Introduction

Mass balance is a primary variable determining the size, height, and shape of ice sheets. One of the fundamental inputs to ice-sheet models is surface mass balance. In order to construct or reconstruct ice sheets the temporal and spatial variation of mass balance must be known. The surface mass balance on an ice sheet is determined by climatic conditions. To determine the mass balance at a given point requires knowing the climatic conditions and elevation of that point. This paper describes a method for determining the temporal and spatial variations of mass balance for ice-sheet reconstructions. Unfortunately, there is no satisfactory equation relating mass balance to specific climate variables. It is equally impossible to specifically determine the climate conditions for the past and the future. However, GCMs and proxy climate records do allow the general climate setting to be identified. Climate settings represent a range of climatic conditions.

Reconstructions of ice sheets have typically relied on mass-balance models based either on mass-balance distribution over the Antarctic ice sheet or Greenland ice sheet, or on the present distribution of precipitation adjusted for ice-age conditions. In reality, the balance gradient of an ice sheet is determined by climate. Thus, climatic setting should be used for reconstructing ice-sheet mass-balance patterns.

Balance Gradient Construction

The balance gradient of a glacier is the change in balance with altitude. Published balance gradients (Table I) for present-day alpine glaciers, where the climate setting is known, cluster into five distinct populations. Cluster analysis indicates that 81% of the alpine glaciers can be accurately assigned to one of the five populations. The five climate settings were chosen to provide the best fit to the data. Each of the five populations is a distinct climatic regime: (I) temperate maritime, (2) sub-polar maritime, (3) sub¬polar mix, (4) polar mix, (5) polar continental (Fig. 1). In addition, a polar desert climatic zone exists over the interior of Antarctica. Each climate zone is typified by temperatures ranging from temperate to polar and by precipitation ranging from maritime to continental (Table II). The sub-polar mix and polar mix climate zones are distinguished by ELA more than by a balance-gradient change. The fact that alpine glacier balance gradients are grouped climatically indicates that the balance gradient is determined by its climate setting (Schytt, Reference Schytt1967). Hence, if the climatic setting can be identified, then the mean balance gradient of local alpine glaciers can be determined.

Table I.

Balance Gradient (B(n)) of Selected Alpine Glaciers. Balance Gradients are reported in cm 100 m⁻¹. Climate Setting, Latitude and Reference for the Balance-Gradient Information are also Listed

Fig. 1.

Balance gradient of alpine glaciers versus latitude. Balance gradient is in cm 100 m⁻¹; this is a standard unit of reporting.

Table II.

Climatic Conditions Associated with each of the five Climate Settings

The climate setting is determined at the margin of the glacier. This method was used for two reasons: (I) this is where most weather records and proxy climate data exist. (2) mass-balance distribution on present-day glaciers is determined by the regional climate setting, which reflects the regional air masses. The air masses are modified by the ice sheet. For this reason, above 2400 m all of the balance curves approach a polar continental climate setting. Below 2400 m, the climate setting at the margin determines the balance gradient. Above 2400 m, the climate setting over ice sheets is polar continental.

Table III.

Sources for Mass-Balance Versus Elevation Data used to fit the Balance Gradients for each Climate Zone

Fig. 2.

The balance curve obtained by application of Equation (1) to available ice-sheet and ice-cap mass-balance data.

Mass-balance variations within a climate zone are determined primarily by elevation. Thus a mass-balance equation must be able to calculate mass balance from elevation. Data of mass-balance changes with elevation from ice caps and ice sheets in each of the five climate zones are used to reconstruct a mean balance gradient for that zone. Data sources are indicated in Table III Data are not used from alpine glaciers because of the steep slopes that cause large fluctuations in orographic precipitation and hence mass balance. A least-squares fit is used to obtain the most representative balance gradient for the cloud of data points from each climate zone. It is not necessary to use least squares to obtain a good fit, but least squares did provide the best fit. Equation (1) is used to obtain best-fit balance gradients:

where h (m) is the altitude of the previous time step, A ₁(m) is ablation at the margin, A ₂(m) is accumulation at the margin, x ₁ m m⁻¹ is the decay exponent of ablation with elevation, and x ₂ m m⁻¹ is the decay exponent of accumulation with elevation.

The constants obtained for each climate zone are shown in Table IV and the balance gradients in Figure 2. Figure 3 shows the balance gradient for polar continental conditions, and the data points used. In constructing ice sheets at a given time step, the elevation of the ice sheet is known at the previous time step. At the first time step the elevation is the bedrock elevation.

Distance from the margin, though an important mass-balance parameter, is secondary to the effect of elevation. The distance of importance is the distance to the primary moisture source measured along the transport path. This distance is seldom known. Thus, although distance from the margin influences mass balance, it cannot be included in the time-dependent finite element, because accurate determination for paleo or future ice sheets is not possible. Even attempting to utilize this parameter in duplicating present-day Antarctic mass balance has proven problematic. On ice sheets the elevation is strongly related to the distance from the edge of the ice sheet. Because the ice-elevation term is squared in Equation (1) the distance from the ice-sheet margin though not directly included is implictly included in Equation (1).

Fig. 3.

The balance curve for the polar continental climate setting, with data points indicated. Dashed lines are the equilibrium lines.

The mass balance of an ice sheet during a glaciation cycle varies depending on the climatic setting. In particular the climatic setting below the ELA changes. The majority of the accumulation zone remains either in a polar continental or a polar desert climate zone. Ice sheets are most susceptible to changes in ablation because the ranges in ablation values are several times larger than for changes in accumulation (Ahlmann, Reference Ahlmann1948; Schytt, Reference Schytt1967; Weidick, Reference Weidick1984). An example is Jakobshavns Isbrae, where peak accumulation is 0.6 m a⁻¹ and ablation at the margin is 6.0 m a⁻¹. Peak ablation is an order of magnitude larger than peak accumulation, and the annual variation of ablation is an order of magnitude larger. This is especially true with respect to ice thickness near the margin. Hence, accurate ice-sheet reconstruction requires knowing the climatic setting in the ablation zone.

Mass-balance changes with time are caused primarily by changes in climatic setting. However, mass balance does change within climate zones, due to changes in the surface heat budget, caused by changes in atmospheric composition, changes in albedo, and changes of incoming solar insolation. The resulting changes in the balance gradient are represented by changes in ELA for each balance gradient. The shape of the balance gradient does not change, only the ELA shifts. That this is actually what happens is demonstrated by changes in climate such as the Little Ice Age. During the Little Ice Age climate zones did not change but ELA were reduced by 150–250 m (Denton and Karlen, 1975).

Table IV.

The Constants used in Equation (1) To Construct the Balance Gradient for each Climate Zone. A ₁ and A ₂ are the Ablation and Accumulation at the Margin, Respectively. X ₁ and X ₂ Determine the changes in Ablation and Accumulation with Elevation

The above method is fully quantitative and is based solely on all available data. With this method the mass-balance distribution can be calculated once the climate setting is determined from proxy records and GCM results. The climate settings and balance gradients used are not ideal; however, they do produce good results based upon all currently available data. The main weakness of this method is in the dome regions, where climate and mass-balance relationships are poorly known.