List of Symbols

• A Total net ablation

• C Total net accumulation

• c Annual cumulative accumulation

• H Heat-flux density

• h Altitude related to the equilibrium line

• Δh Difference in altitude related to the present equilibrium-line altitude

• k Non-dimensional number

• L Specific heat of melting

• T _a Air temperature

• w Cloudiness

• z Altitude

• γ Factor of proportionality

• μ Factor of proportionality

• ρν Absolute humidity

• τ Number of ablation days

1. Introduction

The response of the Greenland ice sheet to the increase of CO₂ and other atmospheric greenhouse gases is of recent interest (United States. Department of Energy, 1985). Climatic warming is closely connected with alterations in the heat balance and the shift of the equilibrium-line altitude. For the calculation of the shift of the equilibrium-line altitude due to climatic perturbations, a concept introduced by Reference KuhnKuhn (1981) was applied and adjusted to the conditions on the Greenland ice sheet (Reference AmbachAmbach, 1989). The equilibrium-line altitude is essentially determined by the heat balance and the annual accumulation. At the equilibrium-line altitude, the heat of melting, supplied by the heat balance during the entire ablation season, equals the heat consumed for melting the annual accumulation including superimposed ice. The heat balance reads

. (1)

τ is the number of ablation days, Η is the daily heat-flux density of melting averaged over the ablation period, k is a factor related to the formation of superimposed ice, i.e. 1≤k≤5/3 (Reference AmbachAmbach, 1985), L is the specific heat of melting and c is the annual accumulation. The subscript zero refers to the steady-state conditions, τ, Η and c are functions of altitude obtained from measurements, and affected by climatic perturbations.

2. Climatic Perturbations

In Kuhn’s concept, the following climatic perturbations were introduced (Reference AmbachAmbach, 1989):

It is true that

(2)

. (3)

The numerical values of μ₁, μ₂, μ₃ and γ are contained in previous work, when μ₁, μ₂ and μ₃ were introduced as constants and γ as a function of altitude (Reference AmbachAmbach, 1989). The data are based on measurements of the heat balance, carried out at 1013 ma.s.l. during the International Glaciological Greenland Expedition (EGIG–1959; Reference AmbachAmbach, 1963). Therefore, the results of the perturbation analysis are strictly valid for the EGIG line only.

Applying perturbation analysis, Equation (1) reads

(4)

This formulation contains the climatic perturbations δr,δH, δc and the respective alterations and due to the shift of the equilibrium line. Here Z is the altitude and Δh is the shift of the equilibrium-line altitude.

3. Shift of the Equilibrium-Line Altitude by Climatic Perturbations

By applying the numerical values of and •c/•z (Reference AmbachAmbach, 1989), the shift of the equilibrium-line altitude is obtained for various perturbations (Table 1). The perturbation δT_a = + 1 K has the biggest effect on the shift of the equilibrium-line altitude, since a rise in temperature leads to an increase in the heat-flux density of melting and a prolongation of the ablation period. Model calculations regarding the greenhouse effect have shown that in higher latitudes a warming of the lower atmosphere by a few degrees is likely (Reference Manabe and WetheraldManabe and Wetherald, 1975). The perturbation δρ_ν = + 0.25gm⁻³ corresponds to a 5% increase in relative humidity at 0°C. The effects of changes in cloudiness are small and are therefore neglected in the following. The inconsiderable influence of cloudiness on the shift of the equilibrium-line altitude is due to the “radiation paradox” (Reference AmbachAmbach, 1974). It implies that shortwave radiation decreases with increasing cloudiness, whereas longwave radiation increases so that a compensation effect occurs. Thus, the net radiation balance increases with cloudiness at an albedo larger than 0.66 and decreases at an albedo smaller than 0.66. The perturbation δc = +50 kg m⁻² is approximately 10% of the annual accumulation at the present equilibrium-line altitude. All perturbations occur with mutual feed-back mechanisms. Very likely, an increase in temperature is coupled with an increase in absolute humidity, cloudiness and precipitation (Reference Letréguilly, Huybrechts and ReehLetréguilly and others, 1991a).

Table 1. Shift of the equilibrium-line altitude by climatic perturbations at the EGIG-line with •c/•z = 0.55kgm⁻²m⁻¹ and •c/•z= –0.073 k m–1

4. Shift of the Equilibrium-Line Altitude and Regional Parameters

The gradients ∂T _a/∂z and ∂c/∂z are regarded as regional parameters. They are hardly influenced by climatic changes. On the Greenland ice sheet, various ablation regimes occur, which are characterized by the values of •T _a/•z and •c/•z. In the following, the limiting values ≤+0.55 kg m–2 m–1 are introduced. With these limiting values and with δΤ_a = 1 Κ, the shift of the equilibrium-line altitude is shown in Table 2. Alterations of •c/•z have a far stronger effect on Δh than alterations of •T _a/•z. Negative values of •c/•z result in a particularly strong shift of the equilibrium-line altitude. Such negative values of •c/•z on the Greenland ice sheet have been reported by Reference BensonBenson (1962). In Figure 1a and b the shift of the equilibrium-line altitude as a function of •T _a/•z and •c/•z is compared for equal perturbations.

Fig. 1. a. Shift of the equilibrium-line altitude by perturbations and as a function of •T _a/•z. b. Shift of the equilibrium-line altitude by perturbations and δc = 100kgm⁻² as a function of ∂c/∂z.

