Global warming and the growth of ice sheets

Climate Dynamics (1994) 9:213-219 C, limu/¢

Uynumla © Springer-Verlag 1994

Global warming and the growth of ice sheets Tamara Shapiro Ledley, Shaoping Chu

Department of Space Physics and Astronomy and the Earth Systems Division of EESI, Rice University, Houston, TX 77251, USA

Received: 27 November 1992/Accepted: 29 March 1993

Abstract. Recent research has suggested that warmer conditions, that may result from increased levels of CO2 in the atmosphere, may induce the growth of the Northern Hemisphere ice sheets (Miller and de Vernal 1992) through the impact of warmer temperature on the water carrying capacity of the atmosphere and thus on precipitation. In this study we examine this possibility by using a coupled energy balance climate-thermodynamic sea ice model. Results indicate that if summer ice albedo is high enough, and there is some mechanism for initially maintaining ice through the summer season, then it may be possible to have ice sheet growth under the conditions of CO2 induced warming.

Introduction

The question of how ice sheet growth begins has been the subject of study for many years. Much of the early work was concerned with how large-scale ice sheet variations occurred over the past 700 thousand years. The geologic evidence showed a dominant 100 ky cycle with the smaller 20 ky and 40 ky superimposed (Hays et al. 1976). Early modeling studies were able to show that the 20 ky and 40 ky cycles were the results of changes in the distribution of solar radiation caused by changes in the Earth's orbital geometry, i.e. the Milan- kovitch theory, but the 100 ky cycle could not be ex- plained (Suarez and Held 1976, 1979). Further studies examined possible mechanisms that might produce the 100 ky cycle such as ice sheet calving in proglacial lakes (Pollard 1982, 1983) and the isostatic response of bedrock (Birchfield and Grumbine 1985; DeBlonde and

This paper was presented at the Second International Confer- ence on Modelling of Global Climate Variability, held in Ham- burg 7-11 September 1992 under the auspices of the Max Planck Institute for Meteorology. Guest Editor for these papers is L. D~imenil

Correspondence to." TS Ledley

Peltier 1991) and internal oscillations of the climate system (Maasch and Saltzman 1990; Saltzman et al. 1984; Kallen et al. 1979). While a number of these studies produced 100 ky glacial cycles, the question of the mechanisms that combine to cause an ice sheet to begin to grow is still open.

The question of what mechanisms are required to produce ice sheet growth has now become important in another climatic context, that of global warming. A recent paper by Miller and de Vernal (1992) presented geologic evidence that suggested that it may be possible for ice sheets to grow under the conditions of greenhouse warming because (1) warmer temperatures will increase the water carrying capacity of the atmosphere and thus precipitation and snowfall rates, and (2) the greatest warming is predicted to occur during the winter, having a relatively small effect on summer melting.

The question of whether an ice sheet can grow under conditions of global warming has been addressed by a number of scientists (Andrews 1991; Oerlemans and van der Veen 1984; Zwally 1989). The modeling work of Oerlemans and van der Veen (1984) suggested that under warmer conditions Greenland would lose mass because the increase in summer melt would dominate the increase in precipitation. However, the Ant- arctic ice sheet would gain mass because temperatures are so cold there that there is no summer melt, and increasing the temperatures as suggested in modeling studies would not be enough to produce melt. There- fore, any increase in temperature would only increase the snowfall rate and, thus, increase the mass of the Antarctic ice sheet. However, observations by Zwally (1989) and Bentley (1989) suggest that both the Green- land and Antarctica ice sheets are growing. Thus, the question of whether an ice sheet can grow under conditions of global warming and the understanding of the mechanisms that would produce this effect is still un- clear.

In this study we use a coupled energy balance climate-thermodynamic sea ice model (CCSI model), which includes a hydrologic cycle and a land surface

214

energy balance to compute snow melt, to examine the possibility that increased levels of CO2 in the atmosphere may induce the growth of Northern Hemisphere ice sheets and to understand the mechanisms that produce the effect.

The coupled climate-sea ice model

The CCSI model is a coupled energy balance climate- thermodynamic sea ice model that treats four parts of the climate system: the atmosphere over land, the atmosphere over ocean, a mixed layer ocean, and a ground layer. Energy fluxes are computed at the top of the atmosphere, at the atmosphere-surface interface, and between latitude zones over land and sea, and are specified in the ocean. The sea ice model (Ledley 1985a, b) is a three layer thermodynamic model, which includes conduction within the ice and snow, penetra- tion of solar radiation into the ice, surface energy bal- ances, leads and sea-ice transport. The hydrological cycle, through which moisture transport and precipitation are computed, has been developed and incorporated into the CCSI model (Chu 1991).

