1. Mohan B. ChandMs By Research in Glaciology,Kathmandu University

2. Introduction Temperature distribution inglaciers and ice sheets deserveattention for both its intrinsicinterest and its relation to otherprocesses. Past variations of surfacetemperature The deformation rate of icedepends sensitively on temperature 3. A glacier, previously frozen to itsbed, starts to slip when its basewarms to the melting point; theterminus then advances.Velocity of seismic waves and theabsorption of radio waves alsovary with temperature 4. Introduction contWhat controls the temperature distribution? Geothermal heat and frictional heating from basal slip warmor melt the base Ice deformation and refreezing of melt water warm theinterior Heat is transferred by conduction, ice movement (advection),and , in some cases, water flow. Geothermal, frictional, and deformational heat sourcestypically concentrate at or near the base of a glacier. 5. Glacier is a thermal insulator and comparatively warm near bed However, glaciers thrive in a diverse range of climaticconditions. Consequently, four main types of temperaturedistribution occur:1) all the ice is below melting point (cold)2) only the bed reaches melting point (cold)3) a basal layer of finite thickness is at melting point(polythermal)4) all the ice is at melting point except for a surfacelayer, about 15 m thick, where temperature fluctuatewith the seasons (temperate) 6. Main points Discuss the general problem of heat transfer in glaciers Thermal properties of snow and ice, the factors controllingtemperatures near the surface, and the characteristics oftemperate glaciers. 7. Thermal parameters of ice and snow Glacier ice is unusually pure Thermal properties at subfreezing temperature are as follows.Yen (1981)The relation of specific heat capacity and thermalconductivity (Kt) of pure ice to the temperature T in Kelvin:C = 152.5 + 7.122T(9.1)Kt=9.828exp(-5.7*10-3T).(9.2) 8. Thermal conductivity also depends on density but the data arewidely scattered, especially for snow with densities less than500 kgm-3 because of snow texture.The Van Dusen (1929) formula gives a lower limit in most casesfor different densities.Kt = 2.1 *10-2 +4.2 *10-4 +2.2 *10-93 .(9.3)Low density hoar layers, however, can be even less conductive(Sturn and Johnson 1992). The Schwerdtfeger (1963) formulagives an upper limit.(9.4) 9. The specific heat capacity of dry snow and ice does not varieswith the density because the heat needed to warm the air andvapor between the grains is negligible. Thermal diffusivity, T,can be calculated for any density and temperature usingT = kT/cKt thermal conductivityC specific heat capacity 10. 9.3 Temperature of surface layersLong period changes in surface temperature can be analyzedby heat conduction theory. Fourierss law of heat conductionstates that the heat flux q at a point in a medium is proportionalto the negative temperature gradient T/z, with z measured inthe direction of the temperature variationThusWhere kT = denotes the thermal conductivity..(9.5) 11. Fig: Variation of temperatureover 10 months at South Pole, atsix depths in the near-surfacefirn. At the end of theexperiment, the depths of the sixthermistors (T1-T6) were about0.2, 0.4, 0.6, 0.8, 1.0, and 1.2meters. Due to the snowfall, thedepths increased by about 0.4 mover the year, beginning aroundhour 3500. temperaturevariability decreases with depth. 12. By the definition of specific heat capacity c, the changein heat in unit time equals c[T /t ]z where isdensity and t the time. It follows that, for constant kT, 13. The following table lists some value of propagation of conductionof a cyclical variation in surface temperature 14. Figure 2: Theoretical seasonal cycle of firntemperatures in central GreenlandFigure show the seasonalvariations in near- surfacetemperatures. Field observationsconfirm that seasonal variationsare undectabale below a depth ofabout 20m. 15. In summer, in most places, heat conduction plays only a minor partin heat transfer through the surface layers. Except in the interiors ofGreenland and Antarctica or on very high mountains the surfacemelts in summer and might receive rain. Percolating water refreezesat a depth, and produce enough heat to raise temperature of 160 g ofsnow of firn by 10C. This process significantly warms the layersnear the surface. 