Q ABS (x,t s ) = S o /4 S(x)a p (x,t s ).

Transcription

1 Lecture 14 ICE The feedback of exanding and contracting ice sheets has often been offered as a lausible exlanation for how the contrasting climates of the glacialinterglacial times can be both (relatively) stable. Icecover is, in general, much more reflective than the surfaces it covers. The annual variation of surface albedo is controlled by snowcover (Figure 9.2). Ocean surfaces at high latitude have albedos of ~ 10% while sea ice at the same latitudes is ~60%. The contrast been coniferous forest and an ice sheet is equally large. 1

2 Ice albedo feedback is strong and ositive. Consider a forcing that cools the surface at high latitude. Such a cooling will roduce an equatorward exansion of the ice fields roducing further cooling. In contrast, consider a forcing during glacial times that drives the surface temeratures higher. Such a forcing would reduce ice coverage, increasing insolation roducing further warming. The influence of ice cover will be most significant during between the vernal and autumnal equinoxes when insolation at high latitudes is large. As we will discuss later, orbital scale forcing of summer insolation at high latitudes is a oular exlanation for the forcing that drives the climate between ice to ice-free. The timescales for accumulation and ablation of ice sheets are quite different. This reflects the temerature deendence of these rocesses. Ablation occurs via absortion of solar radiation, by utake of sensible or latent heat delivered by warm air (and or/rain), and by calving or shedding of icebergs to the ocean or lakes. The mass balance illustrated in the figure above (from Ruddiman) illustrates that there is a relatively narrow temerature range over which the net accumulation is large. Below -20 o C, the growth rate slows as the amount of humidity the air can deliver is reduced. The mass balance becomes highly negative above -10 o C because ablation accelerates and overwhelms accumulation. The temerature where accumulation and ablation balance is known as the equilibrium line. Stable equilibrium for an entire ice sheet occurs when these rocesses are balanced when averaged over the extent of the ice sheet. For most icesheets, net accumulation with little ablation is characteristic of the center while ablation characterizes the margins. Ice flows between these regions in great ice rivers (Kamb). It is obvious from the temerature deendence of the ice mass balance that the critical issue is summer temeratures (and wind). 2

3 The hysical osition of the equilibrium line for ice accumulation deends on latitude and altitude (left, from Ruddiman). The so-called "climate oint" is the osition of the last ermanent ice cover. This osition has shifted by nearly 10 o latitude over a eriod of tens of thousands of years. The extent of ice cover at the LGM is illustrated in the figure below. Much of Siberia and Euroe remained ice free while the Laurentide ice sheet covered much of North America. The lack of ice in the eastern sector is thought to result from the desiccation of the air as it assed over the northern ice. Once ice sheets begin to grow, a ositive feedback (in addition to the albedo feedback) occurs. The "ice elevation feedback" results from the fact that at higher altitude, temeratures are colder. An ice sheet 2 km thick with a 6 o C/km lase rate will be fully 12 o C colder at to than at the margin. Once ice sheets begin to grow, there accumulation rate can increase as the elevation of the ice field increases. Note that even at 0.3 m yr -1 growth rate, nearly 10,000 years is required to grow a 3 km ice sheet. Under a climate forcing, note that the maximum size of the ice sheet reflects the time when the glacier moves from net accumulation to net ablation. This will haen long (thousands of years) after the climate begins to warm. This hase lag can be modeled using a simle sinusoidal forcing function and will be art of future homework. The weight of the ice sheet can force significant alteration to the bedrock. The density of ice is less than 1/3 that of the underlying rock (3.3 g/cm 3 ). Nevertheless, the weight of a 3 km dee ice sheet is enormous and can deress the local bedrock by ~ 1km. Because of the influence of the lase rate on ice sheet growth, such a deression can influence the growth and decay of a large ice field. Bedrock resonds to the forcing by ice with two distinct time constants. Almost immediately, the bedrock sags in an elastic resonse (about 1/3 of the total alteration). On much longer time scales, the slow flow of rock in the softer layer of the uer mantle ( km below surface) roduces a viscous resonse that occurs with a time constant ~3000 years. With the removal of ice, the oosite forcing occurs and today some arts of Canada and Scandinavia are still rebounding from the last glaciation. The slow viscous resonse acts as a ositive feedback for both growth and decay of ice sheets. Finally, getting back to the figure of the rate of ice sheet growth and decay as a function of temerature, it is clear that the retreat of an ice sheet can be substantially faster than its growth. This is undoubtedly art of the exlanation for the 'saw tooth' attern of glaciation evident in the aleoclimate records. 3

