that it makes no sense to use such inaccurate growth rates as basis for a
634
theory of NGG in polar ice. Unfortunately, the limited resolution of most methods
635
of polar ice microstructure analysis imply that the great majority of grain growth
636
rates reported in the literature of polar firn and shallow ice may be impaired by
637
such shortcomings.
638
Furthermore, the sheer fact that grain size data can be fitted with a parabolic
639
growth law is by no means a corroboration of the occurrence of NGG (especially if
640
the growth rates are flawed): Strain-Induced Grain Boundary Migration (SIBM)
641
does not preclude a linear increase of the mean grain cross-sectional area with
642
time, in a regime that may be called Dynamic Grain Growth (DGG, cf. Appendix
643
A). SIBM-driven grain growth data can sometimes be fitted with a NGG law, but
644
in this case the law parameters (activation energy, growth rate, etc.) have no real
645
physical meaning. This explains the low value found for the most reliable grain
646
growth rate, Kall : it does not describe the real velocity of grain boundaries in the
647
NGG regime, simply because NGG cannot control the microstructure evolution
648
of a material undergoing deformation, like polar firn.
649
As pointed out by Azuma et al. (2012) and Roessiger et al. (2011, 2013), the
650
motion of grain boundaries in firn and bubbly ice is strongly affected by a number
651
of influences, including some extraneous to NGG, like stored strain energy and a
652
non-steady-state configuration of the grain-boundary network. Indeed, according
653
to Azuma et al. (2012), the grain boundary migration rate of pure, bubble-free
30
654
ice undergoing true NGG at −45◦ C should be Kfree = 1.6 × 10−1 mm2 /a, which
655
is several orders of magnitude larger than the rates predicted by Paterson (1994)
656
or measured by Kipfstuhl et al. (2009). The reason for the much slower growth
657
rate observed in polar firn cannot be attributed just to pinning by bubbles and
658
other impurities: complex strain-induced boundary motions (SIBM-O) and the
659
formation of new grains by dynamic recrystallization (RRX and SIBM-N) spoil
660
NGG and disguise the real migration rate of the boundaries.
661
An important corollary of the tripartite paradigm is the assumption that grain
662
boundary migration during NGG (i.e. migration driven by the free energy of the
663
grain boundaries) is an efficient softening mechanism that accommodates basal
664
slip deformation. As explained by Pimienta and Duval (1987):
665
In conclusion, grainboundary migration associated with [normal] grain
666
growth is an efficient accommodation process for dislocation glide in
667
fine-grained ices. In consequence the usual transient creep cannot oc-
668
cur and strain energy is always small compared with the driving force
669
for [normal] grain growth.
670
The fact that grain boundary migration is an important recovery mechanism
671
in natural ice is obvious and beyond doubt. On the other hand, considering the
672
fact that grain boundary migration is not a deformation mechanism, its role in
673
the accommodation of deformation is per se controversial (Kocks, 1970; Means
674
and Jessell, 1986; Goldsby and Kohlstedt, 2002; Cahn and Taylor, 2004) and be-
675
comes highly questionable in the case of NGG, seeing that migrating boundaries
676
in the NGG regime should, by definition, move free from the influence of internal
677
stresses and strain heterogeneities.
678
In the case of EDML firn, it is not difficult to show that NGG does not dictate
31
679
the microstructure evolution and that grain boundary migration, if it can be an
680
accommodation mechanism in the first place, is not sufficient to suppress dynamic
681
recrystallization. From Ruth et al. (2007) we calculate two bound estimates for
682
the vertical strain rate (“layer thinning”) of EDML firn at 50 m depth: ε̇total ≈
683
3.2 × 10−11 s−1 and ε̇i.eq. ≈ 7.4 × 10−12 s−1 , see Appendix B. The former (ε̇total )
684
describes the total thinning of the firn layers, including pore-space compression.
685
In contrast, ε̇i.eq. is based on the ice-equivalent depth and consequently excludes
686
any contribution of the pore space. As discussed in Appendix B, the average real
687
strain rate locally experienced by the ice grains in firn, ε̇real , is very difficult to
688
determine with precision, since it depends on the highly variable contribution of
689
the pore space to the strain accommodation. In any case, it should lie between
690
these two extreme strain-rate averages, viz. ε̇total ≥ ε̇real ≥ ε̇i.eq. .
691
In addition to strain rates, in Appendix B we also compute the total vertical
692
strain and the water-equivalent strain at 50 m depth, respectively, εtotal ≈ −30%
693
and εi.eq. ≈ −7%. Thus, from these estimates we conclude that EDML firn at
694
ca. 50 m depth is already deforming in the tertiary creep regime (cf. Sect. 3.1) and
695
should be undergoing dynamic recrystallization (Fig. C.7). These conclusions are
696
in accordance with the experimental observation of dynamic recrystallization in
697
EDML firn by Kipfstuhl et al. (2009).
698
4. Grain and subgrain boundaries
699
As any other polycrystalline material, polar ice consists of connected regions of
700
uninterrupted crystalline lattice known as grains, which are bounded together by
701
grain boundaries. Such crystalline regions are not perfect, though. Localized dis-
702
tortions of the lattice are caused by defects, especially dislocations (Sect. 2), which
32
703
can sometimes arrange themselves in stable structures called subgrain boundaries.
704
By gradually increasing the lattice misorientation across a subgrain boundary,
705
the latter may evolve to a new grain boundary. For this reason, grain and sub-
706
grain boundaries are also named high-angle and low-angle boundaries, respec-
707
tively. These names make evident that the grain-/subgrain-boundary dichotomy
708
is a conceptual simplification, since the transition from low to high misorienta-
709
tion is in fact continuous. As such, the critical misorientation angle that distin-
710
guishes between grain and subgrain boundaries is to some extent a matter of con-
711
vention, which depends on the boundary properties under consideration. In this
712
work we follow Weikusat et al. (2011) by assuming that the lattice misorientation
713
across subgrain boundaries in polar ice is not larger than ca. 5◦ , a result consistent
714
with observations in other minerals (Drury and Urai, 1990; Passchier and Trouw,
715
2005).
716
4.1. Subgrain boundaries
717
Subgrain boundaries are essential features of the ice microstructure, as they are
718
indisputable evidences of heterogeneous strains, intercrystalline incompatibilities,
719
internal stresses and high concentration of geometrically necessary dislocations.
720
They have been observed in ice for at least a century (Tarr and Rich, 1912). By
721
analysing thin sections of bent ice samples, Matsuyama (1920) reported “faint but
722
distinct straight lines” developed within some grains with zigzag boundaries, and
723
the straight lines were observed to sometimes “start from the angular points of
724
these zigzag boundaries.”
725
Nakaya (1958) later recognized that such straight lines were actually subgrain
726
boundaries made up of geometrically necessary dislocations. He performed bend-
727
ing experiments in single crystals with c-axes parallel to the bending load and ob33
728
served the formation of slip bands (cf. Appendix A), which would initially bend
729
with the crystal. This bending of slip bands is the precursor of a particular type of
730
subgrain boundary, by accumulating edge dislocations along several basal-gliding
731
layers in a dislocation wall perpendicular to the slip bands. At already 1◦ of
732
crystal bending, subgrain boundaries can be seen, typically emerging from the
733
high curvature part of slip bands, transforming them into a kink structure, if mis-
734
orientation further increases with ongoing deformation. In the glaciological liter-
735
ature, this process is often called “polygonization” (Alley et al., 1995).
736
The particular type of subgrain boundary described above is known as a basal
737
tilt boundary. In the ideal case it bisects the angle formed by the tilted basal
738
plane and is made up exclusively of basal edge dislocations with Burgers vector
739
b = a (Table D.2). In ice, tilted basal planes or c-axes can be measured using
740
an Automatic Fabric Analyzer (AFA; Wilson et al., 2007) or the formvar etch-pit
741
method (Matsuda, 1979; Barrette and Sinha, 1994; Hamann et al., 2007). Actu-
742
ally, most studies of subgrain boundaries in ice are performed on experimentally
743
deformed specimens (Wilson et al., 1986, this issue; Barrette and Sinha, 1994;
744
Hamann et al., 2007). In the case of naturally deformed ice, as in polar ice sheets
745
or glaciers, the occurrence of subgrain boundaries has often been determined indi-
746
rectly from neighbouring grain misorientation statistics (Alley et al., 1995; Wang
747
et al., 2003; Durand et al., 2008). Only recently, new microscopy methods have
748
allowed the direct and extensive (statistically relevant) observation of subgrain
749
boundaries in naturally deformed ice, e.g. through Microstructure Mapping (µSM;
750
Kipfstuhl et al., 2006). These studies have revealed that, in addition to the clas-
751
sical tilt boundaries characteristic of “polygonization,” other subgrain boundary
752
configurations are also very common in both, naturally and artificially deformed
34
753
ice (Hamann et al., 2007; Weikusat et al., 2009a,b). These configurations (ar-
754
rangements) include boundaries parallel and normal to the basal planes, as well as
755
zigzag combinations of them (Fig. C.10).
756
The observation of such detailed subgrain boundary configurations is only
757
possible because thermal etching (sublimation) is highly sensitive to boundaries
758
with very low-misorientation (0.5◦ ), as proven directly by high-resolution crys-
759
tal orientation measurements, such as X-ray Laue diffraction (Miyamoto et al.,
760
2011; Weikusat et al., 2011) and Electron Backscatter Diffraction (EBSD; Weikusat
761
et al., 2010). These two methods enable complete determination of the crystalline
762
lattice misorientation across the boundary, including both c- and a-axes. A de-
763
tailed knowledge of subgrain boundary misorientation and configuration allows
764
to identify the possible slip systems of its constituent dislocations (Trepied et al.,
765
1980; Prior et al., 1999, 2002; Piazolo et al., 2008). Following this approach,
766
Weikusat et al. (2011) combined µSM with X-ray Laue diffraction to obtain first
767
statistical data about subgrain boundaries and their constituent dislocations in po-
768
lar ice (Table D.2).
769
By recalling the consequences of the low stacking fault energy on the basal
770
plane of hexagonal ice (Sect. 2.1), it may seem paradoxical at first to see in Ta-
771
ble D.2 that almost 30% of all subgrain boundaries in polar ice are composed
772
of non-basal dislocations. The solution of this apparent paradox lies in the high
773
temperatures and low strain rates typical of natural ice deformation, which turn
774
dynamic recovery effective enough to allow the rearrangement of basal and non-
775
basal geometrically necessary dislocations in complex dislocation walls and sub-
776
grain boundaries. Indeed, from the microstructural features observed in polar ice,
777
we conclude that dynamic recovery through the formation of a variety of sub-
35
778
grain boundaries by grain subdivision (cf. Appendix A), as well as the splitting
779
of grains by rotation recrystallization (Sect. 5.1), are fundamental mechanisms
780
of strain accommodation in natural ice. Thus, it follows that geometrically nec-
781
essary dislocations play a decisive role in the accommodation of deformation in
782
polar ice.
783
4.2. Grain boundaries
784
The structure of grain boundaries plays an essential role in the mechanics, re-
785
crystallization, and molecular diffusion of ice, since it determines the energetics,
786
mobility, cohesion, and permeability of grain boundaries. While the structure of
787
low-angle grain boundaries (i.e. subgrain boundaries) in ice is well described by
788
the theory of dislocation arrays (Read and Shockley, 1950; Higashi and Sakai,
789
1961; Suzuki and Kuroiwa, 1972), little is actually known about the structure of
790
high-angle grain boundaries (Higashi, 1978; Hondoh and Higashi, 1978; Petrenko
791
and Whitworth, 1999). For this reason, classical views from metallurgy (Sutton
792
and Balluffi, 1995) are commonly adopted for ice (Goodman et al., 1981; Frost
793
and Ashby, 1982), in particular that the excess volume of grain boundaries ren-
794
der them favourable diffusion paths for interstitials and solutes, in such a manner
795
that the activation energy for diffusion of self-interstitials is expected to be lower
796
within grain boundaries (grain-boundary self-diffusion) than through the ice lat-
797
tice (lattice self-diffusion).
798
Notwithstanding, the density anomaly of water poses an interesting prospect
799
for the structure of grain boundaries in ice: in contrast to metals, water molecules
800
in the grain boundaries of polycrystalline ice could be packed more closely than
801
in the ice lattice (i.e. a negative excess volume), in a sort of amorphous or quasi-
802
liquid state (Clifford, 1967; Kondo et al., 2007). This conjecture is consistent with
36
803
the high molecular disorganization expected within grain boundaries and near free
804
surfaces due to proton disorder (Petrenko and Whitworth, 1999; cf. Sect. 2), as
805
well as with the observation of liquid water veins at the corners and edges of
806
grain boundaries in polycrystalline ice at temperatures close to the melting point
807
(Steinemann, 1958; Barnes et al., 1971; Nye and Frank, 1973; Mader, 1992). An
808
important corollary of such a “dense grain boundary” conjecture is that the be-
809
haviour of grain boundaries in ice could be very sensitive to temperature and im-
810
purity content, causing grain boundaries to possess either a more “liquid” or more
811
“glassy” structure.
812
Unfortunately, direct observation of the molecular structure of ice grain bound-
813
aries has not been possible so far, and grain-boundary diffusion experiments in ice
814
are also very difficult to accomplish. Consequently, grain-boundary migration ex-
815
periments are still regarded as the simplest means of obtaining valuable insights
816
into the structure of ice grain boundaries, seeing that, like the phenomenon of
817
self-diffusion, the migration of grain boundaries involves the jumping of water
818
molecules between lattice and grain-boundary sites, as well as their movement
819
inside the grain boundary.
820
As reviewed in Sect. 3.3 (see also Sect. 3.3 of Part I) the tripartite paradigm
821
states that grain-boundary migration in the upper hundreds of meters of polar ice
822
sheets should occur via Normal Grain Growth (NGG) according to the parabolic
823
law (18). Thus, if the tripartite paradigm were true, the temperature dependence
824
of the grain growth rate K of polar ice could be estimated from grain size versus
825
age data of ice cores extracted from different polar sites. The activation energy of
826
grain growth derived from such analyses (40–50 kJ/mol) has been accepted and
827
widely applied in glaciology. It happens, however, that polar ice is under con-
37
828
tinual deformation and contains many air bubbles. In the past, it was assumed
829
that air bubbles and pores should not significantly affect the migration of grain
830
boundaries (Duval, 1985; Alley et al., 1986b), but recent computer simulations
831
(Roessiger et al., 2013), field observations (Kipfstuhl et al., 2006, 2009) and lab-
832
oratory experiments (Azuma et al., 2012) have proven the contrary. Furthermore,
833
it has been shown that the stored strain energy in polar ice sheets is sufficient
834
not only to keep the ice microstructure out of the quasi-stationary state required
835
for NGG (Faria and Kipfstuhl, 2005; Roessiger et al., 2011), but also to trigger
836
rotation and migration recrystallization in firn and shallow ice (Kipfstuhl et al.,
837
2006, 2009; Faria et al., 2009; Weikusat et al., 2009a,b). Therefore, the tripartite
838
paradigm is generally not valid and the activation energy derived from ice-core
839
grain-size data cannot be the true activation energy of NGG in ice.
840
By using a new technique for producing pure, bubble-free ice, derived from a
841
method introduced by Stern et al. (1997), Azuma et al. (2012) could study the tem-
842
perature dependence of the true NGG rate K of ice. They found that K in bubble-
843
free ice is approximately three orders of magnitude larger than that estimated from
844
ice-core data (Paterson, 1994; cf. Sect. 3.3). Furthermore, an activation energy for
845
NGG of about 110–120 kJ/mol was observed in bubble-free ice at temperatures
846
between −40◦ C and −5◦ C. In contrast, the activation energy for NGG of bubbly
847
ice under the same conditions is circa 40–70 kJ/mol. The similarity between the
848
values of activation energy for grain growth derived from ice-core data and exper-
849
imentally measured in bubbly ice is evident. This fact compared with the apparent
850
activation energy of 50 kJ/mol calculated by Azuma et al. (2012) for the migration
851
of air bubbles in ice, suggest that the slow grain growth observed in polar ice cores
852
is significantly affected by the migration velocity of air bubbles.
38
853
It must be noticed that the true activation energy for NGG in pure, bubble-free
854
ice is approximately twice the activation energy for lattice self-diffusion (Ram-
855
seier, 1967). In the absence of reliable measurements of grain-boundary self-
856
diffusion in ice, and recalling that grain-boundary migration and diffusion involve
857
akin molecular processes (for a deeper discussion see Azuma et al., 2012), we
858
come to the conclusion that the activation energy for grain-boundary diffusion
859
may also be considerably larger than that for lattice diffusion. This result adds
860
support to the dense-grain-boundary conjecture, as suggested by Azuma et al.
