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Chapter
6Temperatures in glaciers andice sheets
Glaciers are divided into three categories, depending on their thermal structure
Cold The temperature of the ice is below the pressure melting temperature through-out the glacier, except for maybe a thin surface layer.
Temperate The whole glacier is at the pressure melting temperature, except forseasonal freezing of the surface layer.
Polythermal Some parts of the glacier are cold, some temperate. Usually thehighest accumulation area, as well as the upper part of an ice column are cold,whereas the surface and the base are at melting temperature.
The knowledge of the distribution of temperature in glaciers and ice sheets is of highpractical interest
• A temperature profile from a cold glacier contains information on past climateconditions.
• Ice deformation is strongly dependent on temperature (temperature depen-dence of the rate factor A in Glen’s flow law; Appendix B).
• The routing of meltwater through a glacier is affected by ice temperature.Cold ice is essentially impermeable, except for discrete cracks and channels.
• If the temperature at the ice-bed contact is at the pressure melting tempera-ture the glacier can slide over the base.
• Wave velocities of radio and seismic signals are temperature dependent. Thisaffects the interpretation of ice depth soundings.
The distribution of temperature in a glacier depends on many factors. Heat sourcesare on the glacier surface, at the glacier base and within the body of the ice. Heat istransported through a glacier by conduction (diffusion), is advected with the movingice, and is convected with water or air flowing through cracks and channels. Heatsources within the ice body are
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Chapter 6 Temperatures in glaciers and ice sheets
• dissipative heat production (internal friction) due to ice deformation,• frictional heating at the glacier base (basal motion)• frictional heating of flowing water at englacial channel walls,• release or consumption of (latent) heat due to freezing and melting.
The importance of the processes depends on the climate regime a glacier is subjectedto, and also varies between different parts of the same glacier.
Polythermal glaciers contain cold(blue) and temperate (red) ice. Thedistribution of the temperature zonesdepends on many factors such as envi-ronmental (temperature, precipitation),bedrock geometry, and ice flow patterns.
a typical for the high Arctic
b many Arctic glacier, e.g. centralflow line of Gornergletscher
c high Arctic, very cold climate (highHimalayas)
d high mountains
e maritime Arctic, e.g. Storglaciären
f a more extreme version of (e)
(from Benn and Evans, 1998)
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Physics of Glaciers I HS 2019
6.1 Energy balance equationThe energy balance equation in the form suitable to calculate ice temperature withina glacier is the advection-diffusion equation which in a spatially fixed (Eulerian)reference frame, and given by
ρC
(∂T
∂t+ v∇T
)= ∇(k∇T ) + P . (6.1)︸ ︷︷ ︸
advection︸ ︷︷ ︸
diffusion︸ ︷︷ ︸
production
The ice temperature T (x, t) at location x and time t changes due to advection,diffusion and production of heat. The material constants density ρ, specific heatcapacity C and thermal conductivity k are given in Table 6.1. For constant thermalconductivity k and in one dimension (the vertical direction z and vertical velocityw) the energy balance equation (6.1) reduces to
ρC
(∂T
∂t+ w
dT
dz
)= k
d2T
dz2+ P. (6.2)
The heat production (source term) P can be due to different processes:
Dissipation In viscous flow the dissipation due to ice deformation (heat release dueto internal friction) is P = tr(ε :σ) = εijσji. Because usually the horizontalshearing deformation dominates glacier flow
Pdef ' 2 εxzσxz .
Sliding friction The heat production is the rate of loss of potential energy as anice column of thickness H moves down slope. If all the frictional energy isreleased at the bed due to sliding with basal velocity ub,
Pfriction = τbub ∼ ρgH tan β ub ,
where τb is basal shear stress and β is the bed inclination.
Refreezing of meltwater Consider polythermal ice that contains a volume frac-tion µ of water. If a freezing front is moving with a velocity vfreeze relative tothe ice, the rate of latent heat production per unit area of the freezing front is
Pfreeze = vfreeze µρwL.
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Chapter 6 Temperatures in glaciers and ice sheets
Quantity Symbol Value Unit
Specific heat capacity of water Cw 4182 JK−1 kg−1
Specific heat capacity of ice Ci 2093 JK−1 kg−1
Thermal conductivity of ice (at 0◦C) k 2.1 Wm−1K−1
Thermal diffusivity of ice (at 0◦C) κ 1.09 · 10−6 m2 s−1
Latent heat of fusion (ice/water) L 333.5 kJ kg−1
Depression of melting temperature (Clausius-Clapeyron constant)- pure ice and air-free water γ 0.0742 KMPa−1
- pure ice and air-saturated water γ 0.098 KMPa−1
Table 6.1: Thermal properties of ice and water.
