1979:
between grain structure and internal stress results in high variations in strain across the same length-scale as the crystals themselves. Additionally, individual ice crystals are not isotropic, and typically are not randomly oriented within the material fabric which undergoes dynamic recrystallization. Grain size and fabric orientation are known to influence the creep of glacial ice, but are dynamic properties which also evolve with the stress regime and are not simple to capture in a model.
1978:
In particular, treatment of the ice as a fluid with bulk properties does not represent and may struggle to capture the cascade of mechanisms which allow the ice to deform at the grain scale in solid state. Glacial ice crystals grow on scales of millimeters up to 10 cm, and constant readjustment
1974:
The use of the word "law" in referring to the Glen-Nye model of ice rheology may obscure the complexity of factors which determine the range of viscous ice flow parameter values even within a single glacier, as well as the significant assumptions and simplifications made by the model itself.
903:
1229:
398:
Under the application of sustained force ice will flow as a fluid, and changes to the force applied will result in non-linear changes to the resulting flow. This fluid behavior of ice, which the Glen–Nye flow law is intended to represent, is accommodated within the solid ice by
1107:
1463:
769:
1137:
988:
470:
787:
677:
With these assumptions, the stress and strain rate tensors here are symmetric and have a trace of zero, properties that allow their invariants and squares to be simplified from the general definitions.
663:
999:
143:
544:
393:
665:. Theoretically, this assumption results from ignoring the third principle invariant of the tensors; physically, this means that the strain rate can only change along the same axes as the
2309:
Larour, E.; Rignot, E.; Joughin, I.; Aubry, D. (2005). "Rheology of the Ronne Ice Shelf, Antarctica, inferred from satellite radar interferometry data using an inverse control method".
673:
While incompressibility is an accurate assumption for glacial ice, glacial ice can be anisotropic and in general the strain rate may respond perpendicularly to the principal stress.
1294:
181:
589:
515:
361:
1689:
1329:
2415:
1915:
1510:
1258:
212:
1883:
1544:
1838:
1791:
1314:
1130:
493:
328:
1941:
1961:
1811:
1760:
1736:
1712:
1615:
1592:
1572:
304:
284:
264:
244:
687:
914:
2451:
Paterson, W. S. B. (1983). "Deformation within polar ice sheets: an analysis of the Byrd
Station and Camp Century borehole-tilting measurements".
432:
2119:
Nye, J. F. (1953). "The flow law of ice from measurements in glacier tunnels, laboratory experiments and the
Jungfraufirn borehole experiment".
2272:
Van Der Veen, Cornelis J.; Whillans, I. M. (1990). "Flow laws for glacier ice: comparison of numerical predictions and field measurements".
898:{\displaystyle {\dot {\epsilon }}_{e}^{2}=II_{\boldsymbol {\dot {\epsilon }}}={\frac {1}{2}}{\dot {\epsilon }}_{ij}{\dot {\epsilon }}_{ij}}
91:
1224:{\displaystyle \mu ={\frac {\boldsymbol {\tau }}{2{\dot {\boldsymbol {\epsilon }}}}}={\frac {1}{2}}\tau _{e}{\dot {\epsilon }}_{e}^{-1}}
666:
2404:
1963:
is also stress dependent, and can reflect different microstructural mechanisms facilitating creep at different stress regimes.
1102:{\displaystyle {\boldsymbol {\dot {\epsilon }}}^{2}={\dot {\epsilon }}_{ij}{\dot {\epsilon }}_{ij}=2{\dot {\epsilon }}_{e}^{2}}
613:
2229:"The flow law of ice: A discussion of the assumptions made in glacier theory, their experimental foundations and consequences"
2533:
2366:
Millstein, J.D.; Minchew, B.M.; Pegler, S.S. (2022). "Ice viscosity is more sensitive to stress than commonly assumed".
17:
1885:. Review of research using a variety of methods and field sites have found the range of plausible values to be around
520:
369:
2243:
2052:
339:
1622:
266:
are scalar constants which have been estimated through a combination of theory and measurements. The exponent
1263:
150:
2623:
2567:
549:
498:
344:
1458:{\displaystyle \mu ={\frac {A^{-1/n}}{2{\dot {\epsilon }}_{e}^{(n-1)/n}}}={\frac {\tau _{e}^{(1-n)}}{2A}}}
416:
1631:
1793:
may be dependent on crystal structure, impurities, damage, or other qualities of the ice. Estimates of
2339:
Goldsby, D.; Kohlstedt, D. L. (2001). "Superplastic deformation of ice: Experimental observations".