The parameterization of moisture transport follows that of Sellers (1973), in which transport by mean meridional and eddy motions are computed separately. Transport by mean meridional motions is a function of surface specific humidity; surface air pressure; surface vapor pressure; precipitable water; and the mean meridional wind velocity, which is a function of surface air temperature gradient. The transport by eddy motions is a function of the surface specific humidity gradient; surface air pressure; surface vapor pressure; precipitable water; and an eddy diffusivity coefficient, which is a function of surface air temperature gradient. The zonal moisture transport is a function of the zonal surface specific himidity gradient; mean zonal wind velocity; surface air pressure; surface vapor pressure; precipitable water; vertical zonal wind gradient with respect to pressure, which is a function of surface temperature gradient; and a term which weights the transport from land to water or vice versa by the effective width of the latitude belt that is land. The moisture convergence in the air over land or sea is computed from the net moisture transports into the air over land or sea divided by the area of land or sea.

The precipitation rate in each latitude zone is deter- mined using a parameterization of zonal precipitation rates developed by Schneider and Thompson (Chu 1991). The parameterization includes contributions from the moisture convergence (discussed already); evaporation-precipitation recycling within a single latitude zone, which is a function of the saturation vapor pressure; and a baroclinicity term, which is a function of the temperature gradient and the saturation vapor pressure.

Recently a land energy balance process has also been incorporated into CCSI model, which computes snowfall accumulation and ablation, and, thus, land "ice" thickness. Note that in the following discussions

Ledley and Chu: Global warming and the growth of ice sheets

the snow is assumed to turn to ice as soon as it hits the ground to obtain land "ice" thickness.

The land surface albedo for direct radiation, C*'l, and diffuse radiation, a?, are set as follwos:

a l = fa l+0 .17 ( Z - 0 . 8 8 ) - 0 . 0 1 7 (T-283.15) T~<283.15 K

[ oq +0.17 (Z-0 .88) T> 283.15 K

e~* = ~e~1-0.017 (T-283.15) T~<283.15 K

[ oq T> 283.15 K

where Z is the solar zenith angle, and ~ = 0.16, is used as a base land albedo (Thompson and Barron 1981). A minimum land ice albedo, of 0.4, was chosen for the control run from observations described by Robock (1980).

The CCSI model computes 336 time steps per year and employs a 10 ° latitude grid, with land-sea resolution in each zone distributed in accordance with current land-sea distribution. A detailed description of the CCSI model may be found in Ledley (1988, 1991).

While this model has many limitations due to its high level of parametrizations and coarse resolution, its relative simplicity allows a detailed examination of the processes involved. Thus, the results do not provide an ultimate solution, but rather provide a focus for future work with GCM models and possibly observational studies.

The experiments

The experiments performed in this study include simulations of present-day conditions, and of the effect of doubling COa. In the two present-day experiments the minimum land ice albedo is set to 0.4, and the initial land ice thickness is set to 0 m, and 10 m in all zones. In the doubling CO2 experiments the initial land ice thickness is 10 m in all zones and the minimum land ice albedo is set to 0.4, 0.55, 0.6, and 0.65 to simulate dif- ferent possible land ice surface conditions. The effect of doubling CO2 in the atmosphere is simulated by increasing the radiative forcing at the tropopause by 4 W m-2 (Schlesinger and Mitchell 1987). The experiment designations and the changes they include is listed in Table 1.

Each experiment is run to equilibrium in the surface air temperature. The criterion for equilibrium is the

Table 1. Experiments

Experiment Change in Minimum Initial number radiative land ice land ice

forcing, albedo thickness, Wm -2 m

P0 0 0.4 0 P10 0 0.4 10 C4 4 0.4 10 C55 4 0.55 10 C6 4 0.6 10 C65 4 0.65 10


temperature change is less than 0.1 K from one year to the next.

