Introduction
One of the agreed objectives of the International Hydrological Decade 1965-74 was to determine the ice, water and energy balances of glaciers in many parts of the world (UNESCO/IASH, 1970, 1973).
The exchanges of energy and mass between a glacier and its surroundings are related as follows:
Conservation of energy implies that the sources of energy available to a glacier are equal to the resulting changes in the energy of the glacier, i.e.
where Fr is the radiative heat flux, Fe the sensible heat flux, F1 the latent heat flux from condensation and evaporation, Fg is a small amount of energy derived from geothermal heat, bottom friction and internal deformation. Ff is the change in energy content of the glacier due to latent heat associated with changes in the mass of snow and ice and Ft the change in energy content of the glacier due to changes in temperature of the snow/ice mass.
Conservation of mass of ice implies that
where Ip is the precipitation in the solid phase, I1 the condensation/evaporation of ice, Ir the change of mass of ice and snow due to calving, snow-drift, avalanches, etc., If the change in mass of ice due to ice/water phase changes, and Im the change in total ice mass.
Between 1969 and 1974 we studied Spartan Glacier in Alexander Island, the most southerly of an internationally selected chain of glaciers extending from Alaska through the Americas to the Antarctic Peninsula. We measured wind speed, temperature and humidity at several levels (Reference Jamieson and WagerJamieson and Wager, in press) and have analysed data covering the period from September 1973 to January 1974. Measurements of wind speed indicated that conditions near the glacier surface were dominated by gravity winds flowing down the glacier. In such circumstances the method developed byReference Deacon Deacon (1949) to calculate sensible and latent heat fluxes (Fe and F1 in Equation (1) could not be used. Reference HolmgrenHolmgren (1971), who studied these conditions in great detail, was also unable to derive a satisfactory theoretical treatment. A further difficulty with the aerodynamic method of estimating energy changes over the glacier was that measurements could only be made at a single point because of the work involved. It was very difficult to extrapolate these measurements satisfactorily to the whole glacier because meteorological conditions were affected by altitude and topography.
Yet another drawback to the aerodynamic method of obtaining the annual energy change was that radiative, sensible and latent heat fluxes had to be measured continuously for the whole year. These quantities had first to be measured, then extrapolated over the glacier, and finally summed for the whole year. Estimates made at Spartan Glacier showed that the mean rate of energy exchange was 1.5 W/m2. The energy gain in summer was about 100 W/m2, and the energy loss in winter was about 50 W/m2. The loss of five days’ records in summer or ten days’ records in winter could have caused an error of the same size as the total change in energy content of the glacier during the whole year.
Although aerodynamic methods can reveal the relative energy contributions from different atmospheric sources, we do not believe that they can be used to determine accurately the total energy change of an entire glacier. Radiation measurements can be made to an acceptable accuracy, although in polar regions much of the incoming radiation is reflected and the energy supplied by turbulent transfer is generally of the same order of magnitude as the net radiation (Reference LiljequistLiljequist, 1957, p. 292;Reference Dalrymple, Dalrymple , Lettau , Wollaston and Rubin Dalrymple and others, 1966, p. 55). For these reasons alternative methods should be considered in order to determine accurately changes in the total energy content of a glacier.
The energy content of a glacier E may usefully be defined as the total energy required to melt it.
where c is the specific heat capacity of ice, θ is the Celsius ice temperature, L is the latent heal of fusion of ice, p is the density of ice, and V is the volume of the glacier.
The corresponding definition of the total mass of the glacier . M is
Because the energy Fr + Fe + Fl + Fg Equation (1) must be supplied from sources external to the glacier, it is logical to treat the energy content E as negative. It also follows from the definition of £ that the heat content of water at o°C is zero.
From >Equation (1), Ft + Fr (=ΔE) is the change in energy content of the glacier. The predicted lifetime of the glacier λ is
where ΔE and ΔM are the measured changes in energy and mass in one year and E and M are the total energy and mass of the glacier. If the glacier is decreasing then λ is the length of time until it vanishes. If (he glacier has been increasing continuously, then it is the length of lime for which it has been in existence.