6.2 Steady state temperature profileThe simplest case is a vertical steady state temperature profile (Eq. 6.2 withouttime derivative) in stagnant ice (w = 0) without any heat sources (P = 0), and withconstant thermal conductivity k. The heat flow equation (6.2) then reduces to thediffusion equation (Poisson equation)
d2T
dz2= 0. (6.3)
Integration with respect to z givesdT
dz= G (6.4)
where G = ∇T is the (constant) temperature gradient. Integrating again leads to
T (z) = Gz + T (0), (6.5)
which means that ice temperature varies linearly with depth.
The heat flux is Q given by
Q = −kdT
dz= −kG . (6.6)
This is the statement of a basic relation of physics: heat (energy) flux is proportionalto the temperature gradient with a material constant k, the heat conductivity.
Choosing for example a temperature gradient of G = 1K/100m = 0.01Km−1 =10mKm−1 gives a heat flux of Q = kG = 0.021Wm−2 = 21mWm−2. For compar-ison, typical geothermal heat fluxes are of the order 40− 120mWm−2.
To transport a geothermal heat flux of 80mWm−2 through a glacier of 200m thick-ness, the surface temperature has to be 8K lower than the temperature at the glacierbase.
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Physics of Glaciers I HS 2019
The above calculations (partly) explain why there can be water at the base of theGreenland and Antarctic ice sheets. Under an ice cover of 3000m and at surfacetemperatures of −50◦C (in Antarctica), only a heat flux of Q = k · 50K/3000m =35mWm−2 can be transported away. Notice that horizontal and vertical advectionchange this result considerably (Section 6.7).
6.3 Ice temperature close to the glacier surfaceThe top 15m of a glacier are subject to seasonal variations of temperature. In thispart of the glacier heat flow (heat diffusion) is dominant. If we neglect advection wecan rewrite Equation (6.2) to obtain the well known Fourier law of heat diffusion
dT
dt= κ
d2T
dh2(6.7)
where h is depth below the surface, and κ = k/(ρC) = 1.09 · 10−6m2 s−1 is thethermal diffusivity of ice. To calculate a temperature profile and changes with timewe need boundary conditions. Periodically changing boundary conditions at thesurface such as day/night and winter/summer can be approximated with a sinefunction. At depth we assume a constant temperature T0
T (0, t) = T0 +∆T0 · sin(ωt) ,T (∞, t) = T0 . (6.8)
T0 is the mean surface temperature and ∆T0 is the amplitude of the periodic changesof the surface temperature. The duration of a period is 2π/ω. The solution ofEquation (6.7) with the boundary condition (6.8) is
T (h, t) = T0 +∆T0 exp
(−h
√ω
2κ
)sin
(ωt− h
√ω
2κ
). (6.9)
The solution is plotted in Figure 6.1 for realistic values of T0 and ∆T on ColleGnifetti (4550ma.s.l., Monte Rosa, Valais). Notice that full ice density has beenassumed for the plot, instead of a firn layer with strongly changing thermal proper-ties, and vertical advection is neglected.
The solution (Eq. 6.9) has some noteworthy properties
a) The amplitude varies with depth h as
∆T (h) = ∆T0 exp
(−h
√ω
2κ
)(6.10)
b) The temperature T (h, t) has an extremum when
sin
(ωt− h
√ω
2κ
)= ±1 ⇒ ωt− h
√ω
2κ=
π
2
75
Chapter 6 Temperatures in glaciers and ice sheets
20 18 16 14 12 10 8 6Temperature ( C)
20.0
17.5
15.0
12.5
10.0
7.5
5.0
2.5
0.0
Dept
h (m
)
1 2 3 4 5 67891012
Figure 6.1: Variation of temperature with depth for the conditions at Colle Gnifetti.Numbers next to curves indicate months (1 corresponds to January).
and thereforetmax =
1
ω
(π
2+ h
√ω
2κ
)(6.11)
The phase shift is increasing with depth below the surface.
c) The heat flux Q(h, t) in a certain depth h below the surface is
Q(h, t) = −k∂T
∂h= (complicated formula that is easy to derive)
For the heat flux Q(0, t) at the glacier surface we get
Q(0, t) = ∆T0
√ωρCk sin
(ωt+
π
4
)Q(0, t) is maximal when
sin(ωt+
π
4
)= 1 ⇒ ωt+
π
4=
π
2=⇒ tmax =
π
4ωwhere tmax is the time when the heat flux at the glacier surface is maximum.