Simulations of present-day climate

300

290

Surface Air Temperature

2 1 5

280

Figure l a shows the seasonal cycles of surface air temperature simulated by the CCSI model, experiment P0, v a70 and observed (Schutz and Gates 1974) over land in the 85°N, 75°N, and 65°N latitude zones for present day. ~ 26o The parameters used for the simulation of the present- m a_ day climate are unchanged from those used in Ledley E 250 (1988, 1991a, b, 1993) with the exception of the incor- ~_ porat ion of the hydrologic cycle. The simulations in these papers, especially Ledley (1988), were compared 240 extensively with observations and the best simulation was chosen for the control run. In the 85°N latitude 230 zone the temperatures are moderate ly well simulated with the simulated temperatures being somewhat 22o warmer than observed during the fall and early winter, and somewhat cooler than observed in the spring and a summer. In the 75°N latitude zone the temperatures are well simulated in the summer, fall, and early winter, and are cooler than observed in the spring. In the 65°N latitude zone the simulated temperatures are 3.0 warmer than observed especially during the spring and summer. During the fall and winter the simulated and

~- 2 .5 observed temperatures are closer, in the 65°N zone, ,, with the simulated temperatures being somewhat warmer than observed in mid-winter.

2.0 Figure lb shows the annual average precipitation ;

rate over land as a function of latitude simulated by the : CCSI model, experiment P0, and observed (Peixoto = 1.5 and Oort 1992). The latitudinal pat tern of precipitation .£ is simulated rather well although there are some dis- ."2_

~- 1,0 crepancies in the subtropics in the Northern Hemis- s phere and in the mid latitudes of the Southern Hemis- p phere. Some of the discrepancies may be due to the a. coarse grid in the model which smooths topography 0.5 significantly.

Overall it is felt that the simulation of the seasonal cycle of surface air tempera ture over land and the annual average precipitation over land by the CCSI mod- b el compares rather well with observations despite some discrepancies which are probably due, in part, to the coarse spatial resolution of the model. This simulation will, therefore, be used as the base cased for comparison with the sensitivity experiments involving the doubling of CO2 and the changes in the minimum ice surface albedo.

In the following discussions we will focus on how the specified changes in environmental conditions as outlined in the experiments listed in Table I affect the climate conditions at 75°N. In the analysis of the results we considered all of the zones north of 60°N. We were interested in the understanding the conditions that would initiate the growth of the ice sheets and found that this was most easily studied at 75°N in our simulations. We did get ice sheet growth on land sur- faces in the 85°N zone, however in that zone there is not enough land for the initiation of large-scale ice

[] []

[ ] . . . ,----. . . . ,

' ' " 2~_ ""'"--.. [3

.......

I I B5N 75N 65N 85N obs 75N obs 65N o b s !

i . . . . . . . . . . . . . . . . . . ~ Z [ ] I

I I I [ I i I I I I I i

D J F M A M J J A S O N

Month

Precipi tat ion Rate

I I I I I

0,0 - 90

0

0 0 0 0 0 0 0

0 = I I I I g

- 60 -30 0 30 60 90

L a t i t u d e

Fig. 1. a Present seasonal cycle of surface air temperature simulated by the CCSI model, experiment P0, and observed (Schutz and Gates 1974) over land in the 85°N, 75°N, and 65°N latitude zones; b present meridional profile of the annual mean precipitation rate in m y-1 over land. Solid line represents simulated results by the CCSI model, and symbols represent the data from Peixoto and Oort (1992)

sheets. While paleoclimatic evidence suggests that most of the ice sheet growth occurred further south (Crowley and North 1991), our simulations at 65°N were too warm for ice sheet growth. Since the initiation of ice sheet growth was probably regional, and here we are mainly interested in examining the mechanisms that would promote the initial of ice sheet growth, the focus of the results at 75°N in our simulation is considered valid.

216

Can CO2 warming produce ice sheet growth?

Figure 2 shows the seasonal cycle of land ice thickness at 75°N for the series of experiments listed in Table 1. It should be noted that in the model simulations it is assumed that as soon as the snow hits the ground it has the density and thus thickness of ice, while in reality the snow would compress over time to produce ice. This may have a warming effect because the albedo of ice is lower than snow. However, the current version of the CCSI model does not include the compression of snow into ice. Since ice sheets build up over long peri- ods of time it is felt the short term process of the compression of snow to ice could be neglected in this study. Thus in Fig. 2, land ice that melts away during the summer represents seasonal snow and not the growth of an ice sheet.

For present day conditions, experiment P0, the CCSI model simulates the steady accumulation of snow from late August to mid-June. Then over the two week period from mid-June to the beginning of July all the snow melts away. This pattern is repeated when the simulation is initialized with a 10 m ice sheet in all latitude zones, experiment PI0. Thus, under present conditions large-scale ice sheet growth does not occur.