16. Figure 9.3: warming of firn bylatent heat of refreezing melt water,at 1600 m elevation on Greenlandice sheet.The firn at 3 to 4 m belowthe surface warms rapidly,by 80C in just 6 days. Thisprocess eliminates thewinters cold wave muchmore quickly than wouldhave been possible by heatconduction alone. 17. At 10 or 15 m, beneath the zone of seasonal variation, equals themean annual air temperature at many cold, dry sites where themaximum air temperature never rises to 00C, but not true ingeneral. Comparisons of air and 10 m firn temperature at polar drysnow locations indicates that , at most sites, the two differ by lessthan 20C,. Firn temperature are colder on average by only 0.70C.Table: Mean annual airtemperature andtemperatureat 10 m depth in dry-snowareas 18. Table: Temperature atdifferent pointsin same glacierIn other cases, the air and firntemperatures differ substantially.Refreezing of surface water warms thefirn and raises its temperatures abovethe mean annual value for the air. Intwo glaciers in the alps, found a00c at 30 m depth,Mean annual air temperature at bothsites was -7 to -8 0c. Such warming isnot effective in the ablation zone,where most surface water escapesfrom the glacier. This explains why, inpolar glaciers, near surfacetemperatures can be lower in ablationzones than accumulation zones,despite their altitude. 19. 9.4 Temperate Glaciers9.4.1 Ice TemperatureTemperate ice is a complex material consisting of ice, water, air,salts, and carbon dioxide. The water content is between 0.1% and2%. Air bubbles typically comprise a few percent of the icevolume. The bulk impurity content is only of the order 10-7 byweight. Temperate Glacier is the glacier of end of the meltingseason.If temperate ice were a mixture of pure ice and pure water,its temperature T at absolute pressure P would beT = T0 P(9.9),where = 7.42 *10-8 Kpa-1 specifies the rate of change ofmelting point with pressure and T0 = 273.16 K=0.01oC denotesthe triple- point temperature of water. 20. Dissolved air at atmospheric pressure lowers the equilibriumtemperature by 2.4 *10-3 oC moreover , solubility increases inproportion to pressure. Equation 9.9 should be therefore bereplaced by T = To P.(9.10)where = 9.8 108K Pa1 or 8.7 104K m1 of ice.Impurities depress the melting melt in proportion to the soluteconcentration in the liquid inclusions. For small concentrations, thisshifts the temperature by sCs/W, for a fractional water contentW by weight, and salt concentration Cs in mol kg1, and s =1.86K kg mol1. Thus the equilibrium temperature of the ice is.(9.11)Curvature of the ice-water interface also influences equilibriumtemperature but by a small amount compared to impurities. 21. Impurities greatly increase the effective specific heat capacity ofice near the melting point. Only part of the heat added to the iceraises the temperature some of it must melt ice to dilute theliquid. The effective specific heat capacity c thus exceeds thevalue for pure ice, c(9.12)Where L denotes the specific latent heat of fusion, and W isrelated to T by eq. 9.11. for typical salt concentrations and T = -0.01oC,, c can be about one hundred times c. Air bubbles havethe same effect as salts. 22. Figure 9.4: Variation of temperature with depth in the temperate Blue Glacier(solid line) and variation expected for pure ice in equilibrium with (a) pure waterand (b) air-saturated water. Impurities depress the melting point in the glacier.Redrawn from Harrison (1975). 23. Harison (1972) proposed a precise definition of temperateice, based on the concept of effective heat capacity:Temperate ice is ice whose effective bulk heat capacity issignificantly greater than that of pure ice monocrystal. 24. 9.4.2 Origin and Effect of WaterTemperate glacier should have heat source and sinks, which isprovided by the freezing of small quantities of water and meltingof small quantities of ice.Accumulation zone- high pressure from snowfall and meltingtemperature decreasesAblation zone- low pressure make melting temperature high 25. In temperate glacier insignificant conduction of heat because ofsmall vertical temperature gradient. The direction of thistemperature gradients prevents heat conduction from the bed.Deformation produces some heat in lower half of the ice issufficient to maintain ice at melting point.The necessary heat must therefore be supplied by the freezing ofthe small quantities of water in the ice.Conti 26. In