the small ice cap instability in seasonal energy balance models

Climate Dynamics (1992) 7 : 205-215

limui¢ Uynumicl

© Springer-Verlag 1992

The small ice cap instability in seasonal energy balance models

$in Huang and Kenneth P Bowman

Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, 105 S. Gregory Avenue, Urbana, IL, 61801 USA

Received February 19, 1991/Accepted September 15, 1991

Abstract. Results from a two-dimensional energy balance model with a realistic land-ocean distribution show that the small ice cap instability exists in the Southern Hemisphere, but not in the Northern Hemisphere. A series of experiments with a one-dimensional energy balance model with idealized geography are used to study the roles of the seasonal cycle and the land-ocean distribution. The results indicate that the seasonal cycle and land-ocean distribution can influence the strength of the albedo feedback, which is responsible for the small ice cap instability, through two factors: the temperature gradient and the amplitude of the seasonal cycle. The land-ocean distribution in the Southern Hemisphere favors the small ice cap instability, while the land-ocean distribution in the Northern Hemisphere does not. Be- cause of the longitudinal variations of land-ocean distribution in the Northern Hemisphere, the behavior of ice lines in the Northern Hemisphere cannot be simulated and explained by the model with zonally symmetric land-ocean distribution. Model results suggest that the small ice cap instability may be a possible mechanism for the formation of the Antarctic icesheet. The model results cast doubt, however, on the role of the small ice cap instability in Northern Hemisphere glaciations.

Introduction

Geologic data reveal that there have been abrupt changes in the past climate (Berger and Labeyrie 1987; Crowley and North 1988). Some examples of abrupt climate changes are the terminations of Pleistocene glaciations (Broecker and Van Donk 1970), the sudden cooling during the last deglaciation in the Younger Dryas pe- riod (Dansgaard 1987), and the formation of the ice sheets on Antarctica and Greenland (Crowley and North 1988). Compared to the time scale of the Earth's history, abrupt may refer to climate changes occurring over a few thousand years.

Offprint requests to: J Huang

Some model studies of the mechanisms of abrupt climate change suggest the importance of multiple equilibrium states of the climate system for given boundary conditions (Budyko 1969; North et al. 1981). In simple energy balance models (EBMs), several equilibrium climate states with different-sized permanent ice caps can be found for identical external forcing. Multiple equilibrium climate states have also been found in a complex ocean-atmosphere general circulation model (GCM), where different modes of the global oceanic thermohal- ine circulation can occur for identical external forcing and boundary conditions (Manabe and Stouffer 1988). Because the multiple equilibria usually occur as a result of an instability, the transition between different states at critical points may be quite abrupt.

It is well known that multiple equilibrium states and abrupt climate changes can occur in annual-mean EBMs. Figure 1 shows the equilibrium ice-line latitude versus solar constant for an annual-mean EBM. The slope of the curve represents the model sensitivity and the sign of the slope indicates the stability of the equilibrium solution. If the ice shrinks as the solar constant increases, the solution is stable, otherwise it is unstable (North et a1.1981). Suppose the initial state is ice-free; that is, the ice line is a 90 ° latitude. Reducing the solar constant does not produce any ice cover until the solar constant decreases to a critical value at about 1.015, at which point the ice line jumps to about 60 ° latitude. Re- versing the process does not reverse the results. If the climate starts with a stable finite-sized ice cap, the ice cap will persist until the solar constant is increased to about 1.03, where the ice line is at about 75 ° latitude. Any further increase of the solar constant makes the climate jump back to the ice-free state. This hysteresis phenomenon is called the small ice cap instability (hereafter referred to as the SICI).

Both EBMs and GCMs exhibit the other instability seen in Fig. 1 for large ice caps that extend into the sub- tropics. This is referred to as the large ice cap instability, but it is not thought to be relevant to the history of the Earth's climate, since the stable equilibrium solution is a completely ice-covered Earth.

206

Latitude

6 0

3 0

Huang and Bowman: The small ice cap instability in seasonal energy balance models

0 i ~ p J i

0 , 95 1 1 .05 1.1 1 .15 1.2 1 ,25 1.3

Normalized solar constant 1,35

Fig. 1. Equilibrium solutions of a one-dimensional annual-mean EBM. The latitude of permanent snow and ice is plotted as a function of normalized solar constant

in the Northern Hemisphere in their model. However, they fixed the albedo for the whole year instead of al- lowing it to change with the seasonal cycle of temperature. In the models of Mengel et al. (1988) and Lin and North (1990), the albedo is allowed to vary seasonally, but they discussed only the cases with zonally symmetric geography. They found that the SICI and abrupt ice line jumps did exist in their models.