Direct Determination Of Energy And Mass Changes
Theoretical considerations
Consider a valley glacier (Fig. l). The line
=
0 represents a surface below which there is very little heat conducted from the atmosphere during the study period. The energy content of the volume below this surface is changed slightly by deformation, by friction on the bed, by geothermal heat and by the heal δ required to melt the small volume lost at the snout,
Fig. 1. Longtudinal section of a valley glacier showing zones with thru associated energy content.
We use the following notation:
Q measured heat content of the volume between the surface
=
0 and the glacier surface ;
δ heat content of the volume of the glacier below the surface
=
0 which is lost by melting at the snout;
D energy supplied to the glacier by internal deformation (negligible near the surface) ;
R energy supplied by friction at the bed of the glacier;
G energy supplied as geothermal heat.
At time T
1 the heat content of the glacier is Q
A+Q
I where Q
A is the heat content of the volume below the surface
=
0 and Q
1 is measured at time T1. At time T
2 the heat content of the glacier is
QA+Q2 + δ+D+R+G
where Q2 is measured at time T2.
The change in heat content of the entire glacier is therefore
Fig. 2. Vertical section near the glacier surface, q is the energy content of the prism bounded by pecked lines. It is calculated from the profiles of p and θ.
Consider a prism extending through the glacier (Fig. 2). The heat required to melt the ice in the prism down to the
=
0 surface is
The total heat content Q of the volume above the surface
=
0 is the surface integral of q over the glacier.
where A is the area of the glacier.
The change of heat content of the surface layers Q2 – Q1 is given by
where
Δq = q2 – q1
Similarly the change in mass of the glacier is
Im = M2 –M1 + Γ
where M1
and M2
are the masses of ice above the surface
=
0 at times T
1 and T2, Γ is the aggregate of mass changes below the surface
=
0 caused by bottom melting, by ablation at the snout, and by freezing of water in crevasses.
where
Δm = m 2 m 1
and
Location of the z0 surface
Measurements are referred to a Z 0 surface, which must be used again for subsequent measurements. The Z 0 surface should be chosen below the level at which there is significant heat flow even at the end of the study period. It may be chosen at a shallower depth in the accumulation area than in the ablation area.
Since the Z 0 surface can be relocated only by reference to stakes drilled into the glacier, it is essential that the stakes remain in place for the entire study period. This may prevent the method being used on glaciers with extensive ablation areas where the stakes cannot be planted deeply enough to remain in place through several ablation seasons. In the accumulation area, the settling of stakes relative to the Z 0 surface must be taken into account.
Practical considerations
In principle p and θ can be measured at any point in the glacier, It is sufficient to consider the volume above the Z 0 surface because Q A is almost constant.
To perform the surface integral for Q we need values of Δ q over the glacier surface. In practice, we must interpolate between sampled points. The required density of points depends on the complexity of the accumulation pattern. We must determine the temperature and density profiles down to the Z 0 surface at each point. The integral for Δ q can then be calculated numerically.
Temperature profiles
The most reliable way to maintain a datum for ice temperatures is to bond sensors to a stake made of an insulating material drilled into the Z 0 surface. The choice between chromel/constantan thermocouples and platinum resistance thermometers depends on cost, sensitivity, stability, and ease of measurement.
Density profiles
Densities obtained at Spartan Glacier by weighing blocks of snow were not sufficiently precise. The blocks crumbled easily because they contained a large amount of superimposed ice. Even on glaciers where the block method is more accurate, gamma-ray transmission measurements (Reference Smith, Smith, Donald and Owens.Smith and others, 1965) would be preferable in that they can determine the densities of a 1 cm layer to an accuracy of 1%.
Evaluation of heat content at sample points
We can now obtain the value of Δ q (= q2 – q1 ) from Equation (5). Functional forms of p and θ may be obtained by fitting suitable curves to the measured values. The integral may then be calculated by numerical methods.
Evaluation of the surface integral
To evaluate the surface integral of Δ q we use an array of equally spaced points covering the glacier. At each array point, we derive a value of Δ q by interpolation between the measured values.