Temperature: tmax =π
2ω
Heat flux: tmax =π
4ω
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Physics of Glaciers I HS 2019
The difference is π/(4ω) and thus 1/8 of the period. The maximum heat fluxat the glacier surface is 1/8 of the period ahead of the maximum temperature.For a yearly cycle this corresponds to 1.5 months.
d) From the ratio between the amplitudes in two different depths the heat diffu-sivity κ can be calculated
∆T2
∆T1
= exp
((h1 − h2)
√ω
2κ
)=⇒ κ =
ω
2
(h2 − h1
ln ∆T2
∆T1
)2
There are, however, some restrictions to the idealized picture given above
• The surface layers are often not homogeneous. In the accumulation area den-sity is increasing with depth. Conductivity k and diffusivity κ are functionsof density (and also of temperature).
• In nature the surface boundary condition is not a perfect sine function.
• The ice is moving. Heat diffusion is only one part, heat advection may beequally important.
• Percolating and refreezing melt water can drastically change the picture byproviding a source of latent heat (see below).
Superimposed on the yearly cycle are long term temperature changes at the glaciersurface. These penetrate much deeper into the ice, as can be seen from Equation(6.10), since ω decreases for increasing forcing periods. A very nice example of thisis shown in Figure 6.9.
In spring or in high accumulation areas the surface layer of a glacier consists of snowor firn. If surface melting takes place, the water percolates into the snow or firn packand freezes when it reaches a cold layer. The freezing of 1 g of water releases sufficientheat to raise the temperature of 160 g of snow by 1K. This process is important to“annihilate” the winter cold from the snow or firn cover in spring, and is the mostimportant process altering the thermal structure of high accumulation areas and ofthe polar ice sheets.
6.4 Temperate glaciersIn a temperate glacier, all heat that is produced at the boundaries or within theglacier is used to melt ice. This means that temperate glaciers contain liquid waterthat is stored in cracks and voids, and also within the ice matrix. The water contentµ of temperate ice is small, usually between 0.1 and 4 volume percent. The wateris stored in veins and triple junctions between the ice grains, where it can slowly
77
Chapter 6 Temperatures in glaciers and ice sheets
percolate through the ice matrix if veins are not blocked by air bubbles. The waterbetween ice grains has an important effect on the deformation properties of ice, andit affects the rate factor A in Glen’s flow law (Duval, 1977; Paterson, 1999)
A(µ) = (3.2 + 5.8µ) · 10−15 kPa−3 s−1
= (101 + 183µ)MPa−3 a−1 , (6.12)
where µ is the percentage of water within the ice.
Pressure melting point temperatureFor pure ice the melting temperature Tm depends on absolute pressure p by
Tm = Ttp − γ (p− ptp) , (6.13)
where Ttp = 273.16K and ptp = 611.73Pa are the triple point temperature andpressure of water. The Clausius-Clapeyron constant is γp = 7.42 · 10−5KkPa−1
for pure water/ice. Since glacier ice contains soluble and isoluble chemicals and airbubbles, the value of γ can be as high as γa = 9.8 · 10−5KkPa−1 for air saturatedwater (Harrison, 1975).
6.5 Cold glaciersIn cold glaciers heat flow is driven by temperature gradients. Usually the base iswarmest due to dissipation, friction and geothermal heat. This has the consequencethat heat is flowing from the base into the ice body, warming up the ice. Advectionof warm or cold ice strongly influences local ice temperature.
Only the highest glaciers in the Alps are completely cold, with a lower limit of about3900ma.s.l.. The most famous of these is Colle Gnifetti (4550ma.s.l.), the highestaccumulation basin of the Gorner-/Grenzletscher system. At temperatures of about−13◦C the ice conserves atmospheric conditions such as impurities and air bubbles.Surface melting only takes place on exceptionally hot days and leads to formation ofice lenses. Many ice cores have been drilled on Colle Gnifetti and were investigatedin the laboratory to obtain the climate history of central Europe.
Figure 6.2 shows two temperature profiles measured in boreholes 200m apart on thesame flow line on Colle Gnifetti. In the absence of advection and density gradientsone would expect straight steady state temperature profiles (cf. Eq. 6.5). The un-equal curvature of the modeled steady state profiles (blue and red curve in Fig. 6.2a)is due to firn density and different advection regimes. This shows that the interpre-tation of temperature profiles to deduce past climate can be quite tricky.
The marked bend towards warmer temperature at 30m depth is due to changingsurface temperatures. Numerical modeling showed that the temperature increase by
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Physics of Glaciers I HS 2019
-14.0 -13.5 -13.0 -12.5 -12.0Temperature [oC]
-100
-80
-60
-40
-20
0
Dep
th b
elow
sur
face
[m]
B 95-2
B 95-1
Q=0 mW/m2
steady state
Tsurf
= -14.0oC var.