In this study, the existence of the SICI in a two- dimensional EBM with a realistic land-ocean distribution and with a seasonally-varying albedo will be examined. A hierarchy of one-dimensional seasonal EBMs with different land-ocean distributions are also to be examined to understand the results of the two-dimensional model. The purpose is to investigate under what conditions the SICI exists in seasonal EBMs, how the heat capacity and land-ocean distribution affect the strength of the albedo feedback and the existence of the SICI, and whether the present land-ocean distribution favors the SICI.

The physical reasons for the existence of the SICI in annual-mean EBMs have been discussed by North (1984). They are determined by the model 's diffusive properties, the longwave cooling parameterization, and the nonlinear albedo feedback. If the solar constant is decreased, the albedo feedback process can make the ice cover grow unstably. The feedback can be conceptually divided into two steps. 1. In response to a reduction in solar radiation, the temperature decreases. 2. The temperature decrease leads to more ice and higher albedo, which causes a further decrease in the absorbed solar radiation (back to 1). The cooling due to the increased ice cover affects nearby regions through the heat transport, which is parameterized as diffusion in simple EBMs. This albedo feedback is responsible for the SICI in EBMs.

Two factors determine the strength of the albedo feedback in annual-mean EBMs: how much the ice cover increases in response to a small cooling, and how much the solar heating decreases as the ice cover grows. The temperature gradient near the ice line determines how much the ice cover grows in response to a cooling; while the difference between the albedos of ice-free and ice-covered areas determines the reduction in solar heating in response to increased ice cover (Held and Suarez 1974). Both a weaker temperature gradient and a larger albedo contrast will cause greater sensitivity.

North (1975) found that the SICI disappears if the equinox solar insolation distribution is used instead of the annual-mean insolation. This follows logically, since at the equinox there is no insolation at the pole, and thus there can be no albedo feedback. The effect of the full seasonal cycle of insolation has not been examined in detail. North et al. (1983) studied the properties of multiple equilibrium climate states in a two-dimensional seasonal EBM with a realistic land-ocean distribution. They found a SICI-like abrupt expansion of snow area

Models and numerical methods

Both one- and two-dimensional EBMs are used for numerical experiments (see North et al. 1981; 1983 for a review). The 2-D version of the model has a realistic present-day land-ocean distribution, and the variables are functions of longitude and latitude. The 2-D model equation can be written:

0T(2, 0, t) c(~, 0)

8t =DV2T+Qs(O,t)[1-c~(T)]-(A +BT), (1)

where surface temperature T(,~, 0, t) is a function of longitude ~, latitude 0, and time t.

The local heat capacity C0~, 0) depends on the surface characteristics. The heat capacity of the ocean, Cw, is assumed to be that of a 75 m-deep mixed layer, which gives a linear radiative relaxation time constant r = C/B of 4.6 years. The heat capacity of land, CL, is set to Cry~ 30, giving a time scale of - 1.8 months. The heat capacity of snow-covered land is assumed to be the same as that of snow-free land. Following North et al. (1983), perennial sea ice is prescribed with an intermediate heat capacity of Cz= Cw/6.5 in those areas where sea ice is presently observed to exist. The value of Ct has been chosen to provide the best simulation of the seasonal cycle in high latitudes in the Northern Hemisphere. These settings are very close to those used by North et al. (1983).

Horizontal heat transport in the model is assumed to be proportional to the temperature gradient, DVT, where D is a constant. The local contribution of the heat transport to the energy budget takes the form of the div- ergence of the transport V'DVT, which is equivalent to diffusion of heat. This diffusive form for the heat transport is an approximation to the heat transport of the atmosphere and ocean and is the principal reason for the computational speed of EBMs compared to GCMs.

Huang and Bowman: The small ice cap instability in seasonal energy balance models 207

Even though the actual instantaneous heat transporta- tion does not necessarily behave diffusively, the long- term average behavior of atmospheric and oceanic ed- dies does tend to smooth out the horizontal temperature field by transporting heat down the temperature gradient. In this study, a constant diffusion coefficient D = 0 . 5 5 W m - 2 ° C - I is chosen for simplicity.

The term QS(O, t) [1 -c~(T)] represents the solar radiation absorbed at the surface, where S(O, t) is the normalized insolation distribution. Following Mengel et al. (1988), we take the co-albedo ( 1 - a ) as a step function of temperature:

1 - a = ao + a2Pz(sin 0) for T > - 2.0 ° C (2)

1 - a = 0 . 4 1 for T < - 2 . 0 ° C

where P2(s in0)=(3 s in20 -1 ) /2 , ao--0.69 and a2= -0 .12 . Once the temperature drops below - 2 . 0 ° C , snow cover is assumed to form. If the temperature rises above freezing, snow or ice is assumed to melt instanta- neously. Snow cover persisting through the summer is referred to as permanent snow or ice cover.

The outgoing infrared radiation is represented as a linear function of the surface temperature, A + B T (Bu- dyko 1969). On the basis of satellite radiation observa- tions, we choose A = 205.0 W m -2 and B = 2 . 0 9 W m - 2 ° C -1 following North et al. (1981).