1849:
681:
The deviatoric stress tensor is related to an effective stress by its second principal invariant:
1888:
1840:, or by comparing measurements of multiple real world glaciers and experiments, or treated as a
1471:
610:
That corresponding components of the two tensors are directly proportional to one another, i.e.
1982:
The Glen-Nye flow law also does not render the full range of ice response to stress, including
1739:
1236:
190:
1966:
Methods to improve estimations of these viscous parameters are an ongoing field of research.
1813:
vary by orders of magnitude and can be derived as a single value from an estimated value for
307:
219:
47:
1862:
1515:
1112:
The viscosity is scalar and cannot be negative (a fluid cannot gain energy as it flows), so
1816:
1769:
1763:
1299:
1115:
478:
313:
8:
2618:
1983:
1920:
764:{\displaystyle \tau _{e}^{2}=II_{\boldsymbol {\tau }}={\frac {1}{2}}\tau _{ij}\tau _{ij}}
604:
400:
75:
71:
35:
2008:
1987:
1946:
1796:
1745:
1721:
1697:
1618:
1600:
1577:
1557:
1316:
can be defined in terms of either the effective strain rate or effective stress alone:
1132:
can be expressed in terms of the invariant effective stress and effective strain rate.
364:
289:
269:
249:
229:
83:
2405:"Discovering the rheology of Antarctic Ice Shelves via physics-informed deep learning"
2464:
2048:
1715:
775:
335:
331:
419:, where the deviatoric stress and strain tensors are related by a viscosity scalar:
2582:
2548:
2511:
2460:
2383:
2375:
2348:
2318:
2289:
2281:
2189:
2162:
2128:
2084:
424:
2180:
Forbes, James D. (1846). "Illustrations of the viscous theory of glacier motion".
2018:
1845:
1617:
has been found empirically to vary with temperature and is often modeled with an
2515:
2379:
2285:
1856:
184:
63:
2552:
2089:
2072:
2612:
983:{\displaystyle {\boldsymbol {\tau }}^{2}=\tau _{ij}\tau _{ij}=2\tau _{e}^{2}}
607:, as volumetric stress is ignored and only the deviatoric stress can do work.
601:, as the single proportionality scalar is the same for all tensor components.
215:
27:
Technical explanation of a rheology model describing the flow of glacial ice
2193:
2166:
2132:
1841:
465:{\displaystyle {\boldsymbol {\tau }}=2\mu {\dot {\boldsymbol {\epsilon }}}}
67:
2586:
2228:
2352:
2322:
79:
55:
2003:
31:
2388:
2294:
1991:
1321:
2013:
598:
404:
51:
223:
59:
2593:
1233:
Here, the Glen–Nye flow law allows us to substitute for either
2502:
Seligman, Gerald (1949). "The growth of the glacier crystal".
2205:
2203:
306:
takes on the units Pa s. The Glen–Nye flow law simplifies the
2534:"The microstructure of polar ice. Part II: State of the art"
2483:
2471:
2432:
2403:
Wang, Yongji; Lai, Ching-Yao; Cowen-Breen, Charlie (2022).
2200:
2532:
Faria, Sérgio H.; Weikusat, Ilka; Azuma, Nobuhiko (2014).
658:{\displaystyle \tau _{ij}\propto {\dot {\epsilon }}_{ij}}
2308:
2153:
Glen, J. W. (1955). "The creep of polycrystalline ice".
1917:
with the most commonly used assumption to be a constant
2568:"Dynamic Recrystallization of Ice in Polar Ice Sheets"
2565:
2365:
2038:
2036:
2034:
1949:
1923:
1891:
1865:
1819:
1799:
1772:
1748:
1724:
1700:
1634:
1603:
1580:
1560:
1518:
1474:
1332:
1302:
1266:
1239:
1140:
1118:
1002:
917:
790:
690:
616:
552:
523:
501:
481:
435:
372:
347:
316:
292:
272:
252:
232:
193:
153:
138:{\displaystyle {\dot {\epsilon }}_{e}=A\tau _{e}^{n}}
94:
2271:
1554:
The Glen–Nye rheology model defines two parameters,
2531:
2402:
2031:
1955:
1935:
1909:
1877:
1832:
1805:
1785:
1754:
1730:
1706:
1683:
1609:
1586:
1566:
1538:
1504:
1457:
1308:
1288:
1252:
1223:
1124:
1101:
982:
897:
781:The same is defined for an effective strain rate:
763:
657:
583:
538:
509:
487:
464:
387:
355:
322:
298:
278:
258:
238:
206:
175:
137:
2610:
2338:
2267:
2265:
2263:
546:is the strain rate tensor. In some derivations,
539:{\displaystyle {\boldsymbol {\dot {\epsilon }}}}
388:{\displaystyle {\boldsymbol {\dot {\epsilon }}}}
2182:Philosophical Transactions of the Royal Society
58:. The Glen–Nye flow law treats ice as a purely
2599:
2489:
2477:
2438:
2334:
2332:
2209:
2042:
2359:
2260:
594:This construction makes several assumptions:
2527:
2525:
2495:
2444:
2341:Journal of Geophysical Research: Solid Earth
415:The constitutive relation is developed as a
2329:
2148:
2146:
2144:
2142:
2066:
2064:
2559:
2222:
2220:
2218:
778:implies summation over repeated indices.