-14.0 -13.5 -13.0 -12.5 -12.0Temperature [oC]
-100
-80
-60
-40
-20
0
Dep
th b
elow
sur
face
[m]
B 95-2
B 95-1
Figure 6.2: Markers indicate temperatures measured in two boreholes on the sameflow line, 200m apart, on Colle Gnifetti. Solid lines are the results of numericalinterpretation of the data with steady state (left) and transient (right) temperatureevolution. Notice that the different curvature of the steady state profiles is due tofirn density and advection. From Lüthi and Funk (2001).
0 500 1000 1500Distance along flowline [m]
3000
3500
4000
4500
Alti
tude
[m a
.s.l]
Tsurf
(t)
3 3
22
1 1
-1
-1
-2-2
-3
-3
-4
-4
-5
-5
-6
-7
-8
-9
-10
-12
-13
-11
q
0 200 400 600 800Distance along flowline [m]
3800
4000
4200
4400
4600
Alti
tude
[m a
.s.l]
-5
-6
-6
-7
-7
-8
-8
-9
-9
-12
-12
-11
-11-1
0
-10
-13
-13
-6
Figure 6.3: Modeled temperature distribution within the Monte Rosa massif. Noticethe highly increased surface temperature on the south (right) face which leads tohorizontal heat fluxes close to steep mountain faces. Thawing permafrost at thebase of the mountain was essential to reproduce the measured heat flux at the glacierbase. From Lüthi and Funk (2001).
79
Chapter 6 Temperatures in glaciers and ice sheets
1◦C around 1990 is mainly responsible for this feature (the profiles were measuredin 1995/96; in 1982 the transient profiles looked like the steady state configuration).
Modeling the advection and conduction of heat in a glacier is usually not sufficient.The geothermal heat flux close to the glacier base is affected by the ice temperature,and therefore by the glacier flow, and the flux of meltwater. At Colle Gnifetti themeasured (and modeled) temperature gradients at the glacier base are different dueto different distance to the south-facing rock wall of Monte Rosa. Figure 6.3 showsthe temperature distribution in the whole massif, with the glacier on top. Noticethat there are areas within the mountain where heat flows horizontally (heat alwaysflows perpendicular to the isotherms). Heat flux at sea level was assumed to be70mWm−2, but it had to be greatly reduced to match the observed fluxes. In themodel this was accomplished with freezing/thawing ice within the rock (permafrost),which is a reaction to climate changes since the last ice age. The lowest calculatedpermafrost margin was at about 2900m elevation.
6.6 Polythermal glaciers and ice sheetsPolythermal glaciers are usually cold in their core and temperate close to the surfaceand at the bed. Heat sources at the surface are from the air (sensible heat flux),direct solar radiation (short wave), thermal radiation (long wave), and the penetra-tion and refreezing of melt water (convection and latent heat). The temperate layerat the bed can be of considerable thickness and is caused by geothermal heat fluxand heat dissipation due to friction and ice deformation. The latter is highest closeto the base because stresses and strain rates are highest there.
Gorner-/Grenzgletscher is the biggest polythermal glacier in the Alps. Cold iceoriginating from Colle Gnifetti (4500ma.s.l) is advected down to the confluencearea at 2500ma.s.l., and all the way to the glacier tongue. Figure 6.4A shows theslow cooling of a borehole in the confluence area after hot-water drilling. Afterequilibration, a cold central core is apparent in Figure 6.4B which was advectedfrom the high accumulation area, and is gradually warming along the flow line(Figure 6.5), mostly due to heat conduction. The cold ice is mostly impermeableto water which leads to the formation of deeply incised river systems and surficiallakes which sometimes drain through crevasses or moulins.
Nice examples of the polythermal structure of the Greenland ice sheet are the tem-perature profiles in Figure 6.6, measured in Jakobshavn Isbræ, Greenland (Ikenet al., 1993; Funk et al., 1994; Lüthi et al., 2002). The drill sites are located 50 kminland of the margin of the Greenland ice sheet, at a surface elevation is 1100ma.s.l.,where ice thickness on the ice sheet is 830m, but about 2500m in the ice streamcenter.