The model Eq. (1) is solved numerically by finite-difference schemes in both time and space. Time marching allows the surface albedo at every grid point to change instantly with the temperature. The finite-difference equation is solved by a multigrid method (Brandt 1977; Bowman and Huang 1991). In these experiments, the finest grid level has 128 grid points in longitude and 64 grid points in latitude. The coarsest grid level has 8 grid points in longitude and 4 grid points in latitude. The time resolution is 48 steps per year. For the parameters we use, it normally takes about 30 years (or 540 time steps) for the model to reach a steady solution.

In applying the finite-difference equation on a sphere we face the problem of converging meridians near the poles. In this study, we use semi-regular grids created by skipping some grid points in high latitudes to keep the physical grid point spacing as uniform as possible. The grid point skipping is made for all the grid levels except the coarsest one. The relative error in the global-mean temperature is about 10-4 on the finest grid. The model also conserves energy to second order.

The 1-D version of the model has the same basic processes and parameterizations as the 2-D version, except all variables are assumed to be zonally symmetric. The co-albedo of snow-free surfaces is set 0.71 independent of latitude, unlike the 2-D model. Other parameters, such as, the long-wave radiation and diffusion coef- ficients, are the same as those in the 2-D model. The 1-D model equation can be solved efficiently with a direct solver. In 1-D model experiments, the time resolution is increased to 384 time steps per year to ensure reasonable temperature change rates for some experiments carried out with very small heat capacities. The resolution in latitude is 128 grid points f rom pole to pole. The 2-D

model produces a simulation identical to the 1-D model when run with zonally-symmetric conditions and the same albedo parameterization.

The model simulation of the present climate closely resembles that of North et al. (1983). In common with their model, the maximum temperature in high latitudes lags the insolation more than is observed, due in part to the simple treatment of sea ice in the model. Tempera- tures in the North Atlantic and Greenland Sea tend to be too cold, probably as a result of the lack of heat transport by ocean currents. The simulated temperature in Antarctica is higher than observed because the surface elevation of Antarctica has not been taken into account in the model. Overall, the comparison between the simulated temperature and the observed data suggests that the 2-D EBM with a realistic land-ocean distribution can simulate the main characteristics of the present surface temperature, including the seasonal cycle.

Presence of the small ice cap instability

The response of permanent (year-round) snow cover to changes of the solar constant is examined to learn whether the SICI exists in both the Northern and South- ern Hemispheres in the 2-D seasonal EBM with a realistic land-ocean distribution. The present-day insolation distribution with eccentricity = 0.017, obliquity = 23.45 °, and perihelion= 102 ° is used in the model. The permanent hemispheric ice cover is calculated for a series of equilibrium solutions with decreasing values of the solar constant, Q, using the previous equilibrium solution as the initial condition for each new solution. This process is then repeated for the same range of solar constants while increasing Q. The presence of the SICI is indicated by more than one equilibrium solution for a given Q, which are achieved from the different initial conditions.

Figure 2 shows the permanent ice cover in each hemisphere as a function of solar constant. Starting from an ice-free state with a large solar constant, in the Southern

0.1

0.2

0 , 3 i i i i i i

0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

Normalized Solar Constant

Fig. 2. The fractional minimum snow area in a year versus the normalized solar constant in the Northern Hemisphere (N.H.) and the Southern Hemisphere (S.H.). Stable equilibrium solutions are shown as solid line and unstable equilibrium solutions are shown as dashed line

208 Huang and Bowman: The small ice cap instability in seasonal energy balance models

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90

60

30

0 . ~ i . 0

0 i i

0.9 1.0 1.1 Normalized Solar Constant

Fig. 4. Same as Fig. 1 except it shows the equilibirum solutions of a one-dimensional seasonal EBM with different values of uniform heat capacity. The values of heat capacity are represented in times of ocean's heat capacity. They are 2.0, 1.0, 0.5, 0.3, 0.2, 0.15, 0.1, 0.08

of the snow line is larger toward the International Date- line and smaller toward the Greenwich meridian.

Idealized experiments with the 1-D EBM

The 2-D model experiments show that the SICI exists in the Southern but not the Northern Hemisphere. In order to understand the influences of the land-ocean distribution and the heat capacity on the existence of the SICI, a series of experiments are carried out with models with idealized, zonally-symmetric, land-ocean distributions. Since all quantities are independent of longitude, the 1-D model can be used. In the 1-D model, a circular or- bit (eccentricity=0.0, obl iqui ty=23.5) is used for simplicity. The experiments are in two groups, the first has a globally-uniform heat capacity and the second has different-sized zonally-symmetric continents (land bands) at different latitudes to represent different prototypical land-ocean distributions.