2522:
2387:
2302:
2293:
2173:
2114:
2112:
2110:
2108:
2106:
2104:
2102:
2100:
2088:
2501:
2450:
2396:
2139:
2061:
2047:(4 ed.). Elsevier. p. 55, 60.
2215:
2073:"Experiments on the deformation of ice"
2043:Cuffey, K.; Paterson, W. S. B. (2010).
1159:
1149:
1007:
920:
908:From this form, we can recognize that:
828:
717:
527:
503:
453:
437:
410:
376:
349:
14:
2611:
2236:IUGG/IAHS Symp. Of Chamonix, IAHS Publ
2179:
2097:
1289:{\displaystyle {\dot {\epsilon }}_{e}}
286:is dimensionless, and the rate factor
176:{\displaystyle {\dot {\epsilon }}_{e}}
1852:for ice flow at a specific location.
584:{\displaystyle \lambda =(2\mu )^{-1}}
517:is the deviatoric stress tensor, and
510:{\displaystyle {\boldsymbol {\tau }}}
356:{\displaystyle {\boldsymbol {\tau }}}
2368:Communications Earth and Environment
2226:
2152:
2070:
403:, and is a dominant mode of glacial
2453:Cold Regions Science and Technology
2118:
1549:
74:, with a viscosity determined by a
24:
2155:Proceedings of the Royal Society A
2121:Proceedings of the Royal Society A
1994:), and transient phases of creep.
1855:Viscous ice flow is an example of
1684:{\displaystyle A=A_{0}e^{(-Q/RT)}}
495:is the viscosity (units of Pa s),
25:
2635:
1844:inferred from observations by a
1010:
831:
591:(units of Pa s) is substituted.
530:
379:
2566:P. Duval, O. Castelnau (1995).
2421:from the original on 2023-06-15
2249:from the original on 2024-01-27
1623:temperature dependence of creep
50:widely used as a model for the
1969:
1676:
1656:
1440:
1428:
1397:
1385:
569:
559:
13:
1:
2541:Journal of Structural Geology
2024:
2465:10.1016/0165-232X(83)90007-1
2311:Geophysical Research Letters
1546:) is sometimes substituted.
218:) are related to the second
46:, is an empirically derived
7:
1997:
1910:{\displaystyle 2<n<4}
417:generalized Newtonian fluid
10:
2640:
2600:Cuffey & Paterson 2010
2516:10.3189/002214349793702601
2490:Cuffey & Paterson 2010
2478:Cuffey & Paterson 2010
2439:Cuffey & Paterson 2010
2380:10.1038/s43247-022-00385-x
2286:10.3189/002214390793701372
2210:Cuffey & Paterson 2010
1505:{\displaystyle B=A^{-1/n}}
147:The effective strain rate
2553:10.1016/j.jsg.2013.11.003
2090:10.3189/S0022143000034067
1253:{\displaystyle \tau _{e}}
334:, which is determined by
310:to a single scalar value
207:{\displaystyle \tau _{e}}
1943:. However, the value of
340:deviatoric stress tensor
2045:The Physics of Glaciers
1859:, which corresponds to
187:) and effective stress
2575:Journal de Physique IV
2194:10.1098/rstl.1846.0013
2167:10.1098/rspa.1955.0066
2133:10.1098/rspa.1953.0161
1957:
1937:
1911:
1879:
1878:{\displaystyle n>1}
1834:
1807:
1787:
1756:
1740:universal gas constant
1732:
1708:
1685:
1611:
1588:
1568:
1540:
1539:{\displaystyle ^{1/n}}
1506:
1459:
1310:
1290:
1254:
1225:
1126:
1103:
984:
899:
765:
659:
585:
540:
511:
489:
466:
389:
357:
324:
300:
280:
260:
240:
208:
177:
139:
42:, also referred to as
2504:Journal of Glaciology
2274:Journal of Glaciology
2077:Journal of Glaciology
1958:
1938:
1912:
1880:
1835:
1833:{\displaystyle A_{0}}
1808:
1788:
1786:{\displaystyle A_{0}}
1757:
1733:
1709:
1686:
1612:
1589:
1569:
1541:
1507:
1460:
1311:
1291:
1255:
1226:
1127:
1104:
985:
900:
766:
660:
586:
541:
512:
490:
467:
390:
358:
325:
308:viscous stress tensor
301:
281:
261:
241:
209:
178:
140:
48:constitutive relation
2353:10.1029/2000JB900336
2323:10.1029/2004GL021693
2227:Glen, J. W. (1958).