The interpolation can be performed by computer programme (Reference DudnikDudnik, 1971). The surface integral becomes
where A is the surface area of the glacier, Δ qj is the jth value of Δ q in the array of J points.
Discussion
Energy changes at Spartan Glacier
From Equation (4) the change in energy content of the glacier is
ΔE = ΔQ+δ+D+R+G.
We have made estimates of the terms using data from Spartan Glacier. The estimates are expressed as energy input per unit time over the whole glacier divided by the surface area of the glacier.
(i) ΔE ≈ 1.5 W/m2 derived from the mass balance, assuming steady temperature conditions in the glacier.
(ii) δ ≈ o.i W/m2 derived from estimates of How and thickness near the snout.
(iii) (D + R) ≈ 0.02 W/m2. All the potential energy gained by the fall of the glacier appears as these two terms. [Mean velocity 25 mm/d (Reference Jamieson and WagerJamieson and Wager, in press) ; mean bottom slope 4° (Wager, in press) ; total mass 700 × 109 kg (Reference Jamieson and WagerJamieson and Wager, in press).]
(iv) G ≈ 0.04 W/m2 (Reference RuncornRuncorn, 1967). Because (D + R + G) must be small for any glacier, the change in energy content of the glacier becomes
ΔE ≈ ΔQ + δ
(v) ΔQ ≈ 1.4 W/m2 from the above equation.
Accuracy of determination of ΔQ
The integral for Δq consists of two parts
in which the smallest detectable energy change is about 106J/m2. This is the result of a temperature change of o. 1 deg at the surface. We assumed the Z 0 surface to be 10 m deep.
and the smallest detectable energy change is again about 106 J/m2. This is the result of a 1 cm change in the surface level zs , assuming a density near the surface of 250 kg/m3.
106 J/m2 would be transferred in 8 d at 1.5 W/m2, the average energy input at Spartan Glacier. It should therefore be possible to measure the change in energy content per year to within 3% averaged over the glacier. Uncertainties in the surface integral increase this error. However, it is likely that ΔC can be determined to within 10% in one year.
A series of measurements of ΔE (=ΔQ + δ) for, say, 10 years would establish the trend and the variation from year to year. Approximate values of ΔE at Spartan Glacier were calculated from the mass balance by assuming steady-state temperature conditions. The results are shown in Table I.
Table I. Annual change in energy of spartan glacier
Free water
Free water in the layers above the Z 0 surface produces errors in both integrals in Equation (5). The specific heat capacity of water is twice that of ice and the heat content is, by our definition, zero. Therefore q must be determined at times when there is no free water in the surface layers, preferably at the end of the accumulation season. The heat content of the glacier is then at a minimum.
Crevasses
Water flowing into crevasses which penetrate the Z 0 surface has zero energy content by our definition and therefore Q A is not changed. However, the corresponding mass, MA, is changed and an error is introduced into the calculation of change of mass of the glacier.
Temperate glaciers
A simplification of Equation (3) is possible for temperate glaciers. The temperature of the whole ice mass is very close to the melting point at the end of the ablation season.
Is therefore zero and
and
ΔE = LΔM.
Changes in energy are due solely to the latent heat associated with changes in mass of ice.
Other ice masses
Although we have discussed a valley glacier the same considerations may be applied to some other ice masses. The method cannot be applied to ice shelves because of the practical difficulties in determining the energy exchange beneath them.
A simple and probably fruitful extension is to large ice sheets (Fig, 3a). No ice flows into a sector bounded by the ice divide and two flow lines (Fig. 3b). The analysis for this volume exactly corresponds with that for the valley glacier.
Fig. 3. Cross-section of on ice sheet showing zones corresponding to those in a valley glacier.
Conclusions
In the past, emphasis has been laid on determining the components of the energy exchange. These are difficult to measure accurately over sufficiently long periods to give useful information about the long-term behaviour of the glacier. We believe that direct methods should be used to determine accurately changes in the energy and mass of glaciers. Only then should attempts be made to relate these changes to meteorological parameters.