Hemisphere the climate stays ice-flee until the solar constant is decreased to 1.032. At that point, the ice sud- denly grows to - 65 ° S latitude (Fig. 3a). This sudden expansion of ice is similar to the SICI in zonally-symmetric, annual-mean models.

Figure 2 also shows the minimum snow area occurring during a year in the Northern Hemisphere as a function of the solar constant. Permanent ice appears when the solar constant decreases to 1.02, but the ice area increases continuously as the solar constant decreases. Extensive numerical experiments have found no evidence for the SICI in the Northern Hemisphere of the model. The continuous increase of the snow area can be seen more clearly in Fig. 3b. The summer snow in the Northern Hemisphere first appears over the east coast of Greenland at Q = 1.02 because the large heat capacity of the ocean keeps summer temperatures low. Sensitivity

Uniform heat capacity

The purpose of the experiments with a globally-uniform heat capacity is to compare the SICI in seasonal and annual-mean models in the simplest possible context. The stability of ice caps in the annual-mean model is determined by the insolation and the meridional gradient of the temperature at the ice margin (Held and Suarez 1974). By making experiments with different values of the heat capacity, we can explore whether other factors can affect the strength of the albedo feedback and thus the SICI in seasonal EBMs. As in the 2-D model, we are interested in only the permanent ice cover occurring during a year.

Figure 4 shows the latitude of the permanent ice margin as a function of solar constant for different values of the heat capacity. The value of the heat capacity is in units of the oceanic heat capacity (Cw). Experiments


were carried out with heat capacities of 2.0, 1.0, 0.5, 0.3, 0.2, 0.15, 0.12, 0.1, 0.08, and 0.05. Because no stable solutions exist for C=0 .05 other than ice-covered and ice-free, results for C= 0.05 are not shown in Fig. 4.

Two principal features can be seen in Fig. 4. First, the SICI exists only in models with a large heat capacity. For the parameter settings of this model, the SICI exists only in models with C= 1.0 and 2.0. The region of instability shrinks as the heat capacity decreases. In an annual mean model, in which the heat capacity can be con- sidered as infinite, the region of unstable polar ice caps has a radius of - 15 ° latitude (see Fig. 1). In the seasonal EBM, the radius of the unstable region is 7.5 ° latitude for C = 2 . 0 and 2.8 ° latitude for C = I . 0 . Second, the large ice cap instability moves poleward as the heat capacity decreases. Stable ice caps extend to 41 ° latitude for C = 1.0. For C--0.08, they can extend only to 63 ° latitude. With further decreases of the heat capacity, the stable region disappears. With a heat capacity of C = 0.05 there are no stable finite-sized ice caps, as the large ice cap instability extends from equator to pole.

In terms of the model stability, the heat capacities can be divided into three categories. For large heat capacity ( C _ 1.0), the model has a SICI. For medium heat capacity (1 .0>C>0 .05 ) , the model has no SICI, only stable finite-sized ice caps. For small heat capacity (C_< 0.05) there are no stable finite-sized ice caps.

In order to explain the dependence of the SICI on the heat capacity, we first look at the temperature gradient at the snow margin in summer for different heat capacities. Three typical heat capacities are chosen to study from the three categories: C= 1.0, 0.15, and 0.05. Fig- ure 5 shows the temperature distribution with latitude on the hottest day for each heat capacity. Compared to the larger values of C, the temperature gradient in summer for C = 0.05 is very small; and the model is very sen- sitive.

3 5 i i i i i i i i

30 / 0 . 1 5 ~

25

2o o 8 /,05

~ 10

5

0

- 5 i i i i i i i i

0 30 60 90 Latitude

Fig. 5. Temperature distributions with latitude at the hottest day for different values of uniform heat capacity: C = I . 0 at Q = 1.0326; C=0.15 at Q = 1.0051; C=0.05 at Q=0.9531

209

However, the temperature gradients, which are similar in the other two cases, cannot explain the qualitative difference between C= 1.0 and 0.15. In order to examine the differences between the strength of the albedo feedback in these two cases, we carry out two successive linear calculations corresponding to the two steps in the albedo feedback loop described in the Introduction. First, given an equilibrium solution to the energy balance equation, the seasonally-varying albedo from the equilibrium calculation is held fixed while the solar constant is reduced slightly. Because the albedo feedback is turned off, the temperature response in general is smaller than for the nonlinear model. Second, the albedo field is adjusted to the new seasonal temperature field and the temperature is again calculated with this new albedo field held fixed. This completes one iteration of the feedback loop. Note that the temperature field is not necessarily in agreement with the albedo field through Eq. (2), since the albedo is not allowed to adjust to the temperature. Repeating this process iteratively will cause the linear solution to converge to the same solution as the fully nonlinear model.

