Mass and Energy Balances of Glaciers and Ice Sheets

165: Mass and Energy Balances of Glaciers and Ice Sheets

J GRAHAM COGLEY Department of Geography, Trent University, Peterborough, ON, Canada

Glaciers exchange energy and mass with the rest of the hydrosphere by snowfall, melting, vapor transfer, and the calving of icebergs. Melting and vapor transfer are significant in both the energy balance and the mass balance, which in consequence are intimately coupled. Glacier energy balances differ from those of other natural surfaces in having small or even negative net radiation. Emission of terrestrial radiation is limited, the surface temperature being no greater than the freezing point, but the surface albedo is always high. The limit on surface temperature, and the year-round tendency for net radiative cooling, means that sensible heat transfer is generally downward, while vapor transfer may be either upward or downward. Once conduction has raised a surface layer to the freezing point, further energy surpluses are used to melt snow or ice. In winter, the energy balance is dominated by radiative cooling. Apart from its close connection with the energy balance, the mass balance is also influenced strongly by glacier dynamics. Glaciers and the flowlines of which they are composed exhibit vertical zonation, with net accumulation at higher and net ablation (mass loss) at lower elevations. This imbalance drives, and is corrected by, the ice flow. The leading methods for the measurement of mass balance are the direct, geodetic, and kinematic methods. Direct measurement involves determining the accumulation and ablation in situ or by equivalent remote sensing, with separate treatment of calving where it occurs. Geodetic measurements require the determination of glacier thickness at two epochs; the change of thickness, approximately equal to the change in surface elevation, gives a volume balance that may be converted to a mass balance if the density of the mass gained or lost can be supplied accurately. In the direct and geodetic approaches, the ice flow is assumed to integrate to zero over any one flowline (correctly, if the entire flowline is measured). Kinematic methods are free of this restriction. They involve measurement of all of the terms in the balance and are therefore more difficult. The need for better understanding of mass balance, at socioeconomic scales from local to global, has stimulated intense study of ways to improve the measurements. Recent and impending methodological advances are coming from radar altimetry, laser altimetry, gravimetry, passive-microwave remote sensing, and interferometry using synthetic aperture radar. A subject requiring increased attention, as the measurements improve in precision and coverage, is improved quantification of the measurement errors. The best current estimates of global average mass balance are equivalent to 0.14–0.44 mm a−1 of sea-level rise, to be compared with the inferred total rate of about 1.9 mm a−1 . This figure is a composite of estimates for “small” glaciers (those other than the ice sheets), whose balance has been growing more negative since the 1960s; the Greenland Ice Sheet, which seems to have a negative balance; and the Antarctic Ice Sheet, for which the sign of the mass balance remains in doubt although its magnitude is probably within a few kg m−2 a−1 (mm a−1 water-equivalent) of zero.

INTRODUCTION from that of a drainage basin or other hydrological unit, the fact that glaciers have a basal energy balance and a Glaciers exchange energy with the atmosphere overlying (typically small) internal energy balance sets them apart. them and with the earth or ocean beneath. While the surface A more obvious distinguishing feature of glaciers is that, energy balance of a glacier is not fundamentally different because they are made of frozen water which is apt to

Encyclopedia of Hydrological Sciences. Edited by M G Anderson.  2005 John Wiley & Sons, Ltd. 2556 SNOW AND GLACIER HYDROLOGY melt, their energy and mass balances are very intimately ordinarily be exposed to the atmosphere, but there may coupled. Mass balance is the glaciological analog of the be a complicating mantle of rocky debris (Nakawo et al., water balance in hydrology. Most glaciers gain mass by 2000). Melting and freezing are regarded as loss and gain snowfall and lose mass by melting, although for some respectively; that is, liquid water is “outside” the column, glaciers, including the largest, the calving of icebergs is being assumed to run off or refreeze in a time shorter than an important term in the balance. the span over which we compute the mass balance (of ice). Glacier energy and mass balances are important in the The column also has internal energy and mass balances; for following unranked respects: example, changes of water phase are not restricted to the surface and the base. • Glacier meltwaters are dominant components of the water balance of semiarid regions downstream of Flowline glacierized mountain ranges (e.g. Su and Shi, 2002). Such regions include the Prairies of Canada, Central A flowline (Figure 1b) is a sequence of ice columns of Asia, and the Himalaya, and much of Andean South infinitesimal cross section arranged so that each column America. They also constitute an important resource for gains mass by flow from an up-ice neighbor and loses hydroelectric power generation, notably in Norway; a mass to a down-ice neighbor. To a good approximation, source of revenue from tourism; and hazards (Richard- flowlines may be identified by beginning at any point where son and Reynolds, 2000). either the slope changes sign – at a flow divide – or the • On the global scale, glaciers exchange mass with the ice thickness drops to zero, and following the direction ocean. As it is currently understood, the water balance of steepest ascent or descent to another such point. The of the ocean fails to add up and an accurate knowledge first column in the sequence has zero flow through one of glacier mass balance is required if we are to explain boundary. Most importantly, the integral of the mass flux the observed contemporary rise of sea level (Church divergence over the entire flowline is zero: a loss by et al., 2001; Munk, 2003). Glacier mass balance also flow from one part of the flowline must be compensated affects the salt balance of the ocean. by a gain somewhere else, which means that we can • Glaciers play a part both in bringing about climatic neglect the flow when estimating the mass balance of change and in helping us to document it. They are the flowline. highly reflective and so reduce the magnitude of net radiation at the Earth’s surface, and as their extents Glacier change so does their influence on the global energy balance and the general circulation. As independent A glacier is a collection of contiguous complete flowlines sources of information about environmental change, through snow and ice that persists on the Earth’s surface they are a valuable supplement to the weather stations for more than one year (Figure 1c). The two largest glaciers from which we derive information about temperature are called ice sheets: the Greenland Ice Sheet and the and other leading climatic variables. Antarctic Ice Sheet. An ice shelf consists of the floating • Gains and losses of glacier mass imply redistribution of parts of two or more glaciers. There are small ice shelves the mass of the Earth, altering its moments of inertia in northernmost Canada and Greenland, but otherwise ice with consequences for the evolution of such geophysical shelves are found only in Antarctica. Ice shelves differ quantities as length-of-day, true polar wander and the from sea ice, which is a few meters thick, in being tens geoid, and with implications for understanding of the to thousands of meters thick. viscosity profile of the Earth’s mantle (Peltier, 1998). Glacier Types and Glacier Zones

Glaciers are at or below their freezing point Tf,whichin DEFINITION OF TERMS the absence of impurities increases from 273.16 K at the −1 A Column of Ice surface at a rate of about 0.67 K km of ice overburden. Cold or polar glaciers are those in which temperature In Figure 1(a), a column extends through ice at the Earth’s T is below Tf except possibly in a surface layer, up to surface. Ice, a soft solid, deforms readily under stress, 10–15 m thick, during summer. Temperate glaciers are so we orient the column with respect to the resulting at T = Tf throughout, except in the surface layer during ﬂow. We assume that net exchanges of energy and mass winter. Polythermal glaciers have, in addition to the surface through the side-walls are negligible, an idealization which layer of seasonal ﬂuctuations, a basal layer at T = Tf and is acceptable for balance studies if not for studies of an intermediate layer in which T

e Divide p e

Grounding line Terminus w p m (ice front)

qin m (b)