2071:Glen, J. W. (1952).
1947:
1921:
1889:
1863:
1817:
1797:
1770:
1764:absolute temperature
1746:
1722:
1698:
1632:
1601:
1578:
1558:
1516:
1472:
1330:
1309:{\displaystyle \mu }
1300:
1264:
1237:
1138:
1125:{\displaystyle \mu }
1116:
1000:
915:
788:
688:
614:
550:
521:
499:
488:{\displaystyle \mu }
479:
433:
411:Viscosity definition
370:
345:
323:{\displaystyle \mu }
314:
290:
270:
250:
230:
222:of their respective
220:principle invariants
191:
151:
92:
18:Glen's flow law
2624:Continuum mechanics
2587:10.1051/jp4:1995317
1984:elastic deformation
1936:{\displaystyle n=3}
1846:numerical inversion
1444:
1409:
1220:
1098:
979:
814:
705:
134:
72:non-Newtonian fluid
36:continuum mechanics
2009:Ice sheet dynamics
1988:fracture mechanics
1953:
1933:
1907:
1875:
1830:
1803:
1783:
1752:
1728:
1704:
1681:
1619:Arrhenius relation
1607:
1584:
1564:
1536:
1502:
1455:
1418:
1366:
1306:
1286:
1250:
1221:
1194:
1122:
1099:
1075:
980:
965:
895:
791:
761:
691:
667:principal stresses
655:
581:
536:
507:
485:
462:
385:
365:strain rate tensor
353:
320:
296:
276:
256:
236:
204:
173:
135:
120:
2602:, p. 66, 72.
2161:(1175): 519–538.
2127:(1139): 477–489.
1956:{\displaystyle n}
1850:momentum equation
1806:{\displaystyle A}
1755:{\displaystyle T}
1731:{\displaystyle R}
1716:activation energy
1707:{\displaystyle Q}
1610:{\displaystyle A}
1587:{\displaystyle n}
1567:{\displaystyle A}
1453:
1411:
1376:
1277:
1204:
1182:
1169:
1165:
1085:
1057:
1035:
1013:
883:
861:
849:
834:
801:
776:Einstein notation
733:
643:
605:Incompressibility
533:
459:
382:
336:tensor invariants
332:dynamic viscosity
299:{\displaystyle A}
279:{\displaystyle n}
259:{\displaystyle n}
239:{\displaystyle A}
226:. The parameters
164:
105:
78:relation between
40:Glen–Nye flow law
16:(Redirected from
2631:
2603:
2597:
2591:
2590:
2572:
2563:
2557:
2556:
2538:
2529:
2520:
2519:
2499:
2493:
2487:
2481:
2475:
2469:
2468:
2448:
2442:
2436:
2430:
2429:
2427:
2426:
2420:
2409:
2400:
2394:
2393:
2391:
2363:
2357:
2356:
2336:
2327:
2326:
2306:
2300:
2299:
2297:
2280:(124): 324–339.