The radiative forcing due to the increase in ice cover can be compared for the experiments with different heat capacities to see the relative strength of the albedo feedback in each case. The initial equilibrium states are chosen to be free of permanent ice with temperatures very close to the freezing point at the pole. If the solar constant is reduced by 1%, the change in the solar heating caused by one iteration of the albedo feedback loop is shown in Figs. 6 and 7. Figure 6 shows that the forcing is located along the snow margin, as expected, where the reduced insolation causes cooling and, in the second step, expansion of the snow cover. Figure 7 shows the

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i i i i i i i i i i i

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Fig. 6. Radiative forcing distribution for decreasing solar constant 1°7o for C= 1.0 and C=0.15. The contour interval is 50Win -2 and the dashed lines are negative


U

-12

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, L v / 0.15

. /

I I I I I I I

3O 60 90

Lat i tude

Fig. 7. The annual mean radiative forcing (unit: W m -2) as a function of latitude for C= 1.0 and C=0.15

annually-averaged forcing at each latitude. The forcing is larger for C = 1.0 and is concentrated in high latitudes (70°-90°), where it ranges f rom about 5 W m -2 to 2 . S W m -2 at the pole. For the experiment with C--0.15, the radiative forcing is distributed over a larger region f rom 400-90 ° latitude, where it ranges f rom 2 W m - 2 to nearly zero at the pole.

The response of the temperature field to the radiative forcing (the second of the two linear calculations) is show in Fig. 8. The induced cooling is much larger for C = 1.0 than 0.15. The max imum polar temperature decrease is - 1 . 2 ° C for C = 1.0 and - 0 . 8 ° C for C=0 .15 . The - 0 . 7 ° C contour line spreads to 85 ° latitude only and lasts less than one month for C = 0.15. For C = 1.0, the - 0 . 7 ° C contour line spreads to 75 ° latitude and lasts about 7 months.

These results suggest that stability is determined in part by the latitudinal range of the seasonal cycle of snow cover, which is determined in turn by the heat capacity. The latitudinal range is small when the heat capacity is large, so the radiative forcing f rom the albedo feedback is distributed over a narrow latitude range. This places the cooling at the critical location: near the pole during the summer. The extreme case is the annual- mean model, where there is no seasonal cycle and the forcing acts at a single latitude, the ice margin. Thus the strength of the albedo feedback is the strongest and the size of unstable ice caps is the largest in the annual-mean model.

As the heat capacity decreases, the latitude range of the seasonal snow cover increases, and the forcing is distributed over a wider range latitudes. The larger the latitudinal extent, the smaller the polar cooling and the weaker the strength of the albedo feedback. As men- tioned in the Introduction, the characteristic length scale of the model is about 15 ° latitude. Therefore, radiative forcing outside the high latitudes has little effect on the polar region through diffusion.

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Fig. 8a, b. The temperature changes caused by the radiative forcing for a C= 1.0; and b C=0.15. The contour line interval is O.I°C

For very small heat capacities, the stability of the model is determined by the instantaneous insolation distribution. As North (1975) has discussed, equinoctial insolation distributions are stable, since there is no radiation at the pole to contribute to an albedo-temperature feedback. Summertime insoaltion, however, is maximum at the pole; and the summer temperature gradients are small. Therefore, the large ice cap instability expands to the pole, and there are no stable equilibrium ice caps.

Zonally-symmetric continents

The experiments with uniform heat capacity indicate that the heat capacity affects the existence of the SICI mainly throught the meridional temperature gradient and the amplitude of the seasonal cycle. In this section,


models with zonally-symmetric continents and oceans are used to investigate the effects of coupling regions with different heat capacities on the temperature gradient and the amplitude of the seasonal cycle and thus the SICI. A single zonally-symmetric continent is assumed to be located in middle to high latitudes. The poleward and equatorward margins of the continents are varied in 10 ° latitude step to investigate the effects of the size and location of the continent on the SICI. The latitudinal width of the continent ranges between 0 ° (no land) and 30 °. Oceans located poleward of the continent are assumed to be covered with sea ice, and have an intermediate heat capacity, as in the preceding section. As the continent moves equatordward, the size of the polar sea ice increases from 0 ° to 40 ° latitude. Experiments are also carried out without continents, but with polar sea ice caps of various sizes.

The model exhibits different types of behavior de- pending on the size and location of the continent. Table 1 summarizes the results of the experiments with different arrangements of continent, ocean, and sea ice. The latitude value in each column of the table indicates the size of polar sea ice. The latitude value in each row of the table presents the location of the equatorward margin of polar sea ice plus continent. The latter P represents a SICI with a jump from the pole to a lower latitude, while C indicates a jump across the continent. The notation PC means that the polar jump and the jump over the continent are combined and only a single jump exists, while a comma separating them indicates that two separate jumps exist. A letter in parenthesis, e.g., (P) means the polar jump is weak. The large ice cap instability is not shown in Table 1, since it exists in every experiment.