In Figure 6.6 (bottom left) temperatures are highest close to the surface and closeto the bed. The thick, very cold layer in between is advected from the inland
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Physics of Glaciers I HS 2019
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
Temperature (◦C)
-300
-250
-200
-150
-100
-50
0
Depth
belo
w s
urf
ace
(m
)
Temp-Depth temp3-08
300 200 100 0 100 200 300Distance (m)
2200
2300
2400
2500
Ele
vati
on (
m a
.s.l.)
ll5
-0.39
-0.30
-0.32 -0.40
-0.50 -0.39
-0.38 -0.41
-0.42
l5
-1.16 -1.29
-1.07 -1.11
-0.34
-0.68
-0.29 -0.33
-0.31
-1.00 -0.98
-1.24 -1.21 -1.15 -1.07
f5
-2.64
-2.47
-2.01
-2.52
-1.15 -1.50
-0.53 -0.70
-0.40
r6
-0.15
-0.65
-0.18 -0.19
-0.20 -0.19
-0.20CTS
2.5
2.0
1.51.0
0.5
Figure 6.4: Left: Cooling of a borehole drilled in the confluence area of Gorner-/Grenzgletscher. Temperatures measured every day after completion of drilling areshown in increasingly lighter colors, and after three months (leftmost yellow curve)(from Ryser, 2009). Right: Cross section through the glacier showing an onion-likestructure of ice temperature, due to advection of cold ice (from Ryser et al., 2013).
200 400 600 800 1000 1200Distance (m)
2200
2300
2400
2500
2600
Ele
vati
on (
m a
.s.l.)
f1
-2.55
-2.43
-2.07
-2.51
-0.73
-1.39
-0.06
-0.24
-0.04
f5
-2.57
-2.43
-1.86
-2.41
-0.94 -1.31
-0.29 -0.47
-0.14
f9
-2.23
-2.12
-1.76
-2.04
-0.81
-1.30
-0.01
-0.31
0.02
f12
-2.07 -1.95
-2.31 -2.16
-0.81
-1.47
0.01
-0.06
0.01
f13
0.16
-1.88
-1.44 -1.71 -1.90 -1.97
f3ED
C
CTS-0.5
-1.0-1.5
-2.0
-2.5
Figure 6.5: Longitudinal section through the glacier showing the avection of cold icewhich is gradually warming due to heat flow on its way towards the terminus (fromRyser et al., 2013).
81
Chapter 6 Temperatures in glaciers and ice sheets
50.0 49.0 48.0Degrees West
69.0
69.2
69.4
Deg
rees
Nor
th
0 10 20 30 km
400
400
400
400
400
600
600
800
800
1000
1000
1200
1200
1400
1400
1600
ED
drill sites 1989
drill site 1995
Jakobshavns Isfjord
-20 -15 -10 -5 0Temperature (oC)
-1500
-1000
-500
0
Dep
th b
elow
sur
face
(m
)
Dsheet
Amargin
Bcenter
mel
ting
poin
t
-4 -3 -2 -1 0Temperature [oC]
-840
-820
-800
-780
-760
Dep
th [m
]
CTSm
eltin
g po
int
Figure 6.6: Top: Location map of the drill sites on Jakobshavn Isbræ, Greenland,where temperature profiles have been measured. Left: Profile A is from the southernside of the ice stream, profile B from the center (the ice is about 2500m thick there)and D on the northern margin. Notice the basal temperate ice at D and also at A.The very cold ice in the middle of the profiles is advected and is slowly warmingup. Right: Closeup of the 31m thick temperate layer at site D. The cold-temperatetransition surface CTS is at freezing conditions. The Clausius-Clapeyron gradient(dashed line) is γ = 0.0743KMPa−1. From Lüthi et al. (2002).
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Physics of Glaciers I HS 2019
100 200 300 400 500Distance along flowline [km]
-1000
0
1000
2000
3000
Alti
tude
[m a
.s.l]
-6-6
-6
-10
-10-10
-14 -14
-14
-18
-18-18
-22
-22
-22
-22
-22
-18
-18 -14
-10
-2
-2
-26
-26
-26
Drill Site D
-20 -15 -10 -5 0Temperature (oC)
1.0
0.8
0.6
0.4
0.2
0.0
Rel
ativ
e de
pth
Sliding ratio
68 %
45 %
acc +30%
57 %
-1.0 -0.8 -0.6 -0.4Temperature [oC]
-805
-800
-795
-790
-785
Dep
th [m
]
0.0 %
0.5 %
1.0 %
2.0 %
32
15
33/34
water content
Figure 6.7: Top: Temperature distribution along a flow line in the Greenland IceSheet, and passing through drill site D. Clearly visible is the advection of cold ice andthe onset of a basal temperate layer far inland at 380 km. Left: Modeled temperatureprofiles for different ratios of basal motion. Right: Modeled temperature profiles fordifferent values of the water content in the temperate ice. From Funk et al. (1994)and Lüthi et al. (2002).