2269:
2258:
2257:
2255:
2254:
2248:
2233:
2224:
2213:
2207:
2198:
2197:
2177:
2171:
2170:
2150:
2137:
2136:
2116:
2095:
2094:
2092:
2068:
2059:
2058:
2040:
1962:
1960:
1959:
1954:
1942:
1940:
1939:
1934:
1916:
1914:
1913:
1908:
1884:
1882:
1881:
1876:
1839:
1837:
1836:
1831:
1829:
1828:
1812:
1810:
1809:
1804:
1792:
1790:
1789:
1784:
1782:
1781:
1766:. The prefactor
1761:
1759:
1758:
1753:
1737:
1735:
1734:
1729:
1713:
1711:
1710:
1705:
1690:
1688:
1687:
1682:
1680:
1679:
1669:
1650:
1649:
1616:
1614:
1613:
1608:
1597:The rate factor
1593:
1591:
1590:
1585:
1573:
1571:
1570:
1565:
1550:Parameter values
1545:
1543:
1542:
1537:
1535:
1534:
1530:
1511:
1509:
1508:
1503:
1501:
1500:
1496:
1464:
1462:
1461:
1456:
1454:
1452:
1443:
1426:
1417:
1412:
1410:
1408:
1404:
1383:
1378:
1377:
1369:
1361:
1360:
1356:
1340:
1315:
1313:
1312:
1307:
1295:
1293:
1292:
1287:
1285:
1284:
1279:
1278:
1270:
1259:
1257:
1256:
1251:
1249:
1248:
1230:
1228:
1227:
1222:
1219:
1211:
1206:
1205:
1197:
1193:
1192:
1183:
1175:
1170:
1168:
1167:
1166:
1158:
1148:
1131:
1129:
1128:
1123:
1108:
1106:
1105:
1100:
1097:
1092:
1087:
1086:
1078:
1068:
1067:
1059:
1058:
1050:
1046:
1045:
1037:
1036:
1028:
1021:
1020:
1015:
1014:
1006:
989:
987:
986:
981:
978:
973:
958:
957:
945:
944:
929:
928:
923:
904:
902:
901:
896:
894:
893:
885:
884:
876:
872:
871:
863:
862:
854:
850:
842:
837:
836:
835:
827:
813:
808:
803:
802:
794:
770:
768:
767:
762:
760:
759:
747:
746:
734:
726:
721:
720:
704:
699:
664:
662:
661:
656:
654:
653:
645:
644:
636:
629:
628:
590:
588:
587:
582:
580:
579:
545:
543:
542:
537:
535:
534:
526:
516:
514:
513:
508:
506:
494:
492:
491:
486:
471:
469:
468:
463:
461:
460:
452:
440:
425:constitutive law
394:
392:
391:
386:
384:
383:
375:
362:
360:
359:
354:
352:
329:
327:
326:
321:
305:
303:
302:
297:
285:
283:
282:
277:
265:
263:
262:
257:
245:
243:
242:
237:
213:
211:
210:
205:
203:
202:
182:
180:
179:
174:
172:
171:
166:
165:
157:
144:
142:
141:
136:
133:
128:
113:
112:
107:
106:
98:
21:
2639:
2638:
2634:
2633:
2632:
2630:
2629:
2628:
2609:
2608:
2607:
2606:
2598:
2594:
2581:(C3): 197–205.
2570:
2564:
2560:
2536:
2530:
2523:
2500:
2496:
2488:
2484:
2476:
2472:
2449:
2445:
2437:
2433:
2424:
2422:
2418:
2407:
2401:
2397:
2364:
2360:
2347:: 11017–11030.
2337:
2330:
2307:
2303:
2270:
2261:
2252:
2250:
2246:
2231:
2225:
2216:
2208:
2201:
2178:
2174:
2151:
2140:
2117:
2098:
2083:(12): 111–114.