861
(2012): when grains grow, the total grain-boundary area must decrease. This
862
leads to fluxes of water molecules across and along the grain boundaries. If the
863
grain boundaries have some sort of “semi-glassy” structure, the activation ener-
864
gies for grain-boundary migration and diffusion must be high, because the water
865
molecules are jammed inside the grain boundaries. On the other hand, if the
866
grain boundaries have a kind of “quasi-liquid” structure, the activation energies
867
for grain-boundary migration and diffusion may be high if the water molecules
868
are aggregated in clusters that must be either thermally activated as a group or
869
broken down to allow self-diffusion (Mott, 1948; Merkle and Thompson, 1973).
870
As a closing remark, it should be noticed that even if the activation energies for
871
grain-boundary migration and diffusion are larger than previously expected, so is
872
also the growth rate K, and consequently the grain boundary mobility, within the
873
temperature range typical of ice sheets (between −80◦ C and 0◦ C). Consequently,
874
grain boundaries in polar ice are very mobile and the grain size evolution turns
875
out to be controlled by second-phase dragging and dynamic recrystallization in a
876
process called Dynamic Grain Growth (DGG; Appendix A). These effects give
877
rise to the well-known apparent correlation of grain size with climate proxies (see
39
878
Part I).
879
5. Dynamic recrystallization
880
In the old glaciological literature, the word “recrystallization” was loosely used in
881
reference to nucleation and growth of new grains favourably oriented for defor-
882
mation; a definition that still can be found in more recent works (Paterson, 1994).
883
Here we adopt a more precise and comprehensive definition of recrystallization
884
as “any reorientation of the lattice caused by grain boundary migration and/or for-
885
mation of new grain boundaries” (cf. Appendix A), which is consistent with its
886
modern meaning in geology (Urai et al., 1986; Drury and Urai, 1990; Passchier
887
and Trouw, 2005).
888
It is worth noticing that metallurgists use a concept of recrystallization simi-
889
lar to the one adopted here, although they often exclude processes driven by the
890
grain boundary energy (Doherty et al., 1997; Humphreys and Hatherly, 2004).
891
This minor difference in terminology reflects the slightly distinct focuses of these
892
two research fields. Metallurgists are frequently concerned with static annealing
893
phenomena, in which recrystallization processes driven by grain boundary energy
894
(usually called “grain growth/coarsening” in metallurgy) occur after the stored
895
strain energy has been consumed by previous static recovery and recrystallization.
896
In contrast, geologists are mostly concerned with dynamic recrystallization pro-
897
cesses, in which strain energy is continually produced during deformation (cf. Re-
898
mark 11). In particular, in the case of natural ice, the increase in mean grain size
899
with age observed in ice cores (see Part I) is clearly influenced by the stored strain
900
energy in a process of Dynamic Grain Growth (DGG; cf. Sect. 4.2 and Appendix
901
A).
40
902
Remark 11. The common etymology of the metallurgical and geological termi-
903
nologies mentioned above may help us to understand their subtle (but consequen-
904
tial) distinction. In the primordial times of research in recrystallization, Alterthum
905
(1922a,b) coined the terms “Bearbeitungsrekristallisation” and “Oberflächen-Rekristallisation,”
906
meaning respectively “work-recrystallization” (namely, driven by the stored strain
907
energy) and “surface-recrystallization” (i.e. driven by the grain boundary energy).
908
It is interesting to perceive how the modern metallurgical terminology evolved
909
giving emphasis on the distinguishing prefixes “work-” and “surface- ,” whereas
910
the current geological terminology emphasizes the common suffix “-recrystallization.”
911
It seems that Alterthum himself had a preference for emphasizing the common
912
suffix, seeing that he considered also the situation when both driving forces (viz. stored
913
strain and grain boundary energies) act together, in a process he named “gemischte
914
Rekristallisation,” that is “mixed recrystallization.”
915
5.1. Rotation recrystallization (RRX)
916
By definition, the formation of a subgrain boundary is related to a slight rotation
917
of the crystalline lattice of a certain portion of the grain, called the subgrain. Such
918
a locallized rotation is usually driven by local distortions of the lattice caused
919
by internal stresses and intercrystalline misfits (cf. Sect. 2.2), which are accom-
920
modated by the subgrain rotation and the resulting concentration of the lattice
921
distortion (i.e. geometrically necessary dislocations) along the subgrain bound-
922
ary (Sect. 4.1). If the driving force for rotation persists, the lattice misorientation
923
across the subgrain boundary increases until the subgrain divides from the parent
924
grain to become a grain in its own. Alternatively, the misorientation across the
925
subgrain boundary may increase by subgrain growth and consumption of neigh-
926
bouring subgrain boundaries in a region with monotonic lattice misorientation
41
927
gradient. In any case, it is the last step of the process, namely the splitting of the
928
parent grain into two or more grains, that we name here rotation recrystallization
929
(RRX; Appendix A).
930
Not all subgrain boundaries evolve to grain boundaries, though. In order to
931
accomplish the creation of a new grain boundary via RRX, the internal stresses
932
causing the subgrain rotation and growth must persist unchanged for a period long
933
enough, and this is often not the case. Instead of developing a single high-angle
934
boundary, the stressed grain often accommodates the internal stresses through
935
the creation of several subgrain boundaries, which offer smoother but more com-
936
plex geometrical possibilities of strain accommodation than a single large-angle
937
boundary could provide (e.g. Figs. C.3, C.7b, C.8b–f and C.10).
938
It is actually not trivial to identify the transformation of a subgrain boundary
939
into a grain boundary via RRX in naturally deformed ice, since natural ice sam-
940
ples provide just a static snapshot of the microstructure evolution. Experience and
941
good sense help in the direct identification of the most conspicuous examples, but
942
direct inspection of grain boundary shapes is not a reliable method for quantify-
943
ing RRX. In the past, RRX has been estimated indirectly from the stabilization of
944
mean grain size (cf. ice-core reviews in Sects. 3.3, 4.2, 4.3, and 5.2 of Part I). This
945
was relatively simple under the assumption of the tripartite paradigm (Sect. 3.3 of
946
Part I; see also Sect. 3.3), since in this case RRX could be inferred from the devi-
947
ation of the observed grain growth data from the theoretical predictions of normal
948
grain growth (NGG) theory (Montagnat and Duval, 2000; Faria et al., 2002; Math-
949
iesen et al., 2004; Placidi et al., 2004). However, if the tripartite paradigm is not
950
valid, as proposed here, then the indirect quantification of RRX from grain size
951
data becomes more difficult, due to the more complex motion of grain boundaries
42
952
during strain-induced boundary migration (SIBM-O), compared to NGG.
953
Alley et al. (1995) have proposed the most reliable method to date for quanti-
954
fying RRX in natural ice. It involves an ingenious analysis of grain boundary mis-
955
orientations, based on the assumption that a grain newly formed by RRX should
956
have a lattice orientation closely related to that of its neighbouring sibling grain.
957
Considering the fact that only c-axes can currently be measured extensively (us-
958
ing an Automatic Fabric Analyzer, AFA; Wilson et al., 2007; see also Sect. 4.3
959
of Part I), this method tends to underestimate RRX. Nevertheless, this underesti-
960
mation may be tolerable, seeing that the fraction of grains formed by RRX about
961
the c-axis is expected to be less than 10%, according to Weikusat et al. (2011),
962
cf. Table D.2.
963
It should be remarked that RRX in ice can start already at very early stages
964
of deformation. As explained in Sect. 3.1, during primary creep (ε . 1%) there
965
occurs the load transfer from easy-glide to hard-glide systems, together with the
966
build up of internal stresses and strain incompatibilities between the grains. All
967
these processes promote the generation of the geometrically necessary disloca-
968
tions needed for subgrain boundary formation and evolution.
969
5.2. Nucleation and migration recrystallization
970
An important contribution of glaciology to geology has been the study of deforma-
971
tion and/or recrystallization of thin polycrystalline sections via transmitted light
972
microscopy. The use of this technique in glaciology can be traced back to the first
973
decades of 20th century (Tammann and Dreyer, 1929; Steinemann, 1958; Rigsby,
974
1960; Wakahama, 1964), and later it found widespread application in structural
975
geology through the use of a number of mineral-analogue materials, including
976
magnesium, camphor, sodium chlorate, and octachloropropane (Burrows et al.,
43
977
1979; Urai et al., 1980; Jessell, 1986; Means, 1989; den Brok et al., 1998).
978
By using this kind of technique, Tammann and Dreyer (1929) managed to
979
monitor the real-time static recrystallization of polycrystalline ice cold-rolled from
980
snow, therefore providing first estimates of two-dimensional grain-boundary mi-
981
gration rates in the temperature range between −2◦ C and −6◦ C. Additionally, they
982
observed grain coalescence and nucleation, and even embarked on an unsuccess-
983
ful attempt of explaining the growth of ice grains during static recrystallization.
984
As mentioned in Sect. 2.1 of Part I, Seligman (1941) accredited to Perutz the
985
interpretation of grain growth in ice during recrystallization as a consequence of
986
grains well-oriented for basal slip having a lower free energy than badly-oriented
987
grains, so that the former should grow at the expenses of those grains that can-
988
not yield to the imposed stresses. This thermodynamic interpretation was subse-
989
quently extended to the nucleation of new grains and tested in experiments and
990
field investigations of recrystallization in temperate and polar (frozen) ice (e.g.
991
Bader, 1951; Rigsby, 1951; Steinemann, 1958; Shoumsky, 1958; Rigsby, 1958b;
992
Kamb, 1959; Rigsby, 1960; Gow, 1963; Kamb, 1964; Wakahama, 1964; Rigsby,
993
1968; Kizaki, 1969; Budd, 1972; Kamb, 1972; Matsuda and Wakahama, 1978).
994
These studies provided a wealth of data, but results were not always fully accor-
995
dant (Remark 12). It became a general consensus that recrystallized ice grains
996
tend to develop irregular shapes (as previously observed by Perutz and Seligman,
997
1939; cf. Sect. 2.1 of Faria et al., this issue) combined with lattice preferred orien-
998
tations (LPOs) that maximize the resolved shear stress on the basal planes. While
999
the LPOs produced by recrystallization in uniaxial compression and extension
1000
seemed compatible with Perutz’ thermodynamic interpretation (viz. large/small
1001
girdles centred around the axis of extension/compression; Kamb, 1972), those
44
1002
produced by simple shear appeared much less intuitive and defied simple expla-
1003
nation. Therefore, owing to the importance of simple shear for the flow of glaciers
1004
and ice sheets, during the 1950–1980’s much attention was dedicated to the un-
1005
derstanding of dynamic recrystallization of ice under simple shear.
1006
Remark 12. The reader revising the literature from the second half of 20th cen-
1007
tury should keep in mind that many glaciologists used to employ the term “recrys-
1008
tallization” in a loose manner, often in reference to recrystallization with nucle-
1009
ation only. Less frequently, the term also included ordinary migration recrystal-
1010
lization without nucleation (SIBM-O, cf. Appendix A). Rotation recrystallization
1011
(RRX) was often ignored in pre-1980 studies.
1012
Rigsby (1958b, 1960) observed much slower recrystallization rates in ice rich
1013
in small air bubbles, and no evidence of mechanical twinning. He reported dif-
1014
ferent LPOs in polar (frozen) and temperate ice: in the case of simple shear the
1015
former exhibited a single maximum perpendicular to the shear plane, while the
1016
latter showed multiple maxima. He interpreted the multiple maxima as the result
1017
of migration recrystallization in a “nearly stress-free environment.” Steinemann
1018
(1958) also found no evidence of mechanical twinning and emphasized the dis-
1019
tinction between the LPOs produced by dynamic and static recrystallization. In
1020
his torsion-simple-shear experiments (420 and 660 kPa at −1.9◦ C) he reported that
1021
dynamic recrystallization generated multiple maxima, while subsequent static re-
1022
crystallization transformed them into a single maximum perpendicular to the shear
1023
plane (these observations were subsequently criticized and re-analysed by Kamb,
1024
1959).
1025
By compiling results from other researchers and from his own investigations,
1026
Kamb (1959, 1964, 1972) concluded that the typical LPOs produced in simple45
1027
shear tests at high temperatures (ca. −5◦ C and above) had a single maximum
1028
perpendicular to the shear plane, sometimes accompanied by a secondary, tran-
1029
sient maximum rotated away from the first in the reverse shear direction. In con-
1030
trast, LPOs found in glacier ice, which was supposedly deforming under simple-
1031
shear conditions similar to those applied to the simple-shear tests, where charac-
1032
terized by four maxima about the normal to the shear plane, ideally forming a
1033
cross/diamond pattern with monoclinic symmetry. Kamb attributed the discrep-
1034
ancy between laboratory and natural deformation to the vast difference in time
1035
scales, so that some sort of lattice-orientation controlling mechanism should be-
1036
come operative at very large strains (ε . 100%). In contrast to Rigsby’s obser-
1037
vations, Kamb (1972) found in his experiments and observations no detectable
1038
influence of air bubbles on recrystallization.
1039
Kizaki (1969) and Budd (1972) proposed that LPOs with multiple maxima
1040
could be produced by ordinary migration recrystallization (SIBM-O, cf. Appendix
1041
A) during dynamic grain growth, so that c-axis distributions with multiple max-
1042
ima should be characteristic of ice with coarse irregular grains, while the c-axes
1043
of fine-grained ice should be either weakly-oriented or clustered in a single max-
1044
imum. Finally, by analysing c- and a-axis orientations in recrystallized ice with
1045
multiple maxima, Matsuda and Wakahama (1978) discovered a common coincident-
1046
lattice relationship between neighbouring grains and speculated that the multiple
1047
maxima could be the result of nucleation via mechanical twinning under a high
1048
shear stress. Such a conjecture was later challenged by Parameswaran (1982)
1049
on the basis of a dislocation model, and by Wilson (1986) through the fact that
1050
twinning as a deformation mechanism has never been observed in ice: rather,
1051
coincident-lattice relationships could be the result of boundary migration during
46
1052
the impingement of growing grains.
1053
Even if mechanical twinning is ruled out as a mechanism of nucleation recrys-
1054
tallization in ice, at least two other nucleation hypotheses are generally considered
1055
by glaciologists. They are named here classical (or spontaneous) nucleation and
1056
pseudo-nucleation (cf. the entry “nucleation” in Appendix A). During classical
1057
nucleation a cluster of water molecules spontaneously form a new embryo, which
1058
evolves to a nucleus that grows as a new strain-free grain. In contrast, during
1059
pseudo-nucleation a microscopic portion of the parent grain undergoes a com-
1060
bination of elementary recovery and recrystallization processes (e.g. boundary
1061
migration, subgrain rotation and growth, etc.; cf. SIBM-N in Appendix A), which
1062
lead to the formation of a little strain-free new grain, called pseudo-nucleus (the
1063
prefix “pseudo-” is used here to emphasize that this nucleus may be larger than
1064
a classical nucleus, but still small enough to undergo complete recovery and be-
1065
come strain-free). Despite recurrent considerations of classical nucleation in the
1066
glaciological literature, it has long been recognized that spontaneous nucleation
1067
as a recrystallization mechanism in single-phase polycrystals is energetically un-
1068
favourable (Cahn, 1970; Urai et al., 1986; Drury and Urai, 1990; Humphreys and
1069
Hatherly, 2004) and there is no evidence that this should be different for ice (Glen,
1070
1974; Wilson, 1986; Kipfstuhl et al., 2009).
1071
During the 1970’s and 1980’s it became increasingly clear that the unsteady
1072
flow of glaciers most likely affected their LPO evolution, making the analysis of
1073
recrystallization structures rather difficult. Therefore, attention slowly turned to
1074
the microstructures of polar ice sheets, which seemed simpler to interpret and were
1075
produced under much more stable flow conditions. A decisive step in this regard
1076
was made by Azuma and Higashi (1985), who empirically discovered that, under
47
1077
common natural conditions, the strain in an ice grain is generally proportional to
1078
the resolved shear stress on its basal plane. Based on this result, they derived
1079
the first successful theoretical model of LPO evolution by lattice rotation in polar
1080
ice (subsequently extended by Frujita et al., 1987; Alley, 1988; Lipenkov et al.,
1081
1989). Later, this model would serve as basis for Azuma’s ice flow model (Azuma,
1082
1994, 1995; Azuma and Goto-Azuma, 1996), which is still today one of the most
1083
popular approaches for describing the anisotropic flow of glaciers and ice sheets.