I f there is no ocean at the pole (as in Antarctica), or the polar ocean is small (10 ° latitude), there is always a polar SICI, as shown in the first two columns in Table 1. Figure 9 shows an example of the seasonal cycle of the snow margin for two values of the solar constant with a continent extending from the pole to 70 ° latitude. The summertime snow line jumps from the pole to 64 ° latitude when changing the solar constant from Q = 1.0066 to Q = 1.0065 (shown in Fig. 9a). There are no stable ice caps smaller than 22 ° latitude in extent; the ice cap always grows to cover the entire continent. The instability of the polar ice caps can be analyzed from the temperature field (shown in Fig. 9b). The temperature gradient at high latitudes in summer is very small, caused by the coupling between the hot continent in the polar region and the cool ocean in mid-latitudes in summer. The polar summer temperature on the continent is so high that the temperature distribution with latitude becomes reversed. Thus, the summer snow appears at about 70 ° latitude first. Because of the small temperature gradient, once the snow appears, it extends to pole. The ocean surrounding the polar continent reduces the annual cycle of temperature at high latitudes, which also favors the existence of the SICI in this kind of land- ocean distribution.

If the size of the polar continent is larger than 40 ° latitude, the SICI merges with the large ice cap instabili-

Table 1. The results of the experiments with non-uniform heat capacity. The latitude value in each column indicates the size of polar sea ice. The latitude value in each row presents the location of the equatorward margin of polar sea ice and continent. The letters P and C represent the polar jump (SICI) and the jump of snow lines across continent, respectively. PC means the two jumps are combined, and (P) means a weak polar jump

Polar sea-ice size

20 ° 30 ° 40 ° 50 ° 0 o 10 o 90 ° P 80 ° PC P 70 ° PC PC P 60 ° PC PC PC P 50 ° PC PC (P), C (P), C 40 ° PC PC C C

N o n e

C None

-=~

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60 Q = 1.0065 ~_._____..~

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Month

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60

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0 I i I J F M A M J J A $ O N D

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Fig. 9. a Snow-line distribution for 70°-90 ° land case for Q = 1.0066 and 1.0065. The shaded area is land surface; b Tem- perature distribution for Q = 1.0066. The contour interval is 2 ° C and the dashed lines are negative

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Fig. 10. a Snow-line distribution for 60°-70 ° land case for Fig. 11. a Snow-line distribution for Q = 0.9901 and 0.99. The shaded area is land surface; b Tempera- Q = 0.9672, 0.9671, 0.967, 0.9668, 0.9667 and 0.9666. The shaded ture distribution for Q = 0.9901. The contour interval is 2 ° C and area is land surface; b Temperature distribution for Q = 0.9672. the dashed lines are negative The contour interval is 2 ° C and the dashed lines are negative

50°-70 ° land case for

ty, as in the g loba l ly u n i f o r m exper iments with small hea t capaci ty , and the snow line j u m p s f r o m the pole to the equa to r .

W h e n the p o l a r ocean (covered by sea ice) is 20 ° or larger , the existence of the S ICI depends on the to ta l size o f sea ice plus cont inent . The SICI exists as long as the p o l a r sea ice plus con t inen t extend no lower t han 60 ° la t i tude , jus t l ike the first two co lumns o f the table . Fig- ure 10 shows an example wi th a 10 ° con t inen t su r round- ing a 20 ° wide po la r sea ice.

As the c o m b i n e d p o l a r sea ice and con t inen t extend to 50 ° la t i tude , the po la r SICI becomes weak. F igure l l a shows a weak S ICI case with a con t inen t f r o m 50 ° to 70 ° la t i tude , sea ice f rom 70 ° to 90 ° l a t i tude and ocean in lower la t i tudes . As the solar cons tan t is reduced , the p e r m a n e n t ice cover j u m p s first f r om the po le to 84 ° , then grows s tably to 77 ° , f r om where it j u m p s across the

con t inen t to 41 °. F r o m the t e m p e r a t u r e f ield (Fig. 1 lb ) , it is f o u n d because the mid - l a t i t ude ocean is suff ic ient ly far f r o m the pole , the m o d e r a t i n g effect o f its hea t capac i ty on the seasonal cycle and t e m p e r a t u r e near the pole are small . There fo re , the a lbedo f eedback is weak and the SICI is weak.

W h e n the po la r sea ice and cont inent extend 40 ° latitude , the SICI d i sappears . The mode l behav io r is s imilar to tha t wi th a g l o b a l l y - u n i f o r m in te rmed ia te hea t capacity. Both the t e m p e r a t u r e g rad ien t and the amp l i t ude o f the seasonal cycle are large.