temperate glacier, water is about 1% of the volume fromvarious source: percolation, conduit flow from the surface, icemelted from deformational hating, melting induce by thepressure changes , and pocket s of water trapped when the iceformed. Near the base of the temperate glacier, shear stress isabout 100 Kpa and the heat of deformation could melt about 1%of the ice in 100 year. 27. Paterson concluded that most of the water trapped when firnbecame ice. Because of low permeability it can not drainawayLarge quantities of water move through temperate glaciers infractures, tubes, and other large passageways. This watercarries some heat into the glacier and portion of it might freeze.It occupy only a small volume of the glacier, their effect ontemperatures is often assumed to be negligible. 28. 9.4.3 Distribution of Temperate GlaciersGlacier of temperate region are temperate glaciersTo be temperate glacier, the previous winters cold wave mustbe eliminated by the end of summerRefreezing of percolating melt water can accomplish thisrapidly in the accumulation zone . In the ablation zone,however, the ice is almost impermeable to water. Paterson(1972b) discussed other processes. 29. Heat conduction is slow and the amount of heat is limitedbecause the surface temperature cannot rise above 0 C. Solarradiation does not penetrate deeply enough. Most of the heat atthe surface warms the surface ice to 0 C and then melts it.Whether all the ice attains 0 C by the end of the summertherefore depends largely on the amount of ablation relative tothe depth of penetration of the cold wave, a function of wintertemperatures and snowfall. In many glaciers the ice likelyremains below melting point in the region of slow ablationimmediately below the equilibrium line. 30. The most likely place to find temperate glaciers is a regionwith a maritime temperate climate where intense summermelting follows heavy winter snowfallsIn polar region, some glaciers may have temperateaccumulation zones as a result of percolating melt water, whilethe ablation zones are cold. 31. 9.4 Steady-State Temperature DistributionsThe temperature distribution in a glacier is neverin a steady state, but heat flows rapidly reduceslarge deviations from a steady pattern 32. 9.5.1 Steady-state Vertical Temperature ProfileConduction transfers heat both vertically and horizontally,but small temperature gradients usually make the latternegligible.Ice moving vertically with velocity carries a heat fluxcT across a plane of unit area, oriented perpendicular to z.this term must be added to q in eq. 9.6. A similar term mustbe included for advection due to ice flow at rate u in thehorizontal direction x. then eq. 9.6 becomes..9.13 33. In thermal steady state , T /t = 0. Because thetemperature would not remain constant if the ice thicknessor velocity changed, the ice sheet is also implicitlyassumed to be in a steady state of flow and geometry. Insteady state, the temperature profile along a given verticalline, fixed in space, remains unchanged as the ice flows by. 34. The term can not be neglected, in general,because, although is small compared with , ,,u is normally much greater than in Robins analysis it isneglected, however; the solution should therefore apply near anice sheet divide, where u is small.9.5.1.1 Steady State with No Horizontal Advection1) Horizontal conduction can be neglected because thehorizontal temperature gradients is small compared with thevertical one.Robins assumptions2) The firn layer is replaced by an equivalent thickness of ice. 35. 3) The heat generated by ice deformation is treated as a flux,additional to the geothermal flux, at the base of the ice. This isa reasonable approximation because, in the slow-flowing partsof ice sheets, most of the shearing occurs near the base.4) The base is colder than melting point. 36. 9.5.1.2 Effect of Horizontal AdvectionHorizontal advection usually exerts a strong influence ontemperature profiles. Temperatures tend to increase in thedirection of glacier flow, because the surface elevationdeclines( 0.4 to 1oC per 100 m along the surface).Glacier flow thus transports colder ice, originating athigher altitudes, into warmer regions-horizontal advectionusually reduces temperatures. 37. 9.6 Measured Temperature ProfilesFigure 9.7: Measured temperature profiles in accumulation zones of polarice sheets and ice caps. For sites with negligible horizontal advection, thenumber in parentheses gives the advection parameter defined by Eq. 9.21. 