If we now double the CO2 in the atmosphere and initialize with 10 m of land ice in all zones, experiment C4, we can see the impact on the seasonal cycle of land ice thickness in Fig. 2. There is a small increase in land ice thickness through the end of May, and then a complete melting away of the ice by mid-June. Thus, doub-


ling CO2 in the atmosphere increases the period of ice free conditions during the summer, eliminating the possibility of ice sheet growth.

If we now increase the minimum albedo of the land ice from 0.4 to 0.55, experiment C55, the maximum thickness of the ice is increased slightly, and the time of the onset of melting is delayed to mid-June, with complete melting occurring by the beginning of July, just as under present conditions. Thus, ice sheet growth does not occur in this case, however, the maximum thickness of the seasonal ice does increase. This indicates that for the right conditions it may be possible for an ice sheet to begin to grow when CO2 is doubled.

In experiments C6 and C65 the minimum land ice albedo is increased to 0.6 and 0.65 respectively. The impact of these changes on the land ice thickness can be seen in Fig. 2. When the minimum albedo is set to 0.6 the simulation shows that the initial land ice thickness of 10 m decreases slowly to about 3.2 m in 120 years. Thus, the warming produced by the doubling of CO2 increases snowfall enough to slow the melt of the land ice, but does not dominate. Therefore, if this case were run for a longer period the simulation of land ice thickness would be similar to that of experiments C4 and C55.

When the minimum land ice albedo is set to 0.65 the increase in snowfall due to the warming is greater than the summer melt, and an ice sheet begins to grow, at- taining a thickness of greater than 30 m after 120 model years. Thus, under the right conditions, CO2 warming can produce ice sheet growth.

30.6

Land Ice Thickness

30.5

30.4

30.3

30.2

g 30.1 if? c/) ¢-

O 22 j--

PO PIO C4 C55 C6 C65

. ~

3.4

3.3

3.2

3.1

0.1

0

D J F M A M d

Month

\ \

--\..~es."

d A S 0 N

Fig. 2. Seasonal cycle of land "ice thickness" at 75°N for the series of experiments listed in Table 1. Integration was done until the seasonal cycle of surface air temperature reached equilibrium. Note the lines for P0 and P10 coincide

Effect of doubling CO2 and albedo changes on surface air temperatures

In the discussion it was shown that for a doubling of CO2 ice sheet growth at 75°N could occur in the CCSI model simulation if a minimum land ice albedo was set to 0.65. Therefore, in that case the air temperature must have been cool enough to either eliminate or significantly decrease the melting of the ice.

Table 2 shows the globally annually averaged temperature for each of the experiments listed in Table 1. While the model simulation of present-day conditions is somewhat cooler than observed, the results do indicate an increase of 2.7 K for a doubling of CO2. When the minimum summer land ice albedo is increased and CO2 is doubled, the globally annually averaged surface air temperature is decreased, but only by 1.1 K from experiment C4 to C65. This means that in experiment C65, ice sheet growth begins with a globally annually averaged surface air temperature 1.6 K greater than that simulated for present conditions, experiments P0 and P10, when ice sheet growth did not occur. There- fore, the changes that allow an ice sheet to begin to grow in experiment C65 must occur regionally and/or seasonally, such as on northward-facing mountain slopes and for cooler summer surface air temperature, which would reduce melt.


Table 2. Globally and annually averaged surface air temperatures

Experiment Temperature, K

P0 285.7 P10 285.7 C4 288.4 C55 288.1 C6 287.8 C65 287.3

300 Surface Air T e m p e r a t u r e - 75N

290

280

270 P -1

260 ID El.

E 250

I'-"

240

230

220

,'~. ":,<<':, ,,-.~ '.k.-~

.~'-~.~. /,','/,4,"

[ J I [ I I I I I I I I I

D d F M A M d J A S O N

Month

Fig. 3. Seasonal cycle of the surface air temperature over land at 75°N for each of the experiments listed in Table 1

Figure 3 shows the seasonal cycles of the surface air temperatures over land for each of the experiments. For present conditions, experiments P0 and P10, the summer surface air temperatures exceed the freezing point, and, thus, produce warming and melting of the land ice, for a period of about one month. Whether the simulation was initialized with 0 m or 10 m of ice makes no difference once the temperatures come to equilibrium. It should be noted that the land ice can melt once it warms to the freezing point even if surface air temperatures are still slightly below the freezing point.

When CO2 is doubled and the albedo is set to 0.4 and 0.55, in experiments C4 and C55, the summer surface air temperatures exceed those of the present-day simulations, and exceed the freezing point by about two months and one and a half months respectively. These high temperatures are enough to warm and melt the ice so that ice sheet growth does not begin.