2069:
2062:
2055:
2041:
2032:
2027:
2019:Ice-sheet model
2000:
1972:
1948:
1945:
1944:
1922:
1919:
1918:
1890:
1887:
1886:
1864:
1861:
1860:
1824:
1820:
1818:
1815:
1814:
1798:
1795:
1794:
1777:
1773:
1771:
1768:
1767:
1747:
1744:
1743:
1723:
1720:
1719:
1699:
1696:
1695:
1665:
1655:
1651:
1645:
1641:
1633:
1630:
1629:
1621:describing the
1602:
1599:
1598:
1579:
1576:
1575:
1559:
1556:
1555:
1552:
1526:
1522:
1519:
1517:
1514:
1513:
1492:
1485:
1481:
1473:
1470:
1469:
1466:
1445:
1427:
1422:
1416:
1400:
1384:
1379:
1368:
1367:
1362:
1352:
1345:
1341:
1339:
1331:
1328:
1327:
1301:
1298:
1297:
1280:
1269:
1268:
1267:
1265:
1262:
1261:
1244:
1240:
1238:
1235:
1234:
1212:
1207:
1196:
1195:
1188:
1184:
1174:
1157:
1156:
1152:
1147:
1139:
1136:
1135:
1117:
1114:
1113:
1093:
1088:
1077:
1076:
1060:
1049:
1048:
1047:
1038:
1027:
1026:
1025:
1016:
1005:
1004:
1003:
1001:
998:
997:
974:
969:
950:
946:
937:
933:
924:
919:
918:
916:
913:
912:
886:
875:
874:
873:
864:
853:
852:
851:
841:
826:
825:
821:
809:
804:
793:
792:
789:
786:
785:
752:
748:
739:
735:
725:
716:
712:
700:
695:
689:
686:
685:
646:
635:
634:
633:
621:
617:
615:
612:
611:
572:
568:
551:
548:
547:
525:
524:
522:
519:
518:
502:
500:
497:
496:
480:
477:
476:
473:
451:
450:
436:
434:
431:
430:
413:
374:
373:
371:
368:
367:
348:
346:
343:
342:
315:
312:
311:
291:
288:
287:
271:
268:
267:
251:
248:
247:
231:
228:
227:
198:
194:
192:
189:
188:
167:
156:
155:
154:
152:
149:
148:
129:
124:
108:
97:
96:
95:
93:
90:
89:
44:Glen's flow law
30:In theoretical
28:
23:
22:
15:
12:
11:
5:
2637:
2627:
2626:
2621:
2605:
2604:
2592:
2558:
2521:
2510:(5): 254–268.
2494:
2482:
2470:
2459:(2): 165–179.
2443:
2431:
2395:
2358:
2328:
2301:
2259:
2214:
2199:
2172:
2138:
2096:
2060:
2053:
2029:
2028:
2026:
2023:
2022:
2021:
2016:
2011:
2006:
1999:
1996:
1971:
1968:
1952:
1932:
1929:
1926:
1906:
1903:
1900:
1897:
1894:
1874:
1871:
1868:
1857:shear thinning
1827:
1823:
1802:
1780:
1776:
1751:
1727:
1703:
1692:
1691:
1678:
1675:
1672:
1668:
1664:
1661:
1658:
1654:
1648:
1644:
1640:
1637:
1606:
1583:
1563:
1551:
1548:
1533:
1529:
1525:
1521:
1512:(units of Pa s
1499:
1495:
1491:
1488:
1484:
1480:
1477:
1451:
1448:
1442:
1439:
1436:
1433:
1430:
1425:
1421:
1415:
1407:
1403:
1399:
1396:
1393:
1390:
1387:
1382:
1375:
1372:
1365:
1359:
1355:
1351:
1348:
1344:
1338:
1335:
1318:
1305:
1283:
1276:
1273:
1247:
1243:
1218:
1215:
1210:
1203:
1200:
1191:
1187:
1181:
1178:
1173:
1164:
1161:
1155:
1151:
1146:
1143:
1121:
1110:
1109:
1096:
1091:
1084:
1081:
1074:
1071:
1066:
1063:
1056:
1053:
1044:
1041:
1034:
1031:
1024:
1019:
1012:
1009:
991:
990:
977:
972:
968:
964:
961:
956:
953:
949:
943:
940:
936:
932:
927:
922:
906:
905:
892:
889:
882:
879:
870:
867:
860:
857:
848:
845:
840:
833:
830:
824:
820:
817:
812:
807:
800:
797:
772:
771:
758:
755:
751:
745:
742:
738:
732:
729:
724:
719:
715:
711:
708:
703:
698:
694:
671:
670:
652:
649:
642:
639:
632:
627:
624:
620:
608:
602:
578:
575:
571:
567:
564:
561:
558:
555:
532:
529:
505:
484:
458:
455:
449:
446:
443:
439:
421:
412:
409:
381:
378:
351:
319:
295:
275:
255:
235:
201:
197:
170:
163:
160:
132:
127:
123:
119:
116:
111:
104:
101:
64:incompressible
26:
9:
6:
4:
3:
2:
2636:
2625:
2622:
2620:
2617:
2616:
2614:
2601:
2596:
2588:
2584:
2580:
2576:
2569:
2562:
2554:
2550:
2546:
2542:
2535:
2528:
2526:
2517:
2513:
2509:
2505:
2498:
2492:, p. 69.
2491:
2486:
2480:, p. 51.
2479:
2474:
2466:
2462:
2458:
2454:
2447:
2441:, p. 55.