Start of flowline End of flowline qout

f m p Precipitation; internal accumulation e Sublimation; condensation w Wind drifting, scouring m Meltwater runoff f Basal freezing q Ice flow (a) (c)

Figure 1 (a) A column of ice, showing leading mass-balance terms. Black arrows: accumulation (gain of mass); white arrows: ablation (loss of mass); grey arrows: throughflow. (b) A flowline considered as a sequence of ice columns. This flowline happens to have a floating terminal section. (c) Plan view of selected flowlines (thick) on a real glacier. Thin lines: contours (100-m interval) can be misleading, for strictly the adjectives apply only to balance. The slush zone (Muller,¨ 1962) is that part of the ice columns. Nevertheless, they are useful when considering percolation zone in which at least some of the added snow is energy and mass balances because the type determines lost from the column, either as meltwater runoff or by slush whether, and if so where, melting and freezing may occur avalanching. The superimposed ice zone represents either beneath the surface. exposed, refrozen percolating meltwater or the product of Because temperature decreases with increasing elevation slush avalanching. Ice is also found at the surface of the at the surface, glaciers exhibit vertical zonation (Figure 2). lowest zone, the ablation zone, but superimposed ice is This concept (Shumskiy, 1955; Benson, 1959) is the newly gained mass while exposed glacier ice implies a loss basis of most remote-sensing studies of energy and mass of mass. balances. Snow is solid condensates and precipitation, The surface volume balance of any column of the flow- including freezing rain, added to the glacier during the line is proportional to z1 − z0 = z(t1) − z(t0) and, when current year. Firn is snow added in previous years. Glacier we can neglect flux divergence and internal accumulation, ice is firn that has been compacted to a density near that of the mass balance is simply this elevation difference mul- pure ice. In the dry-snow zone, the temperature never rises tiplied by average density. The difference, and therefore to the freezing point and there is no melting. In the upper the mass balance, is zero at the equilibrium line over the percolation zone, melting occurs at the surface in summer balance year (see Figure 2) or possibly some longer span. but the meltwater remains within the snow, while in the More generally, the whole flowline or glacier is said to be lower percolation zone some meltwater penetrates to the in equilibrium if the sum of its column balances is zero. underlying firn before refreezing. This constitutes internal Figure 2 hints, unrealistically, that the flowline will grow accumulation, which should be accounted for in the mass continually thicker in the accumulation zone and thinner in 2558 SNOW AND GLACIER HYDROLOGY

e thickness h (m) and water-equivalent “thickness” (ρi/ρw)h AcZ AbZ −2 = −3 d w s (kg m or equivalently mm w.e.), ρi 917 kg m being the density of pure ice. DSZ UPZ LPZ SIZ When discussing glacial contributions to changes of sea t1 r SuZ level a natural unit to use is mm sea-level-equivalent a−1. A glacier mass balance of B kg m−2 a−1 is equal mpd −1 z to −1000(S/So)B/ρw mm s.l.e. a , S being the area of the glacier and S = 362.0Mm2 the area of the ocean and max o t0 ice shelves.

t0 Internal ENERGY BALANCE ftoi accumulation t1 Surface Energy Balance Figure 2 Cross section of a flowline, illustrating glacier The energy balance at the surface of a glacier is zonation in elevation z. Broken cross-hatching indicates snow; firn is grey (the part vulnerable to internal accumulation being hatched), and superimposed ice is Ks + Ls + H + λE + Gs + µMs = 0 (1) dark grey; glacier ice is unshaded. t0, t1: glacier surface at the start and end of the balance year; max: maximum −2 where all the quantities are flux densities in W m . Ks = elevation reached by transient glacier surface between − ↓ ↓ t and t ; ftoi: boundary between firn and ice; mpd: (1 α)K is net solar radiation, K being the incident 0 1 = ↓+ ↑ maximum depth to which meltwater percolates before solar irradiance and α the surface albedo. Ls L L refreezing or reaching the effectively impervious (but not is the net terrestrial radiation, the balance of gains from the necessarily impermeable) barrier ftoi;d:dry-snowline ↑=− 4 upper hemisphere and the surface emittance L εσTs . (surface outcrop of mpd); w: wet snow line; r: runoff limit; The surface emissivity, ε, is the ratio of surface emittance s: snowline; e: equilibrium line. DSZ: dry-snow zone; UPZ, = LPZ: upper and lower percolation zones; SuZ: slush zone; to that of a black body at the same temperature Ts,andσ −8 −2 −4 SIZ: superimposed ice zone; AbZ: ablation zone; AcZ: 5.68 × 10 Wm K is the Stefan–Boltzmann constant. accumulation zone (the set of all zones above e) Snow and ice are usually treated as black bodies, that is, ε = 1. H and λE are turbulent fluxes of sensible and latent the ablation zone. In fact, this pattern of thickness change heat from above respectively, and Gs is the conductive = × 6 −1 and differential loading is what drives the glacier flow. The flux of heat from below; λ 2.835 10 Jkg is the strongest definition of equilibrium is that it is the state that latent heat of sublimation. Finally, µMs is the energy = × 6 −1 prevails when the flow is exactly that required to preserve used for surface melting, with µ 0.335 10 Jkg the the shape of the glacier unchanged, that is, for surface ele- latent heat of fusion and Ms the surface meltwater flux −2 −1 vation z to remain constant everywhere. Figure 2 “works” (kg m s ). General methods for the measurement of terms in the sur- only because of the hidden assumption that the surface z1 face energy balance are described by Oke (1987), and adap- transforms into the surface z0 at the first instant of each new balance year, when snow turns into firn and superimposed tations to the peculiarities of glacier surfaces by Oerlemans ice into glacier ice. (2001). Accurate microclimatological measurements are demanding, and much of the effort in glacier climatology Units is devoted to parameterizing the results of research cam- paigns so that they may be used in wider contexts. Some Glacier energy balances are usually reported in W m−2, important generalizations may however be made readily. which is the standard in climatology. First, the net solar radiation on glaciers is usually Mass balance is a rate per unit of projected horizontal small because the albedo is large (Table 1). At optical area. It is nearly always reported for balance years (begin- wavelengths snow is the brightest of natural surfaces, ning at the start of winter) or their winter and summer com- although its albedo can in fact vary greatly with age, ponents, so the appropriate units are kg m−2 a−1. Several grain size, the abundance of impurities and of liquid other units are in common use. The most common are mm water, the incidence angle of the irradiant flux, and other water-equivalent a−1, numerically identical with kg m−2 a−1 factors (Warren, 1982). Exposed glacier ice is always darker −3 because 1 kg of water, with a density of ρw = 1000 kg m , than snow. Fresh snow may absorb three times less solar is 1 mm deep when distributed over 1 m2. Care is needed radiation than the ice that it covers, and its disappearance when mass and volume balances are discussed together, is followed by a marked shift in the energy balance to a for example when the measured quantity is a change of more absorbent regime in which, other things being equal, elevation or thickness. The thing to avoid is confusing ice melting is accelerated. MASS AND ENERGY BALANCES OF GLACIERS AND ICE SHEETS 2559