In experiment C6 the summer surface air temperatures are lower than at present, reaching the freezing point for a period of less than a month, but does not exceed it, Fig. 3. In this case the ice melts, but at a slow rate. Once summer ice free conditions are attained it is

217

likely that the summer surface air temperature will exceed the freezing point, because the lower albedo of the land surface will permit more solar radiation to be absorbed by the system. Under these conditions ice sheet growth will not occur.

Exper iment C65 is the only case in which the summer surface air tempera ture does not reach the freezing point, and the ice sheet grows. In this case the winter temperatures are warmer than the simulated present conditions, but the late spring and summer temperatures are cooler than the simulated present conditions. Thus, for the snowfall rates simulated in these experiments (see next section) a summer surface air temperature below the freezing point seems to be an important conditon for ice sheet growth.

Effect of doubling COz and albedo changes on precipitation and snowfall

In order get an ice sheet to grow it is necessary to off- set the increased melting that results from the warming induced by the doubling of CO2. This can be accom- plished in part by sufficiently increasing the precipitation and thus snowfall rate. Since the water holding capacity of the atmosphere increases exponentially with temperature, it follows that as the temperature increases with the doubling of CO2 so should the precipitation rate.

Figure 4a shows the seasonal cycle of the precipitation rate over land at 85°N, 75°N, and 65°N for present conditions, experiment P0. As expected the warmer the temperatures the higher the precipitation rate, with the lowest precipitation rate occurring in the winter in the poleward-most latitude zone, and the highest precipitation rate occurring in the summer in the equator- ward-most latitude zone. Figure 4b shows the impact of doubling the CO2 and varying the minimum land ice albedo on the precipitation rate at 75°N. During the winter there is a very small increase in the precipitation rate as compared to present conditions. The increase in precipitation rate is larger during the late spring and early summer, with the largest changes occurring in the case with the lowest minimum albedo, C4, and thus largest warming.

However , as seen in Fig. 5a, an increase in precipitation does not necessarily mean an increase in snowfall. The snowfall rate is computed as a fraction of the precipitation rate such that as the temperature increases the fraction of the precipitation that falls as snow decreases. The fraction of the precipitation that falls as snow is 1.0 when the surface air temperature is at or below 253 K, and is 0 when the surface air temperature is or exceeds 279 K. Thus during the summer when the surface air temperatures, and thus precipitation rates, are the greatest, the snowfall rates may be lower than in the late spring and early fall, and may even be re- duced to zero, Fig. 5a.

When the COa is doubled the snowfall rates increase at 75°N during the winter in proport ion to the increase in the precipitation rate, when all of the preci-

218

1.8

1.6

Precipitation Rate over Land

~" "'" -2%~- ..... t~'" l

~ 1 . 4 f

0 - ~ 1.2

rr r - ] 0

0.8 c~

0 0 .6

0.4

0.2

.,...,.,. '"'"'..%.

... i I ~. .." J ".%

D J F M A M d J A S O N

Month

Change in Precipitation Rate - 75N

"C" 0 .8

E o6 (3) '~ 0.4 rr t - o 0.2 ,.i-,

o £).. o

-0.2 13..

.~ -0.4 O

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.~.~.o_ ~f~o _%Lo 2:Z.Lo.]

~ ' ~ . ; / ' ~ , / "

. 1 I I I I I I I I I I r

b D d F M A M d J A S 0 N

Month

Fig. 4. a Present seasonal cycle of precipitation rates simulated by the CCSI model, experiment P0, over land in the 85°N, 75°N, and 65°N latitude zones; b change in precipitation rates from present conditions (i.e. from experiment P0) at 75°N for a doubling of CO2 and a minimum land "ice" albedo of 0.4 (C4, dashed line), 0.55 (C55, dotted line), 0.6 (C6, double dotted dashed line), and 0.65 (C65, dot dashed line)


2

1.8 f 1.6

~ , 1.4

E 1.2

rr 1

~ ; 0.8

0 0.6

0.4

0.2

Snowfall Rate over Land

85N 75N 65N

P . . ,...-"-. ." .............. .-. -., .,.