1084
Finally, by combining Azuma and Higashi’s (1985) lattice rotation model
1085
and Kamb’s (1972) extension of Perutz’ thermodynamic interpretation of recrys-
1086
tallization, Alley (1988, 1992) managed to merge several ideas about polar ice
1087
microstructure evolution, which were emerging in the ice-core community dur-
1088
ing the 1970’s and 1980’s, into the simple and self-consistent version of the tri-
1089
partite paradigm (cf. Sect. 3.3 of Part I) that many glaciologists still adopt to-
1090
day (when consulting the works by Alley, 1988, 1992, the reader should have
1091
in mind that he used the terms “recrystallization” and “polygonization” as loose
1092
synonyms for “nucleation” and “rotation recrystallization,” respectively). The es-
1093
tablishment of this paradigm brought order to what was a rather chaotic topic,
1094
providing the framework for the development of models of microstructure evolu-
1095
tion and anisotropic flow of ice sheets (Van der Veen and Whillans, 1994; Azuma
1096
and Goto-Azuma, 1996; Gödert and Hutter, 1998; Montagnat and Duval, 2000;
1097
Staroszczyk and Morland, 2001; Faria et al., 2002; Thorsteinsson, 2002).
1098
In spite of being as welcome and needed as it was, today we know that the
1099
tripartite paradigm is fundamentally wrong. Besides the arguments put forward
1100
in Sect. 3.3, recent observations have shown that rotation recrystallization (RRX)
1101
and migration recrystallization with and without nucleation (SIBM-N and SIBM-
48
1102
O, respectively, cf. Appendix A) are widespread phenomena in polar ice sheets
1103
and take place already in firn (e.g. Figs. C.5, C.7, C.8 and C.11; Kipfstuhl et al.,
1104
2006, 2009; Faria et al., 2009, 2010; Weikusat et al., 2009a,b, 2011). Nucleation
1105
is not predominant in polar ice, but newly nucleated grains can be found regularly
1106
in ice-core samples from any depth, and are specially frequent in samples from the
1107
lower firn. Nucleation occurs via SIBM-N through the formation of pseudo-nuclei
1108
(cf. Appendix A) at localized sites characterized by high internal stresses and large
1109
misorientation gradients, like e.g. at grain boundaries, triple junctions, and simi-
1110
lar regions characterized by high concentrations of dislocation walls and subgrain
1111
boundaries. Most frequently the newly nucleated grain seems to grow from the
1112
boundary towards the inside of the parent grain, but nuclei formed at grain bound-
1113
ary bulges or corners that grow over the neighbouring grains are also common
1114
(e.g. Figs. C.3, C.5, and C.8a,b). Much more rare are nucleated islands, which
1115
are new grains or subgrains formed inside a very distorted parent grain, character-
1116
ized by an entangled network of dislocation walls and subgrain boundaries, which
1117
combine to form the boundaries of the new nucleus (Figs. C.5 and C.11).
1118
Ordinary migration recrystallization (SIBM-O; i.e. strain-induced boundary
1119
migration without nucleation of new grains, cf. Appendix A) and grain boundary
1120
pinning are ubiquitous in polar ice. In micrographs, the migration direction of a
1121
moving grain boundary can often be easily identified by the curved shape of the
1122
boundary and the presence of subgrain boundaries and dislocation walls, which
1123
are predominantly found at the convex side of the moving boundary (Figs. C.5,
1124
C.8, and C.11). Polar ice grains are generally irregular in shape, evidencing the
1125
essential role of stored strain energy on the microstructure evolution at all depths.
1126
Pinning is most frequently caused by subgrain boundaries, air hydrates, air bub-
49
1127
bles and firn pores. Particularly interesting is the pinning by microinclusions: in
1128
the upper ice, where the temperature is below ca. −10◦ C, it is difficult to find
1129
evidence of pinning by individual microinclusions, except occasionally in some
1130
grain boundaries in the strongest cloudy bands. Consequently, the explanation for
1131
the typical fine-grained structure of cloudy bands (cf. Fig. A.4 of Part I) remains
1132
uncertain. In contrast, as the temperature rises above −10◦ C in deep ice, most
1133
microinclusions can be found at grain boundaries and at the interfaces between
1134
ice and air hydrates (Fig. C.12). Possible causes of these intriguing phenomena
1135
are analysed in detail by Faria et al. (2010).
1136
5.3. The dynamic recrystallization diagram
1137
As a substitute for the old tripartite paradigm, we propose the dynamic recrys-
1138
tallization diagram in Fig. C.13, which summarizes the various recrystallization
1139
processes that contribute to the microstructure evolution of polar ice, as regions
1140
in the three-dimensional state space S = {ε̇, T, D} of strain rate ε̇, temperature T ,
1141
and mean grain size D.
1142
The main feature of this diagram is the attractor surface D = Dss (ε̇, T ), which
1143
describes the grain size at steady state, Dss , as a function of T and ε̇. This attractor
1144
surface works as follows: in a general situation, the mean grain size D of a piece
1145
of ice evolves according to the kinetic function D = χ(ε̇, T, t). Thus, for fixed
1146
conditions of temperature and strain rate, the mean grain size may evolve in time
1147
by recrystallization, provided that
∂D
∂
=
χ(ε̇, T, t) , 0 .
∂t
∂t
(19)
1148
The explicit form of the kinetic function χ depends on the active recrystallization
1149
processes and cannot be easily determined. However, one thing we know about
50
1150
(19), namely




> 0 (grain growth)





∂D 


< 0 (grain reduction)

∂t 






 = 0 (steady state)
if D < Dss ,
if D > Dss ,
(20)
if D = Dss .
1151
Thus, Dss defines an attractor surface in the state space S which reduces the kinetic
1152
function D = χ(ε̇, T, t) to the steady state relation D = Dss (ε̇, T ) when the mean
1153
grain size achieves its steady-state value.
1154
The derivation of the explicit form of Dss (ε̇, T ) is really straightforward. First
1155
we recall that Dss should obey the empirical relation (2). Second, we combine
1156
this relation with Glen’s flow law (5), setting n = 3 as usual. Finally, using the
1157
Arrhenius-like equation (9) we obtain
Dss (ε̇, T ) =
αϕ 21
ε̇
e−Q/2kB T .
(21)
1158
For the sake of illustration, let us consider the case of a hypothetical ice
1159
core, whose mean grain size evolves with depth as depicted by the green-and-red
1160
curves in Fig. C.13. If the conditions of temperature and strain rate were constant
1161
throughout the core, the mean-grain-size path in S would be a straight, vertical
1162
line hitting the attractor surface Dss and stopping there. This would correspond
1163
to grain growth until the steady-state grain size Dss is achieved. However, in this
1164
hypothetical core we assume that the temperature increases with depth (which
1165
is the expected physical behaviour within an ice sheet) whereas, for simplicity,
1166
the strain rate remains nearly constant. As a consequence, the mean-grain-size
1167
path in S follows not only upwards, but also sidewards, in the direction of higher
1168
temperatures (green part of the curve). Once it hits the attractor surface Dss , it
1169
continues its trajectory towards higher temperatures, without moving away from
51
1170
the surface (red part of the curve). Thus, after the mean grain size achieves its
1171
steady-state value, further grain growth with depth is caused by the increase of
1172
Dss with temperature, as described by (21).
1173
Finally, one could imagine a situation where the attractor surface Dss is shifted
1174
by a sudden change in strain rate or temperature (or impurity content, if we allow
1175
α to depend on it). This situation is not illustrated in the example, but it is not
1176
difficult to realize that in this case the microstructure would turn into a non-steady
1177
state and would start once again to pursue the attractor surface Dss , through a
1178
suitable growth or reduction of grain size.
1179
The zones of influence in S of the different recrystallization mechanisms are
1180
illustrated in Fig. C.14. Owing to the difficulty in visualizing and portraying such
1181
zones in three dimensions, we present here only three cross sections of S. De-
1182
picted are the regions in the state space where a particular process dominates. It
1183
is important to notice, however, that these zones have no sharp boundaries and
1184
they do overlap in most part of S. In fact, the typical situation is that various
1185
processes occur simultaneously and compete with or complement each other. The
1186
only exception is Normal Grain Growth (NGG), which is possible only on the
1187
plane SNGG = {ε̇ = 0, T, D}.
1188
6. Conclusion
1189
Compared to glaciers and other natural ice bodies, polar ice sheets offer many
1190
advantages for the study of natural ice microstructure evolution. In particular,
1191
the history of stress and temperature conditions experienced by a piece of po-
1192
lar ice is generally much longer, simpler and more steady than it would be in a
1193
glacier. This facilitates considerably the interpretation of deformation and recrys52
1194
tallization microstructures. Therefore, polar ice cores have become instrumental
1195
in microstructure investigations of natural ice.
1196
1197
In this work we reviewed our current knowledge of the mechanics and microstructure of natural ice. The main conclusions can be summarized as follows:
1198
• Almost a half-century ago the tripartite paradigm of polar ice microstruc-
1199
ture started to take form (also known as the “three-stage model”; Sect. 3.3
1200
of Part I and Sect. 3.3). It would soon turn into the main cornerstone of our
1201
understanding of natural ice microstructures, establishing a concrete and
1202
sought-after research program on structural glaciology that is still pursued
1203
today. Notwithstanding, in spite of being as welcome and needed as it was, a
1204
large body of evidence has accumulated over the last decade, which reveals
1205
fundamental flaws in that paradigm.
1206
• One fundamental premise of the tripartite paradigm that has to be critically
1207
reconsidered is the belief that only normal grain growth (NGG) can lead to
1208
grain coarsening. As discussed here and in Part I, a typical feature of polar
1209
ice cores is indeed the tendency towards an increase of the mean grain size
1210
with depth and age of the ice (modulated by climate changes). However,
1211
as we learn that microstructures characteristic of dynamic recrystallization
1212
abound in polar ice, we have to face the fact that dynamic recrystallization
1213
can also lead to grain coarsening, through a set of processes collectively
1214
named dynamic grain growth (cf. Appendix A).
1215
• The growth rates and activation energy for grain growth extracted directly
1216
from ice-core data agree well with the rates and energy obtained in grain
1217
growth experiments with bubbly ice, but are in clear disagreement with the
53
1218
real values of these quantities, recently measured in controlled experiments
1219
of normal grain growth in pure, unstrained, bubble-free ice. These conclu-
1220
sions, together with independent results of recent numerical simulations of
1221
normal grain growth in ice, corroborate the dynamic nature of grain growth
1222
in ice sheets, in the sense that it occurs during deformation and is seriously
1223
affected by the stored strain energy, as well as by air inclusions and other
1224
impurities.
1225
• The strong plastic anisotropy of the ice lattice gives rise to high internal
1226
stresses and concentrated strain heterogeneities in the polycrystal, which
1227
demand large amounts of strain accommodation. From the microstructural
1228
analyses of ice cores, we conclude that the formation of many and diverse
1229
subgrain boundaries and the splitting of grains by rotation recrystalliza-
1230
tion are the most fundamental mechanisms of dynamic recovery and strain
1231
accommodation in polar ice. Subgrain boundaries are endemic and very
1232
frequent at almost all depths in polar ice sheets.
1233
• In addition to subgrain formation (i.e. grain subdivision) and rotation recrys-
1234
tallization, microstructural analyses of polar ice cores suggest that strain in
1235
fine-grained, high-impurity ice layers (e.g. cloudy bands) can sometimes be
1236
accommodated by diffusional flow (at low temperatures and stresses) or mi-
1237
croscopic grain boundary sliding via microshear (in anisotropic ice sheared
1238
at high temperatures).
1239
• Evidence of recrystallization with nucleation of new grains is observed at
1240
various depths in the ice sheet, provided that the concentration of strain en-
1241
ergy is high enough (which is not seldom the case). Nucleation seems par54
1242
ticularly frequent in the lower firn layers, where the pore space is still large
1243
enough to weaken the ice matrix, but already small enough to allow consid-
1244
erable interaction between incompatible grains. As in other polycrystalline
1245
materials, nucleation does not happen in the classical sense of spontaneous
1246
embryo formation, but rather through a combination of recovery and re-
1247
crystallization processes (grain boundary migration, subgrain rotation and
1248
growth, etc.) within very localized regions with large misorientation gradi-
1249
ents. For this reason, we call this process nucleated migration recrystalliza-
1250
tion (SIBM-N; cf. Appendix A).
1251
• As a substitute for the tripartite paradigm, we propose a novel dynamic re-
1252
crystallization diagram in the three-dimensional state space of strain rate,
1253
temperature, and mean grain size (Figs. C.13 and C.14). This diagram sum-
1254
marizes the various competing recrystallization processes that contribute to
1255
the evolution of the polar ice microstructure.
1256
Afterword. We dedicate this work to the 60th birthday of Sepp Kipfstuhl, whose
1257
views have inspired many ideas introduced here. Sepp has been a key personal-
1258
ity of European glaciology in the last 30 years, having participated in more than
1259
25 polar expeditions to date (authors’ conservative estimate), including the First
1260
West-German Antarctic Research Overwintering (Georg von Neumeyer Station,
1261
Ekström Ice Shelf, 1981–83) and all European deep-drilling projects in Greenland
1262
and Antarctica since GRIP (cf. Table B.1 of Part I). In the early 1990s he played
1263
a decisive role in the partnership between European GRIP and U.S. GISP2 scien-
1264
tists (Sect. 4.2 of Part I) and since then he has investigated the physical properties
1265
of ice cores, often as the scientist in charge. Through his ingenious approach to
55
1266
observation and legendary devotion to ice, Sepp continues to inspire generations
1267
of scientists and to make ground-breaking findings about the microstructure of
1268
polar ice and firn.
1269
Appendix A. Glossary
1270
Below we summarize the main concepts and definitions used in this work for
1271
discussing ice mechanics and microstructure. They are based on the definitions
1272
put forward by Faria et al. (2009) and are partially inspired by the terms used in
1273
geology and materials science by Poirier (1985), Drury and Urai (1990), Bunge
1274
and Schwarzer (2001), Humphreys and Hatherly (2004), and Passchier and Trouw
1275
(2005).
1276
Clathrate hydrate: Crystalline compound containing guest molecules enclosed in cage-
1277
like structures made up of hydrogen-bonded water molecules. When the guest mol-
1278
ecules form gas under standard conditions, such compounds are also named gas hy-
1279
drates. In particular, air hydrates are formed by atmospheric gases (viz. mainly O2
1280
and N2 ). In natural ice, air hydrates are formed below a critical depth, which is fun-
1281
damentally a function of the overburden pressure and temperature.
1282
Cloudy band: Ice stratum with turbid appearance due to a high concentration of microin-
1283
clusions. Experience shows a strong correlation between high impurity concentration
1284
and small grain sizes in cloudy-band ice.
1285
Crystallite: See grain.
1286
Deformation-related structures: Structural features produced and/or affected by defor-
1287
mation, e.g. dislocations, subgrain boundaries, slip bands, stratigraphic folds, etc.
1288
Diffusion creep: See diffusional flow.
1289
Diffusional flow: Strain caused by diffusional flux of matter through the material. In
56
1290
polycrystals, diffusional flow may involve mass transport through or around the grains.
1291
The former is named lattice diffusion creep (or Nabarro–Herring creep), while the
1292
latter is called grain-boundary diffusion creep (or Coble creep).
1293
1294
Dislocation wall: Deformation-related structure consisting of dislocations arranged in a
two dimensional framework; the precursor of a subgrain boundary (cf. id.).
1295
DML: Dronning Maud Land, Antarctica.
1296
Dynamic grain growth (DGG): Class of phenomenological processes of grain coarsen-
1297
ing in polycrystals during deformation. Several recovery and recrystallization pro-
1298
cesses may be simultaneously active during DGG, all competing for the minimization
1299
of both, the stored strain energy and the grain-boundary energy. The essential fea-
1300
ture of DGG (in comparison to other recrystallization processes) is the monotonic
1301
increase of the mean grain size with time. Owing to its dynamic nature, however, the
1302
diversified kinetics of DGG can generally not be compared with the simple kinetics
1303
predicted for normal grain growth (NGG, cf. id.).
1304
Dynamic recrystallization: See recrystallization.
1305
EDC: EPICA Dome C (a deep-drilling site in Antarctica).
1306
EDML: EPICA DML (a deep-drilling site in Antarctica).
1307
Elementary structural process: The fundamental operation of structural change via re-
1308
covery or recrystallization, e.g. grain boundary migration or subgrain rotation. Sev-
1309
eral elementary processes may combine in a number of ways to produce a variety of
1310
phenomenological structural processes (cf. id.).