The first rows in each co lumn are the cases wi th po la r sea ice coup led with ocean, i .e . , there is no cont inent . A m o d e l wi th u n i f o r m sea ice has no SICI , bu t the S ICI appea r s when the p o l a r sea ice is su r rounded by ocean, because the ocean at the lower la t i tudes suppresses the summer t e m p e r a t u r e and leads to a small t e m p e r a t u r e


gradient in high latitudes. The amplitude of the seasonal cycle is also decreased. When the ocean is too far away from the pole, the SICI disappears and the model be- haves just as a model with a globally-uniform sea ice.

The most interesting behavior of this model is the unstable growth of permanent ice across continents, re- gardless of where the continent is located or how big it is. The instability of the ice lines over continent is related to the uniform temperature field over the continent in summer, which is due to the hot continent in higher latitudes and cool ocean in lower latitudes in summer (see Fig. l l b for the temperature field). The small temperature gradient leads to a strong albedo feedback. Once the ice appears over the continent, it will cover the whole continent.

Thus, the SICI in models with zonally-symmetric continents is not only determined by the local polar heat capacity but also by the land-ocean distribution of the hemisphere. The land-ocean distribution determines the local temperature gradient and amplitude of the seasonal cycle. If the polar region is land covered, the polar SICI always exist. If the pole is covered by sea ice, the existence of the SICI is determined by the distance from the pole to the ocean. Permanent ice cover on zonally- symmetric continents is always unstable.

Interpretation of 2-D model results

In this section, we try to understand the behavior of the 2-D model by using what we have learned from the 1-D model experiments. Because of the zonal asymmetry in the realistic land-ocean distribution, we cannot expect to interpret the 2-D results entirely in terms of the 1-D experiments.

In the Southern Hemisphere, the Antarctic continent is nearly zonally symmetric, extending from the pole to - 7 0 ° latitude. The ice line behavior is essentially identical to that of the 1-D model with a 20 ° land cap at the pole (shown in Fig. 9a). Thus, the SICI in the Southern Hemisphere is due to the small land cap at the South Pole surrounded by ocean, which results in a small summer time temperature gradient at the Pole and a small amplitude seasonal cycle.

In the Northern Hemisphere, the land-ocean distribution is zonally asymmetric, and no evidence for the SICI is found in the 2-D model. The coldest temperatures in summer are not found at the Pole, but near Iceland which is where permanent ice cover first appears in response to a reduction of the solar constant (see Fig. 3b). Although the temperature distribution at 30°W (Fig. 12a) is similar to that in 1-D model with 20 ° land cap at Pole (Fig. 9b) and the polar temperature gradient in summer is small, small ice areas are stable in that region in the 2-D model. The stability of the ice areas is because the initial ice in the Northern Hemisphere is a very small ice patch near Iceland. The radiative forcing caused by a small ice patch is small and thus the albedo feedback is weak in the Northern Hemisphere. By comparison, if a zonally-symmetric land-ocean distribution is put in the 2-D model and if the ice initiates at mid-

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latitude, the initial ice cover would be a zonally symmetric ring, and the albedo feedback would be stronger.

After the ice extends into the polar region from Ice- land, the polar ice lines do not jump from pole to mid- latitudes as in the Southern Hemisphere. Instead, the polar ice expands smoothly with decreased solar constant. Moreover, the ice lines in the Northern Hemis- phere do not jump across the continents. The reason is mainly related to the longitudinal variation of the land- ocean distribution in the Northern Hemisphere. The characteristic length scale of the diffusion is comparable to the size of the oceans and continents, so the longitudinal variation in heat capacity (oceans and continents) is smoothed out and produces an effective medium heat capacity across the Northern Hemisphere.

The effects of the smoothing of the heat capacity can be seen by examining the temperature distribution at two typical longitudes. Figure 12b shows the temperature distribution at 90°E (in the Eurasian continent). At this longitude, the continent extends from 20 ° to 70°N latitude. In the zonally symmetric model, this land- ocean distribution would produce a jump from the pole to the equator because of the small temperature gradient in summer over continent. I n the 2-D model the effect of the continent is diffused by the surrounding ocean, and the temperature distribution in Eurasia is similar to that in the globally-uniform sea ice model. Both the large temperature gradient and the large amplitude of the seasonal cycle are responsible for the lack of the ice line jump across the continent.

Figure 12c shows the temperature distribution at 180 ° longitude. The land-ocean distribution at this longitude is similar to the 1-D case with a land-band at 60°-70 ° latitude (shown in Fig. 10), but the 2-D model does not show the unstable phenomenon in polar region as in the 1-D model (Fig. 10a). The mid-latitude ocean in the 2-D model is not as cool as in the 1-D model because of the transport of heat from nearby continents. Therefore, the temperature gradient in the polar region is not small enough to cause the instability, although the local land band causes somewhat greater sensitivity in eastern Si- beria and Alaska.