Table 1 Typical observed energy balances of glacier surfaces Elevation Locality (m) Period Type α Rs H λEGs µMs References QML 1180 37d (s) B 0.58 50 0 −34 −16 0 Bintanja and Reijmer, 2001 QML 1170 37d (s) S 0.79 2 9 −11 −1 0 Bintanja and Reijmer, 2001 QML 34 2a S 0.87 −1.5 2.3 −0.8 −0.1 0 Reijmer and Oerlemans, 2002 QML 1420 2a S n/a −26 26 0 0 0 Reijmer and Oerlemans, 2002 QML 2892 2a S 0.84 −1.4 2.2 −0.7 −0.1 0 Reijmer and Oerlemans, 2002 W Greenland 1155 60d (s) S 0.77 28 16 −6 −8 −30 Ohmura et al. 1994 SW Greenland 790 512d (s) G ∼0.30 103 62 −6n/a−161 Braithwaite and Zhang, 1999 N Greenland 540 35d (s) G ∼0.48 84 27 −24 −18 −71 Braithwaite and Zhang, 1999 Illimani, Bolivia 6340 21d (w) S 0.82 −12 12 −22 22 0 Wagnon et al. 2003 Pasterze, Austria 2205 47d (s) G 0.20 180 51 11 0 −242 van den Broeke, 1997 Pasterze, Austria 3225 47d (s) S 0.59 65 22 1 0 −89 van den Broeke, 1997 Peyto, Alberta 2240 17d (s) G 0.36 96 51 5 0 −152 Munro, 2001 Peyto, Alberta 2510 17d (s) S 0.73 28 32 5 0 −63 Munro, 2001

−2 Fluxes, in W m , are positive towards the surface (Rs is net radiation, Ks + Ls; melting is negative). Error estimates vary from a few to several tens of W m−2. QML: Queen Maud Land, East Antarctica. Period: s, w denote winter and summer. Type: B, blue ice; G, glacier ice; S, snow.

The longwave (terrestrial) balance is constrained by the because snow fails to accumulate. Apart from scouring fact that Ts ≤ Tf, which limits L↑ to magnitudes no greater (removal as blowing snow), the principal reason for this than about −316 W m−2. in Antarctica is sublimation. Bintanja (1998) estimates by Because the air above glaciers is often warmer than the modeling that sublimation of blowing snow may reach freezing point in summer, and is a heat source fueling −15 W m−2, twice the rate of sublimation of in situ snow intense radiative cooling in winter and at night, the sensible and ice, near the Antarctic coast. heat flux H is generally directed downwards. The latent heat flux λE is often directed downwards also because, Internal Energy Balance even when liquid water is present, the vapor pressure at the The energy balance of a small volume within a glacier surface will be appropriate to saturation at a temperature may be understood (Paterson, 1994) in terms of thermal T near f. On the lower parts of glaciers, the turbulent fluxes diffusion, advection of heat by the ice flow, and energy are enhanced by katabatic drainage of cooled air from high sources due to strain heating, including the compaction of elevations. The katabatic wind, as well as being persistent firn, and the refreezing of meltwater. The strain-heating and directionally constant, can be extremely strong. terms are of order 10−4 Wm−3 or less, and are negligible The heat exchanged with the interior of the glacier drives for balance purposes even when integrated over typical an annual variation of temperature which is confined to column thicknesses, but the refreezing of meltwater can the upper 10–15 m. However, in summertime, once an be significant. It may be expressed as µf c /Z ,wherec is isothermal surface layer at the freezing point has been the surface accumulation rate, f is the fraction of c that established, the heat flux Gs must dwindle to zero, and any refreezes in the firn, and Z the thickness over which it surplus from the atmospheric terms in equation (1) will be refreezes. µf c is of order 0.1–1 W m −2 over the year, but used for melting. This surplus is responsible for most of the Z is at most 10 m so that detectable summertime warming − −1 ablation on most glaciers, exceeding 10 m w.e. a (about of the firn is possible. −100 W m−2 over the year) at lower latitudes. It may also be responsible for advective heat transfer to the interior of Basal Energy Balance the glacier if the meltwater refreezes at depth instead of running off. Glaciers Some representative energy balances are summarized in The energy sources at the bed of a glacier are frictional Table 1. Blue ice is glacier ice exposed at the surface heat and geothermal heat. If the basal temperature is below 2560 SNOW AND GLACIER HYDROLOGY the freezing point, the heat is conducted upwards into the are extremely high. They pertain to areas of only a few tens body of the glacier. The opposite situation, heat flow from of km2 (the square of glacier width at the grounding line), the glacier into its bed, is possible, for example, when but the greatest magnitude, −425 W m−2 or −44 m ice a−1, temperate ice advances over permafrost, but very unusual. at Pine Island Glacier, is a record. The meltwater is buoyant If the bed is at the freezing point the available energy is because it is fresh, and flows upwards along the base of used to melt ice. the rapidly thinning ice shelf to where a lesser pressure Pollack et al. (1993) have compiled measurements of implies a smaller sensible heat flux (higher Tf). Sometimes, −2 geothermal heat flux. Averages are 0.09 W m for Antarc- it enters a regime in which it is actually colder than Tf and tica and 0.04 W m−2 for Greenland, with similar values ice begins to form, accreting as “marine ice” at the base of for other glacierized regions. Only Iceland and the Rocky the shelf. The latent heat flux averaged over all the Antarctic Mountains have geothermal fluxes above 0.10 W m−2, ice shelves, however, is believed to be negative. Jacobs equivalent to −10 kg m−2 a−1 of basal melting. Frictional et al. (1996) estimate it (with an uncertainty of ±50%) as heating derives from the loss of potential energy in the ice −5.4Wm−2,thatis,−500 kg m−2 a−1. column as it moves downslope. Its rate can be expressed as If there is net freezing at the base, it is reasonable (Hol- the product of basal velocity times basal shear stress, yield- land and Jenkins, 1999) to set the temperature gradient in −2 ing typical fluxes of 0.01–1 W m . Although the basal the shelf ice, and the implied heat flux, Gb = ki(∂T /∂z)|b, balance quantities are small, they are only marginally neg- to zero. Where there is net melting, the shelf tempera- ligible given the current accuracy attainable in surface ture gradient is coupled to the dynamics of the shelf ice, mass-balance calculations. They are also heat sources with- but if we neglect this coupling a crude solution is avail- out compensating sinks, so they tend to make cold glaciers able in terms of the temperature difference between surface steadily warmer. and base. Taking typical values, Ts − Tb =−30 K and ther- −1 −1 mal conductivity ki = 2Wm K ,wefindthatGb ranges Ice Shelves from −0.2to−0.02 W m−2 for ice shelves of thickness Assuming that thermodynamic equilibrium prevails (Doake, 300–3000 m. 1976; Holland and Jenkins, 1999), the contact between shelf ice and seawater must be at the freezing point of the seawater, which depends on the pressure of the overlying MASS BALANCE shelf ice and the salinity of the water. But in general the Methods of Measurement seawater and shelf ice at some distance from the contact will not be at Tf, so there is a heat source or sink, and The mass balance b of an ice column is therefore melting or freezing, at the contact. We can write b = c + a + ci + ai + cb + ab + q (3) Hb + Gb + µMb = 0 (2) where the c are accumulation rates (black arrows in where H is the sensible heat flux from the seawater, G Figure 1a), the a are ablation rates (white arrows) and b b = − the conductive flux from the shelf and µMb the latent q qin qout is the flux divergence. Subscripts i and heat flux; fluxes are positive towards the shelf base and b denote the interior and base of the column. Ablation by melting is negative. There are two complications. First, salt calving at the terminus is a special case of equation (3) in is coupled to heat because freezing increases and melting which ai is equal to minus the entire mass of the column. decreases the salinity of the seawater, altering both Tf and The mass balance of a glacier or glacier flowline of area S is the buoyancy of the water. Second, the water flow itself 1 is driven substantially by variations of temperature and B = b ds(4) salinity. S s Holland and Jenkins (1999) envisage a boundary layer When b is assumed to vary only with elevation, as is in the water flow beneath the shelf. At some elevation usual on valley glaciers, the measurements are grouped into below the base, the water is at a temperature To and salinity elevation bands and equation (4) becomes a sum of band determined by the mesoscale ocean circulation, and the averages, each average being weighted by the area of its sensible heat flux depends on the difference between Tf band. and To. Thus, the principal controls on the basal energy balance are the properties “imported” by the mesoscale Direct Measurements water flow. Direct measurements are very difficult, but Direct measurements of column mass balance take the form Rignot and Jacobs (2002) measured basal melting rates indirectly at 23 shelf grounding lines (Figure 1b). These 1 z1 b = c + a = ρ dz(5) rates are well correlated with an indirect estimate of T ,and o t z0 MASS AND ENERGY BALANCES OF GLACIERS AND ICE SHEETS 2561