1311
Note A.1: Recovery and recrystallization are complex physical phenomena that are
1312
better understood if decomposed in a hierarchy of structural processes or mecha-
1313
nisms, here qualified as “elementary” and “phenomenological.” A somewhat sim-
1314
ilar hierarchical scheme for recrystallization has formerly been proposed by Drury
57
1315
and Urai (1990), but with the expressions “elementary/phenomenological process” re-
1316
placed respectively by “basic process” and “mechanism”. We favor here the qualifiers
1317
“elementary/phenomenological” (against the “process/mechanism” scheme) because
1318
these qualifiers facilitate the visualization of the hierarchy and leave us free to use the
1319
terms “process” and “mechanism” as synonyms.
1320
EPF: Expéditions Polaires Françaises.
1321
EPICA: European Project for Ice Coring in Antarctica.
1322
Fabric: See Lattice Preferred Orientation (LPO).
1323
Firn: Sintered snow that has outlasted at least one summer.
1324
GBS: See grain boundary sliding.
1325
GISP2: Greenland Ice Sheet Project 2 (a deep-drilling site in Greenland).
1326
Grain: Connected region in a polycrystalline solid composed of an uninterrupted (al-
1327
though possibly imperfect) crystalline lattice and bounded to other grains by grain
1328
boundaries. Also loosely called crystallite. It should be noticed the difference be-
1329
tween grains of polycrystalline solids (e.g. ice) and the lose particles of crystalline
1330
granular media (e.g. snow).
1331
Grain Boundary Sliding (GBS): Relative slide of a pair of grains by a shear movement
1332
at their common interface. The shear may be completely confined to the boundary, or
1333
occur within a zone immediately adjacent to it.
1334
1335
Grain stereology: Spatial arrangement of grains in a polycrystal, including their sizes
and shapes (cf. orientation stereology and lattice preferred orientation).
1336
Grain subdivision: Phenomenological recovery process of formation of new subgrain
1337
boundaries. It involves the progressive rotation of certain portions of the grain, called
1338
subgrains (cf. id.), as well as the strengthening of dislocation walls through dislo-
1339
cation rearrangement and migration in regions with strong lattice curvature. If the
58
1340
misorientation across the new subgrain boundary increases with time, grain subdivi-
1341
sion may give rise to rotation recrystallization (cf. id.).
1342
GRIP: Greenland Ice-core Project (a deep-drilling site in Greenland).
1343
Inclusion: Locallized deposit of undissolved chemical impurities observed in polar ice,
1344
like air bubbles, clathrate hydrates, or brine pockets. Inclusions not larger than a few
1345
micrometers are often called microinclusions (e.g. dust particles, microbubbles, etc.).
1346
Isotropic ice: In full isotropic polycrystalline ice. Ice with isotropic and homogeneous
1347
orientation stereology (cf. id.). In other words, homogeneous polycrystalline ice with
1348
no LPO (cf. id.).
1349
JIRP: Juneau Ice Field Research Project.
1350
Lattice Preferred Orientation (LPO): Statistically preferred orientation of the crystalline
1351
lattices of a population of grains. In plural (LPOs): the directional pattern of lattice
1352
orientations in a polycrystalline region (cf. orientation stereology). In the glaciologi-
1353
cal literature, LPOs are often called fabric (Paterson, 1994), while in materials science
1354
they are frequently termed texture (Humphreys and Hatherly, 2004). In particular, a
1355
polycrystalline region with a random distribution of lattice orientations is said to have
1356
no LPO (viz. texture-free, random fabric).
1357
LPO: See lattice preferred orientation.
1358
Microbubble: Air bubble not larger than a critical diameter of ca. 100µm in shallow ice.
1359
The critical diameter is usually defined by the typically bimodal size distribution of air
1360
bubbles in natural ice. For deeper ice, the critical diameter reduces with the increasing
1361
overburden pressure. See also inclusion.
1362
Microinclusion: See inclusion.
1363
Microshear: Strong, localized shear across a grain that experiences a highly inhomo-
1364
geneous shear deformation. It culminates with the formation of a new, flat subgrain
59
1365
boundary parallel to the shear plane, called microshear boundary (cf. slip bands).
1366
Microshear is often triggered by grain boundary sliding (cf. id.).
1367
1368
Microstructure: Collection of all microscopic deformation-related structures, inclusions,
and the orientation stereology of a polycrystal.
1369
Migration recrystallization: In full strain-induced migration recrystallization. Class of
1370
phenomenological recrystallization processes based on the elementary SIBM mecha-
1371
nism (cf. id.). If nucleation (cf. id.) is involved in the process, we may call it nucleated
1372
migration recrystallization (SIBM-N), where the suffix “-N” stands for “new grain”.
1373
Otherwise, i.e. if the migration of boundaries occurs without formation of new grains,
1374
we may call it ordinary migration recrystallization (SIBM-O), where the suffix “-O”
1375
stands for “old grain”.
1376
Note A.2: The definition adopted here is based on the concept of “grain-boundary mi-
1377
gration recrystallization” originally described in the pioneering work by Beck and
1378
Sperry (1950). Notice that this definition is not identical to that used by Poirier
1379
(1985) or Humphreys and Hatherly (2004), and it is also quite distinct from some
1380
loose connotations invoked in the glaciological literature. The terms SIBM-N and
1381
SIBM-O are not standard in the literature, but they are nevertheless adopted here be-
1382
cause they describe quite precisely the kind of information obtained from microscopic
1383
analyses of ice core sections. There is unfortunately no one-to-one relation between
1384
SIBM-N/SIBM-O and the expressions “multiple/single subgrain SIBM” used e.g. by
1385
Humphreys and Hatherly (2004).
1386
NBSAE: Norwegian–British–Swedish Antarctic Expedition.
1387
NGRIP: North-Greenland Ice-Core Project, also abbreviated as NorthGRIP (a deep-
1388
1389
1390
drilling site in Greenland).
Normal grain growth (NGG): Phenomenological recrystallization process of grain coarsening in polycrystals, resulting from “the interaction between the topological require-
60
1391
ments of space-filling and the geometrical needs of (grain-boundary) surface-tension
1392
equilibrium” (Smith, 1952). By definition, grain coarsening during NGG is statisti-
1393
cally uniform and self-similar, grain-boundary migration is exclusively driven by min-
1394
imization of the grain-boundary area (and associated free energy), and the grain stere-
1395
ology is close to a configuration of “surface-tension equilibrium” (so-called “foam-
1396
like structure”). Owing to these essential features, NGG is generally regarded as a
1397
static recrystallization process (cf. recrystallization) taking place before/after defor-
1398
mation (cf. dynamic grain growth). Mathematical and physical arguments strongly
1399
suggest that the kinetics of NGG is parabolic with respect to the mean grain radius.
1400
Note A.3: As discussed by Smith (1952), the interest in NGG comes from the fact
1401
that its kinetics depends solely on the properties of the migrating boundaries and is
1402
otherwise independent of the medium or its deformation history. This means that the
1403
theory underlying the NGG kinetics is not restricted to polycrystals: similar coars-
1404
ening phenomena are also observed in foams, some tissues, and many other cellular
1405
media.
1406
Nucleation: Class of phenomenological recrystallization processes involving the forma-
1407
tion of new nuclei (viz. tiny strain-free new grains). Two types of nucleation mech-
1408
anisms can be identified, here called “pseudo-” and “classical nucleation”. During
1409
classical nucleation a cluster of atoms/molecules spontaneously form a new embryo
1410
(the precursor of a nucleus) under the action of high internal stresses and thermally-
1411
activated fluctuations. Despite persistent consideration of this mechanism in the glacio-
1412
logical literature, it is currently acknowledged that it is certainly not relevant for polar
1413
ice (see Note A.4 below). During pseudo-nucleation a special combination of ele-
1414
mentary recrystallization processes (e.g. SIBM, subgrain rotation and growth) takes
1415
place within a small crystalline region with high stored strain energy, giving rise to
1416
a little strain-free new grain called pseudo-nucleus (see Note A.5 below). If pseudo-
1417
nucleation occurs naturally in polar ice, it most likely happens at grain boundaries and
61
1418
other zones of high stored strain energy, e.g. at air bubbles and solid inclusions.
1419
Note A.4: Calculations show (Cahn, 1970; Humphreys and Hatherly, 2004) that clas-
1420
sical nucleation recrystallization is extremely unlikely to occur in single-phase poly-
1421
crystals, owing to the high energies required for the creation and growth of classical
1422
nuclei, except if strong chemical driving forces are present, which is clearly not the
1423
case for polar ice.
1424
Note A.5: The prefix “pseudo-” is used here to emphasize that this nucleus is usually
1425
much larger than the nucleus formed by classical nucleation, but still small enough
1426
to be strain-free. It should be noticed that the distinction between pseudo-nucleation
1427
and a combination of SIBM-O with rotation recrystallization is basically a matter of
1428
scale: in the latter case the new crystallite is large enough to inherit a considerable
1429
amount of internal structures from the parent grain.
1430
1431
Orientation stereology: Spatial arrangement of lattice orientations in a polycrystal, i.e.
the combination of grain stereology and LPO.
1432
Phenomenological structural process: Any combination of elementary structural pro-
1433
cesses that gives rise to general changes in the structure of the polycrystal (cf. ele-
1434
mentary structural process). Examples of phenomenological processes are nucleation
1435
and grain subdivision.
1436
Polygonization: Special type of recovery mechanism for the formation of tilt bound-
1437
aries. It is a particular case of grain subdivision (cf. id.), by restricting it to tilting
1438
(bending) of crystallographic planes. In ice, polygonization is often used in reference
1439
to the bending of basal planes.
1440
Pseudo-nucleus: See nucleation.
1441
Recovery: Release of the stored strain energy by any thermomechanical process of mi-
1442
crostructural change other than recrystallization. The qualifiers dynamic and static de-
1443
note recovery phenomena occurring during and prior/after deformation, respectively.
62
1444
Frequently (especially under dynamic conditions), recovery and recrystallization co-
1445
exist and may even be complementary (e.g. during rotation recrystallization), so that
1446
the distinction between them is sometimes very difficult.
1447
Recrystallization: Any re-orientation of the lattice caused by grain boundary migration
1448
and/or formation of new grain boundaries, therefore including SIBM, RRX, DGG
1449
and NGG (cf. recovery and Note A.6 below). The qualifiers dynamic and static
1450
denote recrystallization phenomena occurring during and prior/after deformation, re-
1451
spectively. Further classification schemes often invoked in the literature include the
1452
qualifiers continuous/discontinuous and continual/discontinual, used to specify, re-
1453
spectively, the spatial homogeneity and temporal continuity of the recrystallization
1454
process. These classifications are, however, not always unique and are therefore of
1455
limited use.
1456
Note A.6: In contrast to the definition adopted here, some authors reserve the term “re-
1457
crystallization” solely for those processes driven by the stored strain energy, therefore
1458
excluding e.g. normal grain growth (NGG, cf. id.) from its definition. Other authors
1459
(especially in the older literature) loosely use “recrystallization” as a synonym for
1460
SIBM-N (cf. migration recrystallization).
1461
Rotation recrystallization (RRX): Phenomenological recrystallization process respon-
1462
sible for the formation of new grain boundaries. It proceeds from the mechanism of
1463
grain subdivision, and as such it involves the progressive rotation of subgrains as well
1464
as the migration of subgrain boundaries through regions with lattice curvature. Notice
1465
that this recrystallization process does not require significant migration of pre-existing
1466
grain boundaries, in contrast to migration recrystallization.
1467
SIBM: See strain-induced boundary migration.
1468
SIBM-N/SIBM-O: See migration recrystallization.
1469
Slip bands: Series of parallel layers of intense slip activity and high amount of intracrys-
63
1470
talline lattice defects (especially dislocations). Slip bands in ice appear always in
1471
groups parallel to the basal planes and are indicative of a nearly homogeneous shear
1472
deformation of the respective grain (cf. microshear).
1473
Static recrystallization: See recrystallization.
1474
Stored strain energy: Fraction of the mechanical energy expended during deformation
1475
that is stored in the material in diverse types of intracrystalline lattice defects, e.g.
1476
dislocations, stacking faults, subgrain boundaries, etc.
1477
Strain-induced boundary migration (SIBM): Elementary recrystallization process of
1478
grain boundary motion driven by minimization of the stored strain energy. It involves
1479
the migration of a grain boundary towards a region of high stored strain energy. The
1480
migrating boundary heals the highly energetic lattice defects in that region, therefore
1481
promoting a net reduction in the total stored strain energy of the polycrystal. See also
1482
migration recrystallization.
1483
1484
Subglacial structure: Any structural feature underneath the ice, ranging from till and
rocks to channels and lakes.
1485
Subgrain: Sub-domain of a grain, delimited by a subgrain boundary and characterized
1486
by a lattice orientation that is similar, but not identical, to that of the rest of the grain.
1487
In ice, the lattice misorientation across a subgrain boundary is limited to a few degrees
1488
(ca. < 5◦ for ice; (Suzuki, 1970; Weikusat et al., 2011)).
1489
Texture: See Lattice Preferred Orientation (LPO).
1490
Tilt boundary: Special type of subgrain boundary in which the misorientation axis is
1491
1492
1493
tangential to the boundary interface.
Twist boundary: Special type of subgrain boundary in which the misorientation axis is
orthogonal to the boundary interface.
64
1494
Appendix B. Deformation of EDML firn
1495
It is a common misconception that the firn zone is one of the least stressed parts of
1496
an ice sheet. In fact, rather the contrary is true. Although the overburden pressure
1497
on firn is much less than on deep ice, it is still large enough to promote the slow
1498
but relentless compaction of the delicate porous structure. Besides, the firn layer is
1499
continually stretched by the flowing ice underneath. These two processes combine
1500
to generate strain rates in firn that are much larger than in bulky ice.
1501
In the snow and shallow firn zones, the dominant metamorphic process is
1502
the rearrangement and packing of old snow particles via boundary sliding (Al-
1503
ley, 1987). As the firn approaches a mass density of ca. 550 kg/m3 (which cor-
1504
responds to a packing fraction of φ = 0.6, very close to that of the maximally
1505
random jammed state, φMRJ ≈ 0.63; Kansal et al., 2002), the dominant sintering
1506
mechanism changes to plastic deformation of the consolidated porous material via
1507
intracrystalline creep (Anderson and Benson, 1963; Maeno and Ebinuma, 1983).
1508
At the EDML site, this critical mass density is reached at around 20 m depth
1509
(Kipfstuhl et al., 2009), although recent computer tomographic analyses suggest
1510
that this transition could start already at 10 m depth, where the firn has an average
1511
mass density of only 475 kg/m3 (Freitag et al., 2008). The creep of firn proceeds
1512
this way for hundreds of years, so that, in the lower half of the firn zone, typical
1513
values of the total vertical strain lie in the range of several tens percent.
1514
From the supplementary material accompanying the work by Ruth et al. (2007),
1515
we estimate that the total vertical strain of the lower firn in the EDML site ranges
1516
between −20% and −50%. It is evident that most of this thinning is actually
1517
caused by the compression of the pore space. This compression, however, cannot
1518
occur without plastic deformation of the ice matrix. It is very difficult to determine
65
1519
with precision the contribution to total vertical strain due to plastic deformation of
1520
the ice matrix alone. In the case of EDML, one possibility is to combine the true
1521
annual layer thickness with the ice-equivalent layer thickness and the estimated
1522
age of the layer (all data provided by Ruth et al., 2007) as follows
ε = ln (1 + εe ) ,
εe =
y0 − y
,
y
(B.1)
1523
where ε and εe are respectively the natural vertical strain and the engineering
1524
vertical strain of the layer, while y and y0 denote the number of years enclosed in
1525
the strained layer and in the reference layer, respectively. Using these formulas we
1526
conclude that the polycrystalline ice skeleton of the lower firn at EDML is already
1527
in the tertiary creep regime (cf. Sect. 3.1), and consequently it could be undergoing
1528
dynamic recrystallization. Indeed, even if we make a very conservative choice for
1529
the reference depth, by assuming that the ice matrix starts to creep only below
1530
20 m depth, we still get εi.eq. ≈ −7% for the ice-equivalent vertical strain at only
1531
50 m depth. For comparison, the total vertical strain of firn at this depth (i.e.,
1532
including pore-space compression) is around εtotal ≈ −30%. Recalling that it
1533
takes about 300 years for the EDML ice to traverse the depth interval 20–50 m,
1534
we conclude that the average ice-equivalent vertical strain rate should be about
1535
ε̇i.eq. ≈ 7.4 × 10−12 s−1 . Likewise, the average total strain rate of the firn layer,
1536
including pore-space compaction, should be around ε̇total ≈ 3.2 × 10−11 s−1 .