2440:
2435:
2417:
2413:
2406:
2399:
2390:
2385:
2381:
2377:
2373:
2369:
2362:
2354:
2350:
2346:
2342:
2335:
2333:
2324:
2320:
2316:
2312:
2305:
2296:
2291:
2287:
2283:
2279:
2275:
2268:
2266:
2264:
2245:
2241:
2237:
2230:
2223:
2221:
2219:
2212:, p. 29.
2211:
2206:
2204:
2195:
2191:
2187:
2183:
2176:
2168:
2164:
2160:
2156:
2149:
2147:
2145:
2143:
2134:
2130:
2126:
2122:
2115:
2113:
2111:
2109:
2107:
2105:
2103:
2101:
2091:
2086:
2082:
2078:
2074:
2067:
2065:
2056:
2054:9780123694614
2050:
2046:
2039:
2037:
2035:
2030:
2020:
2017:
2015:
2012:
2010:
2007:
2005:
2002:
2001:
1995:
1993:
1989:
1985:
1980:
1976:
1967:
1964:
1950:
1930:
1927:
1924:
1904:
1901:
1898:
1895:
1892:
1872:
1869:
1866:
1858:
1853:
1851:
1847:
1843:
1825:
1821:
1800:
1778:
1774:
1765:
1749:
1741:
1725:
1717:
1701:
1673:
1670:
1666:
1662:
1659:
1652:
1646:
1642:
1638:
1635:
1628:
1627:
1626:
1624:
1620:
1604:
1595:
1581:
1561:
1547:
1531:
1527:
1523:
1520:
1497:
1493:
1489:
1486:
1482:
1478:
1475:
1465:
1449:
1446:
1437:
1434:
1431:
1423:
1419:
1413:
1405:
1401:
1394:
1391:
1388:
1380:
1373:
1370:
1363:
1357:
1353:
1349:
1346:
1342:
1336:
1333:
1325:
1323:
1317:
1303:
1281:
1274:
1271:
1245:
1241:
1231:
1216:
1213:
1208:
1201:
1198:
1189:
1185:
1179:
1176:
1171:
1162:
1153:
1144:
1141:
1133:
1119:
1094:
1089:
1082:
1079:
1072:
1069:
1064:
1061:
1054:
1051:
1042:
1039:
1032:
1029:
1022:
1017:
996:
995:
994:
975:
970:
966:
962:
959:
954:
951:
947:
941:
938:
934:
930:
925:
911:
910:
909:
890:
887:
880:
877:
868:
865:
858:
855:
846:
843:
838:
822:
818:
815:
810:
805:
798:
795:
784:
783:
782:
779:
777:
756:
753:
749:
743:
740:
736:
730:
727:
722:
713:
709:
706:
701:
696:
692:
684:
683:
682:
679:
678:
674:
668:
650:
647:
640:
637:
630:
625:
622:
618:
609:
606:
603:
600:
597:
596:
595:
592:
576:
573:
565:
562:
556:
553:
482:
472:
456:
447:
444:
441:
428:
426:
420:
418:
408:
406:
402:
396:
366:
341:
337:
333:
317:
309:
293:
273:
253:
233:
225:
221:
217:
199:
195:
186:
168:
161:
158:
145:
130:
125:
121:
117:
114:
109:
102:
99:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
2595:
2578:
2574:
2561:
2544:
2540:
2507:
2503:
2497:
2485:
2473:
2456:
2452:
2446:
2434:
2423:. Retrieved
2411:
2398:
2371:
2367:
2361:
2344:
2340:
2314:
2310:
2304:
2277:
2273:
2251:. Retrieved
2239:
2235:
2185:
2181:
2175:
2158:
2154:
2124:
2120:
2080:
2076:
2044:
1981:
1977:
1973:
1965:
1854:
1842:scalar field
1693:
1596:
1553:
1467:
1326:
1319:
1232:
1134:
1111:
992:
907:
780:
773:
680:
676:
675:
672:
593:
474:
429:
422:
414:
397:
146:
88:
43:
39:
29:
2242:: 171–183.
2188:: 143–210.
1970:Limitations
80:strain rate
56:glacial ice
2619:Glaciology
2613:Categories
2425:2024-01-27
2412:(Preprint)
2389:1912/29119
2295:1808/17348
2253:2024-01-27
2025:References
2004:Glaciology
214:(units of
183:(units of
32:glaciology
2547:: 21–49.