The flux divergence is ignored because the column for example, both controls are likely to involve more water balances are to be integrated over the glacier, and the than in neighboring unglacierized terrain, and relatively other terms in equation (3) are either ignored or estimated small changes in either can have substantial implications as corrections. If it occurs, calving must be measured for the regional water balance. separately. Standard methods of measurement are described by Østrem and Brugman (1991), and Trabant and March Geodetic Measurements (1999) give a detailed account of a careful protocol for Geodetic measurements of mass balance have until recently fieldwork. Glaciers are dangerous places; safety in the field been used mostly as checks on the reliability of more is discussed by Selters (1999). frequent direct measurements. They rely on pairs of dated A direct measurement of b involves emplacing a stake maps or other representations of z/t to give a volume and/or digging a pit. If the stake is vertical, and does not balance that may be converted to a mass balance by making tilt, bend, or settle, measurements of stake top height above correct assumptions about density. Geodetic measurements the surface at t and t are proportional to z and z , neither 0 1 0 1 require the separate determination of z and z in geocentric of which need be known in an independent coordinate 1 0 coordinates, which introduces a quite different set of system. In the ablation zone (Figure 2), the lost mass may − concerns about accuracy. For example, the quantity which be assumed to have a constant density ρ = 900 kg m 3 should be measured is actually the rate of change of (slightly less than ρ to allow for solid impurities, bubbles, i thickness, h = z − z , and changes in bed elevation, z , intergranular voids, and macropores), so the mass balance b b arising from glacial isostatic adjustment and other causes is b = ρz/t. In the accumulation zone, the density of need to be allowed for. the mass gained must be measured in the walls of snow pits, In the accumulation zone, if the density profile remains augmented with spatially extended surveys of the variability unchanged between t and t ,thenSorge’s Law is said of z by probing or of b by shallow coring and weighing. 0 1 to apply: Density is a function only of depth beneath the The column mass balance should be determinable with surface. However, the compaction rate varies with the rate a standard error of the order of ±50 kg m−2 a−1,and of surface loading by new snow, the temperature and, usually better. Except when the measurement network is possibly, the rate of internal accumulation. Wingham (2000) very dense, an additional error is made by extrapolating modeled the compaction of dry isothermal firn, finding that from points to the whole glacier. Cogley et al. (1996) and the spatial scale of fluctuations in accumulation is of critical Trabant and March (1999) both adopt a standard error of importance. Zwally and Li (2002) modeled the effect ±200 kg m−2 a−1 for elevation-band averages of b,onthe of seasonal fluctuations of temperature and accumulation basis of the ability of single measurements to reproduce loading, reproducing observed fluctuations of z with fair elevation-band averages determined with dense networks. accuracy. Where melting occurs and the resulting meltwater Cogley (1999) showed that the uncertainty in B is not refreezes in the form of ice lenses, the situation is much significantly less than this, because measurements of b at more complicated, and at present it is necessary to invoke different elevations are nearly perfectly correlated. Sorge’s Law arbitrarily. Internal accumulation is a worrisome bias on any cold glacier with a percolation zone. It is impractical to measure it, and models for estimating it are as yet quite crude. Kinematic Measurements Internal ablation (Mayo, 1992) occurs because of the In kinematic measurements, qin or qout or both are measured conversion to heat of the potential energy of meltwater at “gates” (cross-sections), in combination with up-ice or flowing down englacial channels. When the glacier is down-ice measurements of accumulation and ablation. The known to be wet-based, the basal ablation can be estimated advantage is that the complete-flowline assumption can be as a function of the basal heat flux and frictional heating, relaxed. The disadvantage is that qin and qout cannot be although usually both it and internal ablation are neglected. measured inexpensively. They require knowledge of ice Beneath polythermal glaciers, some of it is cancelled out thickness across the gate and of the vertical distribution when meltwater freezes on reaching cold parts of the of ice velocity. In practice, only the surface velocity is bed, although the heat thus released helps to maintain known, and either it must be assumed that the glacier moves the temperate ice at its melting point. Extensive basal entirely by basal sliding, or the velocity profile must be accumulation occurs only beneath some parts of ice shelves. modeled (Kostecka and Whillans, 1988). Hubbard et al. The winter and summer balances, bw and bs,aremea- (2000) modeled glacier flow to generate a map of the flux sured separately on some glaciers. They are defined by divergence. This method is not likely to be applied widely equation (5) with z0 and z1 taken at the endpoints of the because of the amount of boundary-condition information appropriate season, and in most climates they separate the required by the flow model. An important recent advance, two main controls, winter wetness and summer warmth, of discussed below, is the ability to measure q at grounding the annual balance b = bw + bs. This is valuable because, lines by radar interferometry. 2562 SNOW AND GLACIER HYDROLOGY