1537
Admittedly, these are very crude estimates. However, it should be noticed that
1538
almost all the above inaccuracies can be blamed for being too conservative, that is,
1539
for introducing bias against dynamic recrystallization in polar firn. For instance:
1540
1541
• The reference depth is likely to be shallower than the one selected here.
More realistic estimates point to 10–12 m.
66
1542
• In practice, the shallow firn above the reference depth may also experience a
1543
certain amount of intracrystalline deformation, even though boundary slid-
1544
ing is the dominant deformation mechanism in that zone.
1545
• The ice-equivalent estimates do not take into account the contribution of the
pore space to strain accommodation.
1546
1547
• The deformation of firn is know to be extremely inhomogeneous. It is char-
1548
acterized by large strain variability with depth and intense stress concentra-
1549
tions, both influenced by the intricate geometry of the pore space. There-
1550
fore, the stored strain energy is likely to be very high in particular regions
1551
of the ice skeleton, where rotation and migration recrystallization may start
1552
very early.
1553
1554
Thus, we conclude that the real strain rate ε̇real experienced by the ice grains
in firn should be ε̇total ≥ ε̇real ≥ ε̇i.eq. .
1555
The last item above explains also why the c-axis distributions in lower firn are
1556
generally random, with no evident preferred orientations: the stress field within
1557
the ice skeleton is rather complex, with a high spatial variability controlled by
1558
the geometry of the pore space. Therefore, the stresses perceived by the ice on
1559
the grain scale are generally very distinct from the applied macroscopic stress.
1560
Even if preferred orientations are formed on the scale of several grains, the spatial
1561
variability of stress and strain are sufficient to mask any preferred orientations
1562
on the macroscale. Evidently, dynamic recrystallization with nucleation of new
1563
grains can also contribute to suppress the formation of preferred orientations in
1564
firn.
1565
Thus, the fact that the above estimates do support the occurrence of dynamic
67
1566
recrystallization in firn, in spite of all the bias against such a conclusion, just
1567
makes the arguments presented here stronger. Finally, we remark that these con-
1568
clusions are coherent with the experimental observations of dynamic recrystal-
1569
lization in firn by Kipfstuhl et al. (2009).
1570
Acknowledgements
1571
The authors thank Daniel Koehn (Special Issue Editor), Jens Roessiger and an
1572
anonymous reviewer for insightful revisions, as well as Tim Horscroft (Review
1573
Papers Coordinator) and Joao Hipertt (Editor) for managing the submission and
1574
publication process. Thanks go also to Daniela Jansen and Christian Weikusat
1575
for discussions and assistance in the preparation of some figures. Special thanks
1576
to Atsushi Miyamoto for discussions and for kindly providing the micrographs
1577
of Dome F deep ice core. Support from ESF Research Networking Programme
1578
Micro-Dynamics of Ice (Micro-DICE) is gratefully acknowledged. IW acknowl-
1579
edges also financial support by the German Research Foundation (HA 5675/1-1,
1580
WE 4695/1-2) via SPP 1158 and by the Helmholtz Association (VH-NG-802).
1581
References
1582
Ahmad, S., Whitworth, R. W., 1988. Dislocation motion in ice: a study by syn-
1583
1584
1585
1586
1587
chrotron X-ray topography. Philos. Mag. A 57 (5), 749–766.
Alley, R. B., 1987. Firn densification by grain boundary sliding: a first model. J.
Phys. (Paris) 48 (C1), 249–256.
Alley, R. B., 1988. Fabrics in polar ice sheets: development and prediction. Science 240, 493–495.
68
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
Alley, R. B., 1992. Flow-law hypothesis for ice-sheet modelling. J. Glaciol. 38,
245–256.
Alley, R. B., Gow, A. J., Meese, D. A., 1995. Mapping c-axis fabrics to study
physical processes in ice. J. Glaciol. 41 (137), 197–203.
Alley, R. B., Perepezko, J. H., Bentley, C. R., 1986a. Grain growth in polar ice:
I. Theory. J. Glaciol. 32 (112), 415–424.
Alley, R. B., Perepezko, J. H., Bentley, C. R., 1986b. Grain growth in polar ice:
II. Application. J. Glaciol. 32 (112), 425–433.
Alterthum, H., 1922a. Zur Theorie der Rekristallisation. Z. Metallk. 14 (11), 417–
424.
Alterthum, H., 1922b. Zur Theorie der Rekristallisation. Z. Elektrochem. Angew.
Phys. Chem. 28 (15–16), 347–356.
1600
Anderson, D. L., Benson, C. S., 1963. The densification and diagenesis of snow.
1601
In: Kingery, W. D. (Ed.), Ice and Snow. MIT Press, Cambridge, MA, pp. 391–
1602
411.
1603
1604
1605
1606
1607
1608
Andrade, E. N. d. C., 1910. On the viscous flow in metals, and allied phenomena.
Proc. Roy. Soc. London A 84, 1–12.
Ashby, M. F., 1966. Work hardening of dispersion-hardened crystals. Philos. Mag.
14 (132), 1157–1178.
Azuma, N., 1994. A flow law for anisotropic ice and its application to ice sheets.
Earth Planet. Sci. Lett. 128, 601–614.
69
1609
1610
1611
1612
1613
1614
1615
1616
Azuma, N., 1995. A flow law for anisotropic polycrystalline ice under uniaxial
compressive deformation. Cold Reg. Sci. Technol. 23 (2), 137–147.
Azuma, N., Goto-Azuma, K., 1996. An anisotropic flow law for ice-sheet ice and
its implications. Ann. Glaciol. 23, 202–208.
Azuma, N., Higashi, A., 1985. Formation processes of ice fabric patterns in ice
sheets. Ann. Glaciol. 6, 130–134.
Azuma, N., Miyakoshi, T., Yokoyama, S., Takata, M., 2012. Impeding effect of
air bubbles on normal grain growth of ice. J. Struct. Geol. 42, 184–193.
1617
Azuma, N., Wang, Y., Mori, K., Narita, H., Hondoh, T., Shoji, H., Watanabe, O.,
1618
1999. Textures and fabrics in the Dome F (Antarctica) ice core. Ann. Glaciol.
1619
29, 163–168.
1620
Azuma, N., Wang, Y., Yoshida, Y., Narita, H., Hondoh, T., Shoji, H., Watanabe,
1621
O., 2000. Crystallographic analysis of the Dome Fuji ice core. In: Hondoh, T.
1622
(Ed.), Physics of Ice Core Records. Hokkaido University Press, Sapporo, pp.
1623
45–61.
1624
Bader, H., 1951. Introduction to ice petrofabrics. J. Geol. 59 (6), 519–536.
1625
Barnes, P., Tabor, D., Walker, J. C. F., 1971. The friction and creep of polycrys-
1626
1627
1628
talline ice. Proc. Roy. Soc. London A 324, 127–155.
Barrette, P. D., Sinha, N. K., 1994. Lattice misfit as revealed by dislocation etch
pits in a deformed ice crystal. J. Mater. Sci. Letters 13, 1478–1481.
1629
Bartels-Rausch, T., Bergeron, V., Cartwright, J. H. E., Escribano, R., Finney, J. L.,
1630
Grothe, H., Gutierrez, P. J., Haapala, J., Kuhs, W. F., Pettersson, J. B. C., Price,
70
1631
S. D., Sainz-Diaz, C. I., Stokes, D., Strazzulla, G., Thomson, E. S., Trinks, H., ,
1632
Uras-Aytemiz, N., 2012. Ice structures, patterns, and processes: A view across
1633
the icefields. Reviews of Modern Physics.
1634
URL http://link.aps.org/doi/10.1103/RevModPhys.84.885
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
Beck, P. A., Sperry, P. R., 1950. Strain induced grain boundary migration in high
purity aluminum. J. Appl. Phys. 21, 150–152.
Bernal, J. D., Fowler, R. H., 1933. A theory of water and ionic solution, with
particular reference to hydrogen and hydroxyl ions. J. Chem. Phys., 515–548.
Bons, P. D., Jessell, M. W., 1999. Micro-shear zones in experimentally deformed
octachloropropane. J. Struct. Geol. 21, 323–334.
Bryant, G. W., Mason, B. J., 1960. Etch pits and dislocations in ice crystals. Phil.
Mag., Structure and Properties of Condensed Matter 5 (8), 1221–1227.
Budd, W. F., 1972. The development of crystal orientation fabrics in moving ice.
Z. Gletscherkunde Glazialgeol. 8 (1–2), 65–105.
Budd, W. F., Jacka, T. H., 1989. A review of ice rheology for ice sheet modelling.
Cold Reg. Sci. Technol. 16, 107–144.
Bunge, H. J., Schwarzer, R. A., 2001. Orientation stereology—a new branch in
texture research. Adv. Eng. Mater. 13 (1–2), 25–39.
1649
Burg, J. P., Wilson, C. J. L., Mitchell, J. C., 1986. Dynamic recrystallization and
1650
fabric development during the simple shear deformation of ice. J, Struct. Geol.
1651
8 (8), 857–870.
71
1652
1653
Burrows, S., Humphreys, J., White, S., 1979. Dynamic recrystallization. a comparison between magnesium and quartz. Bull. Minéral. 102, 75–79.
1654
Cahn, J. W., Taylor, J. E., 2004. A unified approach to motion of grain boundaries,
1655
relative tangential translation along grain boundaries, and grain rotation. Acta
1656
Mater. 52, 4887–4898.
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
Cahn, R. W., 1970. Recovery and recrystallization. In: Cahn, R. W. (Ed.), Physical
Metallurgy. North-Holland, Amsterdam, pp. 1129–1197.
Clifford, J., 1967. Proton magnetic resonance data on ice. Chem. Commun. (London) 17, 880–881.
Colbeck, S. C., Evans, R. J., 1973. A flow law for temperate glacier ice. J. Glaciol.
12 (64), 71–86.
Cole, D. M., 2004. A dislocation-based model for creep recovery in ice. Philos.
Mag. 84 (30), 3217–3234.
Cole, D. M., Durell, G. D., 2001. A dislocation-based analysis of strain history
effects in ice. Philos. Mag. 81 (7), 1849–1872.
1667
Cuffey, K. M., Conway, H., Gades, A., Hallet, B., Raymond, C. F., Whitlow, S.,
1668
2000a. Deformation properties of subfreezing glacier ice: role of crystal size,
1669
chemical impurities and rock particles inferred from in situ measurements. J.
1670
Geophys. Res. 105 (B12), 27895–27915.
1671
Cuffey, K. M., Thorsteinsson, T., Waddington, E. D., 2000b. A renewed argument
1672
for crystal size control of ice sheet strain rates. J. Geophys. Res. 105 (B12),
1673
27889–27894.
72
1674
Dahl-Jensen, D., 1985. Determination of the flow properties at Dye 3, south
1675
Greenland, by bore-hole-tilting measurements and perturbation modelling. J.
1676
Glaciol. 31 (108), 92–98.
1677
Dahl-Jensen, D., Gundestrup, N. S., 1987. Constitutive properties of ice at Dye 3,
1678
Greenland. In: IAHS Red Book 170, The Physical Basis of Ice Sheet Mod-
1679
elling. International Association of Hydrological Sciences, pp. 31–43.
1680
De la Chapelle, S., Castelnau, O., Lipenkov, V., Duval, P., 1998. Dynamic recrys-
1681
tallization and texture development in ice as revealed by the study of deep ice
1682
cores in antarctica and greenland. J. Geophys. Res. 103, 5091–5105.
1683
den Brok, B., Zahid, M., Passchier, C., 1998. Cataclastic solution creep of very
1684
soluble brittle salt as a rock analogue. Earth Planet. Sci. Lett. 163 (1–4), 83–95.
1685
Doake, C. S. M., Wolff, E. W., 1985. Flow law for ice in polar ice sheets. Nature
1686
314 (6008), 255–257.
1687
Doherty, R. D., Hughes, D. A., Humphreys, F., Jonas, J. J., Juul Jensen, D., Kass-
1688
ner, M. E., King, W. E., McNelley, T. R., McQueen, H. J., Rollet, A. D., 1997.
1689
Current issues in recrystallization: a review. Mater. Sci. Engineer. 238, 219–
1690
274.
1691
Drury, M. R., Humphreys, F. J., 1988. Microstructural shear criteria associated
1692
with grain-boundary sliding during ductile deformation. J. Struct. Geol. 10, 83–
1693
89.
1694
1695
Drury, M. R., Urai, J. L., 1990. Deformation-related recrystallization processes.
Tectonophys. 172, 235–253.
73
1696
1697
Duesbery, M. S., 1998. Dislocation motion, constriction and cross-slip in fcc metals. Modelling Simul. Mater. Sci. Eng. 6, 35–49.
1698
Durand, G., Persson, A., Samyn, D., Svensson, A., 2008. Relation between neigh-
1699
bouring grains in the upper part of the NorthGRIP ice core: implications for
1700
rotation recrystallization. Earth Planet. Sci. Lett. 265 (3), 666–671.
1701
Durand, G., Weiss, J., Lipenkov, V., Barnola, J. M., Krinner, G., Parrenin, F.,
1702
Delmonte, B., Ritz, C., Duval, P., Rothlisberger, R., Bigler, M., 2006. Effect of
1703
impurities on grain growth in cold ice sheets. J. Geophys. Res. 111, F01015.
1704
Durham, W. B., Stern, L. A., Kirby, S. H., 2001. Rheology of ice I at low stress
1705
1706
1707
1708
1709
1710
1711
1712
1713
and elevated confining pressure. J. Geophys. Res. 106 (6), 11031–11042.
Duval, P., 1978. Anelastic behaviour of polycrystalline ice. J. Glaciol. 21 (85),
621–627.
Duval, P., 1985. Grain growth and mechanical behaviour of polar ice. Ann.
Glaciol. 6, 79–82.
Duval, P., Ashby, M. F., Anderman, I., 1983. Rate-controlling processes in the
creep of polycrystalline ice. J. Phys. Chem. 87, 4066–4074.
Duval, P., Castelnau, O., 1995. Dynamic recrystallization of ice in polar ice sheets.
J. Phys. IV (Paris), colloq. C3 5, 197–205.
1714
Duval, P., Montagnat, M., 2002. Comment on “Superplastic deformation of ice:
1715
Experimental observations” by D. L. Goldsby and D. L. Kohlstedt. J. Geophys.
1716
Res. 107 (B4), 2082.
74
1717
1718
1719
1720
Etheridge, D. M., 1989. Dynamics of the Law Dome ice cap, Antarctica, as found
from bore-hole measurements. Ann. Glaciol. 12, 46–50.
Evans, R. C., 1976. An Introduction to Crystal Chemistry, 2nd Edition. Cambridge
University Press, Cambridge.
1721
Faria, S. H., 2001. Mixtures with continuous diversity: general theory and appli-
1722
cation to polymer solutions. Continuum Mech. Thermodyn. 13 (2), 91–120.
1723
Faria, S. H., 2003. Mechanics and thermodynamics of mixtures with continuous
1724
diversity: From complex media to ice sheets. Ph.D. thesis, Darmstadt Univer-
1725
sity of Technology, Darmstadt.
1726
Faria, S. H., 2006a. Creep and recrystallization of large polycrystalline masses.
1727
Part I: general continuum theory. Proc. Roy. Soc. London A 462 (2069), 1493–
1728
1514.
1729
Faria, S. H., 2006b. Creep and recrystallization of large polycrystalline masses.
1730
Part III: continuum theory of ice sheets. Proc. Roy. Soc. London A 462 (2073),
1731
2797–2816.
1732
1733
Faria, S. H., Freitag, J., Kipfstuhl, S., 2010. Polar ice structure and the integrity of
ice-core paleoclimate records. Quat. Sci. Rev. 29 (1), 338–351.
1734
Faria, S. H., Hamann, I., Kipfstuhl, S., Miller, H., 2006a. Is Antarctica like a
1735
birthday cake? Preprint 33/2006, Max Planck Institute for Mathematics in the
1736
Sciences, Leipzig.
1737
Faria, S. H., Hutter, K., 2001. The challenge of polycrystalline ice dynamics.
1738
In: Kim, S., Jung, D. (Eds.), Advances in Thermal Engineering and Sciences
75
1739
for Cold Regions. Society of Air-Conditioning and Refrigerating Engineers of
1740
Korea (SAREK), Seoul, pp. 3–31.
1741
Faria, S. H., Kipfstuhl, S., 2004. Preferred slip band orientations and bending
1742
observed in the Dome Concordia (East Antarctica) ice core. Ann. Glaciol. 39,
1743
386–390.