Discussion

In annual mean energy balance models (EBM), small polar ice caps are unstable to small perturbations. The ice caps either vanish or grow to a stable finite size. It has been suggested that the small ice cap instability may have been the mechanism for the glaciation of Antarcti- ca and the large changes of Northern Hemisphere ice cover during glacial to interglacial cycles (Budyko 1969; Crowley and North 1988; Lin and North 1990).

In this paper, we examine the SICI phenomenon in l- and 2-D EBMs with seasonal cycles, where the land- ocean distribution is represented by the heat capacity of the surface. In a 2-D model with the present-day land- ocean distribution, the SICI occurs in the Southern Hemisphere, but not in the Northern. The reason for the

difference between the two hemispheres is studied with a 1-D seasonal EBM. With the 1-D model, various idealized land-ocean distributions are used to clarify the roles of the heat capacity and the land-ocean distribution.

The strength of the albedo feedback in a seasonal model, and thereby the presence or absence of the SICI, depend on the meridional temperature gradient near the summer ice line (Held and Suarez 1974). If the temperature gradient is weak, a small change in the solar input will produce a large response in the ice cover. In the seasonal model, it is also important to consider the amplitude and phase of the seasonal cycle of temperature with respect to the annual cycle of insolation. In the 1-D model with a uniform surface, the SICI exists only when the heat capacity of the surface is large, as would be the case with an ocean-covered planet. This is consistent with the annual-mean model, since in the limit of infinite heat capacity, the seasonal model should behave like the annual-mean case. For intermediate heat capacities, the SICI does not exist. The larger amplitude of the seasonal cycle distributes the radiative forcing from the albedo feedback widely in time and latitude, and the larger summertime temperature gradient reduces the sensitivity of the model. For a land-covered planet (small heat capacity), there are no stable small ice caps because the small pole-to-equator temperature gradient in summer causes the ice cover to grow to the equator.

In models with non-uniform heat capacity, the existence of the SICI depends on the land-ocean distribution. The SICI exists as long as there is a land cap at the pole. Both the temperature gradient and the amplitude of the seasonal cycle support the SICI. The small temperature gradient in summer is partly due to the land it- self and partly due to the suppression of the summer temperature by the ocean. The warm ocean in winter results in a small amplitude seasonal cycle. The snow lines are always unstable over a zonally-symmetric continent because the summer temperature gradient is small. If the pole is covered by sea ice, the existence of the SICI depends on the distance of the ocean to pole. If the ocean is at the lower latitudes than 50 °, the suppression of summer temperature by the ocean has little effect on the polar temperature. The temperature gradient is as large as that in uniform sea ice model, and the amplitude of the seasonal cycle is large. Both factors do not support the SICI when poleward edge of the ocean is equatorward of 50 °N.

The results in our 1-D models with zonally symmetric continents are similar to those from Lin and North (1990). The ice lines are always unstable across zonally symmetric continents. However, longitudinal variations of the land-ocean distribution in the Northern Hemis- phere cause the absence of the SICI and abrupt ice line jumps. Therefore, the results in models with zonally symmetric geography cannot represent the features of ice line instability in the Northern Hemisphere.

Because of the zonal asymmetry of the land-ocean distribution, the permanent ice initiates as a small patch near Iceland. The small radiative forcing caused by the small ice area expansion leads to a weak albedo feedback. Therefore, there is no abrupt ice line jump across


Greenland, which occurs in 1-D model (Lin and North 1990). When the ice extends to polar regions, the ice lines are still stable. The land-ocean distribution in the Northern Hemisphere does not favor the SICI. The North Pole is covered by sea ice with a medium heat capacity. The longitudinal variation of heat capacity of ocean and land in mid- and low-latitudes is smoothed out and produces an effective medium heat capacity in the Northern Hemisphere. The large temperature gradient and the large amplitude of seasonal cycle do not support any ice line jump in the Northern Hemisphere.

The existence of the SICI in the Southern Hemis- phere is due to the nearly zonally-symmetric continent at the South Pole. The small polar temperature gradient in summer and the small amplitude of the seasonal cycle lead to a strong albedo feedback and a SICI.

In experiments with a 2-D seasonal EBM, North et al. (1983) found a SICI-like ice line jump in the North- ern Hemisphere. This was apparently due to their use o f a fixed albedo with no seasonal cycle. The radiative forcing caused by the albedo feedback is concentrated in fixed locations for the whole year, as in the annual- mean EBM. This enhances the albedo feedback and causes the sudden jump of the ice line in the Northern Hemisphere.

Our model results indicate that the SICI may be a possible mechanism for the sudden formation of the Antarctic ice sheet. The model results cast doubt, however, on the role of the SICI in the abrupt change of the Northern Hemisphere ice sheet. Model results demon- strate the strong influences of the land-ocean distribution on the existence of the SICI, which suggests that continental drift may affect the climate sensitivity and stability.

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