Hydrological Measurements Accurate identification of orbit crossover points is essential, In the hydrological approach, precipitation and evapora- so orbital errors, discussed in detail by Davis et al. (2000), tion over the glacier are estimated along with the runoff become important. Wingham et al. (1998) give a long list of meltwater, and the water balance is solved. This is only of corrections needed before the range differences may be done routinely for Aletschgletscher, Switzerland. Bhutiyani interpreted as elevation changes. It is also assumed that the (1999) has published hydrological estimates for Siachen radar interacts with the glacier by surface backscatter, with Glacier in the Karakoram. The uncertainty in the hydro- extinction and subsurface volume backscatter being unvary- logical method is in practice much greater than in a typical ing. This appears to be a good description of the interaction direct measurement. For example, glacier runoff must be in dry-snow zones, but melting and internal accumulation separated from that contributed by unglacierized parts of the complicate matters. Bamber et al. (2001) showed that the catchment tributary to the discharge measurement station. root mean square (rms) error of radar altimetry with respect to collocated estimates from airborne laser altimetry was Surrogate Observations ∼ 7 m. The laser-altimeter measurements have decimeter- level accuracy (Krabill et al., 1995). Glacier mass balance B is well correlated with the elevation e of the equilibrium line at the end of the balance year Laser Altimetry Laser altimetry is the measurement (Braithwaite, 1984). This offers a means of inferring B of surface elevation by measurement of the two-way travel from less expensive observations of e, possibly from time of a pulse of optical radiation. Repeated measurements space, but, apart from glaciers on which B is already yield the change of elevation, leaving the density of the measured, the only regular reports of e are those of ice column to be supplied, for example, by Sorge’s Law. Chinn (1999) for New Zealand glaciers. B is equally well The position of the laser must be known precisely, and correlated with the accumulation area ratio, which is the it must be possible to reoccupy horizontal positions with area of the accumulation zone divided by the area of an accuracy comparable to the radius of the “footprint” of the glacier (Slupetzky, 1989). These economical surrogates the laser pulse. These problems have been solved by the are valuable for extending knowledge of mass balance integration of GPS measurements into altimeter systems. variability, but they are necessarily quite uncertain. Neither laser altimeters nor radar altimeters can estimate the It is easier to measure the position of a glacier’s terminus flux divergence, so it is necessary to accumulate a glacier- than to measure its mass balance. Terminus fluctuations are wide coverage of column measurements. reported annually for several hundred glaciers, as against The first satellite laser altimeter, GLAS (the Geoscience fewer than 100 reports of mass balance (Haeberli et al., Laser Altimeter System), was launched on ICESat in 1998). Unfortunately, the link between mass balance and January 2003. Airborne laser altimetry was used by Krabill glacier length is indirect. An observed annual change of et al. (2000), who found significant thinning at lower terminus position is a response to balance forcing integrated elevations and overall balance at higher elevations on over the glacier and over some indefinite span much longer the Greenland Ice Sheet over a five-year span. Arendt than one year. et al. (2002) presented balance estimates for 67 glaciers in Alaska, relying on maps drawn from aerial photography Developing and Emerging Technologies flown in the 1950s and on laser-altimetric surveys in the Conventional measurement methods are labor-intensive and 1990s. A notable methodological conclusion is that errors uncertain, but the need for better and more comprehensive in the geodetic balance are dominated by the errors in the estimates of mass balance has stimulated vigorous research old maps. The old maps offer the chance of extending the into alternatives. Some of these are discussed briefly here. historical span of the measurements, but their inaccuracy limits significantly what can be achieved. Radar Altimetry Radar altimeters, mounted on air- craft or satellites, emit pulses towards the nadir (the point Gravimetry Velicogna and Wahr (2002) have studied on the surface vertically beneath the sensor) and “track” the the joint resolving power of GLAS, GPS measurements, and waveforms of the return pulses. Tracking involves predic- the Gravity Recovery And Climate Experiment (GRACE). tion of future ranges (distances to the surface, convertible Launched in March 2002, GRACE is a pair of satellites to elevations z) from ranges detected in the immediate past. flying about 200–250 km apart along-track. Each satellite Poor predictions, as when the surface is steep or undulating, ranges to the other using microwave phase measurements. result in loss of “lock” on the surface. A cross-track interfer- When nongravitational accelerations are accounted for, the ometer on the future CryoSat radar-altimetry mission will residual fluctuations in range may be used to map the address this problem, which at present compromises the gravity field with a horizontal resolution of a few hundred glaciological use of radar altimetry outside the interiors of kilometers once every 30 days. With the aid of GLAS the two ice sheets. In those regions, however, repeated radar and GPS estimates of change in elevation, GRACE geoid altimetry has transformed our knowledge of accumulation. changes are interpreted as a result of glacial isostatic MASS AND ENERGY BALANCES OF GLACIERS AND ICE SHEETS 2563 adjustment and of changes in ice mass. Errors in the latter determine the start and end of the melt season on Alaskan of the order of ±16 kg m−2 a−1 may be expected over glaciers. 250-km spatial and 5-year temporal scales. This is quite Active-microwave sensors (scatterometers and imaging large by comparison with the minimum accumulation in the radars) measure the backscatter from pulses which interior of East Antarctica, but GRACE offers a substantial they themselves emit. Like radiometers, scatterometers improvement over current coverage. have poor spatial resolution (tens of kilometers or worse), although it can be improved by temporal Ice-penetrating Radar and Ice Cores Radars averaging. Drinkwater et al. (2001) reported standard errors − − with wavelengths in the meter range (frequencies of of 50–70 kg m 2 a 1 for a linear regression between 10–500 MHz) can yield information on the depth to reflec- climatological estimates of accumulation and NSCAT tive horizons within the ice. The vertical separation between scatterometer data in Greenland. In the percolation zone of any two such horizons, if both horizons are isochronous western Greenland, Wismann (2000) showed an excellent and of known age, is a measure of volume balance. The correlation between seasonally integrated reduction of same reasoning applies to horizons identified in ice cores. backscatter, with respect to a reference wintertime average, Only the accumulation at the core site can be measured, and positive degree-days. Because of its buried ice lenses, but the resolution in time is likely to be much better, at resulting from internal accumulation, the percolation zone least in the upper core where annual layers are recogniz- is one of the most radar-bright surface types on Earth when able. When a well-dated core record can be tied to extensive cold and dry, and one of the darkest when wet. radar surveys, a great increase in coverage results. It would Synthetic aperture radars (SARs), unlike scatterometers, be valuable if this capacity, at present largely restricted to can resolve surface features as small as 5–100 m. They century and longer timescales (e.g. Siegert, 2003), could cover much less ground per megabyte of data, so that com- be extended to shorter and more recent time spans and plete one-time coverage of an ice sheet is a major under- to smaller glaciers. Here, the aim is to find summer sur- taking (e.g. Jezek, 2002). In SAR studies of small glaciers, faces – the crusts that separate balance years. Palli¨ et al. the emphasis, to date, has been on delineating zones and on (2002) estimated errors for a traverse in Svalbard with a searching for the equilibrium line (e.g. Demuth and Pietron- 50-MHz radar. Most of the uncertainty was due to reflector- iro, 1999). Cogley et al. (2001) demonstrated a different tracking error. For 13-year-old (Chernobyl) and 36-year-old approach to SAR observations, relying on browse images (nuclear test) layers, with accumulation rates of 670 and to identify the seasonal course of melting as a function of 580 kg m−2 a−1, errors were ±134 and ±49 kg m−2 a−1. elevation. There was a clear dependence of change in image brightness upon positive degree-days accumulated between Microwave Emission and Backscatter Passive- same-day image pairs from summer days. In a similar way, microwave radiometers sense the emission of surfaces at Nghiem et al. (2001) were able to map the extent of melt which they are pointed. Microwave emission from glaciers and refreezing in Greenland, relying on diurnal variations measured by the Quikscat scatterometer. (Matzler,¨ 1987) is always reported in terms of the brightness Microwave estimates of accumulation in the dry-snow temperature T = εT − T ,whereε is the emissivity of B a zone are already among the best available, and estimates of the medium (a mixture of ice, air, and possibly water), T melting in the percolation zone have shown great promise. is its physical temperature and T , often neglected, is the a However, the ablation zone lacks a distinctive seasonal brightness temperature of the downwelling sky radiation. In microwave signature. dry snow, the emissivity is determined by volume scattering at interfaces such as grain boundaries and larger structures Interferometric Synthetic Aperture Radar In a such as buried hoar layers. Grain growth rate depends on suitable configuration (e.g. Madsen and Zebker, 1998), two both temperature and accumulation rate, and it is possible to SAR images of a ground target can be used to estimate exploit this (Zwally, 1977) to model accumulation rate as a its elevation z by interferometry. More important payoffs function of TB and T . Estimates of c from the Zwally model of InSAR can be realized once the effect of topography compare well with estimates from surface measurements is removed: the measurement of surface velocity and the (Vaughan et al., 1999). mapping of grounding lines. When the images are separated Scattering at air-water interfaces is much more effective by one day, as in the ERS tandem mission, 1995–1996, than at air-ice interfaces, and, when melting occurs at the errors in surface velocity are of the order of meters/year. surface, volume scattering ceases to be significant and the Vertical motions can be resolved as well, and near to emissivity approaches unity. Abrupt changes are observed grounding lines the contribution of tidal flexure to the in TB as meltwater comes and goes, and have been used vertical motion can be evaluated. The downstream change to estimate the extent and duration of melting over the ice in flexure is largest just downstream of the grounding line, sheets (Abdalati and Steffen, 2001; Mote, 2003). Ramage so the number of interference fringes is greatest there. and Isacks (2003) have exploited the same behavior to Accurately located grounding lines are valuable because 2564 SNOW AND GLACIER HYDROLOGY at the grounding line the ice thickness is a function of z if sastrugi and snow dunes have been migrating across the by hydrostatic equilibrium, and the discharge q may be core site. Global estimates of small-glacier mass balance estimated by integrating thickness times velocity across rely on observations spanning 1–50 years from at most a the gate. Speckle tracking (Gray et al., 2001) is a less few hundred glaciers, and some of the unmeasured glaciers accurate but more robust way of measuring velocity. It is are thousands of kilometers from the nearest measured analogous to feature tracking in optical imagery, although glacier. Information on time and space scales of variability the conditions for interferometry must still be satisfied. is essential in each of these contexts if the measurements are to be interpreted meaningfully, that is, if they are to Undersampling be given accurate error bars. This statistical problem is Sparse information is a problem common to nearly all comparable in magnitude with the observational challenges mass-balance methods. We wish to know the balance at being addressed by new technologies. M points in space during N spans of time, but have measurements only at a smaller number m of points for a Results smaller number n of spans. In whatever way m, n and M, N are related, interpolation or extrapolation will be required. Small Glaciers How do spatial and temporal variability affect our ability Measurements of mass balance are reported to the World to interpolate or extrapolate? This question can appear Glacier Monitoring Service (WGMS), Zurich, which pub- in different guises. Altimeter measurements of elevation lishes biennial bulletins (Haeberli et al., 2001) and quin- change over a few years, and interferometric estimates of quennial summary volumes (Haeberli et al., 1998). Not all ice discharge over a few days, need to be compared with measurements find their way into the WGMS database. estimates of accumulation over much longer spans. In ice Cogley and Adams (1998) and Dyurgerov (2002) are cores, measurements can be well resolved in time, but among those who have published more complete com- they may measure spatial as well as temporal variability pilations. Dyurgerov’s is the most comprehensive, and