83
Chapter 6 Temperatures in glaciers and ice sheets
parts of the ice sheet (see Figure 6.7 top). The lowest 31m are at the pressuremelting temperature Tm = −0.55◦C. A quick calculation with Equation (6.13) andreasonable values for the ice density and gravity gives
Tm = Ttp − γp(p− ptp) = Tp − γp(ρgH − ptp)
= 273.16K− 7.42 · 10−5KkPa−1(900 kgm−3 · 9.825m s−2 · 830m
)' 273.16K− 0.545K ' 272.615K = −0.54◦C . (6.14)
The strong curvature of the temperature profiles where they are coldest indicatesthat the ice is warming (explain why!). Upward heat flux from the temperate zoneclose to the base is very high. The temperature gradient just above the CTS (800mdepth) in profile D is 0.048Km−1. The upward heat flux therefore is (per unit area)
Q = kdT
dz' 2.1Wm−1K−1 · 0.048Km−1 ' 0.1Wm−2 . (6.15)
This heat has to be produced locally, since heat fluxes in the temperate zone below800m depth are downward (away from the CTS), but very small due to the Clausius-Clapeyron temperature gradient. The only way to produce this heat is by freezingthe water contained within the temperate ice. From a flow model we know that theCTS moves with respect to the ice by vfreeze ∼ 1ma−1. With a moisture content ofµ = 1% (by volume) we obtain a heat production rate of
P = µ vfreeze ρwL ' 0.01 · 1ma−1 · 1000 kgm3 · 3.33 · 105 J kg−1
= 3.55MJ a−1m−3 ' 0.113Wm−3 (6.16)
which matches up nicely with the heat transported away by diffusion (Eq. 6.15). Adetailed analysis with the flow model shows that the most likely value for the watercontent is about 1.5% (Fig. 6.7, bottom right). Another interesting result form themodel is that an ice column passing through drill site D has already lost the lowest40m through melting at the glacier base.
6.7 Advection – diffusionClose to the ice divides in the central parts of an ice sheet, the ice velocity is mainlyvertical down. Under the assumptions of only vertical advection, no heat generation,and a frozen base we can calculate the steady state temperature profile with thesimplified Equation (6.2)
κd2T
dz2= w(z)
dT
dz(6.17)
The boundary conditions are
surface: z = H : temperature Ts = const
bedrock: z = 0 : basal heat flux − k
(dT
dz
)B
= −kGb = const
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Physics of Glaciers I HS 2019
For the first integration we substitute f = dTdz
so that κf ′ = wf =⇒ f ′
f= 1
κw with
the solutiondT (z)
dz=
(dT
dz
)B
exp
(1
κ
∫ z
0
w(z) dz
). (6.18)
Now we make the assumption that the vertical velocity is w(z) = wsz/H, with thevertical velocity at the surface equal to the net mass balance rate ws = −b. Withthe definition L2 := 2κH/b we get
T (z)− T (0) = Gb
∫ z
0
exp
(− z2
L2
)dz (6.19)
which has the solution
T (z)− Ts =
√π
2LGb
[erf( zL
)− erf
(H
L
)]. (6.20)
The so-called error function is tabulated and implemented in all mathematical soft-wares (usually as erf), and is defined as
erf(x) = 2√π
∫ x
0
exp(−x2) dx.
We now introduce the useful definition of the Péclet Number which is a measure ofthe relative importance of heat transport by advection and diffusion
Pe =advective heat transportdiffusive heat transport =
wsH
κ. (6.21)
The shape of the temperature profiles in Figure 6.8 therefore only depends on thePéclet Number. Depth and temperature can be scaled by the dimensionless variables
scaled distance above bed ξ =z
H,
scaled temperature Θ =k(T − Ts)
GH,
such that the quantity L of Equation (6.20) is L2 =2H2ρiCp
Pe .
One application of this theoretical solution is shown in Figure 6.9 where deviationsfrom the theoretical curve (calculated with a model) are interpreted as signature ofpast climate variations of the last 8000 years (Dahl-Jensen et al., 1998).
85
Chapter 6 Temperatures in glaciers and ice sheets
0.0 0.2 0.4 0.6 0.8Scaled temperature Θ
0.0
0.2
0.4
0.6
0.8
1.0
Sca
led
dept
hξ
0.00.5
13
510
2030
Figure 6.8: Dimensionless steady temperature profiles in terms of the dimensionlessvariables ξ and Θ for various values of the Péclet number Pe (next to curves). Athigh Péclet numbers the advection dominates, and cold ice is transported down fromthe surface into vicinity of the bed.