1744
1745
Faria, S. H., Kipfstuhl, S., 2005. Comment on “Deformation of grain boundaries
in polar ice” by G. Durand et al. Europhys. Lett. 71 (5), 873–874.
1746
Faria, S. H., Kipfstuhl, S., Azuma, N., Freitag, J., Hamann, I., Murshed, M. M.,
1747
Kuhs, W. F., 2009. The multiscale structure of Antarctica. Part I: inland ice.
1748
Low Temp. Sci. 68, 39–59.
1749
1750
Faria, S. H., Kipfstuhl, S., Lambrecht, A., in preparation. The EPICA-DML deep
ice core. Springer, Heidelberg.
1751
Faria, S. H., Kremer, G. M., Hutter, K., 2003. On the inclusion of recrystallization
1752
processes in the modeling of induced anisotropy in ice sheets: a thermodynam-
1753
icist’s point of view. Ann. Glaciol. 37, 29–34.
1754
Faria, S. H., Kremer, G. M., Hutter, K., 2006b. Creep and recrystallization of
1755
large polycrystalline masses. Part II: constitutive theory for crystalline media
1756
with transversely isotropic grains. Proc. Roy. Soc. London A 462 (2070), 1699–
1757
1720.
1758
1759
Faria, S. H., Ktitarev, D., Hutter, K., 2002. Modelling evolution of anisotropy in
fabric and texture of polar ice. Ann. Glaciol. 35, 545–551.
76
1760
1761
Faria, S. H., Weikusat, I., Azuma, N., this issue. The microstructure of polar ice.
Part I: highlights from ice core research. J. Struct. Geol.
1762
Fischer, D. A., Koerner, R. M., 1986. On the special rheological properties of
1763
ancient microparticle-laden Northern Hemisphere ice as derived from bore-hole
1764
and core measurements. J. Glaciol. 32 (112), 501–510.
1765
Fowler, A., 2001. Modelling the flow of glaciers and ice sheets. In: Straughan, B.,
1766
Greve, R., Ehrentraut, H., Wang, Y. (Eds.), Continuum Mechanics and Applica-
1767
tions in Geophysics and the Environment. Springer, Heidelberg, pp. 201–221.
1768
Freitag, J., Kipfstuhl, S., Faria, S. H., 2008. The connectivity of crystallite ag-
1769
glomerates in low density firn at Kohnen station, Dronning Maud Land, Antarc-
1770
tica. Ann. Glaciol. 49, 114–120.
1771
1772
1773
1774
1775
1776
Frost, H. J., Ashby, M. F., 1982. Deformation-mechanism Maps. Pergamon, Oxford.
Frujita, S., Nakawo, M., Mae, S., 1987. Orientation of the 700m Mizuho core and
its strain story. Proc. NIPR Symp. Polar Meteo. Glaciol. 1, 122–131.
Fukuda, A., Hondoh, T., Higashi, A., 1987. Dislocation mechanisms of plastic
deformation of ice. J. Phys. (Paris) 48, 163–173.
1777
Gammon, P. H., Kiefte, H., Clouter, M. J., Denner, W. W., 1983. elastic con-
1778
stants of artificial and natural ice samples by Brillouin spectroscopy. J. Glaciol.
1779
29 (103), 433–460.
1780
1781
Gifkins, R. C., 1976. Grain-boundary sliding and its accommodation during creep
and superplasticity. Metall. Trans. A 7, 1225–1232.
77
1782
Gillet-Chaulet, F., Gagliardini, O., Meyssonnier, J., Montagnat, M., Castelnau, O.,
1783
2005. A user-friendly anisotropic flow law for ice-sheet modeling. J. Glaciol.
1784
51 (172), 3–14.
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
Gilra, N. K., 1974. Non-basal glide in ice. physica status solidi (a) 21 (1), 323–
327.
Glen, J. W., 1952. Experiments on the deformation of ice. J. Glaciol. 2 (12), 111–
114.
Glen, J. W., 1955. The creep of polycrystalline ice. Proc. Roy. Soc. London A
228, 519–538.
Glen, J. W., 1968. The effect of hydrogen disorder on dislocation movement and
plastic deformation of ice. Phys. Kondens. Mater 7, 43–51.
Glen, J. W., 1974. The physics of ice. Cold regions science and engineering monograph II C2a, U. S. Army CRREL, Hanover, NH.
Glen, J. W., 1975. The mechanics of ice. Cold regions science and engineering
monograph II C2b, U. S. Army CRREL, Hanover, NH.
Glen, J. W., Perutz, M. F., 1954. The growth and deformation of ice crystals. J.
Glaciol. 2, 397–403.
Gödert, G., Hutter, K., 1998. Induced anisotropy in large ice sheets: theory and
its homogenization. Continuum Mech. Thermodyn. 13, 91–120.
Goldsby, D. L., Kohlstedt, D. L., 1997. Grain boundary sliding in fine-grained ice.
Scripta Mater. 37, 1399–1406.
78
1803
1804
Goldsby, D. L., Kohlstedt, D. L., 2001. Superplastic deformation of ice: experimental observations. J. Geophys. Res. 106, 11017–11030.
1805
Goldsby, D. L., Kohlstedt, D. L., 2002. Reply to comment by P. Duval and M.
1806
Montagnat on “Superplastic deformation of ice: experimental observations”. J.
1807
Geophys. Res. 107 (B11), 2313.
1808
1809
1810
1811
1812
1813
1814
1815
Goodman, D. J., Frost, H. J., Ashby, M. F., 1981. The plasticity of polycrystalline
ice. Philos. Mag. 43, 665–695.
Gow, A. J., 1963. Results of measurements in the 309 meter bore hole at Byrd
Station, Antarctica. J. Glaciol. 4 (36), 771–784.
Gow, A. J., 1969. On the rates of growth of grains and crystals in south polar firn.
J. Glaciol. 8 (53), 241–252.
Gundestrup, N. S., Hansen, B. L., 1984. Bore-hole survey at Dye 3, south Greenland. J. Galciol. 30, 282–288.
1816
Hamann, I., Weikusat, C., Azuma, N., Kipfstuhl, S., May 2007. Evolution of ice
1817
crystal microstructures during creep experiments. J. Glaciol. 53 (182), 479–489.
1818
Higashi, A., 1978. Structure and behaviour of grain boundaries in polycrystalline
1819
ice. J. Glaciol. 21 (85), 589–605.
1820
Higashi, A., Fukuda, A., Shoji, H., Oguro, M., Hondoh, T., Goto-Azuma, K.,
1821
1988. Lattice defects in ice crystals. Hokkaido University Press, Sapporo,
1822
Japan.
1823
1824
Higashi, A., Sakai, N., 1961. Movement of small angle boundary of ice crystal. J.
Fac. Sci. Hokkaido Univ. Ser. 2 Phys. 5 (5), 221–237.
79
1825
1826
Hirth, J. P., Lothe, J., 1992. Theory of Dislocations, 2nd Edition. Krieger Publishing Company, Malabar, FL.
1827
Hobbs, P. V., 1974. Ice Physics. Clarendon, Oxford.
1828
Hondoh, T., 2000. Nature and behavior of dislocations in ice. In: Hondoh, T.
1829
(Ed.), Physics of Ice Core Records. Hokkaido University Press, Sapporo, pp.
1830
3–24.
1831
1832
Hondoh, T., 2009. An overview of microphysical processes in ice sheets: toward
nanoglaciology. Low Temp. Sci. 68, 1–23.
1833
Hondoh, T., Higashi, A., 1978. X-ray diffraction topographic observations of the
1834
large-angle grain boundary in ice under deformation. J. Glaciol. 21 (85), 629–
1835
638.
1836
Hondoh, T., Higashi, A., 1983. Generation and absorption of dislocations at large-
1837
angle grain boundaries in deformed ice crystals. J. Phys. Chem. 87 (21), 4044–
1838
4050.
1839
1840
1841
1842
Hondoh, T., Iwamatsu, H., Mae, S., 1990. Dislocation mobility for nonbasal glide
in ice measured by in situ x-ray topography. Phylos. Mag. 62, 89–102.
Hooke, R. L., 1973. Structure and flow in the margin of the Barnes Ice Cap, Baffin
Island, N.W.T., Canada. J. Glaciol. 66, 423–438.
1843
Hooke, R. L., 1981. Flow law for polycrystalline ice in glaciers: comparison of
1844
theoretical predictions, laboratory data, and field measurements. Rev. Geophys.
1845
Space Phys. 19 (4), 664–672.
80
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
Hooke, R. L., 2005. Principles of Glacier Mechanics, 2nd Edition. Cambridge
University Press, Cambridge.
Hudleston, P. J., 1977. Similar folds, recumbent folds, and gravity tectonics in ice
and rocks. J. Geol. 85, 113–122.
Humphreys, F. J., Hatherly, M., 2004. Recrystallization and Related Annealing
Phenomena, 2nd Edition. Pergamon, Oxford.
Hutchinson, W. B., 1976. Bonds and self consistent estimates for creep of polycrystal materials. Proc. Roy. Soc. London A 348 (1652), 101–127.
Hutter, K., 1980. Time-dependent surface elevation of an ice slope. J. Glaciol.
25 (92), 247–266.
Hutter, K., 1981. The effect of longitudinal strain on the shear stress of an ice
sheet: in defence of using stretched coordinates. J. Glaciol. 27 (95), 39–56.
Hutter, K., 1982. Dynamics of glaciers and large ice masses. Ann. Rev. Fluid
Mech. 14, 87–130.
1860
Hutter, K., 1983. Theoretical Glaciology. Reidel, Dordrecht.
1861
Hutter, K., Legerer, F., Spring, U., 1981. First-order stresses and deformations in
1862
1863
1864
1865
1866
glaciers and ice sheets. J. Glaciol. 27 (96), 227–270.
Iliescu, D., Baker, I., Chang, H., 2004. Determining the orientations of ice crystals
using electron backscatter patterns. Microsc. Res. Tech. 63, 183–187.
Jacka, T. H., 1984. The time and strain required for development of minimum
strain rates in ice. Cold Reg. Sci. Technol. 8 (3), 261–268.
81
1867
1868
Jacka, T. H., Li, J., 1994. The steady-state crystal size of deforming ice. Ann.
Glaciol. 20, 13–18.
1869
Jacka, T. H., Li, J., 2000. Flow rates and crystal orientation fabrics in compression
1870
of polycrystalline ice at low temperatures and stresses. In: Hondoh, T. (Ed.),
1871
Physics of Ice Core Records. Hokkaido University Press, Sapporo, pp. 83–102.
1872
Jessell, M. W., 1986. Grain boundary migration and fabric development in exper-
1873
1874
1875
imentally deformed octachloropropane. J. Struct. Geol. 8 (5), 527–542.
Jones, S. J., Chew, H. A. M., 1983. Creep of ice as a function of hydrostatic
pressure. J. Phys. Chem. 87 (21), 4064–4066.
1876
Kamb, B., 1972. Experimental recrystallization of ice under stress. In: Heard,
1877
H. C., Borg, I. Y., Carter, N. L., Raleigh, C. B. (Eds.), Flow and Fracture
1878
of Rocks. No. 16 in Geophysical Monograph. American Geophysical Union,
1879
Washington, DC, pp. 211–241.
1880
1881
Kamb, W. B., 1959. Ice petrofabric observations from Blue Glacier, Washington,
in relation to theory and experiment. J. Geophys. Res. 64 (11), 1891–1909.
1882
Kamb, W. B., 1964. Glacier geophysics. Science 146 (3642), 353–365.
1883
Kansal, A. R., Torquato, S., Stillinger, F. H., 2002. Diversity of order and densities
1884
in jammed hard-particle packings. Phys. Rev. E 66 (4), 041109–1–041109–8.
1885
Kipfstuhl, S., Faria, S. H., Azuma, N., Freitag, J., Hamann, I., Kaufmann, P.,
1886
Miller, H., Weiler, K., Wilhelms, F., 2009. Evidence of dynamic recrystalliza-
1887
tion in polar firn. J. Geophys. Res. 114, B05204.
82
1888
Kipfstuhl, S., Hamann, I., Lambrecht, A., Freitag, J., Faria, S. H., Grigoriev,
1889
D., Azuma, N., 2006. Microstructure mapping: A new method for imag-
1890
ing deformation-induced microstructural features of ice on the grain scale. J.
1891
Glaciol. 52 (178), 398–406.
1892
Kirby, S. H., Durham, W. B., Stern, L. A., 1991. Mantle phase changes and deep-
1893
earthquake faulting in subducting lithosphere. Science 252 (1991), 216–225.
1894
Kizaki, K., 1969. Ice-fabric study of the Mawson region, East Antarctica. J.
1895
1896
1897
Glaciol. 8 (53), 253–276.
Kocks, U. F., 1970. The relation between polycrystal deformation and singlecrystal deformation. Metall. Trans. 1, 1121–1143.
1898
Kondo, T., Kato, H. S., Kawai, M., Bonn, M., 2007. The distinct vibrational sig-
1899
nature of grain-boundary water in nano-crystalline ice films. Chem. Phys. Lett.
1900
448 (1–3), 121–126.
1901
Ktitarev, D., Gödert, G., Hutter, K., 2002. Cellular automaton model for recrystal-
1902
lization, fabric and texture development in polar ice. J. Geophys. Res. 107 (B8),
1903
EPM 5–1–EPM 5–9.
1904
1905
Legrand, M., Mayewski, P. A., 1997. Glaciochemistry of polar ice cores: a review.
Rev. Geophys. 35, 219–143.
1906
Lemke, P., Ren, J., Alley, R. B., Allison, I., Carrasco, J., Flato, G., Fujii, Y.,
1907
Kaser, G., Mote, P., Thomas, R. H., Zhang, T., 2007. Observations: changes in
1908
snow, ice and frozen ground. In: Solomon, S., Qin, D., Manning, M., Chen, Z.,
1909
Marquis, M., Averyt, K. B., Tignor, M., Miller, H. L. (Eds.), Climate Change
83
1910
2007: The Physical Science Basis. Contribution of Working Group I to the
1911
Fourth Assessment Report of the Intergovernmental Panel on Climate Change
1912
(IPCC). Cambridge University Press, Cambridge.
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
Lile, R. C., 1978. The effect of anisotropy on the creep of polycrystalline ice. J.
Glaciol. 21 (85), 475–483.
Lipenkov, V. Y., Barkov, N. I., Duval, P., Pimienta, P., 1989. Crystalline texture of
the 2083m ice core at vostok station, antarctica. J. Glaciol. 35 (121), 392–398.
Liu, F., Baker, I., Dudley, M., 1993. Dynamic observations of dislocation generation at grain boundaries in ice. Philos. Mag. A 67, 1261–1276.
Liu, F., Baker, I., Dudley, M., 1995. Dislocation–grain boundary interactions in
ice crystals. Philos. Mag. A 71, 15–42.
Lliboutry, L., 1969. The dynamics of temperate glaciers from the detailed viewpoint. J. Glaciol. 8 (53), 185–205.
Lliboutry, L., 1976. Physical processes in temperate glaciers. J. Glaciol. 16 (74),
151–158.
1925
Louchet, F., 2004. Dislocations and plasticity in ice. C. R. Physique 5, 687–698.
1926
Mader, H. M., 1992. The thermal behaviour of the water-vein system in polycrys-
1927
talline ice. J. Glaciol. 38 (130), 359–374.
1928
Maeno, N., Ebinuma, T., 1983. Pressure sintering of ice and its implication to the
1929
densification of snow at polar glaciers and ice sheets. J. Phys. Chem. 87 (21),
1930
4103–4110.
84
1931
Marshall, H. P., Harper, J. T., Pfeffer, W. T., Humphrey, N., 2002. Depth-varying
1932
constitutive properties observed in an isothermal glacier. Geophys. Res. Letters
1933
29 (23), 61–1–61–4.
1934
Mathiesen, J., Ferkinghoff-Borg, J., Jensen, M. H., Levinsen, M., Olesen, P., Dahl-
1935
Jensen, D., Svensson, A., 2004. Dynamics of crystal formation in the Greenland
1936
NorthGRIPice core. J. Glaciol. 50 (170), 325–328.
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
Matsuda, M., 1979. Determination of axis orientations of polycrystalline ice. J.
Glaciol. 22 (86), 165–169.
Matsuda, M., Wakahama, G., 1978. Crystallographic structure of polycrystalline
ice. J. Glaciol. 21 (85), 607–620.
Matsuyama, M., 1920. On some physical properties of ice. J. Geol. 28 (7), 607–
631.
McConnel, J. C., 1890. On the plasticity of an ice crystal. Proc. Roy. Soc. London
49 (296–301), 323–343.