0.12 0

100 No. of annual B 0.08 )

2 200

300

Area (Mm 0.04

400

(a) 0.00 1500 500 1000

) 500 1 − a

2 0 −

−500

−1000

Balance (kg m −1500

−2000

−2500 −90 −60 −30 0 30 60 90 (b) Latitude

Figure 3 (a) Zonal distribution of small glaciers (shaded histogram, left axis; Antarctica excluded) and annual mass-balance measurements (thick line, right axis). (b) Average annual balances of measured glaciers, 1961–1990 (open circles: record length n < 5 years; solid circles: n ≥ 5 years) MASS AND ENERGY BALANCES OF GLACIERS AND ICE SHEETS 2565 is available in spreadsheet form on CD-ROM. That of 0.6 Cogley and Adams (1998; “CA” hereafter), covering only annual mass balance, may be obtained from http: 0.4 //www.trentu.ca/geography/glaciology.htm. Most reported measurements are direct measurements. Conven- 0.2 tional geodetic measurements are unlikely to alter the pic- 0.0 ture greatly, although recent laser-altimetric measurements (Arendt et al., 2002) have established more conﬁdently that T anomaly (K) NH −0.2 Alaska is a substantial contributor to sea-level rise. − 2 (a) 0.4 Small glaciers occupy between 0.6 and 0.7 Mm , consist- 200 2 2 ing of 0.539 Mm plus about 0.070 Mm in Greenland and ) 1 −

an undetermined extent in Antarctica not belonging to the a 0 2 ice sheet. At present (2004) the CA dataset contains mea- − surements from 310 glaciers dating back to 1885, although −200 continuous measurements began only in the 1940s and a worldwide picture is only available after 1960. The spa- − tial distribution of measured glaciers is uneven (Figure 3a), 400 with high northern latitudes, Patagonia, and Tibet underrep- resented at the expense of less remote regions (Europe and −600 western North America). Glaciers with calving terminuses B (kg m Global average (b) −800 (19 of the 310) are not well represented. Most records are 600 short. The modal length is 1 year, only 51 are 20 years or longer, and in no one year have as many as 100 glaciers 400 been measured (Figure 4c). To set against this evidence of 2 sparse coverage, there are about 2500 km of ice per mea- 200 sured glacier, to be compared with about 30 000 km2 of land per station for temperature climatologies. Moreover, B of annual No. 0 CA note that three quarters of the small-glacier ice has at (c) 1940 1950 1960 1970 1980 1990 2000 2010 least one annual balance measurement within 400 km, and that the decorrelation distance for balance time series is Figure 4 (a) Northern Hemisphere surface temperature about 600 km. anomalies; (b) small-glacier mass balance from the CA Spatial variations in mass balance are difficult to iden- dataset, with shaded confidence region; crosses show effect of correcting for spatial bias; and (c) annual tify, at least at zonal resolution (Figure 3b). Allowing for measurements contributing to each pentadal average uncertainties, most measurement series have averages indis- balance (Jones and Moberg, 2003.  American Meteoro- tinguishable from zero, but together they give a global logical Society. http://www.cru.uea.ac.uk/cru/data arithmetic average of −165 ± 34 kg m−2 a−1 for the refer- /temperature/) ence period 1961–1990. (Here, and below, uncertainties are twice the standard error.) The error estimate is some- balances were matched to regional anomalies. Figures 4(a) what optimistic, and the balance estimate is biased by the and 4(b) help to justify the modeling of mass balance as neglect of internal accumulation in cold glaciers and of a function of temperature (Wild et al., 2003), and more internal ablation in temperate glaciers, and other factors importantly show that two independent measures agree including possibly the spatial unevenness of the measure- in identifying the late twentieth century as a period of ment network. significant global change. As records grow longer, however, the evolution of mass balance presents an increasingly coherent picture Greenland Ice Sheet (Figure 4b). The world’s small glaciers were close to The ice sheets are too big for an integrated measurement of equilibrium in the 1960s and have been losing mass since mass balance to be practical. Instead the aim is to compile then at a growing rate. When spatial bias is corrected the results of separate evaluations of each component. with an interpolation algorithm, the global average for For Greenland, accumulation is the best-known com- 1961–1990 increases to −123 kg m−2 a−1, which is a best ponent. There are up to 400 column measurements, and estimate. three recent analyses interpolate to unmeasured parts of Mass balance is well correlated with temperature the ice sheet in different ways, hand contouring (Ohmura (Figure 4a; r =−0.79 for the spatially corrected balance). et al., 1999) and kriging (Calanca et al., 2000; Bales et al., The association would no doubt be closer if regional 2001). Figure 5 (Cogley, 2004) is constructed with another 2566 SNOW AND GLACIER HYDROLOGY