Figure 6.9: Left: The GRIP and Dye 3 temperature profiles [blue trace in (A) and(C)] are compared to temperature profiles [red trace in (A) and (C)] calculated underthe condition that the present surface temperatures and accumulation rates have beenunchanged back in time. Right: The reconstructed temperature histories for GRIP(red curves) and Dye 3 (blue curves) are shown for the last 8 ky BP (A) and thelast 2 ky BP (B). The two histories are nearly identical, with 50% larger amplitudesat Dye 3 than found at GRIP. From Dahl-Jensen et al. (1998)
86
Bibliography
Benn, D. and Evans, D. (1998). Glacier and Glaciations. Arnold, a member of theHodder Headline Group, 338 Euston Road, London NW1 3BH.
Cuffey, K. and Paterson, W. (2010). The Physics of Glaciers. Elsevier, Burlington,MA, USA. ISBN 978-0-12-369461-4.
Dahl-Jensen, D., Mosegaard, K., Gundestrup, N., Clow, G. D., Johnsen, S. J.,Hansen, A. W., and Balling, N. (1998). Past temperatures directly from thegreenland ice sheet. Science, 282:268–271.
Duval, P. (1977). The role of water content on the creep rate of polycristalline ice. InIsotopes and impurities in snow and ice, pages 29–33. International Associationof Hydrological Sciences. Publication No. 118.
Funk, M., Echelmeyer, K., and Iken, A. (1994). Mechanisms of fast flow in Jakob-shavns Isbrae, Greenland; Part II: Modeling of englacial temperatures. Journalof Glaciology, 40(136):569–585.
Harrison, W. D. (1975). Temperature measurements in a temperate glacier. Journalof Glaciology, 14(70):23–30.
Hutter, K. (1983). Theoretical glaciology; material science of ice and the mechanicsof glaciers and ice sheets. D. Reidel Publishing Company/Tokyo, Terra ScientificPublishing Company.
Iken, A., Echelmeyer, K., Harrison, W. D., and Funk, M. (1993). Mechanisms offast flow in Jakobshavns Isbrae, Greenland, Part I: Measurements of temperatureand water level in deep boreholes. Journal of Glaciology, 39(131):15–25.
Lüthi, M. P. and Funk, M. (2001). Modelling heat flow in a cold, high altitudeglacier: interpretation of measurements from Colle Gnifetti, Swiss Alps. Journalof Glaciology, 47(157):314–324.
Lüthi, M. P., Funk, M., Iken, A., Gogineni, S., and Truffer, M. (2002). Mecha-nisms of fast flow in Jakobshavns Isbrae, Greenland; Part III: measurements of
87
Appendix BIBLIOGRAPHY
ice deformation, temperature and cross-borehole conductivity in boreholes to thebedrock. Journal of Glaciology, 48(162):369–385.
Paterson, W. S. B. (1999). The Physics of Glaciers. Butterworth-Heinemann, thirdedition.
Ryser, C. (2009). The polythermal structure of Grenzgletscher, Valais, Switzerland.Masterarbeit, Abteilung für Glaziologie, VAW (unpublished), ETH-Zürich.
Ryser, C., Lüthi, M., Blindow, N., Suckro, S., Funk, M., and Bauder, A. (2013).Cold ice in the ablation zone: its relation to glacier hydrology and ice watercontent. Journal of Geophysical Research, 118(F02006):693–705.
Smith, G. D. and Morland, L. W. (1981). Viscous relations for the steady creep ofpolycrystalline ice. Cold Regions Science and Technology, 5:141–150.