McConnel, J. C., Kidd, D. A., 1888. On the plasticity of glacier and other ice.
Proc. Roy. Soc. London 44 (266–272), 331–367.
Means, W. D., 1989. Synkinematic microscopy of transparent polycrystals. J.
Struct. Geol. 11 (1–2), 163–174.
Means, W. D., Jessell, M. W., 1986. Accommodation migration of grain boundaries. Tecronophys. 127, 67–86.
85
1951
Meier, M. F., 1958. Vertical profiles of velocity and the flow law of glacier ice. In:
1952
IAHS Red Book 47, Physics of the Movement of Ice. International Association
1953
of Hydrological Sciences, pp. 169–170.
1954
1955
1956
1957
1958
1959
1960
1961
Meier, M. F., 1960. Mode of flow of Saskatchewan Glacier, Alberta, Canada.
Professional Paper 351, U.S. Geological Survey, Washington, DC.
Mellor, M., Testa, R., 1969a. Creep of ice under low stress. J. Glaciol. 8 (52),
147–152.
Mellor, M., Testa, R., 1969b. Effect of temperature on the creep of ice. J. Glaciol.
8 (52), 1131–145.
Merkle, K. L., Thompson, L. J., 1973. Atomic-scale observation of grain boundary motion. Mater. Lett. 48 (3–4), 188–193.
1962
Miyamoto, A., Weikusat, I., Hondoh, T., 2011. Complete determination of ice
1963
crystal orientation and microstructure investigation on ice core samples en-
1964
abled by a new x-ray laue diffraction method. J. Glaciol. 57 (201), 67–74, awi-
1965
n18929.
1966
Montagnat, M., Castelnau, O., Bons, P. D., Faria, S. H., Gagliardini, O., Gillet-
1967
Chaulet, F., Griera, A., Lebensohn, R., Roessiger, J., 2013. Multiscale modeling
1968
of ice deformation behavior. J. Struct. Geol. this issue.
1969
Montagnat, M., Duval, P., 2000. Rate controlling processes in the creep of po-
1970
lar ice, influence of grain boundary migration associated with recrystallization.
1971
Earth Planet. Sci. Lett. 183, 179–186.
86
1972
1973
1974
1975
Morland, L. W., Staroszczyk, R., 1998. Viscous response of polar ice with evolving fabric. Continuum Mech. Thermodyn. 10, 135–152.
Mott, N. F., 1948. Slip at grain boundaries and grain growth in metals. Proc. Phys.
Soc. 60 (4), 391–394.
1976
Nakaya, U., 1958. The deformation of single crystals of ice. In: IAHS Red Book
1977
47, Physics of the Movement of Ice. International Association of Hydrological
1978
Sciences, pp. 229–240.
1979
1980
Norton, F. H., 1929. The Creep of Steel at High Temperatures. McGraw-Hill, New
York.
1981
Nye, J. F., 1953. The flow law of ice from measurements in glacier tunnels labo-
1982
ratory experiments and the Jungfraufirn experiment. Proc. Roy. Soc. London A
1983
219, 477–489.
1984
1985
Nye, J. F., 1957. The distribution of stress and velocities in glaciers and ice-sheets.
Proc. Roy. Soc. London A 239, 113–133.
1986
Nye, J. F., Frank, F. C., 1973. Hydrology of the intergranular veins in a temperate
1987
glacier. In: IAHS Red Book 95, Symposium on the Hydrology of Glaciers.
1988
International Association of Hydrological Sciences, pp. 157–161.
1989
1990
1991
1992
Okada, Y., Hondoh, T., Mae, S., 1999. Basal glide of dislocations in ice observed
by synchrotron radiation topography. Philos. Mag. A 79 (11), 2853–2868.
Ostwald, W., 1929. Ueber die rechnerische darstellung des strukturgebietes der
viskosität. Kolloid-Z. 47 (2), 176–187.
87
1993
1994
1995
1996
1997
1998
1999
2000
Parameswaran, V. R., 1982. Fracture criterion for ice using a dislocation model. J.
Glaciol. 28 (98), 161–169.
Passchier, C. W., Trouw, R. A. J., 2005. Microtectonics, 2nd Edition. Springer,
Berlin.
Paterson, W. S. B., 1977. Secondary and tertiary creep of glacier ice as measured
by borehole closure rates. Rev. Geophys. Space Phys. 15 (1), 47–55.
Paterson, W. S. B., 1991. Why ice-age ice is sometimes “soft”. Cold Reg. Sci.
Technol. 20 (1), 75–98.
2001
Paterson, W. S. B., 1994. The Physics of Glaciers, 3rd Edition. Pergamon, Oxford.
2002
Pauling, L., 1935. The structure and entropy of ice and of other crystals with some
2003
randomness of atomic arrangement. J. Amer. Chem. Soc. 57, 2680–2684.
2004
Peltier, W. R., Goldsby, D. L., Tarasov, L., 2000. Ice-age ice-sheet rheology: con-
2005
straints from the Last Glacial Maximum form of the Laurentide ice sheet. Ann.
2006
Glaciol. 30, 163–176.
2007
Perutz, M. F., 1949. The flow of ice and other solids. in: Joint meeting of the
2008
british glaciological society, the british rheologists’ club and the institute of
2009
metals. J. Glaciol. 1 (5), 231–240.
2010
2011
Perutz, M. F., 1950. In: Glaciology–the flow of glaciers. The Observatory
70 (855), 64–65.
2012
Perutz, M. F., Seligman, G., 1939. A crystallographic investigation of glacier
2013
structure and the mechanism of glacier flow. Proc. Roy. Soc. London A 172,
2014
335–360.
88
2015
2016
2017
2018
Petrenko, V. F., Whitworth, R. W., 1999. Physics of Ice. Oxford University Press,
Oxford.
Pettit, E. C., Waddington, E. D., 2003. Ice flow at low deviatoric stress. J. Glaciol.
49 (166), 359–369.
2019
Piazolo, S., Montagnat, M., Blackford, J. R., 2008. Sub-structure characterization
2020
of experimentally and naturally deformed ice using cryo-EBSD. J. Microsc.
2021
230 (3), 509–519.
2022
2023
Pimienta, P., Duval, P., 1987. Rate controlling processes in the creep of polar
glacier ice. J. Phys., Colloq. C1, Suppl. 3 48, 243–248.
2024
Placidi, L., Faria, S. H., Hutter, K., 2004. On the role of grain growth, recrystal-
2025
lization, and polygonization in a continuum theory for anisotropic ice sheets.
2026
Ann. Glaciol. 39, 49–52.
2027
Placidi, L., Greve, R., Seddik, H., Faria, S. H., 2010. Continuum-mechanical,
2028
anisotropic flow model, based on an anisotropic flow enhancement factor
2029
(CAFFE). Continuum Mech. Thermodyn. 22 (3), 221–237.
2030
Placidi, L., Hutter, K., 2006. Thermodynamics of polycrystalline materials treated
2031
by the theory of mixtures with continuous diversity. Cont. Mech. Thermodyn.
2032
17 (6), 409–451.
2033
Poirier, J.-P., 1985. Creep of Crystals. Cambridge University Press, Cambridge.
2034
Prior, D. J., Boyle, A. P., Brenker, F., Cheadle, M. C., Day, A., Lopez, G., Peruzzo,
2035
L., Potts, G. J., Reddy, S., Spiess, R., Timms, N. E., Trimby, P., Wheeler, J.,
2036
Zetterström, L., 1999. The application of electron backscatter diffraction and
89
2037
orientation contrast imaging in the sem to textural problems in rocks. American
2038
Mineralogist 1741-1759, 84.
2039
Prior, D. J., Diebold, S., Obbard, R., Daghlian, C., Goldsby, D. L., Durham, W. B.,
2040
Baker, I., 2012. Insight into the phase transformations between ice Ih and ice II
2041
from electron backscatter diffraction data. Scripta Mater. 66 (2), 69 – 72.
2042
Prior, D. J., Wheeler, J., Peruzzo, L., Spiess, R., Storey, C., 2002. Some garnet
2043
microstructures: an illustration of the potential of orientation maps and mis-
2044
orientation analysis in microstructural studies. Journal of Structural Geology
2045
24 (6-7), 999 – 1011.
2046
2047
2048
2049
2050
2051
2052
2053
Ramseier, R. O., 1967. Self-diffusion of tritium in natural and synthetic ice
monocrystals. J. Appl. Phys. 38 (6), 2553–2556.
Read, W. T., Shockley, W., 1950. Dislocation models of crystal grain boundaries.
Phys. Rev. 78 (3), 275–289.
Rigsby, G. P., 1951. Crystal fabric studies on Emmons Glacier Mount Rainier,
Washington. J. Geol. 59 (6), 590–598.
Rigsby, G. P., 1958a. Effect of hydrostatic pressure on velocity of shear deformation on single ice crystals. J. Glaciol. 3 (24), 271–278.
2054
Rigsby, G. P., 1958b. Fabrics of glacier and laboratory deformed ice. In: IAHS
2055
Red Book 47, Physics of the Movement of Ice. International Association of
2056
Hydrological Sciences, pp. 351–358.
2057
2058
Rigsby, G. P., 1960. Crystal orientation in glacier and in experimentally deformed
ice. J. Glaciol. 3 (27), 589–606.
90
2059
2060
2061
2062
Rigsby, G. P., 1968. The complexities of the three-dimensional shape of individual
crystals in glacier ice. J. Glaciol. 7 (50), 233–251.
Roessiger, J., Bons, P. D., Faria, S. H., 2013. Influence of bubbles on grain growth
in ice. J. Struct. Geol. this issue.
2063
Roessiger, J., Bons, P. D., Griera, A., Jessell, M. W., Evans, L., Montagnat,
2064
M., Kipfstuhl, S., Faria, S. H., Weikusat, I., 2011. Competition between grain
2065
growth and grain-size reduction in polar ice. J. Glaciol. 57 (205), 942–948.
2066
Rosen, J., 1995. Symmetry in Science. Springer, New York.
2067
Rosen, J., 2005. The symmetry principle. Entropy 7 (4), 308–313.
2068
Russell-Head, D. S., Budd, W. F., 1979. Ice-sheet flow properties derived from
2069
bore-hole shear measurements combined with ice-core studies. J. Glaciol.
2070
24 (90), 117–130.
2071
Ruth, U., Barnola, J. M., Beer, J., Bigler, M., Blunier, T., Castellano, E., Fis-
2072
cher, H., Fundel, F., Huybrechts, P., Kaufmann, P., Kipfstuhl, S., Lambrecht,
2073
A., Morganti, A., Oerter, H., Parrenin, F., Rybak, O., Severi, M., Udisti, R.,
2074
Wilhelms, F., Wolff, E., 2007. “EDML1”: a chronology for the EPICA deep ice
2075
core from Dronning Maud Land, Antarctica, over the last 150 000 years. Clim.
2076
Past 3, 475–484.
2077
2078
2079
2080
Schulson, E. M., Duval, P., 2009. Creep and Fracture of Ice. Cambridge University
Press, Cambridge.
Seligman, G., 1941. The structure of a temperate glacier. Geogr. J. 97 (5), 295–
315.
91
2081
Seligman, G., 1949. The growth of the glacier crystal. J. Glaciol. 1 (5), 254–268.
2082
Sharp, R. P., 1954. Glacier flow: a review. Geol. Soc. Amer. Bull. 65, 821–838.
2083
Shearwood, C., Whitworth, R. W., 1991. The velocity of dislocations in ice. Philo-
2084
2085
2086
sophical Magazine A 64 (2), 289–302.
Shoji, H., Langway, Jr., C. C., 1984. Flow behavior of basal ice as related to
modeling considerations. Ann. Glaciol. 5, 141–148.
2087
Shoumsky, P. A., 1958. The mechanism of ice straining and its recrystallization.
2088
In: IAHS Red Book 47, Physics of the Movement of Ice. International Associ-
2089
ation of Hydrological Sciences, pp. 244–248.
2090
Smith, C. S., 1952. Grain shapes and other metallurgical applications of topology.
2091
In: Metal Interfaces. American Society for Metals (ASM), Cleveland, OH, pp.
2092
65–108.
2093
2094
2095
2096
2097
2098
2099
2100
Smith, G. D., Morland, L., 1981. Viscous relations for the steady creep of polycrystalline ice. Cold Reg. Sci.Technol. 5 (2), 141–150.
Song, M., 2008. An evaluation of the rate-controlling flow process in newtonian
creep of polycrystalline ice. Mater. Sci. Eng. A 486, 27–31.
Staroszczyk, R., Morland, L. W., 2001. Strengthening and weakening of induced
anisotropy in polar ice. Proc. Roy. Soc. London A 451 (2014), 2419–2440.
Steinemann, S., 1954. Results of preliminary experiments on the plasticity of ice
crystals. J. Glaciol. 2, 404–412.
92
2101
Steinemann, S., 1958. Experimentelle Untersuchungen zur Plastizität von Eis.
2102
Beitr. Geol. Schweiz, Hydrol. 10, 1–72, also as Ph.D. Thesis, Swiss Federal
2103
Institute of Technology (ETH) Zurich.
2104
Stephenson, P. J., 1967. Some considerations of snow metamorphism in the
2105
antarctic ice sheet in the light of ice crystal studies. In: Oura, H. (Ed.), Physics
2106
of Snow and Ice. Vol. 1. Hokkaido University Press, Sapporo, pp. 725–740, pro-
2107
ceedings of the International Conference on Low Temperature Science, 1966,
2108
Sapporo, Japan.
2109
Stern, L. A., Durham, W. B., Kirby, S. H., 1997. Grain-size-induced weakening of
2110
H2 O ices I and II and associated anisotropic recrystallization. J. Geophys. Res.
2111
102 (B3), 5313–5325.
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
Sutton, A. P., Balluffi, R. W., 1995. Interfaces in Crystalline Materials. Clarendon,
Oxford.
Suzuki, S., 1970. Grain Coarsening of Microcrystals of Ice. (III). Low Temperature Science Ser. A 28, 47–61.
Suzuki, S., Kuroiwa, D., 1972. Grain-boundary energy and grain-boundary groove
angles in ice. J. Glaciol. 11 (62), 265–277.
Talalay, P. G., Hooke, R. L., 2007. Closure of deep boreholes in ice sheets: a
discussion. Ann. Glaciol. 47, 125–133.
Tammann, G., Dreyer, K. L., 1929. Die Rekristallisation leicht schmelzender
Stoffe und die des Eises. Z. Anorg. Allg. Chem. 182 (1), 289–313.
93
2122
2123
Tarr, R. S., Rich, J. L., 1912. The properties of ice—experimental studies. Z.
Gletscherkunde 6 (4), 225–249.
2124
Taylor, G. I., 1938. Plastic strain in metals. J. Inst. Metals 62, 307–324.
2125
Thompson, D. E., 1979. Stability of glaciers and ice sheets against flow perturba-
2126
tions. J. Glaciol. 24 (90), 427–441.
2127
Thorsteinsson, T., 2002. Fabric development with nearest-neighbor interaction
2128
and dynamic recrystallization. J. Geophys. Res. 107 (B1), ECV3–1–ECV3–13.
2129
Thorsteinsson, T., Kipfstuhl, J., Miller, H., 1997. Textures and fabrics in the GRIP
2130
ice core. J. Geophys. Res. 102, 26583–26599.
2131
Trepied, L., Doukhan, J. C., Paquet, J., January 1980. Subgrain boundaries in
2132
quartz theoretical analysis and microscopic observations. Phys. Chem. Miner.
2133
5 (3), 201–218.
2134
Treverrow, A., Budd, W. F., H., J. T., Warner, R. C., 2012. The tertiary creep of
2135
polycrystalline ice: experimental evidence for stress-dependent levels of strain-
2136
rate enhancement. J. Glaciol. 58 (208), 301–314.
2137
Urai, J. L., Humphreys, F. J., Burrows, S. E., 1980. In-situ studies of the deforma-
2138
tion and dynamic recrystallization of rhombohedral camphor. J. Mater. Sci. 15,
2139
1231–1240.
2140
Urai, J. L., Means, W. D., Lister, G. S., 1986. Dynamic recrystallization of miner-
2141
als. In: Hobbs, B. E., Heard, H. C. (Eds.), Mineral and Rock Deformation: Lab-
2142
oratory Studies. Geophysical Monograph 36. American Geophysical Union,
2143
Washington, pp. 161–199.
94
2144
2145
2146
2147
Van der Veen, C. J., Whillans, I. M., 1994. Development of fabric in ice. Cold
Reg. Sci. Technol. 22, 171–195.
Waddington, E. D., 2010. Life, death and afterlife of the extrusion flow theory. J.