1992:crevasses
1660:−
1487:−
1435:−
1420:τ
1392:−
1374:˙
1371:ϵ
1347:−
1334:μ
1322:viscosity
1320:Glen–Nye
1304:μ
1275:˙
1272:ϵ
1242:τ
1214:−
1202:˙
1199:ϵ
1186:τ
1163:˙
1160:ϵ
1150:τ
1142:μ
1120:μ
1083:˙
1080:ϵ
1055:˙
1052:ϵ
1033:˙
1030:ϵ
1011:˙
1008:ϵ
967:τ
948:τ
935:τ
921:τ
881:˙
878:ϵ
859:˙
856:ϵ
832:˙
829:ϵ
799:˙
796:ϵ
750:τ
737:τ
718:τ
693:τ
641:˙
638:ϵ
631:∝
619:τ
574:−
566:μ
554:λ
531:˙
528:ϵ
504:τ
483:μ
457:˙
454:ϵ
448:μ
438:τ
423:Glen–Nye
380:˙
377:ϵ
350:τ
318:μ
196:τ
162:˙
159:ϵ
122:τ
103:˙
100:ϵ
76:power law
68:isotropic
2416:Archived
2244:Archived
2014:Rheology
1998:See also
599:Isotropy
405:ice flow
363:and the
52:rheology
1848:of the
1762:is the
1738:is the
1714:is the
338:of the
224:tensors
60:viscous
2374:(57).
2051:
1990:(i.e.
1742:, and
1694:where
1468:where
1324:of ice
1296:, and
774:where
475:where
427:of ice
330:, the
84:stress
38:, the
2571:(PDF)
2537:(PDF)
2419:(PDF)
2408:(PDF)
2317:(5).
2247:(PDF)
2232:(PDF)
993:and
401:creep
2049:ISBN
1902:<
1896:<
1870:>
1574:and
246:and
82:and
34:and
2583:doi
2549:doi
2512:doi
2461:doi
2384:hdl
2376:doi
2349:doi
2345:106
2319:doi
2290:hdl
2282:doi
2190:doi
2186:136
2163:doi
2159:228
2129:doi
2125:219
2085:doi
1625::
1594:.
1260:or
86::
54:of
2615::
2577:.
2573:.
2545:61
2543:.
2539:.
2524:^
2506:.
2455:.
2414:.
2410:.
2382:.
2370:.
2343:.
2331:^
2315:32
2313:.
2288:.
2278:36
2276:.
2262:^
2240:47
2238:.
2234:.
2217:^
2202:^
2184:.
2157:.
2141:^
2123:.
2099:^
2079:.
2075:.
2063:^
2033:^
1986:,
1718:,
407:.
395:.
216:Pa
70:,
66:,
62:,
2589:.
2585::
2579:5
2555:.
2551::
2518:.
2514::
2508:1
2467:.
2463::
2457:8
2428:.
2392:.
2386::
2378::
2372:3
2355:.
2351::
2325:.
2321::
2298:.
2292::
2284::
2256:.
2196:.
2192::
2169:.
2165::
2135:.
2131::
2093:.
2087::
2081:2
2057:.
1951:n
1931:3
1928:=
1925:n
1905:4
1899:n
1893:2
1873:1
1867:n
1826:0
1822:A
1801:A
1779:0
1775:A
1750:T
1726:R
1702:Q
1677:)
1674:T
1671:R
1667:/
1663:Q
1657:(
1653:e
1647:0
1643:A
1639:=
1636:A
1605:A
1582:n
1562:A
1532:n
1528:/
1524:1
1498:n
1494:/
1490:1
1483:A
1479:=
1476:B
1450:A
1447:2
1441:)
1438:n
1432:1
1429:(
1424:e
1414:=
1406:n
1402:/
1398:)
1395:1
1389:n
1386:(
1381:e
1364:2
1358:n
1354:/
1350:1
1343:A
1337:=
1282:e
1246:e
1217:1
1209:e
1190:e
1180:2
1177:1
1172:=
1154:2
1145:=
1095:2
1090:e
1073:2
1070:=
1065:j
1062:i
1043:j
1040:i
1023:=
1018:2
976:2
971:e
963:2
960:=
955:j
952:i
942:j
939:i
931:=
926:2
891:j
888:i
869:j
866:i
847:2
844:1
839:=
823:I
819:I
816:=
811:2
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757:j
754:i
744:j
741:i
731:2
728:1
723:=
714:I
710:I
707:=
702:2
697:e
669:.
651:j
648:i
626:j
623:i
577:1
570:)
563:2
560:(
557:=
445:2
442:=
294:A
274:n
254:n
234:A
200:e
185:s
169:e
131:n
126:e
118:A
115:=
110:e
20:)
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