Accumulation (kg m−2 a−1 )

Figure 5 Accumulation on the Greenland Ice Sheet. The rate is below 100 kg m−2 a−1 in the northern interior and approaches 1400 kg m−2 a−1 in the southeast (Reproduced from Cogley (2004) by permission of American Geophysical Union) MASS AND ENERGY BALANCES OF GLACIERS AND ICE SHEETS 2567

Standard error of accumulation (kg m−2 a−1)

Figure 6 Standard error of accumulation on the Greenland Ice Sheet. Errors (for an assumed 30-year span) are as low as 8 kg m−2 a−1 in the interior and reach several hundred kg m−2 a−1 in places near the margin, where the interpolation algorithm falters for lack of information (Reproduced from Cogley (2004) by permission of American Geophysical Union) 2568 SNOW AND GLACIER HYDROLOGY interpolation algorithm. It is broadly similar to the other relationships to predict calving velocity as a function of maps, but the algorithm has the advantage of generat- ice thickness or water depth. The latter were taken mostly ing formal estimates of the error at each interpolation from bathymetric charts. The estimates range from −73 to point (Figure 6). The result is an accumulation estimate of −132 kg m−2 a−1 with an uncertainty estimated as ±70%. 299 ± 23 kg m−2 a−1, very close to the Ohmura, Calanca, Rignot et al. (1997) used InSAR to locate the grounding and Bales estimates, 297, 290, and 305 kg m−2 a−1 respec- lines of 14 outlet glaciers of the northern Greenland Ice −2 −1 tively. Uncertainty is small in the interior of the ice sheet, Sheet and to measure qout as 136 kg m a from an area where measurements are relatively abundant, and grows of 0.332 Mm2. The surface mass balance, estimated from rapidly towards the edge where measurements are few (or a map of accumulation and a simple degree-day model nonexistent because the balance is negative and no snow of melting, was 113 kg m−2 a−1, so the total balance was survives). B =−25 kg m−2 a−1. The calving flux at the ice front was The spatial and temporal variability of accumulation has several times smaller than the grounding-line flux, requir- of late received considerable attention (e.g. McConnell ing that basal melting of the floating ice be of the order − − et al., 2001), stimulated by advances in geodetic measure- of thousands of kg m 2 a 1. Rignot et al. (2001) found − − ment technology which have led to improved estimates of B =−2kgm 2 a 1 for a different but overlapping set of surface elevation change (e.g. Krabill et al., 2000; Davis north Greenland glaciers. On 8 of 12 outlet glaciers, the et al., 2000). The variability is such that the patterns of ele- grounding line retreated inland over 1992–1996. This is vation change determined by altimetry can be understood consistent with the laser-altimetric observations of near- mostly in terms of short-term fluctuations of the compaction coastal thinning, which also imply that the margin should rate and the accumulation rate (Braithwaite and Zhang, be retreating where it is on land. There is some limited 1999) and spatial “glaciological noise”. evidence for this (Sohn et al., 1998). According to the altimetry, average elevation change in Not all of the balance components are accompanied by the interior of the ice sheet is close to zero, implying detailed error estimates, but it seems likely that the mass b = c + q 0 because ablation a is near to zero. The balance of the Greenland Ice Sheet still cannot be distin- kinematic measurements of Thomas et al. (2001) support guished reliably from zero. Krabill et al. (2000) estimated it − − this conclusion. The altimetry shows, however, that parts as −27 kg m 2 a 1, but this figure awaits analysis of errors of the ablation zone are thinning rapidly. and more complete documentation. Nevertheless priorities The ablation zone occupies 10–15% of the Greenland for future study are clear. Radar interferometry and laser Ice Sheet. Surface measurements are too few for a coherent altimetry both suggest that the closest attention should be picture to be drawn from them, and the most comprehensive given to lower altitudes of the ice sheet, where surface mea- current understanding of surface ablation derives from surements are fewest and the energetics of melting and the observations of melt extent and duration (Abdalati and dynamics of thinning need to be better understood. Steffen, 2001) and from modeling. Mote’s model (2003), which builds on passive-microwave observations of melt Antarctic Ice Sheet − − duration, yields a 12-year average of −155 kg m 2 a 1 for The Antarctic Ice Sheet is seven times the size of the meltwater runoff from the whole ice sheet. Wild et al. Greenland Ice Sheet: 12.3 Mm2 of conterminous grounded (2003) parameterized ablation as a function of temperature ice as against 1.7 Mm2, plus about 1.6 Mm2 of ice shelf and and surface elevation to obtain an equivalent estimate of ice rises. − − −152 kg m 2 a 1. Accumulation reaches less than 25 kg m−2 a−1 in the inte- The estimates given so far imply a net surface mass rior of East Antarctica and exceeds 2500 kg m−2 a−1 in − − balance of about 150 kg m 2 a 1. Zwally and Giovinetto the mountains of the Antarctic Peninsula (Turner et al., (2000), using passive-microwave and thermal infrared 2002). Vaughan et al. (1999) compiled surface observa- satellite observations and a parameterization of meltwa- tions of accumulation and extrapolated them using a model ter runoff, estimated this quantity as 128 kg m−2 a−1,and based on microwave brightness temperature (Zwally, 1977) summarized earlier estimates ranging between 97 and as a background field. They estimated accumulation to be 211 kg m2 a−1. 149 kg m−2 a−1 over the grounded ice and 166 kg m−2 a−1 It remains to evaluate the losses due to calving, or prefer- over the entire ice sheet. Giovinetto and Zwally (2000) con- ably the discharge qout across the grounding line. Many toured the surface observations by hand and constructed outlet glaciers, particularly in north Greenland, have float- from the contour map a grid of interpolates by eye. The ing terminal sections. We would prefer the grounding-line resulting estimate of accumulation over the entire ice sheet flux because the floating ice has already made its contribu- was 159 kg m−2 a−1, to which they applied somewhat con- tion to sea-level rise, and because the basal mass balance jectural bulk adjustments totalling −10 kg m−2 a−1 for loss of the floating terminuses is difficult to evaluate. Bigg by melting and deflation (wind scouring and sublimation) (1999) estimated ablation due to calving using empirical in coastal areas. MASS AND ENERGY BALANCES OF GLACIERS AND ICE SHEETS 2569