Glaciol. 56 (200), 973–996.
2148
Wakahama, G., 1964. On the plastic deformation of ice. V. Plastic deformation
2149
of polycrystalline ice. Low Temp. Sci. A 22, 1–24, in Japanese with English
2150
summary.
2151
Wang, Y., Kipfstuhl, S., Azuma, N., Thorsteinsson, T., Miller, H., 2003. Ice-
2152
fabrics study in the upper 1500 m of the Dome C (East Antarctica) deep ice
2153
core. Ann. Glaciol. 37, 97–104.
2154
Wang, Y., Thorsteinsson, T., Kipfstuhl, J., Miller, H., Dahl-Jensen, D., Shoji, H.,
2155
2002. A vertical girdle fabric in the NorthGRIP deep ice core, North Greenland.
2156
Ann. Glaciol. 35, 515–520.
2157
2158
2159
2160
Weertman, J., 1983. Creep deformation of ice. Ann. Rev. Earth Planet Sci. 11,
215–240.
Weertman, J., Weertman, J. R., 1992. Elementary Dislocation Theory. Oxford
University Press, New York.
2161
Weikusat, I., de Winter, D. A. M., Pennock, G. M., Hayles, M., Schneijdenberg,
2162
C. T. W. M., Drury, M. R., June 2010. Cryogenic EBSD on ice: preserving a
2163
stable surface in a low pressure SEM. J. Microsc. 242 (3), 295–310.
2164
Weikusat, I., Kipfstuhl, S., Azuma, N., Faria, S. H., Miyamoto, A., 2009a. Defor-
95
2165
mation microstructures in an Antarctic ice core (EDML) and in experimentally
2166
deformed artificial ice. Low Temp. Sci. 68, 115–123.
2167
Weikusat, I., Kipfstuhl, S., Faria, S. H., Azuma, N., Miyamoto, A., 2009b. Sub-
2168
grain boundaries and related microstructural features in EDML(Antarctica)
2169
deep ice core. J. Glaciol. 55 (191), 461–472.
2170
Weikusat, I., Miyamoto, A., Faria, S. H., Kipfstuhl, S., Azuma, N., Hondoh, T.,
2171
2011. Subgrain boundaries in Antarctic ice quantified by X-ray Laue diffrac-
2172
tion. J. Glaciol. 57 (201), 85–94.
2173
Wilen, L. A., DiPrinzio, C. L., Alley, R. B., Azuma, N., 2003. Development,
2174
principles, and applications of automated ice fabric analyzers. Microsc. Res.
2175
Technique 62, 2–18.
2176
2177
2178
2179
Wilson, C. J. L., 1979. Boundary structures and grain shape in deformed multilayered polycrystalline ice. Tectonophys. 57 (2–4), T19–T25.
Wilson, C. J. L., 1982. Texture and grain growth during the annealing of ice.
Textur. Microstr. 5, 19–31.
2180
Wilson, C. J. L., 1986. Deformation induced recrystallization of ice: the applica-
2181
tion of in situ experiments. In: Hobbs, B. E., Heard, H. C. (Eds.), Mineral and
2182
Rock Deformation: Laboratory Studies. Geophysical Monograph 36. American
2183
Geophysical Union, Washington, pp. 213–232.
2184
2185
Wilson, C. J. L., Burg, Mitchell, 1986. The origin of kinks in polycrystalline ice.
Tectonophysics 127, 27–48.
96
2186
Wilson, C. J. L., Peternell, M., Piazolo, S., Luzin, V., this issue. Microstructure
2187
and fabric development in ice: lessons learned from in situ experiments and
2188
implications for understanding rock evolution. J. Struct. Geol.
2189
Wilson, C. J. L., Russell-Head, D. S., Kunze, K., Viola, G., March 2007. The
2190
analysis of quartz c-axis fabrics using a modified optical microscope. J. Mi-
2191
crosc. 227 (1), 30–41.
2192
Wilson, C. J. L., Russell-Head, D. S., Sim, H. M., 2003. The application of an
2193
automated fabric analyzer system to the textural evolution of folded ice layers
2194
in shear zones. Ann. Glaciol. 37, 7–17.
2195
2196
Wilson, C. J. L., Zhang, Y., 1996. Development of microstructure in the hightemperature deformation of ice. Ann. Glaciol. 23, 293–302.
2197
Zhang, Y., Wilson, C. J. L., 1997. Lattice rotation in polycrystalline aggregates
2198
and single crystals with one slip system: a numerical and experimental ap-
2199
proach. J. Struct. Geol. 19 (6), 875–885.
97
2200
Appendix C. FIGURE CAPTIONS
Figure C.1: The crystalline lattice of ice Ih. Red and white spheres represent oxygen and hydrogen
atoms, respectively, while grey rods symbolize hydrogen bonds. Top: view along the c-axis.
Bottom: view along an a-axis. The hexagonal symmetry of the lattice is highlighted by the yellow
dashed line (after Faria and Hutter, 2001).
Figure C.2: Schematic representation of possible slip systems in ice (after Hondoh, 2000; Faria,
2003). Cf. Table D.1.
98
Figure C.3: Mosaic image showing examples of several microstructural features in a sublimated
sample of Antarctic ice (EDML, 1656 m depth). Recognizable are slip bands (SB), grain boundaries (GB), subgrain boundaries (sGB), and [decomposed] air hydrates ([d]AH). Sublimation polishes the ice sample surface through thermal etching, forming as by-product observable etch
grooves at points where grain or subgrain boundaries meet the surface (Kipfstuhl et al., 2006).
In contrast, slip bands are volume features, which appear as series of parallel fringes that are only
observable in sections with a certain thickness (several hundreds of micrometers), when the c-axis
of the sheared grain lies nearly parallel to the sample surface plane (within a few degrees of misorientation). Air hydrates inside the sample appear as bright inclusions. If they lie on the surface,
however, they decompose and appear dark, because they are not stable at atmospheric pressure and
high temperatures. Completely unfocused structures are sublimation-etched features at the bottom
side of the sample, visible through the transparent ice matrix. The dark circular object on the top
right is a deposit or imperfection on the surface, while the curved shadow at the right border is part
of a bubble in the silicone oil that preserves the ice surface.
Figure C.4: Schematic representation of extended basal dislocations combined with non-basal
dislocation segments in ice. (a) A dislocation with an initially arbitrary shape soon evolves into
the more stable “terraced” configuration illustrated here, which combines long basal and short
non-basal segments. (b) Glissile screw dislocation dipole with Burgers vector a = (1/3) < 112̄0 >
led by a glissile non-basal edge segment. (c) Sessile edge dislocation dipole with Burgers vector
c =< 0001 > or a + c = (1/3) < 112̄3 > led by a glissile non-basal screw segment. After Hondoh
(2000).
99
Figure C.5: Typical manifestations of internal stresses and heterogeneous strains in an Antarctic
EDML sample from 556 m depth (bubbly ice). Air bubbles appear black. Width of each micrograph: 2.5 mm. Top left: Classical example of migration recrystallization (SIBM-O; cf. Appendix
A). Many subgrain boundaries and dislocation walls irradiating from a bulged grain boundary,
which is migrating to the left towards the region with high stored strain energy. The illumination
is especially favourable in this image for revealing the 3D-shape of the bulging grain boundary:
one can identify the bulged shadow produced by the grain boundary groove at the bottom surface
of the sample, as well as a grain boundary edge emanating from the triple junction on the left
towards the bottom of the sample. Top right: Another classical example of SIBM-O (centre), as
well as of grain subdivision (top left). Notice the elongated (sub-)grain island (centre top) nucleated in the region of high stored strain energy. Centre left: Well-developed subgrain island (left)
in a region of highly heterogeneous strain, characterised by many entangled dislocation walls and
subgrain boundaries. Centre right: Bending of a large grain and simultaneous consumption of the
irregular tilt boundary by a smaller grain (bottom right). Again, the 3D-shape of the smaller grain
can be visualized by the defocused curve/shadow produced by the groove at the bottom surface
of the sample (notice the cusp pointing in the direction of the “tilt boundary”). From the visible
slip bands, the misorientation across the irregular tilt boundary is & 7◦ . Bottom left: Large, welldeveloped subgrain island (bottom) near a jagged subgrain boundary. Notice also the tiny subgrain
island at the centre top. Bottom right: Classical examples of nucleated migration recrystallization
(SIBM-N; cf. Appendix A). A newly nucleated grain (top right) grows into the highly strained
region in the centre, characterized by numerous subgrain boundaries and dislocation walls. At the
same time, the bulge on the top left seems to be in the process of becoming a new grain by rotating
itself with respect to its parent grain, as indicated by the roughly vertical subgrain boundaries at
the top left. The unfocused shadows on the left are grain boundary grooves on the bottom surface
of the sample.
100
Figure C.6: Typical creep curves obtained in laboratory tests for initially isotropic (black) and optimal anisotropic (blue) ice. The evolution of the LPOs in the case of unconfined vertical compression is also outlined. Capital letters delimit the various deformation stages. AB: “instantaneous”
elastic strain. BC: transient primary creep (ε̈ < 0). CD: minimum secondary creep (ε̈ = 0). DE:
accelerating tertiary creep (ε̈ > 0). EF: steady tertiary creep (ε̈ = 0). For initially isotropic ice
(black), the strain rate first decelerates to a minimum value (ε̇min at εmin ≈ 1%) prior to accelerating to the stable tertiary creep rate (ε̇max at εmax ≈ 10%). In contrast, the optimal anisotropic
ice (blue) decelerates much less and reaches the stable tertiary creep rate already at the end of
secondary creep (εmin = εmax ≈ 1%), without passing through the phase of accelerating tertiary
creep, because it already has fully developed LPOs compatible with the stress regime. (based on
Budd and Jacka, 1989; Treverrow et al., 2012).
Figure C.7: Dynamic recrystallization of polar firn. Dark patches depict the pore space, while dark
lines are grain boundary grooves on the sample surface. Some straight vertical lines are remaining
scratches from microtoming (sublimation of firn samples must be performed with moderation, in
order to preserve the original geometry of the pore space). Scale bars: 1 mm. Left: EDML firn
sample from 40 m depth. Grain boundaries seem straight and smooth, although some subgrain
boundaries (faint lines) are visible, indicating some points of internal stress concentration. Notice also how much pore space exists for accommodating strain incompatibilities. Right: EDML
firn sample from 70 m depth. Grain interaction is much stronger at this depth, causing heterogeneous strains and high internal stresses that manifest themselves in the forms of grain subdivision
(subgrain boundaries), rotation recrystallization (RRX), migration recrystallization (SIBM-O) and
nucleation (SIBM-N); cf. Appendix A.
101
Figure C.8: Dynamic recrystallization in the bubbly-ice zone of various ice cores. In these examples we can identify bulged and cuspidate grain boundaries (SIBM-O; cf. Appendix A), subgrain
boundaries, nucleated grains (SIBM-N) at triple junctions or at grain boundaries as two-sided
grains. Grain boundary pinning by air bubbles or subgrain boundaries is also evident. Scale bars:
1 mm. Top: Two examples from Dome F core, 175 m depth. Centre: Two examples from EDML
core, 304 m depth. Bottom: Two examples from EDC core, 685 m depth.
Figure C.9: Evolution of techniques for displaying the microstructure of natural ice. From left
to right: Seligman’s pencil rubbing on paper (Seligman, 1949, scale bar: 5 cm); thin section
between crossed polarizers (scale bar: 1 cm); digital mosaic trend representation of the azimuth
(color) and colatitude (brightness) of c-axes in a thin section, produced by a modern Automatic
Fabric Analyzer (AFA; see e.g. Wilen et al., 2003; Wilson et al., 2003, scale bar: 1 cm); digital
mosaic image of a thick section consisting of ca. 1500 high-resolution micrographs, produced by
the method of Microstructure Mapping (µSM; see e.g. Kipfstuhl et al., 2006, scale bar: 1 cm).
Notice that the first and last methods do not reveal c-axis orientations, but reproduce the precise
shape of grain boundaries as they meet the ice surface. In contrast, the two intermediate methods
do display c-axis orientations, but show only the depth-integrated shape of grain boundaries across
the thickness of the sample.
Figure C.10: Mosaic image of an Antarctic ice sample (EDML, 2176 m depth) produced via
Microstructure Mapping (µSM; Kipfstuhl et al., 2006). Abbreviation as in Fig. C.3. Grain and
subgrain boundaries appear as dark and grey lines, respectively. Polygonal or dash-shaped objects
are post-drilling relaxation voids called plate-like inclusions (PLI). Blue arrows show examples
of different types of subgrain boundaries: p=parallel to basal planes, n=normal to basal planes
(Nakaya type) and z=zigzag type.
102
Figure C.11: Dynamic recrystallization in the bubble-free-ice zone of various ice cores. In these
examples we can identify bulged and cuspidate grain boundaries (SIBM-O; cf. Appendix A), subgrain boundaries, nucleated grains (SIBM-N) at triple junctions or at grain boundaries as two-sided
grains. Grain boundary pinning by air hydrates or subgrain boundaries is also evident. Top: Two
examples from EDML core, 1885 m depth (scale bars: 1 mm). Notice the pinning by air hydrates
in both images. Whether the isolate pearl-shaped grain in the left image is a true grain island
(cf. Fig. C.5) or just the cross section of a protruded grain is not clear. Centre: Two examples
from EDC core, 2061 m depth (scale bars: left 1 mm, right 2 mm). A large two-sided grain can be
seen in the left image. The fact that it does not show internal structures and is bulging towards a
region rich in dislocation walls and subgrain boundaries suggests that it has nucleated via SIBM-N
(cf. Appendix A). In the right image, complex subgrain boundary formations and severe bulging
and pinning of grain boundaries are evident. Bottom: Grain subdivision, rotation recrystallization
(RRX), migration recrystallization (SIBM-O) and nucleation (SIBM-N) in Antarctic ice samples
from EDC core (left; 2061 m depth) and EDML core (right; 1885 m depth). Scale bars: 2 mm.
In particular, notice the small, two-sided, square-shaped grain at the top of the right image, which
seems to have just nucleated via SIBM-N.
Figure C.12: Microinclusions (tiny black dots) accumulated at a grain boundary of deep Antarctic
ice (EDML core, 2656 m depth; scale bar: 3 mm). By moving the focal point into the sample, the
focused microinclusions reveal the 3D shape of the grain boundary, which penetrates the sample
in a slope towards the bottom of the image.
103
Figure C.13: State space for the dynamic recrystallization diagram. The blue surface DSS represents the steady-state region of constant grain size, for a given strain rate and temperature. Below
this surface there is the zone of grain growth, while above the surface there is the zone of grain
reduction. The small panel on the right illustrates the case of a hypothetical deep ice core: the
green curve describes the increase of mean grain size with depth up to the steady state size DSS .
Further grain-size increase with depth is caused by the higher temperatures at the bottom of the
ice sheet, and is represented by the red line that follows the DSS surface towards higher values of
temperature. For more information, see the description in the main text.
Figure C.14: Cross sections of the dynamic recrystallization diagram shown in Fig. C.13, including
the zones of major influence of different recrystallization mechanisms (cf. Appendix A): rotation
recrystallization (RRX), migration recrystallization without nucleation (SIBM-O), migration recrystallization with nucleation (SIBM-N) and normal grain growth (NGG). The latter occurs only
when ε̇ = 0.
104
2201
Appendix D. TABLES
Table D.1: Possible slip systems in ice. After Hondoh (2009).
slip plane
slip system
basal
(0001) h1120i
primary prismatic
{1100} h1120i
{1100} h0001i
{1100} h1123i
secondary prismatic
{1120} h0001i
primary pyramidal
{1011} h1120i
{1011} h1123i
secondary pyramidal
105
{1122} h1123i
Table D.2: Subgrain boundaries in polar ice. The vectors a and c denote the translation vectors
of the ice unit cell. Dislocation data from Hondoh (2000) and subgrain boundary statistics from
Weikusat et al. (2011).
subgrain boundary
component dislocation
type
misorient. axis
frequency
type
Burgers vector b
length b
basal tilt
a
39%
edge
a = (1/3) < 112̄0 >
4.52 Å
27%
edge
c =< 0001 >
7.36 Å
a
a + c = (1/3) < 112̄3 >
8.63 Å
a = (1/3) < 112̄0 >
4.52 Å
non-basal
tilt
basal twist
c
7%
other
arbitrary
27%
106
screw
diverse and mixed
Figure 1
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Figure 2
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Figure 3
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Figure 7
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Figure 8
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Figure 9
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Figure 10
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Figure 11
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Figure 12
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Figure 13
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Figure 14
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