4 The ice albedo feedback was first incororated in simle models by the Russian scientist, Mikhail I. Budyko of the Leningrad Geohysical Observatory. Contemoraneously, WD Sellers ublished a similar result. Both models roduced very sensitive climates such that small forcings (such as changes in solar illumination) could drive the climate from comletely ice covered to ice free. These models assumed that everything about the climate could be characterized by the surface temerature, and that the only indeendent variable was latitude (comare to ice extent illustrated above at the LGM). The models were based on simle conservation of energy and such models are now dubbed "energy-balance climate models". These models asked about how growth and retreat of the ice cas would influence the climate forcing. The steady-state model balances three terms, insolation, IR emission, and horizontal transort of energy by the atmoshere and ocean. Q ABS (x,t s ) - F (x,t s ) = F ao (x, T s ) where x = sin φ (sine of latitude), Q ABS is the absorbed radiation, F ao is the meridional transort in the oceans and atmoshere. For each term, a arameterization is used to characterize the latitudinal variability. The absorbed solar radiation, Q ABS, is written as the roduct of the solar constant, S o, a function that describes the latitudinal deendence of the solar flux, S(x), and the absortivity for solar radiation, a (x,t s ) = 1- α (x,t s ): Q ABS (x,t s ) = S o /4 S(x)a (x,t s ). In these models, the annual mean insolation is given by something like: s(x) = P 2 (x) where P 2 (x) = ½(3x 2-1) the Legendre olynomial of order 2 in x. This attern of insolation can be comared with that shown in Figure

5 The emitted longwave flux is secified as a linear function of surface temerature, e.g.: F + ( x, Ts ) = A BT s The coefficients can be chosen to match the resonse illustrated in Figure 9.1. The transort term ( F ao ) can be secified in different ways. Budyko assumed a linear form, such that at every latitude the transort relaxes the temerature back towards its global-mean value: Sloe = B F ao ) = γ ( T T s ) Budyko s The coefficient γ is chosen so as to reroduce the observed meridional heat transort observed for the existing temerature gradient. Albedo feedbacks is introduced by assuming that ice forms when the temerature falls below some critical value and that when this haens, the albedo resonds instantaneously. As discussed above, this critical temerature is ~ -10 o C. So: o α = α, T < 10 C o where α α = αicefree, Ts > 10 C ice = 0.62 and α icefree = 0.3. With these assumtions, Budyko's model equilibrium condition becomes: A + BT s + γ(t s -T s (average)) = S o /4 s(x)a (x,x i ) The absortivity is a function of only latitude and the osition of the ice line, x i. Next, we define a new variable, I, the ratio of the local terrestrial radiation to the global average insolation: ( A + BTs ) I = (1/ 4) S Substitution into the equilibrium model yields: I + δ ( I I) = s( x) a ( x, xi ) Where δ = (γ/b). If δ is large, then meridional transort is efficient comared to longwave cooling, and the equator-to-ole gradient in I will be small. ice s o 5

6 The global area average of this exression is: I = s( x) a ( x, x ) dx The global average value of the terrestrial emission divided by the insolation is equal to the global average of the roduct of the absortivity and the distribution function for insolation. Thus the global average will be high when the absortivity is high where the insolation is high. Since I is related to T s linearly, the global mean temerature follows in the same manner. The albedo increase associated with icecover will have its greatest influence as this ice extends equatorward (no surrise). The Budyko model can be solved for the latitude of the ice boundary as a function of solar constant for articular values of δ. The solution is obtained by secifying the osition of the iceline and then solving for I at the iceline latitude. Because the albedo secification is discontinuous at the iceline, I and T s are discontinuous and the solution for x i as a function of S o is not unique. To remove this discontinuity, we solve the for the albedo on each side of the ice line and use the average at the line itself. i The general solution is shown in Figure 9.5. The model is highly non-linear (driven by the albedo contrast). For many values of the solar constant, three solutions are found. For S o above 1.2 (normalized for the value of S o that uts x i at 72 o N), the solution is a stable, ice free world. For So below 0.95, only one stable solution is resent, namely an ice-covered world. In between, we have interesting behavior. If the ice line stays at latitude > 45 o, the solution is stable - increases in S o roduce retreat of the ice. However, once the ice moves equatorward of 45 o latitude, the solution is unstable, and we lunge into a snowball Earth. Note that only a small change in S o is required (5%). 6

7 As designed, Budyko's model is highly sensitive to small solar forcing changes. The answer is, however, quite sensitive to the choice of arameters. In Figure 9.6 the solutions are shown for varying δ. For larger values (more efficient transort), x i is more sensitive to S o. This results because large δ imlies a small temerature gradient with latitude. Budyko used a value of B = 1.5 W m -2 K -1 whereas more recent estimates are closer to 2.2 Wm -2 K -1. The resulting value of δ in Budyko's calculation was 2.6 and this high value contributed to the great sensitivity of his model. Finally, as illustrated in Figure 9.7, the assumtion of single values for lanetary albedo for ice and icefree conditions in a oor assumtion and one that the model results are quite sensitive to. As we have seen, albedo at high latitudes is also associated with the larger average SZA and the resence of clouds. In figure 9.7 the albedo for icefree conditions is taken to be a (x) = a o +a 2 P 2 (x) and a (x) = b o for ice conditions with δ = 1.9, b o =.7, and a o = The value of a 2 range from 0 to -.32 (ice-free and icecovered albedo equal at ole). In Budyko's model, a 2 = 0 whereas a more realistic value is ~