Although ablation by meltwater runoff is small in Antarc- presented is not the least significant contribution made by tica, not all of the ice sheet has a positive surface bal- this work. ance. Bintanja (1999) reviewed the widely distributed When pooled, the Rignot–Thomas measurements give a areas of blue ice, where there is no snow at the sur- balance of −3 ± 4kgm−2 a−1. If the Vaughan accumulation face. Mass balance minima range from −350 kg m−2 a−1 field is used in place of the Giovinetto–Zwally field, the in northern coastal areas where melting is significant to balance becomes 6 ± 4kgm−2 a−1. It is not clear how much smaller magnitudes (∼−50 kg m−2 a−1) at higher eleva- weight should be given to the reservations that led Rignot tions where sublimation is low because temperatures are and Thomas to prefer the Giovinetto–Zwally field, and, by low. Winther et al. (2001) estimated the extent of blue analogy with the error analysis for Greenland (Figure 6), icetobe0.12–0.24Mm2. Over about half of this extent the standard error of accumulation may not be as small as of the negative balance is due to melting and over the their chosen 5%. If it is larger, or if equal weight is given other half it is due to scouring and sublimation. These to the two accumulation fields, then the Rignot–Thomas results suggest that blue-ice areas reduce the mass balance results resemble all earlier estimates of Antarctic mass of the grounded ice sheet by perhaps 0.5–4 kg m−2 a−1, balance in showing no significant difference from zero. which is negligible given the present accuracy of accu- A further source of uncertainty, not considered by Rignot mulation estimates at the ice sheet scale. Surface ablation and Thomas, is the difference in time span between the may not be negligible, however, in some of the smaller accumulation maps, based on measurements covering up to basins. several decades, and the quasi-instantaneous 1996 InSAR Rignot and Thomas (2002) reported ice fluxes obtained at measurements of discharge. Too little is known about the grounding lines by InSAR and coupled these estimates with temporal variability of the discharge of ice streams and the Giovinetto–Zwally estimates of accumulation to yield outlet glaciers for us to be confident that the InSAR the nearest approach to date to a whole-glacier estimate of snapshots are good estimators of multidecadal averages; this B for catchments in Antarctica. The 33 catchments, labelled is a question requiring more systematic attention. in Figure 7, cover 7.2 Mm2. Their ice discharges range from Radar-altimetric surveys of thickness change (Wing- a negligible 3 kg m−2 a−1 (from the inactive Ice Stream C) ham et al., 1998; Davis et al., 2001) agree with the to more than 1000 kg m−2 a−1 (Smith and DeVicq Glaciers InSAR/accumulation estimates in finding no significant in the Amundsen sector). Their balances (accumulation change in the interior of East Antarctica. (This is a puz- minus discharge) range from 131 kg m−2 a−1 (Ice Stream C) zle awaiting an explanation, for in a warmer world, as in to well below −500 kg m−2 a−1 (the small outlet glaciers in Figure 4(a), the atmosphere should deliver more snow to the Amundsen sector). The spatially variable picture thus Antarctica.) In the Amundsen sector of West Antarctica, the

Weddell Sea Larsen Ice Shelf Filchner Ice Shelf Amery Ice Shelf

Bellingshausen Sea

SMI Amundsen KOH Sea DVQ LAN Ross sea 0 1.5 km/year

Figure 7 Major basins of Antarctica. The underlying field is the balance velocity (the column-averaged velocity qout/ρi which makes b + qout zero in equation 3) (Reprinted from Rignot and Thomas, 2002. Mass balance of polar ice sheets. Science, 297, 1502–1506.  2002 American Association for the Advancement of Science). A color version of this image is available at http://www.mrw.interscience.wiley.com/ehs 2570 SNOW AND GLACIER HYDROLOGY two methods also agree in finding a substantial negative measurement technology is the subject of vigorous research balance. Shepherd et al. (2002) measured inland migra- and is improving rapidly. In the near future, the mass tion of the grounding lines and rapid thinning of Pine balance of the two ice sheets will become known with an Island, Thwaites, and Smith Glaciers (Figure 7). They uncertainty small enough to state with confidence whether estimated the balances of Thwaites and Smith Glaciers they are growing or shrinking. At present the weight of as −22 ± 8kgm−2 a−1 and – 233 ± 34 kg m−2 a−1 respec- evidence, some of it circumstantial, implies that both have tively for 1991–2001, while Rignot and Thomas (2002) negative mass balance, but the errors are not adequately gave −123 ± 92 kg m−2 a−1 and −698 ± 222 kg m−2 a−1. quantified and for the Antarctic Ice Sheet, in particular, The discrepancies obviously require investigation, but may a conclusion about the sign of the mass balance would be consistent with other evidence suggesting that the thin- be premature. For the small glaciers, the picture is more ning began rather abruptly at some time during the 1990s. clear. Their balances are negative on average and have been An abrupt onset would invite the speculation that the thin- growing more negative since the 1960s, but here also there ning is due to changes in ocean circulation leading to is scope for a more thorough assessment of errors. increased basal melting near the grounding line (Rignot and The best balance estimates, in sea-level equivalents, are: Jacobs, 2002). for small glaciers, as calculated above, 0.21 mm s.l.e. a−1 The mass balance of the ice shelves is difficult to assess, with a poorly quantified uncertainty of the order of 0.06 mm mainly for lack of complete estimates of its components. s.l.e. a−1; for the Greenland Ice Sheet (Krabill et al., 2000), The grounding-line flux estimates of Rignot and Thomas 0.13 mm s.l.e. a−1 with no estimate of uncertainty; and for − (2002) sum to 754 Gt a 1. If they account for the same the Antarctic Ice Sheet (Rignot and Thomas, 2002; Vaughan fraction of shelf nourishment as the fraction of grounded et al., 1999), between 0.10 and −0.20 mm s.l.e. a−1 with an ice from which they come, 59.3%, the total grounding- uncertainty of at least 0.13 mm s.l.e. a−1. The sum of these − line flux would be 1271 Gt a 1. Vaughan et al. (1999) estimates, 0.14 to 0.44 mm s.l.e. a−1, is a small proportion estimated the surface accumulation on the ice shelves as of the contemporary rate of sea-level rise, 1.84–1.91 mm − 478 Gt a 1. Several of the most northerly shelves have s.l.e. a−1 (Peltier, 2001; but see also Miller and Douglas, disintegrated in recent years, and surface melting has been 2004, and references cited therein). implicated in these events, but quantitatively it makes little contribution to the balance; Jacobs et al. (1992) gave a − −1 − −1 crude estimate of 36 Gt a .Theyalsogave 2016 Gt a REFERENCES for the rate of calving, based mainly on iceberg censuses. From oceanographic observations, Jacobs et al. (1996) − Abdalati W. and Steffen K. (2001) Greenland ice sheet estimated the basal balance to be −756 Gt a 1.Thetwo − −1 melt extent: 1979–1999. Journal of Geophysical Research, inputs and the three outputs sum to 1059 Gt a ,or 106(D24), 33983–33988. − −2 −1 708 kg m a . Arendt A.A., Echelmeyer K.A., Harrison W.D., Lingle C.S. and Although there are scattered in situ and remotely sensed Valentine V.B. (2002) Rapid wastage of Alaska glaciers and measurements, at present the only realistic way to get a their contribution to rising sea level. Science, 297, 382–386. broad view of the basal mass balance of an ice shelf Bales R.C., McConnell J.R., Mosley-Thompson E. and Csatho B. is to model it using an ocean circulation model. 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