0
Appendix
AList of Symbols
Latin lettersSymbol Description Units
A softness parameter, a constant in Glen’s flow law MPa−3 a−1
b specific mass balance rate kgm−2 a−1
bi specific volumetric mass balance rate ma−1
B(T ) temperature dependence of viscosityC specific heat capacity J kg−1K−1
E Young’s modulus of elasticity MPag acceleration due to gravity ms−2
g vertical gradient of balance rate ∂bi/∂z a−1
h vertical coordinate, depth below surface mH ice thickness mk heat conductivity Wm−1K−1
n exponent in Glen’s flow lawp pressure MPaP heat production WQ heat flux Wm−2
q ice flux, water flux m3 s−1
t time (seconds, years) s, aT temperature ◦Cubal balance velocity ma−1
u, v, w components of the velocity vector v ma−1
v velocity vector, v = (u, v, w) m a−1
w.eq. water equivalentx, y, z space coordinates mx position vector, x = (x, y, z) mz vertical coordinate, pointing upwards mzb bedrock elevation mzELA equilibrium line altitude mzs surface elevation m
1
Appendix A List of Symbols
Greek lettersSymbol Description Units
α surface slope tanα = dzsdx
◦
β bed slope tan β = dzbdx
◦
ε strain rate tensor with components εij a−1
η shear viscosity MPa · aγ Clausius-Clapeyron constant [ ∼ 0.074KMPa−1 ] KMPa−1
κ thermal diffusivityν elastic Poisson ratioρi density of ice [ 900− 917 kgm−3 ] kgm−3
ρw density of water kgm−3
σe effective uniaxial stress [σe := (32σ(d)ij σ
(d)ij )
12 =
√3τ ] MPa
σm mean stress [σm := 13σkk] MPa
σ stress tensor with components σij MPa
σ(d) stress deviator tensor [σ(d)ij := σij − 1
3σkkδij = σij − σmδij] MPa
τ effective shear stress [τ := (12σ(d)ij σ
(d)ij )
12 = 1√
3σe] MPa
A 2
Appendix
BUseful quantities
Quantity Symbol Value UnitMechanical properties
Density of water (0◦C) ρw 999.84 kgm−3
Density of bubble free ice (0◦C) ρi 917 kgm−3
Young modulus of ice E 8.7 · 109 PaShear modulus of ice µ 3.8 · 109 PaPoisson ratio of ice ν 0.31Creep activation energy (<−10◦C) Q 78 kJmol−1
Thermal properties
Specific heat capacity of water Cw 4182 JK−1 kg−1
Specific heat capacity of ice Ci 2093 JK−1 kg−1
Thermal conductivity of ice (at 0◦C) k 2.1 Wm−1K−1
Thermal diffusivity of ice (at −1◦C) κ 1.09 · 10−6 m2 s−1
Latent heat of fusion (ice/water) L 333.5 kJ kg−1
Depression of melting point (Clausius-Clapeyron constant)- pure ice and air-free water γp 0.074 KMPa−1
- pure ice and air-saturated water γa 0.098 KMPa−1
Constants
Gravity acceleration g 9.81 m s−2
Triple point temperature Ttp 273.16 KTriple point pressure ptp 611.73 PaGas constant R 8.31 Jmol−1K−1
Avogadro number NA 6.023 · 1023Boltzmann constant kb 1.3807 · 10−23 JK−1
Stefan-Boltzmann constant σsb 5.67 · 10−8 Wm−2K−4
Solar constant (radiation) Qsolar 1368 Wm−2
1
Appendix B Useful quantities
Flow law parameter
T (◦C) A ( s−1 Pa−3) A ( a−1MPa−3) AP ( s−1 kPa−3) AP ( a−1MPa−3)0 2.4 · 10−24 75.7 (6.8 · 10−15) (215)-2 1.7 · 10−24 53.6-5 9.3 · 10−25 29.3 (1.6 · 10−15) (50.5)-10 3.5 · 10−25 11.0 (4.9 · 10−16) (15.5)-15 2.1 · 10−25 6.62 (2.9 · 10−16) (9.2)-20 1.2 · 10−25 3.78 (1.7 · 10−16) (5.4)-30 3.7 · 10−26 1.17 (5.1 · 10−17) (1.6)-40 1.0 · 10−26 0.315 (1.4 · 10−17) (0.44)-50 2.6 · 10−27 0.082 (3.6 · 10−18) (0.11)
Table B.1: Flow law parameter A recommended by Cuffey and Paterson (2010), andthe older values AP recommended by Paterson (1999).
It is common to assume that the flow law parameter A can be split into a constantrate factor at a reference temperature A0 and a parameter absorbing the temper-ature dependence B(T ) (e.g. Hutter, 1983; Paterson, 1999). At temperatures be-low −10◦C the rate factor is of Arrhenius type with an activation energy of about60 kJmol−1 (Paterson, 1994). A double exponential fit derived by Smith and Mor-land (1981, eq. 21) is often used
B(T ) = 0.9316 exp(0.32769T ) + 0.0686 exp(0.07205T ) , T ≥ −7.65◦C, (B.1)B(T ) = 0.7242 exp(0.59784T ) + 0.3438 exp(0.14747T ) , T < −7.65◦C, (B.2)
where T is the Celsius temperature. This parameterization is almost identical tothe values given in Table B1 (Paterson, 1999, p. 97).
The rate factor A in Glen’s flow law is also affected by the percentage of water µwithin the ice (Duval, 1977; Paterson, 1999)
A(µ) = (3.2 + 5.8µ) · 10−15 kPa−3 s−1
= (101 + 183µ)MPa−3 a−1 , (B.3)
B.1.
B 2