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-,... .... ...1. "--. ............ .. • : t / t : \ ," -

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r I I I I I I I I I I

D J F M A M d d A S O N

Month

Change in Snowfall Rate - 75N

0.8

0 .6

v • 0.4

rr 0.2

0 O C

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J

I I I I I I L I I I I

D J F M A M J J A S 0 N

Month

Fig. 5. a Present seasonal cycle of snowfall rates simulated by the CCSI model, experiment P0, over land in the 85°N, 75°N, and 65°N latitude zones; b change in snowfall rates from present conditions at 75°N for a doubling of CO2 and a minimum land "ice" albedo of 0.4 (C4, dashed line), 0.55 (C55, dotted line), 0.6 (C6, double dotted dashed line), and 0.65 (C65, dot dashed line)

pitation that falls is snow. Thus, during the winter in experiments C4, C55, C6, and C65 the snowfall rates increase compared to the present conditions, and the largest increases occur when the min imum summer ice albedo is the lowest, and thus the temperatures are the highest (Fig. 5b). However , during the summer the only times, in the experiments performed, that the snowfall rates increase compared to present condit ions are when the summer max imum surface air temperature is at or be low the freezing point. This only occurs

in experiments C6 and C65 when the min imum surface albedos are set to 0.6 and 0.65 respectively (Fig. 5b). In experiment C6 the increase in snowfall is not enough to overcome the summer melt of ice, thus, the initial 10 m of ice melts s lowly and will eventually melt away. However , in experiment C65 the increase in summer snowfall rates coupled with the decrease in summer surface air temperatures compared to present, decreases land ice melt enough to permit an ice sheet to grow.


Summary and discussion

T h e resul t s d e s c r i b e d h e r e ind ica t e tha t if a t m o s p h e r i c CO2 is d o u b l e d and the m i n i m u m ice sur face a l b e d o is set to 0.4, the i nc rea se in snowfa l l w o u l d n o t b e e n o u g h by itself , o r wi th an in i t ia l ice shee t 10 m thick, to over - c o m e the me l t i ng a w a y of the snow and ice du r ing the s u m m e r , and s ta r t ice shee t growth . This is b e c a u s e the inc rease in s u m m e r t e m p e r a t u r e s p r o d u c e s an inc rease in the me l t ra te , and a d e c r e a s e in the f r ac t ion of p rec i - p i t a t i o n tha t falls as snow. T h e s e two fac tors d o m i n a t e ove r the i nc rea se in the snowfa l l du r ing the w in t e r due to w a r m e r t e m p e r a t u r e s to e l i m i n a t e the poss ib i l i ty of the g rowth of an ice sheet .

H o w e v e r , it was f o u n d tha t if the m i n i m u m a l b e d o of the ice sur face is r e l a t i ve ly high, on the o r d e r of 0.65, so tha t a b s o r b e d so lar r a d i a t i o n is r e d u c e d e n o u g h to m a i n t a i n s u m m e r air t e m p e r a t u r e s b e l o w the f r eez ing po in t , and if s o m e m e c h a n i s m can b e f o u n d to in i t ia l ly m a i n t a i n s u m m e r ice, as was s imu- l a t ed by spec i fy ing an ini t ia l 10 m th ick ice sheet , t hen it m a y be pos s ib l e to induce ice shee t g rowth u n d e r the cond i t i ons of d o u b l i n g a t m o s p h e r i c CO2. Such a s i tua- t ion m a y be phys i ca l ly r ea l i zab le . Fi rs t , the h ighe r surface a l b e d o m a y occur if we c o n s i d e r tha t the snow at the ice surface , e spec ia l ly newly fa l len snow, has a h ighe r a l b e d o t han the b a r e ice surface , as is a s s u m e d in the s imu la t ions in this s tudy. Second , the in i t i a t ion o f ice shee t g rowth will p r o b a b l y occur on a r e l a t i ve ly smal l -sca le , as the resu l t of loca l or r e g i o n a l effects , r a t h e r t han the coa r se scale used in this mode l . This m a y occur on high p l a t e a u s o r on n o r t h w a r d fac ing s lopes (Bi rchf ie ld et al. 1982) w h e r e the i nc iden t so la r r a d i a t i o n m a y b e r e d u c e d b y the g e o m e t r y of the l and surface.

Thus , t he resul t s p r e s e n t e d h e r e sugges t t ha t the g rowth o f an ice s h e e t m a y occur if the a t m o s p h e r i c CO2 is d o u b l e d .

Acknowledgements. This work has been supported by NASA Training Grant NOT-30072. This work has also been supported by two grants from the National Science Foundation, ATM- 8904437 and DPP-8922415. This material is also based in part upon work supported by the Texas Advanced Technology Pro- gram under grant 003604009. Computer time was supplied by the National Center for Atmospheric Research.

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Global warming and the growth of ice sheets