Table 2. Shift of the equilibrium-line altitude for δT_a = 1 K and various combinations of ∂c/∂z and ∂T_a/∂z

5. Altitudinal Profile of Net Ablation

The heat of melting of snow, superimposed ice and glacier ice is represented as a function of altitude at the EGIG line without any perturbation (Fig. 2a). The difference between the dashed line and the solid line corresponds to the melting of snow and superimposed ice, whereas the solid line stands for the altitudinal profile of the heat of melting of glacier ice, corresponding to the net ice ablation. The net ice ablation starts at the equilibrium-line altitude (Δ = 0) and increases towards the ice margin (Δh = −600 m) to about 3000 kg m⁻². The nonlinear profile results from both the increase in the heat-flux density of melting and the longer ablation period at lower altitudes.

Fig. 2. a. Heat of melting as a function of altitude with δΤ_a = 0 for the entire ablation period. The dashed range corresponds to melting of snow and superimposed ice, the solid line corresponds to net ablation. b. As Figure 2a with δΤ_a = + 1 Κ. Arrows indicate the increase in net ablation.

For comparison, the altitudinal profile of the heat of melting was calculated for the perturbation δΤ_a = + 1 Κ (Fig. 2b, dashed line). The corresponding shift of the equilibrium-line altitude is indicated on the horizontal axis and the augmented ice net ablation due to this perturbation is shown by arrows. The increase in ice net ablation varies along the profile between 300 kg m⁻² (Δh = 0) and 750 kg m⁻² (Δh = −600m). Assuming a constant slope along the profile, the averaged relative increase in net ice ablation amounts to 45%.

6. Final Remarks

Th extension of the perturbation analysis to the entire Greenland ice sheet is of speculative character, since the required data are broadly unknown. Equilibrium-line altitude, ablation area and climatological conditions vary considerably from one region to another. Therefore, it is proposed to subdivide the Greenland ice sheet into four climatic zones (Reference OerlemansOerlemans, 1991; Reference Oerlemans, van der Wal and ConradsOerlemans and others, in press).

The gradient •c/•z is of decisive importance for the stability of the mass balance (Reference AmbachAmbach, 1988). Negative values of •c/•z which are significant for the sensitive response of the equilibrium line on the Greenland ice sheet have been reported by Reference BensonBenson (1962). Along a profile at 70°Ν between 2000 m ≤ z ≤ 3100 m a.s.l. the averaged value amounts to , along a profile at 77° Ν between 1000m ≤ z ≤ 2100 m a.s.l. it amounts to . The latter value is of particular importance since the present-day equilibrium line in the Thule area is situated only slightly below this range of altitude. When the equilibrium line is shifted by climatic warming into an area of negative values of •c/•z the instability criterion becomes effective. In this case, the shift of the equilibrium line due to perturbations in temperature is significantly enlarged by a factor of 1.8 compared with the condition at the EGIG line.

With the perturbation δΤ_a = +1 Κ, an increase of the net ice ablation of 45% was obtained at the EGIG line. If this figure is considered to be representative for the entire Greenland ice sheet, the annual increase of total net ablation results in 0.45A₀ km³ year⁻¹ water equivalent, where AQ is the present annual total net ablation of the entire Greenland ice sheet. With A ₀ = 300 km³ year⁻¹ (Reference WeidickWeidick, unpublished), the annual change in total net ablation amounts to 135 km³ year water equivalent for δΤ_a = + 1 Κ. The related sea-level rise depends significantly on the share of refreezing of meltwater in the ablation area. If no refreezing occurs, the sea-level rise results in 0.37 mm year⁻¹, which is the same figure obtained by Reference Oerlemans, van der Wal and ConradsOerlemans and others (in press) by means of a more detailed analysis. A previous work by Reference Ambach, Kuhn and OerlemansAmbach and Kuhn (1989) yielded a value of 0.34 mm year⁻¹. Reference BindschadlerBindschadler (1985) took into consideration the response of the ice margin, when a perfect plastic ice sheet is assumed. For two scenarios, the corresponding sea-level rise results in 0.40 and 0.54 mm year⁻¹. However, a considerable degree of uncertainty must be accepted.

Another criterion for the stability of the Greenland ice sheet is the condition, when the total net ablation (A) equals the total net accumulation (C). Neglecting possible refreezing of meltwater and applying data from the EGIG-line on to the entire Greenland ice sheet, the condition A = C is fulfilled at a warming of δT_a ∼ +2K (Reference Ambach, Kuhn and OerlemansAmbach and Kuhn, 1989). In this case, the surface balance is zero; due to continuous iceberg discharge, the shrinking of the Greenland ice sheet is accelerated. Even if some parameters in the model are varied, the condition A = C results in between 1 and 3 Κ (Table 3). Thus, the Greenland ice sheet reacts very sensitively to climatic warming.

Table 3. Perturbations in temperature which yield the condition A = C, dependent upon various combinations of gradients •c/•z and •T _a/•z as well as initial values A₀, C₀

In view of a three-dimensional thermomechanic model applied by Reference Huybrechts, Letréguilly and ReehHuybrechts and others (1991), the run-off is increased by 40% (exactly 42%) for T _a = +1Κ as compared to 45% obtained in the present paper. The surface balance becomes negative for a warming above +2.7 K. This conclusion is in good agreement with the condition A = C for δT_a ≈ +2 Κ, which is derived in the present paper. The entire Greenland ice sheet will disappear for δT_a = +6 Κ after 20 000 years (Reference Letréguilly, Huybrechts and ReehLetréguilly and others, 1991a). Reference KoernerKoerner (1989) described observational evidence for intensive melting of the Greenland ice sheet in the last interglacial, although, according to Rech and others (1991) and Reference Letréguilly, Huybrechts and ReehLetréguilly and others (1991b), the summit of the central Greenland ice sheet seems to have survived the warm climate due to isostatic uplift.