38. 9.7 General Equation of Heat Transfer9.7.1 Derivation of Equation 39. 9.7.1.1 Heat Sources1) Ice deformationDeformational heat production concentrates where bothdeviatoric stresses and strain rates are highest usually in basallayers but also in lateral shear margins.2) Firn Compaction,3) Freezing of water: Refreezing liberates latent heat4) Heat of sliding friction5) Geothermal heat: Depends on tectonic setting and age 40. 9.7.2 Boundary and Basal ConditionAt the surface, temperature is prescribed as a function of time.In contrast, neither the temperature nor the heat flux at the baseof the glacier can be prescribed except in special cases.At the bottom of the domain, temperature or heat flux is heldconstant. To prescribe the latter, the temperature gradient isusually set equal to the heat flux divided by the thermalconductivity of the substrate material. This assumes a negligibletransport of heat by any circulating fluids. 41. 9.8 Temperature Along a Flow LineDahl-Jensen (1989) calculated how the steady temperaturedistribution varies along a flow line in an ice sheet.The assumptions are steady-state, two-dimensional flow, no firn,constant thermal conductivity, internal heating only from shear.The bed is assumed to be horizontal. At the surface, as elevationdrops along the flow line, the specific balance decreases and thetemperature increases at prescribed rates. 42. The main feature of temperature along flow line are:1. Basal temperature increases with distance from the icedivide because both surface temperature and heat ofdeformation increase.2. Basalmelting starts at X =0.625. (Here X denotes thedistance along the flowline expressed as a fraction of totallength.) The basal temperature gradient, hitherto increasingwith X, starts to decrease because some heat goes tomelting. 43. 3. A temperate basal layer starts to form at X = 0.75, still in theaccumulation zone. It first thickens with increasing X but thenthins to zero as deformational heating declines near the terminus.(Along a fast-flowing outlet glacier, in contrast, heating wouldpresumably remain important all the way to the front.)4. Horizontal advection produces a minimum in the verticalprofile of temperature. This cold spot strengthens and persistsalong the flow line, as far as the outer ablation zone. 44. 5. The temperature profile near the terminus resemblesthat predicted for an ablation zone by Robins simpleanalysis (the curve for =3 in Figure 9.5).Figure 9.8: (a) Theoreticaltemperature profiles alongan ice sheet flow line. Thenumber on each curve isthe distance as a fractionof the flow-line length.The equilibrium line is at0.91. (b) Closer view ofprofiles near the margin. Atemperate layer developsat the bottoms of profiles0.83 through 0.98. 45. 9.8.1 ObservationsA series of temperature profiles along a flow line has never beenmeasured in an ice sheet. However, Figure 9.9 shows temperatureprofiles measured along the centerline of White Glacier, a polarglacier on Axel Heiberg Island, Canada. Conditions in this valleyglacier, which is 15 km long, differ from those assumed in thetheoretical analysis in several ways:1. The glacier is not in a steady state.2. The bed has an average slope of 6 .3. The ice thickness varies with distance in an irregular way. 46. 4. The 10m ice temperature does not increase steadily with X.5. The ablation zone is proportionately much larger.Nevertheless the data show most of the predicted features:1. An increase in basal temperature with distance down-glacier.2. A temperate basal layer, in this case restricted to the ablationzone, that does not extend to the terminus.3. Profiles of the predicted shape near the terminus. 47. 4. A temperature minimum extending into the ablation zone. However, theminimum appears in the first profile, only 1.2 km from the head of theglacier. Thus the cool spot probably represents, in part, a remnant of lowtemperatures during the Little Ice Age and not just the effects of horizontaladvection (Blatter 1987). 48. Figure 9.9: Temperature profiles in White Glacier, at variousdistances along the flow line. Depth to bed is indicated ineach case. Note change in temperature scale between 9.9 and12 km. Data from Blatter (1985). 49. 9.9 Time-Varying Temperature 50. 9.10 Temperature in Ice Shelves